Theoretical and practical aspects of the

Vibroseis method

G.J.M. Baeten

Theoretical and practical aspects of the

Vibroseis method

ISBN 90-9002817-X Druk: ALEVO - Delft April 1989

Cover page: l.V.I. "birdwagen MARK IV off-road vehicle with 27 ton vibrator. (courtesy: I.V.I.)

Theoretical and practical aspects of the Vibroseis method

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector

Magnificus, prof. drs. P.A. Schenck, in het openbaar te verdedigen ten overstaan van een commissie

aangewezen door het College van Dekanen op donderdag 11 mei 1989 te 14.00 uur

door

Guido Jozef Maria Baeten mijningenieur geboren te Ens

L TR diss 1719

Dit proefschrift is goedgekeurd door de promotor Prof. A.M. Ziolkowski M.A., Ph.D., M.Sc. (Econ.)

The research reported in this thesis has been performed at the Delft University of Technology, Department of Mining and Petroleum Engineering, Section Applied Geophysics. Part of this research has been financially supported by Industrial Vehicles International, Inc., Tulsa, Oklahoma, U.S.A.

Table of contents

Notations and conventions 1

Chapter 1 An introduction to Vibroseis 4

1.1 The mechanics of the seismic vibrator 4 1.2 The shape of the signal emitted by a seismic vibrator 6 1.3 The control over the Vibroseis source signal 13

1.3.1 Engineering aspects 13 1.3.2 Geophysical aspects 14 1.3.3 Combined engineering and geophysical aspects 14

1.4 The use of Vibroseis in seismic exploration 16 1.5 Objectives of the research in this thesis 19

Chapter 2 The wavefield of a surface source in a horizontally layered elastic medium 21

2.1 Introduction 21 2.2 Configuration 21 2.3 Basic relations 22 2.4 Decomposition into upgoing and downgoing wavefields 24 2.5 The local reflection of the wavefield at a layer interface 28 2.6 The propagation of the wavefield within a single layer 31 2.7 The global reflection response 32

2.7.1 Backward recursion 32 2.7.2 Forward recursion 33

2.8 The particle velocity and traction components at an arbitrary point in the medium 36

2.9 The transformation to the space domain 40 2.10 The elastic halfspace 43 2.11 The horizontally layered acoustic medium 44 2.12 Far field relations 47 2.13 Conclusions 52

Chapter 3 The seismic vibrator 53

3.1 The force exerted on the baseplate 53 3.2 The baseplate behaviour 57

3.2.1 Uniform displacement 58 3.2.2 Uniform traction 60 3.2.3 The mass-loaded boundary condition 61 3.2.4 The flexural rigidity method 63

3.2.4a Classical plate theory 65 3.2.4b The magnitude of the flexural and

torsional rigidity components 72 3.2.4c Dynamic terms 75 3.2.4d The boundary conditions 76 3.2.4e Solution of the differential equation 77 3.2.4f Physical interpretation 82

3.3 Conclusions 85

Chapter 4 The combined earth-vibrator model 87

4.1 The earth model 87 4.1.1 Theory 87 4.1.2 Amplitude and phase of the earth's Green's function 89 4.1.3 The asymptotic behaviour of the earth's Green's function 91

4.2 The combined earth-vibrator model 92 4.2.1 Uniform displacement 93 4.2.2 Uniform traction 94 4.2.3 The mass-loaded boundary condition 95 4.2.4 The flexural rigidity method 96

4.3 The numerical procedure 98 4.4 Numerical results 99

4.4.1 The distribution of traction and displacement over the baseplate 99

4.4.2 The radiation impedance function 112 4.4.3 The average power factor 115 4.4.4 The radiated power distribution in the far field 117 4.4.5 Synthetic vibrator measurements 120 4.4.6 Vibrator arrays 123

4.5 Conclusions 128

Chapter 5 Test of the theory by experiment 129

5.1 The data set 129 5.2 The earth model 131

5.2.1 The velocity profile 132 5.2.2 The density profile 133 5.2.3 Modelled and measured downhole response 135

5.3 The vibrator model 140 5.3.1 Uniform displacement 144 5.3.2 Uniform traction 145 5.3.3 The mass-loaded boundary condition 146 5.3.4 The flexural rigidity method 148

5.4 The feedback signal on the seismic vibrator 152 5.4.1 Baseplate acceleration 153 5.4.2 Reaction mass acceleration 156 5.4.3 The weighted sum method 158 5.4.4 The flexural rigidity feedback signal 162

5.5 Conclusions 169

Chapter 6 The marine vibrator source 171

6.1 Introduction 171 6.2 Configuration 172 6.3 Vibroseis: land and marine 173 6.4 The marine vibrator model 174

6.4.1 The earth model 174 6.4.2 The marine vibrator model 176

6.4.2a The mechanical model 17 6 6.4.2b The marine vibrator casing 179 6.4.2c Combined mechanical model and

vibrator casing model 180 6.5 The combined earth-vibrator model 182 6.6 The numerical procedure 183

6.7 The distribution of pressure and displacement at the surface of the vibrator 184

6.8 The average power factor 187 6.9 Far field relations 187 6.10 Test of the theory by experiment 194 6.11 Conclusions 197

Concluding remarks 198

Appendix A Determination of Green's matrices for acoustic and elastic media 202

A. 1 The horizontally layered elastic medium 202 A. 1.1 Expressions for the Green's matrices in the

wavenumber domain 202 A. 1.2 Transformation to the space domain 209

A.2 The elastic halfspace 213 A.3 The horizontally layered acoustic medium 215 A.4 Far field relations 218

Appendix B Solution of the differential equation describing the plate bending 226

B. 1 The response to a point force 226 B. 1.1 Derivation of the integral relation describing

the plate deflection 227 B. 1.2 Calculation of the integral relation describing

the plate deflection 228 B.2 The response to a pressure distribution 232

B.2.1 Summation method 233 B.2.2 Circle approximation 234

Appendix C The numerical procedure for the combined earth-vibrator model 240

C. 1 The earth's Green's function 240 C.1.1 Numerical calculation of the the earth's Green's function 240 C.l.2 Check on the numerical calculation of the

earth's Green's function 245 C.2 The complex power balance 247

C.2.1 General theory 247 C.2.2 Application of the power balance to the elastic halfspace 248 C.2.3 Application of the power balance to the baseplate

of the vibrator 251

References 254

Summary 256

Samenvatting (summary in Dutch) 259

Acknowledgements 262

Curriculum vitae 263

Notations and conventions

In this thesis, vectors and matrices are typed bold-faced. To locate a point in space, orthogonal Cartesian coordinates X\,x2 andx3 are employed with respect to a given orthogonal Cartesian reference frame with origin O and the three mutually perpendicular unit vectors i}, i2 and i3; in the given order, ij, \2 and i3 form a right-handed system. The unit vector i3 points downwards. The vector x = X\ ij + x2 i2 + *3 «3 denotes the three-dimensional position vector in the xhx2, x-i space.

Occasionally, the subscript notation for (Cartesian) vectors and tensors is used. Latin subscripts are to be assigned the values 1, 2 and 3, while Greek subscripts are to be assigned the values 1 and 2; for repeated subscripts, the summation convention holds.

Use is made of the temporal and spatial Fourier transform pairs. Quantities in the time domain are denoted by lowercase symbols, quantities in the frequency domain by the corresponding capitals, and quantities in the wavenumber domain are denoted by a circumflex (A) on top. In most cases, the parameters "co" and "/" are omitted for brevity. The temporal Fourier transform pair is defined as

f(t)=(2n) I F(ti) exp(-im) doo , 1 F(a>) = I ƒ « exp(icor) dr .

The spatial Fourier transformation pair is given by

-2 r°° r°°^ F(x}, x2, (0) = (In) I I F(kx, k2, a>)txp[i(k-ix-l+k2x2)]dk-[dk2 ,

F(k} , k2, a)- I I F(x: ,x2, co) c%p[-i(kix1 +k2x2)]dxidx:

A list of major symbols is given below.

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List of major symbols

Latin symbols

03, ar accelerations of baseplate and reaction mass, respectively (ms - 2) cp compressional (P) wave velocity (ms - 1 ) cR Rayleigh wave velocity (ms - 1) cs shear (S) wave velocity (ms - 1) Cjjpq stiffness tensor (Nm - 2 ) D flexural rigidity (Nm) etj strain tensor

j applied force delivered by the drive mechanism of the vibrator (N) Gfcx Green's function for flexural rigidity method (s2kg"') Hn , Hn Bessel functions of the third kind (also called Hankel functions), of

order n / „ , Kn Modified Bessel functions of order n ï j , ï2> b Cartesian unit vectors ./„ Bessel function of the first kind of order n Mp, Mr, Mh masses of baseplate, reaction mass and holddown mass, respectively

(kg) kp, ks, kR P - S - and Rayleigh wavenumber (m_ 1) t time coordinate (s) f,- traction vector (Nm - 2 ) t;j stress tensor (Nm - 2 ) Hi particle displacement vector (m) » 3 , u r , uh displacements of baseplate, reaction mass and holddown mass,

respectively (m) v (- particle velocity vector (ms~]) x position vector .V], x2: v3 Cartesian coordinates Yn Bessel function of the second kind (also called Weber's function) of

order n

. - > .

Greek symbols

A Rayleigh denominator (s4m~4) A , \x Lamé elastic parameters (Nnr 2 )

7 , ys vertical P - and S wave slownesses ( snr ' )

p volume density of mass (kgirr3) o surface density of mass of the baseplate (kgni-2) v Poisson's ratio co angular frequency (s_1)

Chapter 1 An introduction to Vibroseis

In seismic exploration, the use of a vibrator as a seismic source has become widespread ever since its introduction as a commercial technique in 1961. This chapter explains the principles on which the Vibroseis™ method is based. The mechanism which allows the seismic vibrator to exert a pressure on the earth is explained. The two basic features of the force function generated by the seismic vibrator are discussed: first, the signal generated by a seismic vibrator is not impulsive, but has a duration of several seconds; second, the monitoring system of the seismic vibrator allows control over the outgoing vibrator signal. The choice of the signal that is to be monitored on the vibrator in order to control the outgoing signal is a controversial issue. A brief review of existing methods is given. Finally, the advantages and disadvantages of Vibroseis over most impulsive sources are discussed, and the objectives of the present investigations are explained.

1.1 The mechanics of the seismic vibrator

An example of a seismic vibrator is shown in Figure 1.1.

rs' Registered trademark of Conocolnc.

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The vibrator is a surface source, and emits seismic waves by vibrating a plate which is kept in contact with the earth.

Figure 1.2 The force-generating mechanism of the seismic vibrator source.

The driving force applied to the plate is supplied either by a hydraulic system, which is the most common system in use, or an electrodynamic system, or by magnetic levitation, the latter being a new development in the field of seismic vibrator technology. The direction in which the plate vibrates can also vary: P wave vibrators (where the motion of the plate is in the vertical direction) as well as S wave vibrators (vibrating in the horizontal direction) are used. Finally, a marine version of the seismic vibrator has been developed.

For all these vibrator types, the general principle which governs the generation of the driving force applied to the plate (usually referred to as the baseplate) can be described by the configuration shown in Figure 1.2. A force F is generated by a hydraulic, electrodynamic or magnetic-levitation system. A reaction mass supplies the system with the reaction force necessary to apply a force on the ground. The means by which this force is actually generated is illustrated in Figure 1.3, in which the principle of the hydraulic drive method is shown. By pumping oil alternately into the lower and upper chamber of the piston, the baseplate is moved up and down. The fluid flow is controlled by a servo valve. The driving force acting upon the baseplate is equal and opposite to the force acting upon the reaction mass, as can easily be

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Reaction mass Piston

/Fluid pressure from

servo valve

Baseplate

Figure 13 Schematic view of the generation of the driving force for a hydraulic vibrator.

inferred from Figure 1.3. In general, the peak force is such that the accelerations are in the order of several g's, so that an additional weight has to be applied to keep the baseplate in contact with the ground. For the hydraulic and electrodynamic vibrators, the weight of the truck is used for this purpose. This weight, commonly referred to as the holddown mass, is vibrationally isolated from the system shown in Figure 1.3 by an air spring system with a low spring rate (shown in Figure 1.4), and its influence on the actual output of the system is usually neglected. The resonance frequency of the holddown mass is in the order of 2 Hz, the lowest frequency of operation in Vibroseis seismic surveys for exploration purposes being usually not less than 5 Hz.

1.2 The shape of the signal emitted by a seismic vibrator

The signal emitted by the seismic vibrator is not impulsive, but typically has a duration of 10 sec. This seems to be in contradiction with the fact that seismic exploration methods aim to detect the impulse response of the earth. This apparent contradiction can be clarified by taking a closer look at the properties of an impulse and its earth response.

A perfect impulse at time f=0 contains all frequency components with equal amplitude and zero phase. This is illustrated in Figure 1.5.

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STILTS VIBRATOR ACTUATOR

AIR SPRINGS

(a) (ft)

Figure 1.4 (a) schematic view of the Vibroseis truck with the air springs, the baseplate and the vibrator

actuator (reaction mass), and (b) detailed view of the middle pan of the truck.

'lime t = 0

(fl)

Amplitude Phase

-► frequency frequency

TIME DOMAIN FREQUENCY DOMAIN

(b)

Figure 1.5 The notion of a perfect impulse, (a) in the time domain, and (b) its corresponding frequency domain version.

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In practice, one cannot generate a perfect impulse because this would require an infinite amount of energy; the best one can achieve is to emit a bandlimited impulse, resulting in a finite-amplitude wavelet whose time duration is small compared with any time interval of interest.

The Vibroseis source emits a bandlimited, expanded impulse; The band limitation has two aspects: at the low frequency end, it is dictated by the mechanical limitations of the system (the amount of travel or "stroke" of the reaction mass, and the amount of hydraulic flow that can be supplied by the pump) and the size of the baseplate. The high frequency limit is determined by the mass and stiffness of the baseplate, the compliance of.the trapped oil volume for a hydraulic vibrator and mechanical limitations of the drive system.

The notion of an "expanded" impulse can be explained in terms of the amount of energy per unit time, further referred to as energy density. In an impulsive signal, all energy is concentrated in a very short time period, leading to a very high energy density. In the Vibroseis method, the same amount of energy can be transmitted over a longer time, so that the energy density of the signal is reduced considerably. This reduction in energy density can be achieved by delaying each frequency component with a different time delay, while keeping the total energy contained in the signal constant. Thus, instead of emitting a signal with a flat amplitude spectrum and a zero phase spectrum, a signal is created which has the same flat amplitude spectrum in the bandwidth of interest, but has a non-zero phase spectrum. The frequency-dependent phase shifts cause time delays which enlarge the duration of the signal. However, the total energy of the signal is determined only by its amplitude spectrum. Since the amplitude spectrum of the expanded impulse is equal to the amplitude spectrum of the true, non-expanded impulse, in this way a signal is created which has the same amount of energy as is contained in the zero phase impulse.

The effect which the increased time duration of the emission has on the recorded seismogram needs to be eliminated. This can be achieved by controlling the phase function of the emitted signal. Then, the signal received at the geophone can be corrected for the non-zero phase spectrum of the source wavelet by performing a cross-correlation process. To clarify this point, let the source wavelet be denoted by s(t). If the convolutional model is adopted to describe the response at the geophone, x(t), the following expression is obtained in the absence of noise:

*(») = s(0 * S(0 , (1-D

where g{t) denotes the impulse response of the earth, and "*" denotes a convolution. The validity of the convolutional model for the Vibroseis configuration is discussed in Chapter 2. Transforming equation (1.1) to the frequency domain yields

X(co) = S(a) C(ffl) . (1.2)

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If the received signal x(t) is cross-correlated with the source signal s(t), the signal c(t) is obtained which, in the frequency domain, is given by

2 C(o))=X(fl))S*(©) =|S(©)| G(co) , (i.3)

since cross-correlation of x(t) with s(t) in the time domain corresponds to a multiplication in the frequency domain of X(a) with the complex conjugate of S(co). In this equation, the complex conjugate is denoted by the superscript "*". Since the amplitude spectrum of S(co) is flat over the frequency band, and zero outside this frequency band, it follows that by cross-correlating the measured seismogram x(t) with the source function s(t) the (scaled) bandlimited impulse response of the earth is obtained. The scaling factor is simply the (constant) amplitude spectrum of the source signal squared. In the time domain, the cross-correlated seismogram is a convolution of the earth's impulse response with the autocorrelation of the sweep. In Vibroseis applications, the measured seismogram x(t) is ususally referred to as "vibrogram", whereas the seismogram after cross-correlation, c(t), is denoted as "eorrelogram".

Therefore it is possible to emit a signal with a flat amplitude spectrum and an arbitrary phase spectrum, and still obtain the exact (bandlimited) impulse response of the earth, provided the phase spectrum of this source wavelet is known. It may be observed that this cross-correlation is merely a special deconvolution process, in which we exploit the fact that the amplitude spectrum is constant, thus avoiding the design of deconvolution filters.

This completes the general framework into which the vibrator's source signal is placed. The detailed shape chosen for the vibrator signal in practice is now described. A signal q(t) (usually referred to as the "sweep") is generated by the sweep generator of the seismic vibrator. The signal s(t) introduced in equation (1.1) is different from the sweep q(t), for reasons that are explained when the control óver the vibrator source is discussed in Section 1.3. The generated sweep q(t) has the general form

q(i) = a{t)sin [2*0( r ) ] . (1.4)

a(t) is a taper function which is usually chosen to be a linear or cosine roll-off taper, and typically has a length of 250 msec at each end of the sweep. The taper is applied only at the beginning and at the end of the sweep to reduce Gibbs' oscillations. The sweep typically has a duration T of 10 sec. The function 0(0 determines the frequency of the sine wave as a function of time; more specifically, the derivative of 0(0 is equal to the instantaneous frequency ƒ ""'(0 of the emitted sine wave:

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dm dl

= e-(o =/"'(o . (1.5)

As an example, a sweep is considered where the instantaneous frequency varies linearly with time, as illustrated in Figure 1.6:

r\t)=h+hzf±t, (1.6)

in which f0 is the starting frequency of the sweep, ƒ, its end frequency and T the sweep duration.

Figure 1.6 The instantaneous frequency of a linear sweep as a function of time.

This sweep with a linearly-varying instantaneous frequency is commonly referred to as a linear sweep, and is given by

qit) =a(f)sin[2ff(/o +jf-±j/°t )t) (1.7)

If the first sweep frequency fa is smaller than the last frequency ƒ,, the sweep is called an upsweep; if/0 is larger than/,, it is called a downsweep.

Although the relation between the sweep and its instantaneous frequency as a function of time is readily established, the Fourier transform of the sweep cannot be detennined analytically. At this point, a clear distinction must be made between the concept of "instantaneous frequency", which is the single frequency contained in the signal at a single time instant, and the concept of

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a frequency component as used in conjunction with a Fourier-transformed time function. In the latter case, the spectrum of q(t) can in general not be obtained analytically; however, Rietsch (1977a, 1977b) has shown using the method of stationary phase that the spectrum Q( ƒ) of q{t) can be approximated by

QV) = A(f) exp[-2ai(0(T)- /T) + i J ] , (1.8)

in which the different sign convention for the temporal Fourier transform adopted by Rietsch and the different sweep definitions (Rietsch uses a cosine function instead of a sine function to define the sweep) are taken into account. In this equation, T is the stationary point, given by the solution of the equation

0 •(*)-ƒ=() , (1.9)

and/if/) is given by

*(/>= a(T) ,7- o.io) 2 [ 0 " ( T ) ] 2

In equation (1.8), an upsweep is assumed; this, however, is not essential. The meaning of this equation can be illustrated using the previous example of the linear upsweep. For this signal, the stationary point T is given by

*=——lf-fo] . (1.11) / l 'JO

and it can easily be verified that the phase spectrum of the linear upsweep is a second degree polynomial in/. Since the second derivative of 6(t),

e r ( T > - ^ . a.n)

is a constant, the amplitude spectrum of the sweep is a scaled version of the taper function A(f), which is flat over most of the bandwidth. Sometimes a nonlinear sweep is taken as a source signal, for instance to account for frequency-dependent absorption (Hargrove et al, 1984).

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50 100

Frequency (Hz) (b)

S 150000 -

50 100

Frequency (Hz) (c)

Figure 1.7 An 8 sec, 10-100 Hz upsweep with a taper length of 250 msec, (a) the sweep in the time domain; the frequency range for this Figure is 1-5 Hz for display purposes, (b) the amplitude spectrum of the sweep, (c) the phase spectrum of the sweep, in degrees, and (d) the autocorrelation of the sweep.

In Figure 1.7 the concepts outlined above are illustrated for the example of a linear upsweep. An 8 sec, 10-100 Hz linear upsweep is used with a taper length of 250 msec. Figure 1.7a shows the sweep q(t). Because the oscillations in the sweep are too rapid to yield a clear picture, the frequency limits for this figure are 1-5 Hz. Figures Mb and 1.7c show the amplitude and phase, respectively, of Q(f). It can be observed from these figures that the phase indeed is a quadratic function of frequency, and that the amplitude spectrum of the sweep is constant over the bandwidth, apart from some Gibbs' oscillations at the beginning and end frequencies. Finally, Figure Md shows the autocorrelation of the sweep.

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1.3 The control over the Vibroseis source signal

1.3.1 Engineering aspects

The Vibroseis source is designed in such a way that the operator has, at least in principle, perfect control over the emitted wavelet. It follows from the previous discussion that knowledge of the source wavelet emitted by the vibrator is essential. The Vibroseis method presents the opportunity for this kind of signal control. This is achieved by the system shown in Figure 1.8.

feedback signal f(0

compensator inputforce modification

vibrator Pre-determined sweep input q(t)

Figure 1.8. The feedback system of a seismic vibrator.

The pre-determined sweep input q(t) (also called the pilot sweep) is compared with a feedback signal f(t), which is a signal measured on the vibrator. In principle, any vibrator motion that can be measured can serve as a feedback signal. The main function of the feedback system is to produce a feedback signal which resembles the pre-determined sweep as close as possible. Amplitude and phase differences between these two signals are corrected for by a change in the time-varying input force. This correction for differences between the feedback signal and the pre-determined sweep is necessary because the vibrator electronics can, for the example of a

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hydraulic vibrator, only control the oil flow in the hydraulic piston, which in general is not equal to the feedback signal chosen on the vibrator.

Until recently, the control system of the vibrator was designed to correct only for phase differences between die pre-determined sweep and the vibrator feedback signal. Since the early 1980's, amplitude control techniques have been in use which allow the vibrator to operate as close to the maximum holddown weight as possible without decoupling.

1.32 Geophysical aspects

In the previous section, the control of the vibrator source by means of a feedback system was discussed. It was shown that the problem of ensuring that the pre-determined sweep q(t) and the feedback signal f(t) are equal is an engineering problem.

The geophysical problem of the vibrator is the determination of the far field wavelet from measurements on the vibrator. For conventional geophysical applications, the wavelet s(t) that appears in the convolutional model is of interest. In the convolutional model, it is assumed that the wavelet s(t) does not change its shape while propagating, and is the same in all directions. The first demand, that 5(0 remains constant, implies that s(t) is the far field wavelet; the second demand, the source being non-directional, implies that the source dimensions must be small compared with a wavelength. The latter condition is usually satisfied when a single vibrator is considered. It should be noted that the term "far field wavelet" usually refers to the far field particle velocity, since in normal land seismic applications geophones are used for signal detection.

We intend to show that it is theoretically possible to calculate the far field wavelet from measurements on the vibrator. This calculated wavelet is called svib(t). The band-limited impulse response of the earth is obtained by cross-correlating the received seismogram x(t) with this far field vibrator signal svih(t). The frequency domain expression for the seismogram after cross-correlation, c(t), is thus given by

C{co)=X(co)Svib*(.co) = 5(w) Svib' G(co) . (1.13)

If s(t) and svib(t) are the same, the desired zero-phase impulse response g(t) is obtained. Whether they are the same or not depends on the validity of the theory.

1.3.3 Combined engineering and geophysical aspects

The two aspects of vibrator control that are discussed in the two previous sections can be summarized as follows:

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1. Engineering problem: make the pre-determined sweep q(t) and the feedback signal f(t) equal. 2. Geophysical problem: determine svib(t) from measurements on the vibrator such that svib(t)

is equal to the far field wavelet s(t).

In all present applications of the Vibroseis technique, the feedback signal f(i) is different from the far field vibrator signal svib(t). One may wonder why it is not normal practice to make f(t) equal to svib(t). There are several reasons for this discrepancy between feedback signal and far field vibrator signal:

1. There is still discussion about the actual shape of the far field wavelet s(t), and, consequently, about the choice of the far field vibrator signal svib(t).

2. The far field vibrator signal svib(t) can only be calculated approximately from measurements. 3. Often, only the phase information of the feedback signal is considered for source control

purposes. 4. Even if the far field vibrator signal is known, and can be determined exactly from

measurements on the vibrator, practical reasons may prevent the feedback signal f(t) from being chosen equal to the far field vibrator signal svlb(t). For example, it was stated in the previous section that vibrators are usually operated as close to the maximum holddown-weight as possible without decoupling, to maximize the energy output. This condition need not necessarily have any consequences for the control of the far field wavelet shape.

The possible discrepancy that may exist for practical reasons between the feedback signal f(i) and the far field vibrator signal svib(t) (point 4 above) is not essential; in the remainder of this thesis it is assumed that the feedback signal on the vibrator is chosen such that it attempts to predict the far field vibrator signal, and thus the far field wavelet.

In practice, the measured seismogram x(t) is cross-correlated in real time with the pre­determined sweep q(t). Only the resulting correlogram c(t), in the frequency domain given by

C(a>) = X(co) Q\co) , (1.14)

is recorded. It is thus assumed in practice that the feedback system of the vibrator ensures that the feedback signal f(t), which is a measure of the actual vibrator motion, and the pre­determined sweep q(i), which is the output of a sweep generator, are equal. One of the reasons for this assumption is that the feedback signal contains the effect of harmonic emissions due to the non-linear behaviour of the vibrator hydraulics.

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The question of which feedback signal should be used to predict the far field wavelet has been a controversial issue since it was first raised by Lerwill in 1981. To establish the correct feedback signal, two questions have to be addressed:

1. What is the relation between the shapes of the near field and the far field wavelet? 2. Given this relation, how can the far field wavelet shape be related to measurements on the

vibrator?

So far, three different feedback signals have been in use in Vibroseis surveys: baseplate acceleration, reaction mass acceleration, and a weighted sum of baseplate acceleration and reaction mass acceleration. All three methods produce different answers to the two questions stated above.

Using baseplate acceleration as a feedback signal, it is assumed that

1. The far field particle velocity is in phase with the near field particle velocity.

2. The near field particle velocity is equal to the baseplate velocity.

Using the reaction mass acceleration as a feedback signal, it is assumed that

1. The far field pressure or panicle velocity is in phase with the near field pressure. 2. The near field pressure is proportional to the reaction mass acceleration.

Using a weighted sum of baseplate acceleration and reaction mass acceleration as a feedback signal, it is assumed that

1. The far field displacement is in phase with the integrated surface traction. 2. The integrated surface traction is equal to a weighted sum of baseplate and reaction mass

acceleration.

1.4 The use of Vibroseis in seismic exploration

One may wonder why it is not normal practice in seismic exploration to use an impulsive source, since, after all, it is the earth's impulse response we are after. As can be seen in Figure 1.9, the most well-known impulsive seismic source, dynamite, is indeed used very often in land seismic surveys. There are, however, some distinct disadvantages related to the use of an impulsive source like dynamite.

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1962 1967 1972 1977 1982 1987

Year &0Ö3 Vibroseis I I dynamite

Figure 1.9 The contribution of Vibroseis and dynamite to the total number of crew months spent in land petroleum exploration, in %, for the years 1962-1987*.

First of all, due to the high energy density of the dynamite explosion, severe harm can be done to the environment. In any case, the destructive nature of the dynamite source prohibits its use in densely populated areas. Second, a hole has to be drilled for every shotpoint in which the dynamite charge is placed. Third, the high energy-density of the explosion results in a non­linear zone surrounding the explosion. Although the ignition time of the dynamite itself is short compared with any time duration of interest in seismic exploration, this nonlinear zone results in a distorted wavelet. It has been shown by Ziolkowski and Lerwill (1979) that the high frequency content of the signal decreases when the charge size is increased (the low frequency content increases). This yields a trade-off between penetration and resolution: a large charge size has better penetration, but lacks high frequencies. Another disadvantage of the creation of a nonlinear zone around the dynamite explosion is that effectively a wavelet is transmitted into the earth that is not an impulse, and that has a shape which is unknown and cannot be measured directly.

Foi 1962-1971, the figures for Vibroseis must be interpreted as maximum numbers, since only a division between dynamite and non-dynamite was made for these years. Also, marine surveys were included in the 1962-1973 figures, but the contribution of marine surveys never exceeded 10 % in these years. No figures were available for 1969. Data arc from the Geophysical Aclivity Reports in Geophysics (1963-1981) and Geophysics: The Leading Edge of Exploration (1982-1988).

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The Vibroseis source has some distinct advantages over the dynamite source. First, the emitted signal contains an amount of energy that is (roughly) comparable to the energy contained in a dynamite signal: Janak's (1982) results from a land seismic source study indicate that a 2.5 kg dynamite charge buried at 5 m depth delivers an amount of energy of 9.5 x 106

Joules, whereas a single Litton model 311 P wave vibrator emitting a 10-79 Hz sweep (the drive level is not mentioned in the report) delivers 1.2 x 106 Joules. Because of the use of an expanded impulse, the energy density of the source wavelet in the Vibroseis technique is much less than the energy density of the dynamite wavelet. Therefore, destructive effects are much less severe. Second, Vibroseis provides us with a direct means to measure and control the outgoing wavelet. Third, there is no need to drill holes when using Vibroseis.

There are, however, also some disadvantages connected with the use of Vibroseis as a source. First, a single vibrator in general does not deliver a sufficient amount of energy required for seismic exploration purposes, so that arrays of vibrators have to be used. Typically, 4 vibrators vibrate at each vibration point simultaneously (Figure 1.10).

Figure 1.10 A seismic survey using an array of 4 vibrators.

Second, as vibrators are surface sources, large amounts of Rayleigh waves are generated. The generation of Rayleigh waves can be suppressed in a dynamite survey by placing the charge at the bottom of the weathered layer. In Vibroseis surveys, the Rayleigh waves have a very high amplitude and are an undesired feature on the seismogram. Third, the Vibroseis method can be

-18-

employed only in areas which are accessible to the seismic vibrator trucks, whose weight may exceed 20 tons. Fourth, correlation noise limits the ratio between the largest and smallest detectable reflections.

In spite of all its disadvantages, the Vibroseis method is now a standard method in the seismic exploration for hydrocarbons. In 1987, Vibroseis was used more often in land seismics than dynamite (the contribution of Vibroseis to the total number of crew months spent in land seismic petroleum exploration was 49 %, whereas the contribution of dynamite was 48.3 %). The operational advantages of the Vibroseis method over the conventional dynamite survey result in an average cost per kilometre of Vibroseis which is only two-thirds of the cost per kilometre for a dynamite survey (figures for 1987). Also, the average number of kilometres that can be covered per crew month is 30 % higher for Vibroseis surveys than it is for dynamite surveys (figures for 1987). This cost-effectiveness and efficiency, together with the increasing importance of signal control in the search for higher resolution of seismic data and the non­destructive character of the method explains the increasing popularity of Vibroseis.

7.5 Objectives of the research in this thesis

In this section, the objectives of the research described in this thesis are explained.

The cross-correlation of the received seismogram with the sweep (which can be either the pre­determined sweep input q(t) or the feedback signal/(/J) does not yield a seismogram in which the earth's impulse response is convolved with a zero phase wavelet. The reason is that the sweep and the far field wavelet s(t) are different. This thesis is devoted to

(a) explaining why this is so, (b) what can be done about it, (c) tests on real data.

The contents of this thesis is now discussed in more detail.

The first issue that is investigated is the propagation of the wavefield emitted by the seismic vibrator into the earth. Two questions are of particular interest:

1. For forward modelling purposes, how can the wavefield at any point in the earth be described, given the vibrator motion?

2. What vibrator motion should the received seismogram be deconvolved with in order to obtain the desired impulse response of the earth?

-19-

These two closely related questions are investigated in Chapter 2, where expressions are derived for the wavefield in a plane-layered elastic medium, and for the wavefield in the far field of an elastic halfspace. It is shown that the far field displacement is essentially equal to the reaction force exerted by the earth on the baseplate, or, in other words, to the ground force.

Once this choice of the far field vibrator signal has been established, it may seem straightforward to define the appropriate feedback signal in terms of measurements that can be made on the seismic vibrator. However, the complexity of the vibrator behaviour prohibits a simple connection to be made between the far field vibrator signal (the ground force, or, more precisely, the time derivative of the ground force), and vibrator signals that can be measured in practice (the accelerations of the baseplate and the reaction mass). Measurements show that the traction directly underneath the baseplate is not distributed uniformly over the plate. The ground force is equal to the integration of the traction over the entire baseplate area. Since the ground force has to be determined from measurements on the vibrator, a thorough understanding of the nature of this non-uniform traction distribution is required, as well as a good model for the relation between the traction underneath the baseplate on the one hand, and the accelerations of baseplate and reaction mass on the other hand. First, a mechanical model of the seismic vibrator is developed, which is described in Chapter 3. Since the motion of the vibrator is a result of the interaction between the vibrator and the earth, the coupled earth-vibrator model is investigated in Chapter 4. In this chapter, the distributions of traction and displacement on the baseplate are investigated using different baseplate models. Also, the power output of a vibrator and the radiation impedance of the baseplate is discussed. Chapter 5 addresses the question of which vibrator motion yields the best estimate for the ground force. In this chapter, theoretical results are compared with measurements made on a seismic vibrator, and in a borehole containing a downhole geophone array.

The question of which feedback signal must be chosen on the vibrator is also important in the marine environment. Since the marine vibrator differs in many aspects from the land vibrator, a special chapter (Chapter 6) is devoted to the marine Vibroseis configuration.

•

-20-

Chapter 2 The wavefield of a surface source in a horizontally layered elastic medium

2.1 Introduction

A seismic vibrator is a surface seismic source which transmits waves into the earth by applying a force at the surface of the earth. In this chapter, the wavefield generated by an arbitrary distribution of tractions on the surface of an horizontally layered elastic earth is derived. It is not necessary to specify any characteristics of the vibrator; the theory can be applied to P and S wave vibrators, and to single vibrators as well as to arrays. In Chapter 3, the origin and nature of these surface tractions are specified.

Expressions are derived for the components of particle velocity and stress within or at the surface of a horizontally layered, elastic medium on top of which the surface source is acting. Each medium is assumed to be homogeneous, isotropic and perfectly elastic. The source is described by the distribution of traction components it generates at the surface, hereafter referred to as the surface traction components.

The well-known reflectivity method is used to derive the expressions for the wavefield in the medium. In this method, the propagation of the wavefield through the different layers is described mathematically by means of reflection matrices, containing the reflectivity properties of the individual layers as well as the propagation effects within each layer. The reflection matrix has to be calculated for each layer separately; this can be achieved by means of a recursive scheme. The reflectivity method is modified to include the surface tractions, resulting in an expression for the components of stress and displacement at any point in the medium in terms of the surface tractions. All derivations are made in the frequency - wavenumber domain.

The relations for an elastic half space are obtained from the expressions derived for the layered medium. These lead to expressions for the shape of the far field wavelet.

2.2 Configuration

A stack of A' plane elastic homogeneous isotropic layers is sandwiched between a lower halfspace and an upper surface on which the surface traction components are acting. Each layer j (J=\,...., N) has a thickness /( and Lamé elastic parameters A ■, u ■ and density p associated with it. The bottom of layer y' coincides with depth level .r3=.v, . The configuration is shown in Figure 2.1.

-21-

Bull Su hl

rfare /

Surface TraciionComponents

/ /

Layer 1 A, . /i , . p , *l

.1

Layer 2 A 2 , / i 2 , p2 h2

Layer N lK.fs,pN *v

Layer A'+ 1 AA . , . p M , p ^ . ,

Lwfr halfspace

— ' S . !

—"3.2

— x y A-i

- » 3 . A '

Figure 2.1 The eanh model, consisting of/V plane elastic homogeneous isotropic layers, bounded by a halfspace at the bottom and a surface at the top. On the surface, a distribution of tractions is present, as illustrated for points A and B.

2.3 Basic relations

The integral equations that relate surface traction components and the components of stress and particle velocity in each layer are derived from the equations that govern the wave motion. The relevant layer indices are, for the moment, omitted for simplicity; they are attached to the resulting equations in Section 2.5 when the coupling of the wave fields at the layer boundaries is included.

In the absence of body forces, the linearized equation of motion in the frequency domain is given bv

djTtj+pco t/, = 0 , (2.1)

-22-

in which 7^ denotes the stress tensor of rank two, p the mass density of the medium, co the angular frequency and Ui the displacement vector. The components of stress and strain are connected by means of the constitutive relation. For a perfectly elastic solid, the constitutive relation is

* ij — Cijpq kpq < (2.2)

where c^ is the stiffness tensor, and E is the strain tensor. For isotropic, homogeneous them

be written as

media, the number of independent elements in the stiffness tensor reduces to two, and c- can

C ÜP? = A 5ij 5Pq

+ »<>Sip 5k + 8iq SjP> • (2-3)

In the absence of volume sources, the strain tensor E is given by

E„ = ±(BpUq+dfJp). (2.4)

To solve these equations, the procedure is followed as outlined by Kennett (1974, 1979, 1983), Ursin (1983) and du Cloux (1986). First, the linear wave equations are subjected to a two-dimensional Fourier transformation with respect to the space coordinates x1 and x2- Then, the stress components which are not continuous across a layer interface (7"n , T12. ^21 anc* r 2 2 ) are eliminated. This yields the relation

33F = -ico E F , (2.5)

where F is a (6x1) column vector containing the field components:

F = [ F 1 , F 2 ] T , (2.6)

F ^ l V j , ? , , ^ ] 7 , (2.7)

F 2 = r 7 i 3 , P 1 , V 2 ] T , (2.8)

in which the superscript "T" denotes the transpose, E is the system matrix, \7,are the components of the particle velocity and 7",- are the components of the traction:

-23-

Ti = Tij rij = Ta (2.9)

The system matrix E has the following structure:

E = (2.10)

in which the submatrices E, and E2 are given by

E ,=

A+2/i

k>\ Xp2

A+2/i

Ap,

X+2/u

Ap2

X+2n 4/j(A+/i)p] 2

p W92 A+2jz

H(3X+2n)p]p2 P

X+2/i

H(3X+2n)plp2

A+2/J

4p(X+n)p2 2 W i

X+2p. X+2n

(2.11)

p

Pi

Pi

P\

1

0

P2

0

1

(2.12)

where

Pa=-co

(2.13)

2.4 Decomposition into upgoing and downgoing waveficlds

To solve equation (2.5), the wavefield is decomposed into upgoing and downgoing waves by diagonalizing the system matrix E. The diagonalization can be achieved by calculating the eigenvalues of the system matrix, and subsequently determining an independent set of eigenvectors. The decomposition is given by the equation

-24-

E = P Q £ P - (2.14)

In equation (2.14), Q£ is a diagonal matrix with the following structure:

Q £ = Q o o - Q

Q is a diagonal matrix containing the eigenvalues of the matrix E:

Q = diag^, y^y,),

with

rP.s=\—-p %

(Re,lm)yp s>0

(2.15)

(2.16)

(2.17)

and

2 2 2 P =P\+P2- (2.18)

cp and cs denote the velocity of the compressional and shear waves, respectively, and are given by

\X + 2H %

. c, = -%

(2.19)

The composition matrix P contains the eigenvectors of the system matrix. There is some freedom in the scaling of these eigenvectors, but the choice of a set of independent eigenvectors is by no means essential. Ursin's (1983) scaling is adopted because of the resulting simplicity of the inverse of the composition matrix. Ursin obtains

P = 4ï

Pi

P 2 - P i (2.20)

The components of Pj and P2 are found to be

-25-

p , =

M ,PI

Mï) :A

& -2p2nm

and

P,=

PM 2

'Ah P) I r / 2 | J _ :

JP-W Pi V*2 \c* I

p{pyfA\^-2P

-pApyfA

-pi{pyf/2

m

1 / \A 2W —

\PI

*f / \A

AP)

% PiMn

p\i \A

P4M'%

-PMnV\-XA M

(2.21)

(2.22)

Ursin (1983) derived similar expressions for the components of the eigenvector matrices, but his corresponding expression for Pj contains an error in the second and third element of the

- j - 2 p I is l

As a result of this choice of the eigenvectors, the following symmetry property is obtained:

second column; the term| 2p is missing

, - i 1 Pi (2.23)

Now the wavefield can be decomposed by applying the transformation

-26-

F = P \V . (2.24)

Substituting equation (2.24) into equation (2.5), and using that from equation (2.14) it follows that

Q £ = P E P , (2.25)

the relation

a 3 W = -iü> Q£W (2.26)

is obtained and, since Q £ is a diagonal matrix, the desired decomposition has been achieved. The vector W can be expressed as

(2.27) -

w = UP D

where

UP = [£//>,, UP2, UPj] (2.28)

is a vector containing the three components of the upgoing wavefield, and

D = [D1,D2,53]T (2.29)

is a vector containing the three components of the downgoing wavefield. Ü P and D do not correspond to physical quantities such as displacement and traction. Rather, they represent the upgoing and downgoing wavefield potentials, which makes a physical interpretation of these components difficult. When applying the boundary conditions at layer interfaces in a later section, these potentials must be transformed into the physical quantities panicle velocity and traction by applying the composition matrix P.

However, the introduction of these wave potentials leads to a simple solution of equation (2.26), given by

UP = A expH'wQA-3), (2.30)

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and

^ ^DOWN D = A e\p(icoQx3). (2.31)

^ UP ^ DOWN

Here, A and A are vectors containing the amplitude factors for the upgoing and downgoing wavefields, respectively, and Q is the diagonal matrix containing the vertical P and S wavenumbers (see equation (2.16)).

The wavefield in the layer can now be constructed from these upgoing and downgoing waves by applying the composition matrix P (equation (2.24)):

F = P W , (2.32)

^ ^ UP DOWN

where the vector \V contains the unknown amplitude factors A and A . These amplitude factors can be determined using the boundary conditions at the interfaces of each layer. In the next section it is shown that application of the boundary conditions at the layer interfaces relates the upgoing and downgoing waves at the layer boundaries by means of a scattering matrix containing local reflection and transmission matrices.

2.5 The local reflection of the wavefield at a layer interface

In the previous section, relations for the wavefield in each individual layer were derived using the procedure of Kennett. As the coupling of these wavefields across layer boundaries is now discussed, the relevant subscripts are attached to the field quantities and medium parameters to indicate to which layer they refer. This notation does not interfere with the use of the summation convention: in the remainder of this chapter, the summation convention is only used in conjunction with Greek subscripts, whereas the layer indices are denoted by Latin subscripts.

Assuming firm contact at each interface, the components of particle velocity and traction are continuous across each interface. Since the components of particle velocity and traction are contained in the vector F , this boundary condition can be expressed as

F;(bottom) = F;-+](top) , (2.33)

-28-

in which the index "bottom" indicates that the relation holds at the bottom of the relevant layer, and the index "top" refers to the top of the relevant layer. Using equation (2.24), the formulation of this boundary condition in terms of upgoing and downgoing waves is obtained:

PjWj (bottom) = P;+1W;+1(t0p) . (2.34)

Using this equation, the scattering matrix containing the local reflection and transmission matrices for the upward and downward going waves can be introduced, yielding

U P ; (bottom)

D;+i(top)

DOWN UP

DOWN UP v.y+i rJ.j+i

Dj (bottom)

UP j+1(top) (2.35)

This equation expresses the concept of upgoing and downgoing waves being reflected and transmitted at each layer interface, as illustrated in Figures 2.2 and 2.3.

Dj.(bottom)

UP,-+1 (top)

Laf-erj +1

r rS „ N DOWN* ,, . .UP rrp; U P ; (bottom) = Tj J + 1 Dj (bottom) + t;- ;+1 UP 7 + , (top)

Figure 2.2 The composites of ihe upgoing wavcficld al the bollom of layer j .

-29-

\

~ \ \ .DOWN \ hj+i

f rup^«v / r;.y+l \

/ UP,+1 (top)

R , . .DOWNfï D>+, (top) = t;- ;+1 D7

Layer]

Layerj +1

D7+1 (top)

(bottom) + ry,.,-+iUPj+i(top)

Figure 2.3 The composites of ihe downgoing wavefield at the top of layer j +1.

The reflection matrices in equation (2.35) are denoted "local" because they describe only the reflection and transmission response at a single layer interface, without accounting for the contribution of the other layers to form the total wavefield. These contributions are considered in Section 2.7, where "global" rather than "local" reflection matrices are introduced.

The elements of the local reflection and transmission matrices can be identified with the reflection and transmission of compressional (P), vertically polarized shear (SV) and horizontally polarized shear (SH) waves, as was shown by du Cloux (1986). They are given by

DOWN DOWN _ rPPJ.j+\ rPSV,j,j+\ u

DOWN DOWN _ rSVP,j,j+l rSVSV,j,j+] u

DOWN u u rSHSH.j,j+\

= r P L + i P i . ; + P l . , + i P 2 . , r 1 [ P l . / + i P 2 . J - P l y + i F i , , ] . (2.36)

which denotes the local reflection matrix for the downgoing waves incident at the interface from layer/,

DOWN

-30-

UP r ; .>+i "

UP UP „ rPP,j,j+) rPSV,j,j+l U

UP UP „ rSVP,j,j+l rSVSV.j,j+\ u

„ „ U P u u rSHSH,j,j+\

.PL-PI .M + PT.>P2.M -1 T

] [ P ; , P T

(2.37)

which denotes the local reflection matrix for the upgoing waves incident at the interface from layer y+1,

DOWN 7.7+1

0 DOWN DOWN tpPJJ+l lPSV,j,j+\ DOWN DOWN „

tsvpj.j+i tsvsv.jj+i u

0 0 DOWN 'silSll.j.j+i

2 [ P l y P ] . J + 1 + P1 ' ,yP2,;+1]"1 .

(2.38)

which denotes the local transmission matrix for the downgoing waves incident at the interface from layer), and

UP V. 7+1

UP fpp. 7. UP

'SVP, j

0

7+1

j'+i

UP 'PSV, j , UP

lSVSV, j

0

7+1

.7+1

0

0 UP

hnsii. j 7+1

2[P2 ,y + lP l . ;+P l , ; + lP 2 . 7 ]

(2.39)

which denotes the local transmission matrix for the upgoing waves incident at the interface from layer y+1. In these equations, use has been made of the symmetry properties of the inverse of the decomposition matrix (equation (2.23)).

2.6 The propagation of the wavefield within a single layer

The above equations describe the behaviour of the wavefield when it strikes an interface. Before the wavefield in the medium can be determined, propagation of the wavefield within a layer also has to be described. This relation, which is a simple phase-shift operation, can be easily inferred from equations (2.30) and (2.31), yielding

-31-

e\p[-i(oQj(x3-x3)]

t\p[-iü)Qj(xj-x3)] UPj(4) (2.40)

where Q is defined in equation (2.16), and in which x3 and x3 denote the x3 coordinates of two points that are located within layer). With the aid of the equations derived in this section, the recursive formulae can be calculated, to facilitate the determination of the wavefield at any point in any layer.

2.7 The global reflection response

2.7.1 Backward recursion

Now a backward recursion procedure for the problem is performed according to the standard reflectivity method. A global reflection matrix is defined that relates upgoing and downgoing waves at the bottom of a layer j+1:

— up ^ UPy+1(bottom) = R;-+1 Dy+](bottom) (2.41)

and thus (using equation (2.40)),

UP UPy+](top) = exp(io>Q;+,/iy+j) R ;+, exp(ifi>Qy+1/i;+1) Dy+1(top). (2.42)

The boundary conditions at the interface of layers; andj+1 are the continuity of the three components of particle velocity and traction. These are expressed by equation (2.34), which can be written as (see equation (2.35))

-—- DOWN ~ UP -~~^ UPy (bottom) = r, y+] D ; (bottom) +ty y+1 UP;+1(top)

v DOWN *"* UP "—"" D,+ )(top) = ty,y+1 Dy(bottom) + ry y+1 UPy+,(top) .

(2.43)

(2.44)

Combination of equations (2.42), (2.43) and (2.44) yields

-32-

.TS „ x f DOWN „UP .. „ . VT>UP .. ry , „ U P ; (bottom) = ( r ; . J + 1 +tyii+1expOö)Q;+,^+i)Ry+1exp(ift)Qy+i/i>+1) x

r UP UP ~\ DOWN 1 ""* x [I-rJiy+iexp(jwQJ+]/!J+i)R;+1exp(iüjQJ+1/i;+1)J t y ; + I j D, (bottom), (2.45)

up so that the global reflection matrix of thej'th layer, R, , can be determined using the recursive formula:

„ U P DOWN .UP ,. „ , . „ U P .. „ , . Rj = Th /+i + lJ. y'+i expaö)QJ+1AJ+1) Rj+1 exp(iCt>Q,+1/i;+1) x

r* UP ,■ ^ . V„UP ,• /-, , a „DOWN X [ ^ • " y . y + l ^ P ^ Q M ^ + ^ ^ + l ^ P f ' ^ Q y + l V ^ J Vw+l ' (2-46)

The recursion is initialized by using the knowledge that there are no upgoing waves in the lower halfspace, and thus

R ^ ! = 0 • (2.47)

2.7.2 Forward recursion

In the forward recursion, the source is included by accounting for the surface traction components at the top of the configuration. This is why a new formulation of the forward recursion scheme is needed, which differs from the recursion that can be found in the literature, where either a halfspace or a free surface is present at the top of the configuration (e.g. du Cloux 1986).

From equation (2.24), it follows that

Tj^BjVPj+A^Dj , (2.48)

in which T ■ is a vector containing the three components of the traction,

T i =( r 3 J , r 1 < , - ,T 2 ) / ) T . (2.49)

the matrix B ,• is given by

-33-

B'-7?

, 1

1.7 (2.50)

where P ■ denotes the q row of matrix P „ ,, and

1 V2

~ r 2 J P 2

p 3

(2.51)

Equation (2.48) is applied at the surface of the configuration to obtain a relation between the surface tractions and the upgoine and downgoing wavefields. First the vector ST is defined, containing the traction components at the surface of the earth:

ST = ( 5 7 3 , 5 7 - , , ST2)7 = ( T ^ T , , f 2 ) * „• (2.52)

Since the vector ST contains all characteristics of the surface source, the wavefield needs to be expressed in terms of this vector. For this purpose, equation (2.48) is applied at the top of the configuration, and subsequently rewritten as

Dj(top)= - A i B 1 U P 1 ( t o p ) + A , S T , (2.53)

in which

A,=-V2

/i,4 ] " i

y -!__•> 2 %

--pyP,Mr,^ '-** -ip \CsJ I

Pi

-2P2Ys,i(Pi7P,J Vl

ViPjJL--** (p^y^-pH-j—zp \(pjsj/2 %

■yC/'.r,,/^.4.

(2.54)

and

-34-

A,B,=

A,

-(-i— 2p2\ + *p\A7sA -4P(-L-2P2\YPI1YJ'

Vs.l ) \Cs,l 1 ,]/2

-4p|-i— *pyp.xr,.y2 i-r-V| - V r „ 7 , , *.1 / \ C s . l

i , A % / i 1/ ; , .»2 2\ . 2

* 1

(2.55)

where A1 is the so-called Rayleigh denominator, and is given by

A\ = {-T~ 2p2\ +Ap\V'.i- (2-56)

The components of the slowness vector that appear in these expressions are defined in Section 2.4. In view of the structure of equation (2.53), the global reflection matrices relating the downgoing and upgoing waves at the top of layer j are defined as:

^ nnwN -—- ST -—• D, (top) = R; U P; (top) + R f A, S T . (2.57)

ST The matrix A] is not included in the reflection matrix Ry because Aj is dependent on both px and p2, and not simply on the slowness p. In the transformation to the space domain, the

DOWN ST separation between A] and the reflection matrices R,- and R, (which depend only onp) is essential when cylindrical coordinates are introduced.

Using the same procedure as applied in the backward recursion ( applying the boundary conditions at each layer interface using equations (2.43) and (2.44), and successively eliminating D • (top) and U Py (top)), the following recursive scheme is obtained:

„DOWN .DOWN , . , . . . „ D O W N .. „ , . R>+, =«j,y+i expOo)Qjhj)Rj expQcoQ/ij) x

r T DOWN .. „ , .„DOWN .. , . , . -i~ IT UP X L I _ rJ.M cxp(i(oQjhj)Rj txpdwQjhj) \ <;.^i + r/.>+i' (2.58)

-35-

D S T .DOWN , . _ , . f , „DOWN .. „ , . R J + 1=tyy+ , txp(icoQjhp [ I +R, expOwQjhp

. ST x [ l - r^avQuQjhpR^ctXiwQjhp] r™» exP(/a> Q, h]) } R f

(2.59)

The recursion can be initialized using equations (2.53) and (2.57):

„DOWN R i = - AiB] , (2.60)

and

ST R i = I (2.61)

2.8 The particle velocity and traction components at an arbitrary point in the medium

To obtain the wavefield at any given point in the medium, the forward and backward recursion have to be coupled to each other at the given depth level. As the expressions derived so far are all in the wavenumber domain, the horizontal spatial coordinates of the observation point only enter the calculations when the wavefield is transformed back to the space domain. An expression is wanted for the wavefield in the k layer, at depth X3 * . The vertical distance of the top of layer k to the observation point is denoted by hk :

, obs obs hk = x2.k - * 3 , * - i - ( 2 . 6 2 )

as illustrated in Figure 2.4. From the backward recursion, the following expression is obtained for the upgoing wavefield

at depth x? J (equations (2.40) and (2.41)):

T T S , obs . „ , ofc.s UP , . „ ., .obs■ * obs. , , , „ UPA(.v3it) = exp[ /oQ t ( / i t -A t )]Rk exp[icoQk(hk-hk )]Dk(x3,k) . (2.63)

-36-

Observation point , , obs

depth x3k

, obs hk

, , obs hk~hk

Layer k - 1

Layer k

Layer k + 1

c3.*-l

c3.*

Figure 2.4 The observation point in layer k, and the notation associated with its depth level.

obs From the forward recursion, the corresponding downgoing wavefield at depth level x3 k is obtained (equations (2.40) and (2.57)):

£i , obs ^ r . .~ , obs, „DCWN . . „ ,obs,rTt^ , obs. Dk(x3,k) = txp[icoQkhk ]Rk txp[icoQkhk ]UPk(xXk) +

+ exp[icoQkhlbs]R!?A1ST. (2.64)

obs From these two equations, the upgoing and downgoing wavefields at depth level x3<k, --* ^ obs ^s> obs " **' U P* (x3j(k) and Dk (x3 k ) , can be expressed in terms of the surface traction vector ST:

T>k(xl*ks)=[l-exp[i<oQkh°k

bs]Rk ™ exp\io>Qkhk]Rk txp[icoQk(hk- h°kb*)] ] x

x txp\ia>Qkh0k s]Rk A] ST, (2.65)

and

-37-

u p i (*3.t) = expL/6)QA Vik-lt°k")] R)ïFzxp[i(üQk (hk- hf")] x

x [ i - e x p l / ü J Q i / ï f ' l R ^ ^ ' e x p l / C ü Q ^ J R ^ e x p t / ö Q ^ / ^ - ^ ^ ) ] ] x

x exp[iü)QA / if*]Rf A,ST. (2.66)

Symbolically, these equations can be written as

D t ( ^ ) = R D A 0 « : ^ ) A 1 S T , (2.67)

and

Ö P ^ U ) = RL' t Cx-3^) A, ST , (2.68)

in which the global downgoing and upgoing reflection matrices RD^and RUj. are introduced, and are given by

- l _ _ obs. r . r- r\ i °bs, _ DOWN r . .-. , Tr-lfP ,. „ ., , r f i . , l

RD*U3.t)= [ I - e x p [ i w Q t / i t ]R* expDcoCMJRt expliü)Q t(/i t-/i t )J J x

x exp[ /a)Q t / ! f s ]Rf , (2.69) and

R U t (*3?/) = expfüuQ* (/(*- / / f ■')] R^expfiwQi ( / i t- /if*)] x

x [l-exp[/üjQA/j^ 5 ] R t ' txp[iwQkhk]Rk txp[icoQk (hk- h°k ")] ] -ï

x

x exp[/ö)Q t / jf- ']Rf , (2.70)

respectively. The global downgoing and upgoing reflection matrices RD t and K U t directly relate the upgoing and downgoing wavefields to the surface traction components.

-38-

Now the wavefield in terms of its particle velocity and traction components can be determined by applying the composition matrix P to equations (2.67) and (2.68). This composition, expressed by equation (2.24), results in

F~ / °*J\ r O < obs. if, , obs. -2, , obs ^-7 i(*3.*) =iV3(x3,k),T1(x^k),T2(x3k)]

= = P l t 4 [ R U t C x # ) + R D t C x # ) ] A , ST , (2.71)

F" , obs s r £, , obs. f) . obs. rj , obs. , T 2 C*3.t ) = [ r 3 Cx3.t) > vi C*3.*). v2 C*3l*) ]

= P 2 . t [ R U 4 ( x ^ ) - R D J l ( x ^ ) ] A 1 g T . (2.72)

From equations (2.71) and (2.72), it follows that the components of particle velocity and traction at any depth level, in the wavenumber-frequency domain, are given by the multiplication of the surface tractions, and a matrix that will be referred to as the Green's matrix G „ ( a = l , 2 ) . Thus,

r i obs \ p! i obs \ cr Fa(*3 ,*) = G a (x 3 i i t )ST, (2.73)

in which the Green's matrices are given by

G ,(*36;) = P U [ R U k ( x f j ) + RDk(x°3

bks)] A , , (2.74)

and

G2(x?Sk) = y= P2>* t RUtOr^) - RBk(xf;) ] A , . (2.75)

Expressions are now derived for the components of particle velocity and traction in the spacc-frequency domain in terms of the tractions acting at the surface of the medium. The reflection matrices appearing in these expressions are computed by means of the recursive scheme described above.

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2.9 The transformation to the space domain

In the previous section, it is shown that the components of particle velocity and traction in layer k at depth level .r3 k' , contained in the vectors Fj and F 2 , can be written in the frequency-wavenumber domain as a multiplication of a Green's matrix and the traction distribution at the surface of the earth:

FB(*3Ü') = G a ( x # ) S T , (2.76)

in which

p . obs. ~ obs C; obs C; . obs. , T F lfr3,* ) = f V3C*3.* ) . Flfr3.it ) - Tl(x3,k ) J . (2,77)

ifr3.*) = P i , J R U * f r 3 , * ) + R D * f r 3 , i ) ] A i . (2.78)

F2fr3,t ) = [ F3fr3,* ) , Vi(x3 i t) , V2fr'3,* ) ] , (2.79)

G jfrf*) = yy P 2.* [ R U*fr3ft ) - RD t(i3 J;) ] A j , (2.80)

and

ST = [ 7 3 , 7 ] , 7 2 ] a l ^ 3 = 0 • (2.81)

The global reflection matrices relating the downgoing and upgoing waves to the surface tractions, RD^.and RU^, respectively, are given by equations (2.69) and (2.70), in which the global reflection matrices can be computed by a recursive scheme given by equations (2.46), (2.58) and (2.59). The remaining two matrices, P„ k and A,, depend only on the medium

parameters of layer k and layer 1, respectively. They are given by equations (2.21), (2.22) and (2.54).

Since a multiplication in the wavenumber domain corresponds to a convolution in the space domain, the components of traction and displacement at position x{ are given by

ob ? f°° [°° r, , obs obs obs F a ( x * ' ) = l I C'u(.x3.k'xl,k-xl'x2,k-x2)SJ(x1,X2)dx1d.X2 , (2.82)

-40-

where the Green's matrices in the space domain follow from

n . obs obs obs « <x(*3,*> x \,k ~ xh x2.k~x2 > ~

= (2K)~ J J Ga{x03_s

k,ki,k2)cxp[ikp(x°p^-xp)]dkldk2. (2.83)

To appreciate the symmetry present in the configuration, and to exploit this symmetry to arrive at a more elegant wavenumber-space transformation, the following substitutions are made. First, since the Green's matrices are derived in terms of slownesses pa, the substitution

ka=capa (2.84)

is made, and the analysis is restricted to positive frequencies. The time domain expressions can be obtained from Appendix A, equation (A.47). Then, the cylindrical symmetry in the earth model is accounted for by substituting

\p\=P cos(£) \ . - , • (2.85) | p 2 = psinCr)

2 2 ' / wherep = (p} + p 2 ) » an^L

x°,k~xi = rcos(0) obs . tA (2.86)

x2,k~x2= ''Sin(0)

in which r= [ (x°ks-X\) + (x2,k~x2) ] T h e notation employed for the space

coordinates is illustrated in Figure 2.5. As the desired integration over the angle % cannot be recognized if the recursive scheme

depends on X' it is first noticed that, as could be expected for a cylindrically-symmetric earth model, the reflection matrices are indeed independent of X- Furthermore, it can easily be verified that the symmetry present in the local reflection and transmission matrices is also present in the reflection matrices RUt and RDt:

-41-

RU, = | Ru pp Ru psy O RU SVP RUSvsv O

O O RU SHSH (2.87)

and

RD PP RD, RD svp

RDpSV O RDsvsv O

Dr, 'SHSH

(2.88)

Earïh Surface

Layeri

Layer k ■. 1

Figure 2.5 The intioducüon of cylindrical coordinates.

-42-

The correspondence of the global reflection matrices with the compressional (P), vertically polarized shear (SV) and horizontally polarized shear (SH) waves is extensively discussed in du Cloux (1986). From equations (2.87) and (2.88), it can be seen that the SH waves do not interact with the P and SV waves. This property is exploited by splitting up the wavefield into two parts: a part that can be identified with horizontally polarized (SH) shear waves, and a part that can be identified with compressional (P) waves and vertically polarized (SV) shear waves. Two Green's matrices G a and G are therefore defined, which satisfy

„ , obs a. „PSV obs a. „SHSH obs . . Ga(x3k,r,6) = Ga (x3ik,r,0) + Ga (x3tk,r,6) . (2.89)

From equations (2.74)-(2.75) for the Green's matrices in the wavenumber domain, it follows that the X dependence of the Green's matrices is contained entirely in the matrix Aj. Because the (recursively calculated) reflection matrices do not depend on X> the integration over X can now be performed analytically, yielding

Ga U ^ , r , 0 ) = { G„ {x03°k

s,p,r,e)dp , (2.90) ■'o

and

GrV£V,0)= f~ GS"\03bk\p,r,e)dp . (2.91)

•'o

Explicit expressions for the components of the Green's matrix are given in Appendix A.

2.10 The elastic halfspace

Use will frequently be made of the relations between the different components of the wavefield and the surface tractions in an elastic halfspace. Expressions for the elastic halfspace are now obtained from the results derived in the previous section. From equations (2.69)-(2.70), it can be verified that when no contrast is present in the medium, the global reflection coefficients RU, and RDj at depth level x3 are given by

/ R U 1 C X 3 * ' ) = O

] R D 1 ( . ^ ) = exp(/o;Q1xf) ( 2 9 2 )

-43-

Substitution of these values in the expressions for the Green's matrices for the layered elastic medium (see Appendix A, section (A.2)), and omitting all layer indices, yields

_ • , obs. f ° ° f °° „ . obs obs obs F a(x ) = j J G a (x 3 ,x] -xhx2 -x2)ST(xh x2)dxldx2 , (2.93)

where, applying the substitutions (2.85) and (2.86), the Green's matrices follow from

U a U3 ,r,ti)-(ja (x3 ,r,V)+\j,a (x3 ,r,o) , (2.94)

with

r,PSV obs „. f " ^PSV, obs . . . G„ (A3 ,r,0)= G a (A-3 ,p,r,8)dp , (2.95)

•'n

and

G„ (A3 ,r,0)= G a (.v3 ,p,r,6)dp . (2.96)

Again, explicit expressions for the Green's matrix constituents are given in Appendix A.

2.11 The horizontally layered acoustic medium

The earth is often approximated as an acoustic medium, although in reality it is elastic. This approximation, which basically assumes that the effects of shear waves and associated wave conversions can be neglected, leads to a ver)' simple formulation of the problem, reducing the reflection matrices derived for the elastic medium to scalar variables, and also reducing the number of field quantities to be taken into account from 6 in the elastic case to 2 in the acoustic case (namely V'j, and 73 which denotes minus the pressure in the medium). A short review of

the wavefield in the acoustic medium is now given to indicate the changes to be made in the vectors and matrices used in the derivation of the wavefield in the elastic medium.

First of all, the dimensions of all matrices and vectors used in the elastic case are reduced in the acoustic case: (3x3) matrices become scalars, (6x6) matrices become (2x2) matrices, (6x1) vectors become (2x1) vectors and (3x1) vectors become scalars.

For the acoustic case, the vector F is defined as

-44-

F = [ F 1 ) F 2 ] T1 (2.97)

where now

F 2 = ?3 ■ ( 2-98)

The elements of the system matrix E are found to be

(2.99)

The eigenvalue matrix in the acoustic case is a scalar containing the vertical P-wavenumber.

0=7 , , . (2.100)

and the elements of the decomposition matrix P, containing the eigenvectors of the system matrix, are scaled in such a way that the symmetry properties valid in the elastic case also hold in the acoustic case:

(2.101)

Performing the same steps used when calculating the response for the elastic medium, the global reflection coefficients for the acoustic case are given by the following expressions:

DOWN DUI'

'' = , w DUP. „ . — ' (2-102)

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UP „DOWN ^ D O W N , rjj+i+Rj exp(.2ico<ypJhj)

i+l . D O W N _ D O W N • . . . ' (2.103)

and

DOWN , . , , . S T ST_ fo+i expdcoYpjhpRj ;+1 , DOWN' DOWN ... " (2.104)

l-rjJ+1 Rj exp{2ia)ypjhj)

These recursions are initialized by the conditions

R N + 1 = 0 „ DOWN , * i = ' • (2.105)

*f =1 The components of the wavefield in the medium in the space domain, contained in the vector F , are now given by

_ . Obs. f°° f°° r-AC, obs 0bs °bs ^ rrr, -. J j FcMk ) = ) I Ga Cv3jl, xhk-x], x2 Jl-x2)ST(xhx2)dxldx2 , (2.106)

where 5 7 is a scalar containing the vertical component of the traction at the surface:

ST = T2(x3 = 0). (2.107)

In reality, surface seismic sources cannot exist at the surface of an acoustic environment. The application of the acoustic model is therefore restricted to an approximation of the elastic case, in which wave conversions and their effect on the unconverted P wave are neglected.

The Green's functions, after the introduct ion of polar coordinates, again follow from an integration over the radial wavenumberp:

AC obs 0 , f°° - S . A C . obs „. .

The superscript "AC" is used to indicate that the Green's functions refer to an acoustic medium. Explicit expressions for the acoustic Green's functions are given in Appendix A.

-46-

2.12 Far field relations

Far field relations are obtained for the wavefield in an elastic halfspace. The far field particle velocity in the elastic halfspace can be obtained by taking the observation point to infinity. The steps in the computation are summarized as follows:

1. Starting from the integral representation for the wavefield in cylindrical coordinates (equations (A.68)-(A.71)), the contour for the calculation of the Green's matrix constituents, corresponding to an integration over all radial slownesses p, is deformed. The resulting integral no longer suffers from a singularity at the Rayleigh wavenumber; the contribution of the Rayleigh wave now follows from a separate, analytic expression. What remains, then, is to calculate the far field body wave contribution.

2. The Green's matrix (the resulting integral over all radial slownesses) can be approximated by integrating over the so-called "path of steepest descent" (Bath and Berkhout 1984), as the observation point tends to infinity. This contour deformation allows an analytic expression for the far field Green's matrix constituents to be obtained.

3. The spatial convolution of the surface tractions with the far-field Green's matrix constituents can be written in a more convenient form by introducing cylindrical and spherical coordinates, respectively.

The notation used in the following is illustrated in Figure 2.6. The final result is that the surface wave contribution in the far field, expressed in terms of its

cylindrical particle velocity components, is given by

exp[/ (r0kR--)] V, = A n co% f f rrexpt-ifc/; r, cos(t^) ]dA +

'A ■ Js

txj>[i{r0kK )] + Ari y aft}} T2txvl-ikRrtcos{\i/)]dA , (2.1091

cxp[i(r0kg )] VB= A0e coAJj Teexp[-ikflrlcos(yï)]dA , (2.110)

-47-

and

V 3 = ^ 3 r

txp[i(r0kK )] 4

co%\j Trexp[-ikfirtcos(\ir)]dA +

+ A

exp[i'(r0*rt )] 4

33" w3/2 J J T3 exp[-il'^j r, cos(y/) ] d/i . (2.111)

Figure 2.6 The notaiion used in the derivation of the far field relations.

-48-

In these equations, Arr, ArV Age, Ajr and v433 are simple amplitude scaling factors (see Appendix A). S denotes the area on which the surface tractions are acting (the remainder of the surface is assumed to be traction-free), and kR represents the Rayleigh wavenumber:

k* = TR' (2.112)

in which cR denotes the velocity of the Rayleigh waves. The body wave contribution in the far field, expressed in its spherical particle velocity components, follows from

exp(/«o*o) {[ VR = -iwARr -^ £- JJ Trcxpl-ikprtcos(y)] dA +

txp(iR0kD) rr -icoAR3 j p— ]\ r 3 exp[-ikpr,cos(y/) ] dA , (2.113)

exp(iRnk,) rr V^= -ia>A fy VK

R ° s' j J r rexp[-j*, r, cos(y) ] dA +

txp(iR0k.) rr -iü)A p ^ J J ^3 txp[-iks r, cos(y/)} dA , (2.114)

and

exp(i/?0*j) cxp(iKnKs) rr Ve=-io)Aeg ^ JJ Tetxp[-iksrlcos(,y/)]dA , (2.115)

where A Rr, AR3, A^., A^3 and A ee are frequency-independent directivity functions, and kp

and ks denote the P and S wavenumber, respectively. Explicit expressions for these directivity functions can be found in Appendix A.

As can be seen from equations (2.109)-(2.111) and (A.108)-(A.112), the surface waves decay exponentially with depth, and with the reciprocal of the square root of the radial distance

'/, from the source. The amplitude increases with frequency as a» .

The contributions of the individual surface traction components to the respective body wave components of the far field particle velocity (equations (2.113)-(2.115)) consist of five terms: a differentiating term, a trivial phase delay, a 1/R scaling, a frequency-independent directivity function and a surface integral over the surface traction component, weighted with an exponential function. As pointed out in Appendix A, this exponential factor in the surface integrals becomes unity when the dimensions of the source are small compared with a

-49-

wavelength, as is usually the case for a single vibrator. Then the far field particle velocity components are given by the time derivative of a weighted sum of the integrated surface traction components, in which the weighting factors are frequency-independent directivity functions. The integrated surface traction components represent the components of the ground force F c . The ground force is the force exened by the earth on the baseplate of the seismic vibrator, and is equal and opposite to the integrated surface traction. In the absence of shear stress at the surface of the medium, the radial and tangential components of the far field particle velocity, corresponding to the propagation of P- and S waves, respectively, are given by

VR = ARI\\ ~'0)T3dA (2.116)

and

' 4 = A$fj -UoT3dA (2.117)

in which the trivial phase delay and the 1/R scaling are omitted. The behaviour of the directivity functions ARi and A g$ as a function of £, is illustrated in Figure 2.7.

Angle (degrees!

IC 40 6C

Angle (degrees)

Figure 2.7 The directivity functions AR3 (solid line) and A s(dashed line) for the far field particle velocity, as a funciion of the angle £ with the vertical, (a) amplitude, and (b) phase in degrees. The earth model has a P wave velocity of 400 ms-1, an S wave velocity of 150 ms-' and a density of 1700 Kgnr3.

-50-

The directivity function for the compressional wave, ARi, is zero when £ = 90°. The second

zero for £= arcsinj S 1 cannot be present since it corresponds to a zero Poisson's ratio.

Further, it is observed that ARi is real and negative. The directivity function for the vertically polarized shear wave, A , is zero when

£ = 0° or £ = arcsin — . (2.118) #

Since the shear wave velocity of the medium is always less than the compressional wave velocity, the directivity function corresponding to the propagation of shear waves always has two zeroes. Also, A £3 is real and positive for small angles of incidence, but becomes complex when

$ > arcsin— . (2.119)

In conclusion, in the absence of shear stress at the surface, the far field particle velocity components are given by the time derivative of the integrated surface traction. Therefore, it is the time derivative of the ground force that is needed to determine these components. The polarity of the far field panicle velocity components can be inferred from Figure 2.7 and equations (2.116) - (2.117): the radial component has the same polarity as the ground force, and the tangential component has opposite polarity (the ground force is positive downwards). Returning to the convolutional model described in Chapter 1, it follows that the convolutional model holds for the Vibroseis configuration if the wavelet s(t) is chosen to be the time derivative of the ground force. However, some restrictions apply to this conclusion. The ground force (or, more precisely, the time derivative of the ground force) is no longer representative for the shape of the far field particle velocity components if:

1. the dimensions of the source are not small compared to a wavelength. In practice, this occurs

a. if vibrator arrays are used, or b. if a single vibrator is operated at high frequencies, or vibrating on a low-velocity medium.

In either case, the radiated wavefield has a frequency-dependent directivity pattern, and the earth impulse response can only be restored by applying directional deconvolution. The far field panicle velocity emitted by a vibrator array can be obtained by vector summation of the ground force contributions of the individual vibrators.

-51-

2. the angle of incidence is not small. In this case, the phase of the far field shear wave is distorted with respect to the ground force due to the frequency-independent shear wave directivity factor, which becomes complex after a certain critical angle. This restriction may not be quite as severe as the previous one, since it is normally the shape of the wavelet at near-vertical incidence that is of primary importance in seismic exploration.

The same reasoning can be applied to the configuration where only shear stresses are acting on the surface. In this case, the ground force is given by the integrated shear stress at the surface.

2.13 Conclusions

In this chapter, expressions have been derived for the calculation of the wavefield at any point in a layered elastic medium, due to the presence of a traction distribution at the surface. To obtain these expressions, the standard reflectivity method is modified to include the presence of these surface tractions in the model. Using this approach, the wavefield in the medium can be obtained by a spatial convolution of the surface tractions and a Green's matrix. The Green's matrix can be obtained by a recursive scheme in the wavenumber-frequency domain.

Using the equations for the elastic halfspace, it has been shown that the far field panicle velocity components are equal to the time derivative of a weighted sum of the ground force components, in which the weighting factors are frequency-independent directivity functions. Therefore, the time derivative of the ground force should be used as a feedback signal for the seismic vibrator. However, this result is no longer valid if the source dimensions are not small compared to a wavelength (that is, if an array of vibrators is used instead of a single vibrator, or if the frequency of operation of a single vibrator becomes large), since then the wavefield has a frequency-dependent directivity pattern.

for the amplitude The amplitude spectrum of the ground force is required to behave as

spectrum of its time derivative to be flat. Therefore, the amplitude of the ground force should increase towards the low frequency end. This increased output at the low frequencies can be achieved (a) by using vibrator arrays, and (b) vibrate for longer at the low frequencies (see also Anstey 1980).

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Chapter 3 The seismic vibrator

In this chapter, the distribution of traction and displacement directly underneath the baseplate of a seismic vibrator is investigated. For this purpose, a vibrator model is developed which consists of two parts: (a) a mechanical model which describes the dynamic input force acting on the baseplate, and (b) a model which describes the behaviour of the baseplate under the combined action of this dynamic force and the reaction force of the earth.

In the mechanical model of the vibrator, the motions of the reaction mass, holddown mass and baseplate are related to a model containing springs, masses and a dashpot. Using this model, an expression is obtained for the force exerted by the vibrator's drive system on the baseplate.

For the baseplate behaviour, several models are discussed, each model involving different assumptions about the baseplate properties. The conventional assumptions of uniformly distributed traction or uniformly distributed displacement, both directly underneath the baseplate of the seismic vibrator, are evaluated. Field measurements show that the distributions of traction and displacement under the plate are non-uniform. Two new models which yield non-uniform distributions of both traction and displacement are presented. The first of these assumes that the applied force is distributed uniformly over the baseplate. The formulation of this model then follows from the condition that the total force acting on the plate is identical with the inertial force related to the acceleration of the mass of the baseplate. The second new model contains a more rigorous treatment of the plate stiffness; the analysis of the plate behaviour in this model is referred to as the flexural rigidity method. Due to the complexity of this analysis, the largest part of this chapter is devoted to the description of the flexural rigidity method.

3.1 The force exerted on the baseplate

The mechanism by which the seismic vibrator applies a force to the baseplate is very complicated, and differs for different vibrators. In this section, the applied force is described using a simplified mechanical model for a hydraulic P wave vibrator, developed by Lerwill (1981). Other, more complicated models for the drive mechanism of the vibrator can easily be incorporated in the remainder of this section, as they only modify the expression for the applied force.

Lerwill (1981) introduced a model of a compressional wave vibrator which describes the different components of the vibrator in terms of masses, springs and dashpots. The model, shown in Figure 3.1, contains three masses. These are the holddown mass, which represents the weight of the truck and is used to keep the baseplate in contact with the ground; the reaction mass, which allows the vibrator to exert a force on the baseplate; and the baseplate, which is in

-53-

contact with the eanh's surface. The input force /, which is supplied by the vibrator's hydraulic system, is not the same as the force ƒ exerted on baseplate and reaction mass due to the compressibility of the oil pumped in the cylinder. The suspension 5, represents the means to support the reaction mass in its neutral position. The connection between the truck and the baseplate by means of isolated air bags is represented by the dashpot K and suspension s2. Gravity forces are not included in the analysis because they represent a static load, and do not affect the dynamic behaviour of the seismic vibrator.

Holddown mass

h

K

h

h

f2

Reaction mass

T.

ƒ,

Baseplate

n

f

<b /

Figure 3.1 The mechanical model of Ihc Vibroscis truck.

The displacement of the three masses is taken positive downwards. Using this convention, the forces due to the dashpot and suspensions are found to be:

ƒ, = ,v, ( K , . - H 3 ) (3.1)

ƒ 2 = ^ ( " / . - " s ) ' (3.2)

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f 3 = K 5 F ( H A ~ W 3 ) . (3.3)

in which uh, ur and w3 denote the displacement of the holddown mass, the reaction mass and the baseplate, respectively. As is shown below, the displacement of the baseplate is not uniform over the plate, so that the baseplate displacement «3 in the above equations should be interpreted as the displacement at the point on the baseplate where the relevant forces are applied.

The equations of motion of the holddown mass and the reaction mass are

,2 a ur -fi-f=Mr——, (3.4) dt

d'uh -fi-fi = Mh—r- • (3.5)

The total force applied on the top of the baseplate, ƒ "rp 'e , is now given by

fapP"Cd = ƒ + ƒ ] + f2+f3 . (3.6)

and thus, using equations (3.4) and (3.5),

r>»m-Mti±-MjJ±. (3.7) dt dt

The holddown mass is vibrationally isolated from the baseplate by the system shown in Figure 1.4. If the influence of the holddown mass is neglected, the model as shown in Figure 3.2 is obtained. This model leads to the following set of equations:

ƒ ] = . v , ( , • / , - i / 3 ) , ( 3 . 8 )

and

and the resulting force applied on top of the baseplate is equal to minus the reaction mass times its acceleration:

.2 a H, J =fi+f =-Mr -

dt (3.10)

Reaction mass

/ i

/ i

ƒ t Ö

/

Baseplate

Figure 3.2 The mechanical model of the Vibroseis truck i f the influence of the holddown mass is neglected.

Since it is convenient to work in the frequency domain, a Fourier transfomiation is performed on the equations pertaining to the two mechanical models. For the mechanical model including the holddown mass, the following set of equations is thus obtained:

F, = Si(U,-U3) , (3.11)

F2 = s2(Uh-U3) (3.12)

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F3 = -icoK (Uh-U3) , (3.13)

2 - F j - F = -a>MrUr , (3.14)

2 -F2-F3 = -coMhUh. (3.15)

The resulting force applied to the baseplate is given by

2 2 Fapp',ed =F +F1+F2 + Fi= eoMrUr+coMhUh. (3.16)

If the influence of the holddown mass is neglected, the two equations

Fx = j , ( l / r - l / 3 ) (3.17)

and

2 - F , - F = -coMrUr (3.18)

are obtained. The force applied to the baseplate is now given by

F°pl"ied=F} + F =w2MrUr . (3.19)

Numerical values for the mechanical parameters of the mechanical model were given by Lerwil! (1981).

3.2 The baseplate behaviour

Because of the complicated mechanical structure of the baseplate, the problem of describini: its behaviour under a dynamically applied load can only be solved in an approximate way. An extra complication arises because the seismic vibrator is vibrating on the surface of the eanh. which presents a complex load impedance to the baseplate of the vibrator.

The coupling of the vibrator to the earth is discussed in the next chapter. In this section, the behaviour of the baseplate when a force is exerted on the plate is investigated. The earth's complex load impedance is, at this stage, treated as an extra external force acting upon the plate.

-57-

Four different approaches to the problem of formulating the baseplate behaviour under a dynamic load are discussed. The conventional approach is to assume that either the displacement or the traction is distributed uniformly underneath the baseplate (Sections 3.2.1 and 3.2.2, respectively). As measurements show that neither of these assumptions is satisfied in reality, two new models are proposed to describe the baseplate behaviour. The first model assumes that the thickness of the baseplate can be neglected, but not its mass (Section 3.2.3). The plate stiffness is included in this model by assuming that the force applied to the baseplate by the vibrator's drive system is distributed uniformly over the baseplate. As the latter assumption was based more on intuition than on scientific arguments, a second model is developed with a more rigorous treatment of the plate stiffness (Section 3.2.4).

3.2.1 Uniform displacement

The boundary condition that the displacement directly underneath the surface source is constant has been extensively investigated in the literature (Bycroft 1956; Awojobi and Grootenhuis 1965; Robertson 1966; van Onselen 1982; Tan 1985). The assumption of uniform displacement is equivalent to assuming that the baseplate acts as a rigid body. Thus, the forces acting on the plate are used only to accelerate the mass of the plate, so that the additional equation to determine the magnitude of the constant displacement is given by the integrated equation of motion of the plate. This yields the following two equations:

L^U], x2) = UT, = constant, (3.20)

and

FapPiud+jj T^x^X2)dXxdXl = -a? MpU3> (3 21)

where S denotes the baseplate area. Measurements of the baseplate acceleration underneath the baseplate of a seismic vibrator

show, however, that the displacement is not distributed uniformly underneath the baseplate. Figure 3.3 shows measurements of amplitude and phase of the baseplate acceleration under the plate, of a TR2/X2 vibrator, at a frequency of 70 Hz. All values for the baseplate acceleration are relative to the average acceleration of the baseplate. The measurements were provided by Sallas et al, who presented them at the 1985 SEG Workshop in Monterrey. A constant-frequency signal was emitted using the TR2/X2 vibrator. The vibrator was phase-locked to the "weighted sum" ground force approximation. This "weighted sum" method provides an estimate for the

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ground force by summing the accelerations of the baseplate and the reaction mass, weighted by the mass of the baseplate and the reaction mass, respectively. This approximation can be deduced from the integrated equation of motion (3.21), in which the displacement is assumed uniform. In this equation, the applied force F"PPhed is given by minus the product of mass and acceleration of the reaction mass (equation (3.19). The vibrator was positioned over a thin (l/8th inch) neoprene mat on sand. The measurements underneath the baseplate were recorded by bolting sensor plates to the bottom of the vibrator baseplate, on which accelerometers and load cells were mounted. The baseplate contained a total of 128 sample points for both acceleration and force.

Figure 33 Modulus and phase of the baseplate acceleration underneath the plate at 70 Hz, relative to the average acceleration of the baseplate (from: Sallas et al, 1985). (a) modulus in dB, (b) phase in degrees.

From these figures, it is observed that the acceleration is distributed nonuniformly over the baseplate.The phase differences lie in the range from 0° to 30°, and the amplitude of the baseplate acceleration drops off by 3 dB or less. Although these differences are moderate, it is

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argued in Section 3.2.4, when, the bending stiffness of the plate is discussed, that even small variations of the baseplate acceleration over the plate area affect the plate behaviour.

It is observed that the assumption of uniform displacement does not correspond to reality. It can thus be concluded that the assumption of uniform displacement directly underneath the baseplate is inadequate to describe the behaviour of the baseplate of a seismic vibrator.

3.2.2 Uniform traction

Another boundary condition which has been investigated in the literature is the condition that the traction is uniformly distributed directly underneath the baseplate (Miller and Pursey 1954; van Onselen 1980; de Hoop 1970). As is shown in Chapter 4, the assumption of uniformly distributed traction leads to a simplification in the coupled earth-vibrator model which is very attractive from a computational point of view.

The assumption of constant traction:

T 3(*i> *2 ) = T3 = constant, (3,22)

has to be supplemented by an additional equation to specify the magnitude of the traction. The equation of motion integrated over the whole baseplate will be used for this purpose:

FappUcd+ATi = -a> [[ o{xl,x2)U3(x1,x2)dx1dx2 , (3.23) JJs

where S is the area under the plate, A denotes the magnitude of S and a denotes the surface density of mass of the baseplate. As was the case for the displacement, measurements of the traction underneath the baseplate of a seismic vibrator show that the traction is also distributed non-uniformly underneath the baseplate. Figure 3.4 shows measurements of the amplitude and phase of the vertical component of the traction under the plate, at a frequency of 70 Hz. The measurements are again taken from Sallas et al (1985); details are given in the previous section. All load cell measurements are relative to the weighted sum feedback signal. From Figure 3.4, it is observed that the traction is distributed non-uniformly over the baseplate. Amplitude and phase differences are more pronounced than was the case for the baseplate acceleration measurements. Phase differences exceeding 60° exist, and the amplitude of the traction drops off by more than 12 dB from the centre of the plate towards the plate edges. Therefore, the assumption of uniform traction does not correspond with reality. It can thus be concluded that the assumption of uniform traction directly underneath the baseplate is inadequate to describe the behaviour of the baseplate of a seismic vibrator.

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f^'a r-jzr-\ 1

r V .y

/ff-\ \

"0,

)

/

r

P \ /.

. II.ÜBi'AL J Df

w

COLOUR INTERVAL: IC DIGRUS

Figure 3.4 Modulus and phase of the traction underneath the plate at 70 Hz, relative to the weighted sum ground force approximation (from: Sallas et al, 1985). (a) modulus in dB, (b) phase in degrees.

3.2.3 The mass-loaded boundary condition

The first model to be developed which allows both traction and displacement to vary over the baseplate area uses the equation of motion of the baseplate in its local rather than in its global (integrated) form. In this model, the thickness of the plate is neglected, but not its mass.

Consider an elemental area of such a plate of infinitesimal thickness with side lengths <£c, and dx2, and with a certain force df^pp 'c imposed on it. This force is used not only to exert a force on the ground but also to accelerate the mass of this elemental area. The latter contribution is given by Newton's second law of motion, which can be expressed as

.2 ,r applied ..ground " ",-

dfi =dft + o——dxidx2, dt

(3.24)

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in which djfrou denotes the components of the force exened by the plate on the ground, and

a is the surface density of mass of the plate. Since

djground =_t.dXidx2t (3.25)

where r,- denotes the traction at the surface of the earth, and since equation (3.24) must hold for

each elemental area on the baseplate, then in general

applied. . . . , d Ui(x],X2) l; (XitX2) + / , ( * 1 > * 2 ) = 0(X1,X2)

dt (3.26)

where

,r applied applied , , dfi =-'i dx-[dx2. (3.27)

In the frequency domain, this yields

T°ppUe\xl ,x2) + 7 , - (*! ,* 2 ) = - o f t , , x2)co l/(- ( j^ , x 2 ) (3.28)

I' applied

(a)

applied

Figure 3.5 (a) the force applied on lop of the baseplate, and (b) the assumed distribution of this force when the thickness of the plate is neglected.

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Although it is not known what the distribution of the force applied on top of the baseplate would be when its thickness is neglected, this force is assumed to be distributed uniformly over the baseplate because the baseplate is stiff. For the case of a vertical applied force, this assumption is illustrated in Figure 3.5, in which

—.applied „applied -T3 = P ' . (3.29) The assumption of a uniformly distributed applied force is crucial in the derivation of the mass-loaded boundary condition. To answer the question whether this assumption accounts correctly for the plate stiffnesss, a more rigorous treatment of the plate stiffness is given in the next section where the flexural rigidity method is discussed.

Use of the assumption of uniformly distributed applied force yields

—applied -.applied „applied *" i Ji = " pi = ~A— • (3.30)

in which A denotes the magnitude of the baseplate area. Using equation (3.28), the mass-loaded boundary condition can be formulated as

—applied 0

- ^ — +7 ' ; ( i 1 , . t 2 ) = - o ( x , , x 2 ) u ) Ui(xl,x2) . ( 3 . 3 1 )

3.2.4 The flexural rigidity method

In this section, the distribution of traction and displacement on the baseplate is investigated using classical plate theory.

The analysis of plates goes back to the early 1800's, when Cauchy, Poisson, Navier, Lagrange and Kirchhoff developed the first theories to describe the deflection of a plate under a static load. Since then, plate analyses have been developed and applied in many different fields, such as the design of bridges, ships and aircraft runways. The assumption of a uniformly distributed applied force that was made in the development of the mass-loaded boundary condition led to a closer investigation of the flexural rigidity (or bending stiffness) of the baseplate, and consequently to the application of existing plate theories to the Vibroseis configuration.

The theory of plates starts with the basic linear equations describing the wave motion in an elastic medium. Using these equations, expressions for the baseplate deflection are derived by retaining only the most important terms for the components of stress and strain. Since this

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approximate solution inevitably brings along restrictions on the validity of the theory, the derivation of this approximate solution and its underlying assumptions are discussed in detail.

Consider the problem of a plate under a static load, in which the plate has different elastic properties in two orthogonal directions. A plate which has these properties is called an onhotropic plate. The introduction of this natural orthotropy (that is, onhotropy which is a result of different physical properties of the material itself) allows, under certain assumptions, extension of the theory for naturally onhotropic plates to structurally orthotropic plates (plates that are reinforced or composed of different materials). This concept of so-called elastic equivalence can be used to include the structure of the. baseplate in the calculation of the baseplate deflection. The structure of the baseplate used in the land vibrator experiment is illustrated in Figure 3.6.

A -►.

B -*

(a)

Figure 3.6 (a) top view of the baseplate used in the land vibrator experiment, and (b) cross-section A-B (enlarged).

The time-varying load which acts on a baseplate is incorporated into the theory by making all quantities time-variable, and by introducing an extra term that accounts for the inertial force related to the acceleration of the mass of the baseplate (Gorman 1982). The resulting differential equation describing the baseplate deflection is equivalent to the formulation of the mass-loaded boundary condition, except for terms accounting for the flexural rigidity of the plate.

The differential equation describing the baseplate deflection is normally solved using a series approximation (Navier 1823; Levy 1899). It is shown here how the differentia] equation can be solved analytically for the case of an impulsive load applied to a plate of infinite extent. The baseplate deflection due to an arbitrary load distribution can be obtained by spatially convolving the force distribution with the response of the system to an impulsive load. For the Vibroseis

top plate

bottom plate I-beam stiff eners

(«

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configuration, the force distribution consists of the force exerted by the vibrator's drive system on the top of the baseplate, and of the reaction force exerted by the earth on the bottom of the plate.

32.4a Classical plate theory

Consider a homogeneous, perfectly elastic plate on which a vertical load is acting. The differential equation governing the baseplate deflection can be obtained by making the following assumptions (Timoshenko and Woinowsky-Krieger 1959; Troitsky 1967):

1. The middle surface of the plate (Figure 3.7) is a neutral surface; thus, it remains unstrained during bending. This assumption is valid if the deflections are small compared with the plate thickness, and if the edges of the plate are free to move in the plane of the plate so that reactive forces at the edges are normal to the plate.

middle surface before deformation

\

" (

middle surface after deformation

- ► * !

"3

Figure 3.7 The middle surface of the plate, before and after deformation (after: McFarland el al, 1972).

2. Normals lo the middle surface before deformation remain normal to the same surface after deformation. This assumption is equivalent to assuming that distoriions of transverse sections caused by the existence of transverse shear strain make a negligible contribution to

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the displacements. This assumption is valid-if deflections are small compared with the thickness of the plate, but becomes unreliable in edge regions.

3. The effect of normal stresses in the plate caused by the applied load is disregarded. This assumption is valid if the thickness of the plate is small compared with the plate dimensions in its plane, but becomes unreliable for highly concentrated transverse loads.

4. The plate material is assumed to be perfectly elastic and homogeneous, to obey Hooke's law, and to possess different elastic properties in the two orthogonal directions Xj and x2.

With the aid of these assumptions, the governing differential equation for the baseplate deflection can be derived. The derivation can be divided into four sequential steps:

1. determination of the strain in terms of the baseplate deflection, 2. determination of the stress, using the stress-strain relation as given by Hooke's law, 3. determination of the resulting forces and moments per unit length in the x^ and x2 direction,

and 4. determination of the equilibrium conditions in the x3 direction for these resultant forces and

moments per unit length.

1. The strain components

Let £ denote the perpendicular distance of a point in the plate measured from the middle surface (Figures 3.7 and 3.8). Since it is assumed that normals to the middle surface remain normal after deformation (Assumption 2), the displacements ux and u2 can be obtained from geometrical considerations (see Figure 3.8):

ui(.xl,x2) = -^sin(ei) , (332)

u2(xh x2) = - %sin(92) , (3 33)

or, if the deflections are small (Assumptions 1 and 2).

•h = -Z— - (3.34)

u2 = - $-— . (3.35) 3A 2

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Figure 3.8 The displacement in the x\ and xi direction of a point in the plate a distance E, from the middle surface (after: McFarland el al, 1972).

in which the indices for the displacement components are omitted for brevity. The components

of the strain in the x} and x2 direction are now found to be

2

3*1 ax\ ' (3.36)

2 3«2 _ 3 M3

e72 = = - £ d x 2 OX

2 ' 2

(3.37)

ox 2 o*\ o.Vjdrj (3.38)

The definitions of e12 and <?2i differ by a factor 2 from the definitions for these components of the strain tensor introduced in Chapter 2, equation (2.4). This different definition (3.38) is chosen to be consistent with the notation used in the literature on plate analysis.

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2. The stress components

The stress components acting on an elementary volume of the plate are illustrated in Figure 3.9. Vertical stresses are neglected (Assumption 3).

/

,„«r" '23

/ ' U

'12 , „ x2 T

*3

Figure 3.9 Stresses acting on an elementary volume of the plate.

In this case, Hooke's law for an orthotropic plate can be written as (Assumption 4):

h\ = Eiieu + Ene22, (3.39)

r 2 2 = £ 2 1 e n + £ 2 2 e 2 2 '

hl-h\- Gi2 e i 2 •

(3.40)

(3.41)

The elastic constants in this equation are given by

£ n = -V,EX T-T V' £ ,

r - ** r 2 ' r _ ' 2

. £ 2 2 > £ 1 2 ' L2\ 1 _ V x Vx ] - Vx Vx l ~ Vx Vx ] - Vx Vx

■* 1 ■* 7 X -. X -> X 1 X 2 X 1 X

(3.42)

Here, Ex , Ex , vx and vx denote the components of Young's modulus and Poisson's ratio in

the xl and x2 directions, respectively. It can be shown (e.g. Allen 1969; Troitsky 1967) that

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v * l£ * . = v x , £ ' 2 ' (3.43)

from which it follows that £ ] 2 = £21 • G\2 ' s o n e °f , n e three shear moduli of the orthotropic material. Since distortions due to the transverse shear strains e13 and e^ are neglected, the other two shear moduli, G13 and G23, do not enter into the derivation.

Combining equations (3.36)-(3.41), the stress components can be expressed in terms of the baseplate deflection:

2 2 d u-i 3 Mo

'n = - S (.Eu—^ + E12—f), (3.44) dx^ dx2

2 2 3 Mo 3 Ho

' 2 2 = - 1 ( ^ 1 2 — Y + E 2 2 — - f ) , (3.45) 3JCJ 3;t2

2

r 1 2 =/ 2 1 = - 2 £ G,2 — . (3-46) dx-[dx2

3. The resulting forces and moments

Expressions are obtained for the moments and forces per unit length in the .\j and x2

directions by integrating the moments and forces acting on an elementary volume over the thickness of the plate.

The orientation of the moments and forces is illustrated in Figure 3.10. The following expressions for the bending and twisting moments Wj, m2, mn and m2l are obtained:

/■A/2

m, = I

f ",2 . ' 2 = I b'2

•^ 1,/T

2 3 u-,

dx]

2 3 «1

c)A'i

+ £1 2

+ £22

dx 2 -A/2

d v 2 •'-/1/2

d£ = -

dc =-

2 2 3 ii\ ti it-* (D]—± + Dn—f), 3.ï] 3 . \ 2

(3.4

2 2 3 u-i 3 //-!

3.Vj 6x2 -A/2

(3.48)

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mi2=m21

du, 'hl2 r u 3 M, r ' 2

1 = - $t12dS=2Gl2—-M § d £ = 2 D dxldx2S-hl2

2

(3.49)

where the flexural rigidities DltD2, D12 and £)2i and the torsional rigidity D G are introduced,

which are given by

3 3 3 3 Euh Ej7h E\->h Gy2h

£ > , = — — , D , = ———, D 1 7 = D , , = — — — , D r = ——— 1 12 ' 2 12 ' 1 2 2 1 12 ' G 12 (3.50)

Figure 3.10 The resultants of moments and forces, per unit length in the X\ and JC2 directions.

Since the stress components given by equations (3.44)-(3.46) are odd functions of the distance from the middle surface £, the corresponding components of the resulting forces per unit length in the.Vj and x2 direction vanish:

f A/2 / l l = tnd£=0,

- ƒ

-A/2

A/2 t22dl = 0 .

A/2

(3.51)

(3.52)

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/•A/2 / i 2 = / 2 i = l ti2dl;=0. (353)

•'-/i/2

Note that the force resultants f^ and ƒ23 do not vanish; however, no explicit expressions are needed for these components.

4. Equilibrium conditions

The final result can now be obtained by considering the equilibrium conditions of a differential plate element. A vanishing resultant force in the A'3 direction is required, and equilibrium with respect to rotation about the x] and x2 axes:

—-+—- + P =0 , (3.54) ax 1 dx2

— - — — / i 3 = 0 , (3.55) oX] ox2

dm2 dm12 r /23 = 0 , (3.56)

ox2 dxl

in which pin denotes the vertical applied force per unit area. If the expressions (3.55)-(3.56) are substituted for the resulting forces in the x3 direction in equation (3.54), this yields

2 2 2 a m\ » o m12 d m2 in

—T~2T~T~ —7 p = (3-57) 3JCJ axiax2 dx2

Using expressions (3.47)-(3.49) for the resulting moments, the differential equation governing the plate deflection is obtained:

4 . 4 4 3 «3 d «3 d «3 ,,,

D' TT+ 2 W T ^ V ^ T - t ^ • (3-58) O.Ï ] C U ] CiY 2 OX 2

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in which the effective torsional rigidity H is introduced:

H=Dn+2DG. (3.59)

3.2.4b Tlie magnitude of the flexwal and torsional rigidity components

Usually, the value of the shear modulus G12.(and thus of the torsional rigidity DG) is unknown for orthotropic materials (Troitsky, 1967). Timoshenko and Woinowsky-Krieger (1959) state that all values of the torsional rigidity DG based on pure theoretical considerations should be regarded as a first aproximation, and they recommend a direct test to obtain more reliable values of G12- However, assuming pure elastic deformations, constant plate thickness and small deflections, the effective torsional rigidity H can be obtained from the flexural rigidities Dy andD2 by (Huber, 1914):

H = VSTTJ; . (3.60)

The differential equation (3.58) describes the deflection of a homogeneous, orthotropic plate. In reality, the baseplate is not homogeneous, but has a complicated structure. For the baseplate used in the land vibrator experiments, the basic structural components of the baseplate are formed by I-beam stiffeners in the plate (Figure 3.6). This baseplate structure can be incorporated in the theoretical framework discussed so far by using the method of elastic equivalence, as first proposed by Huber (1914). In this method, the components of the flexural and torsional rigidity are conceived as applying to a homogeneous orthotropic plate which is equivalent to the actual plate-stiffener combination. As the term "equivalence" can be based on several different physical quantities (e.g., equivalence of strain energy, equivalence of displacements), this method of elastic equivalence is only meaningful when various equivalence criteria lead to the same results. Both experimental results (Hoppmann 1955) and theoretical investigations (Huffington 1955) indicate that orthotropic plate theory is applicable to stiffened plates provided that the ratios of stiffener spacing a to plate boundary dimensions l,w are small

a a enough (y , — « 3) to ensure approximate homogeneity of stiffness. Since in the experiments described in this thesis the ratio of stiffener spacing a to smallest boundary dimension of the plate iv is smaller than 0.05, differential equation (3.58) can be applied to the structurally orthotropic plate of Figure 3.6.

What remains, then, is to determine the magnitude of the components of the flexural rigidity (the torsional rigidity can be approximated from the flexural rigidity components using equation (3.60)). The baseplate used in the aforementioned experiments can be described as a structural

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sandwich panel, or, more precisely, as a corrugated-core sandwich. Libove and Hubka (1951) derived exact expressions for the flexural rigidity components of such a plate. Allen (1969) used their results to arrive at approximate expressions for the flexural rigidity of a plate with stiffeners in the xi direction. If the stiffness of the core cannot be taken as infinite, differential equation (3.58) is no longer valid but has to be supplemented by extra terms accounting for the deformation of the core.

¥ a top plate

J __»-

i

t

f -

d

n

b

stiffener

.JL 1 ^ —

f bottom plate

i

r

Figure 3.11 The structure of the baseplate used in the land vibrator experiments, and the notation employed in this section.

However, Allen states that the stiffness of a corrugated sandwich panel in the vertical direction, and in the direction of the corrugations, can be taken as infinite. The effect of core shear deformation due to a finite stiffness of the core in the direction perpendicular to the corrugations will be neglected. In his analysis, Allen also assumes that the orthotropic behaviour of the plate is induced by the structural properties of the plate, that is, the'plate - and core material themselves are isotropic, with Young's modulus £ƒ and Ec, respectively. Allen obtains the following expression for the flexural rigidity in the .r, direction, Df

D, ^qr^Ef + Ec/c , (3.61)

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in which lc represents the second moment of area of the cross section of the core per unit width in the x2 direction, q denotes the thickness of the faces, r denotes the centre line to centre line distance between the two faces, and Er and Ec denote Young's modulus for the faces and the core material, respectively. For the vibrator baseplate used in the land vibrator experiments, the flexural rigidity in the x^ direction, D j , can now be calculated as follows. The value of Young's modulus for steel lies in the range from 19.7xl010 to 21.2xl010(Nm~2), depending on the specific type of steel used (CRC Handbook of Chemistry and Physics 1971). A constant value of 20xl010 for Young's modulus E of both stiffeners and faces is assumed. The moment of area of a stiffener in the xt direction, per unit width in the x^ direction, is calculated as follows.

Let a denote the width of the stiffener, b the height, d the thickness of the middle part and / the thickness of the top and bottom part (Figure 3.11). Numerical values for the dimensions of plate and stiffener used in the experiment are given in Table 3.1.

thickness of middle pan of stiffener d 4 mm thickness of top part of stiffener t 4 mm width of stiffener a 50 mm height of stiffener (excl. top) b 90 mm thickness of plate q 14 mm length of plate / 1840 mm width of plate w 1100 mm

Table 3.1 Numerical values for the dimensions of plate and stiffener used in the experiment.

The total moment of area of the stiffener is found to be

1 3 2 3 1 2 2 I =j^db + ja + jb at + abt . (3.62)

Substituting the numerical values for the different plate dimensions in this equation, and taking into account that the moment of area must be divided by the centre line to centre line distance a of the stiffeners, reveals that / per unit width equals 2.25 x 10~5 (m3). This yields a contribution to the flexural rigidity D, of the stiffeners of 4.5 x 10 (Nm). The contribution of the plate itself to D] equals

D\ =jEqr , (3.63)

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where q denotes the plate thickness, and r the centre line to centre line distance between the two plates (Figure 3.11). This yields a contribution of the two plates \oDx of 17.5x10 (Nm). The total flexural rigidity of the sandwich panel in the x: direction, Dj, is therefore 2.2x10 (Nm).

Since the contribution of the core to the total flexural rigidity of the sandwich panel in the.Aj direction, Dj, is small, the flexural rigidity of the panel in the x2 direction is assumed to be equal toZ)]. The following values are then obtained for the flexural rigidity components of the baseplate used in the land vibrator experiment:

DX=D2 = D =2.2x 107 (Nm), (3.64)

and, using equation (3.60),

H = D = 2.2x\01 (Nm). (3.65)

Since all components of the flexural rigidity of the baseplate used in the land vibrator experiment are thus assumed equal, the differential equation (3.58) can be simplified to:

4 4 4 d u-i d Hi 5 « i

D ( - + 2 — + -dx i dxxdx2 dx2

3.2.4c Dynamic terms

(3.66)

Finally, the inertial force that occurs in the bending of the baseplate of a seismic vibrator is accounted for. Because a non-static load is acting on the top and on the bottom of the baseplate, the mass of the baseplate is accelerated. First, a temporal Fourier transform is performed on differential equation (3.66). Therefore, all variables are now denoted by their corresponding capitals. The extra force term can be included in the differential equation using Newton's second law of motion, with the following result (Love 1929; Gorman 1982):

4 4 4

n , a u 3 si;, a y , , 2 D{—T-+2-T-T + —t]-oco V ^ P ' (3.67)

dx} dx}dx2 dx2

in which 0"denotes the surface density of mass of the baseplate.

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The pressure applied to the vibrator baseplate, /"", consists of two terms: the traction at the surface of the earth, T$, and the pressure PaPPlied applied at the top of the baseplate by the drive system, which is given by (see equation (3.19)):

Fapplied _o>MrVr F — — , x)tx2 s S

applied / A A r ( * i , x 2 ) - \ . (3.68)

0 , elsewhere

in which AF denotes the magnitude of the area SF over which the force on top of the baseplate is exerted. Thus, the differential equation applied to the vibrator plate finally becomes

4 4 4 d U-> d U-i d U-> 2 applied

D{-^-+2-T^ + —^-)-aco U, = Papped+T,, ( 3 6 9 ) 3*] dxidx2 3x2

in which popped JJ^ a n c j j ^ ^ functions of position and frequency.

3.2.4d The boundary conditions

To complete the formulation of the baseplate deflection problem, the governing differential equation (3.69) must be supplemented by boundary conditions at the edges of the plate. The edges are located at JCJ=0, xt = * j , JC2=0 and x2 = x2- The notation used in the following is illustrated in Figure 3.12. Since all edges are free, there are no bending moments and twisting moments, and also no vertical shearing forces acting on the plate edges. Thus,

A"! = 0 or A'I = * ! : M1 = M 1 2 = F 1 3 = 0 e ■ (3.70)

x2 = 0 or x2 = x2 '■ M2= M2l = F 2 3 = 0

Two boundary conditions at each edge are sufficient for the complete determination of the baseplate deflection. The redundancy in conditions (3.70) was explained by Kelvin and Tait (1883), who showed that, due to the assumptions underlying the theory, the conditions of vanishing twisting moment and vanishing resultant shearing force at the plate edge are equivalent.

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f

A—*X} S

■* 2 •* 2 ~ * ^

If" / 1^23

/ r T 2 - '0

Figure 3.12 The notation used in the discussion of the boundary conditions.

The boundary conditions are thus formulated as:

X] = 0 or X] = X ] : A / ] = F ] 3 = 0

A 2 = 0 or jr2 = ; t 2 : M 2 = F 2 3 = 0

(3.71)

5.2.4e Solution of the differential equation

The differential equation (3.69), subjected to the boundary conditions (3.71), cannot be

solved analytically. To obtain an approximate solution to the problem, a plate of infinite extent

is considered. The differential equation is then solved by first calculating the response of the

baseplate to a point force. This response is denoted by the Green's function Gflcx. Thus, Gf" satisfies the equation

, / 7 M ,/!>*, ^flex. d GJ'"(x,, x, ) d G ' (*„ jr , ) 3 GJ (xit Xn) D{ —!—— +2 . . " + — — 1 + 2 2

3.Vj3.v2 D.v-

flex -o(X], jr2)co G ( A , , X 2 ) = S(X,-jr'j, X 2 - - -*2)• (3.72)

-77-

Note that the frequency spectrum of the point force is assumed to be of unit magnitude and zero phase; however, the frequency dependence of the source function enters the problem linearly, and from the solution obtained the actual response of a non-white source signal can be obtained merely by multiplying it with the source spectrum.

In Appendix B, it is shown that the solution to this differential equation is given by

flex 1 O O G"ex(r) = - J _ [ H0 (or) - HQ (far)), (3.73)

8VoD ica

in which H0 denotes the Hankel function of order zero, D is the flexural rigidity of the baseplate, CTis the surface density of mass of the baseplate (which is assumed constant in the derivation), a is given by

f 2\'4 oca

a = l - ^ - l . (3.74)

and r is the distance from the observation point xa to the integration point x'a, given by

i / r=l(xa-xa)(xa-xa)Y2>0. (3.75)

Although the Hankel function is singular at r = 0, the limiting value for &a as r tends to zero exists, and is given by

C/ 7"(r = 0) = T = L — . (3.76)

8VoD ico

In Figure 3.13, the amplitude and phase of the function F(ar), as a function ofar, are shown. F(ar) is given by

( i ) ( i )

F(ar) = H0 (or)- HQ (iar) . (3.77)

The shape of the Green's function G^" can easily be derived from this figure.

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Figure 3.13 The function F, defined by equation (3.77), as a function of the quantity or (denoted by "Omega"). (a) modulus, and (6) phase in degTces.

Equation (3.73) can be rewritten to obtain an expression which is more convenient from a computational point of view, with only real arguments of the modified Bessel function K0

(Abramowitz and Stegun, 9.6.4):

G^V.'od»-1 (D 2/

r— [H0 (.ar) + -K0{ar)] 8Vc© ico n

(3.78)

The correctness of this solution is investigated by a discussion of three issues:

1. mathematical correctness, 2. causality of the solution, 3. errors introduced by neglect of the boundary conditions (3.71).

1. Mailiematical correctness The correctness of the solution is checked by substituting the solution back in the differential equation. At the source point, the differential equation is integrated over a small circular area, including the source point, to obtain a finite contribution of the delta pulse source function. Both checks (validity of the solution outside the source point, and in the small region comprising the source point) show that the solution is mathematically correct.

2. Causality If the solution is causal in the time domain, the real and imaginary parts in the frequency domain are related by the Hubert transform (see Bracewell 1978, p.271). The causality of

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the solution (3.78) cannot be checked analytically; instead, a numerical check is performed. First, the real part of the Green's function for the particle velocity (the Green's function for the particle displacement is singular at zero frequency, and the Green's function for the plate acceleration converges very slowly) is calculated. It should be noted that the term -iocorresponds to an integration in the time domain, and need not be taken into account when considering causality. The frequency-dependence of the Green's function for the particle velocity, G/7"-1", is given by

Gflex<\co) = K ( J0(c*fco ) + i[Y 0(rfa ) + -K 0(cVÖ>) ] } , (3.79) n

in which K and c are real constants, for a given position on the plate. In the calculations, K was set to 1, and c was chosen to be 0.1. The real part of this expression is Fourier transformed, the time domain result is set to zero for negative time and doubled for positive time, and transformed back to the frequency domain. The resulting imaginary part then is equal to the Hilbert transform of the real part. This imaginary part is compared with the analytic imaginary part of the Green's function given by equation (3.79). The analytic and calculated imaginary parts are shown in Figure 3.14, from which it is observed that the two parts are virtually equal. From this, it is concluded that the obtained solution satisfies the condition of causality.

3. Boundary conditions The boundary conditions are that the shear force and moments vanish at the plate boundary. The effect of the neglect of these boundary conditions is investigated by considering a circular plate loaded at the centre. Because a point force yields an infinite moment at the plate centre, a constant force of unit magnitude is applied in a small circular region with radius 0.01 m at the plate centre. This allows for a quantitative evaluation of the decrease of the bending moment towards the plate edge. The bending moment in polar coordinates is denoted by Mn whereas the resulting shear force in the vertical direction at the plate edge is

W0OC 40000

Frequency (Hz)

Figure 3.14 Analytic imaginary part of the solution (3.79) (dotted line), and the Hilbert transform of the real pan of the solution (solid line).

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denoted by Fr3. The displacement of the plate due to this applied force is given by equations (B.53) and (B.54). From these expressions for the displacement, the bending moment and shearing force are calculated according to the following expressions (Timoshenko and Woinowsky-Krieger, 1959):

Mr=-D[ 2

17 vdU,

+ T £ ] • (3.80)

d .d U 3 1 dU3 — + 2 r dr (3.81)

0.5 Distance (m)

The bending moment and shearing force as a function of the distance from the centre of the force application are shown in Figure 3.15. From this figure, it is observed that the shearing force drops off very rapidly with distance, whereas the decay of the bending moment is less rapid and has a value of approximately 0.2 in the range of baseplate radii of interest (0.5 - 1 m).

Figure 3.15 The normalized bending moment (solid line) and normalized shear force (dashed line) as a function of distance from the centre of the force application.

The displacement of the baseplate is obtained by spatially convolving the actual input pressure with the Green's function:

■' -'s 1 -M x\, X'>-.T'i)dx\dx'i . (3.82)

where the input pressure P,n is given by

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Pin(x],x2) = PappUe\x\,x2) + T,(x\,x2) , (3.83)

in which pappUed \s defined in equation (3.68). This yields the following relation between baseplate displacement, applied pressure and traction directly underneath the plate:

^3(*i, x2) = f f { PappUe\x\ ,x'2) + T3 Or', ,x2)} Gfl" (x,-x\ ,x2-x2) dx\dx2. J J S

(3.84)

The convolution of the Green's function with the applied pressure needs to be performed only in the restricted source region SF. In Appendix B, two methods of performing the spatial convolution given by equation (3.84) are discussed.

3.2.4f Physical interpretation

In this section, a physical interpretation is given of the differential equation (3.69) describing the plate behaviour:

4 4 4 3 U3(xi,x2) 3 U3(xltx2) 3 U3(Xi,x2) •.,?,,, x

D{ +2 — + ) -oixltx2)co U3(xi,x2) =

dx^ 3x]3jr2 3*2

= PappUed(x1,x2) + T^xl,x2) , (3.85)

together with a physical description of its solution (3.84)

Vfri • *2> = ff I PapP"ed(x\ ,x'2) + T3Cx', ,x2)) Cflex(xi -x\,x2-x2)dx\dx2. JJS

(3.86) in which

Crx(r, ft» = j=— I H0 (ar) - HQ (iar) ] , a 8 7 ) 8Vc4> ico

and where <ris now assumed constant.

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If the differential equation (3.85) is compared with the formulation of the mass-loaded boundary condition for the vertical components of displacement and traction (see equation (3.31)):

-a{x^x2)co U3(xltx2) = Papphe\x^x2) + T3(x^x2) , (3.88)

two things can be observed. First of all, for the mass-loaded boundary condition the applied pressure PaPPn^ j s assumed to be distributed uniformly over the baseplate, whereas in the flexural rigidity method the applied pressure is a function of position, and is zero outside the contact area between the hydraulic piston and the baseplate. Second, apart from this different definition, the two equations (3.85) and (3.88) are equivalent except for the term

4 4 4 d U3 d U3 d U3

D^^L+2-T\ + —T) ■ 0.89) dx1 3x^X2 dx2

This term can be interpreted as being the force (or, more precisely, the force per unit area) that is associated with the bending of the baseplate. This force is referred to as the bending force. Thus, as in the mass-loaded boundary condition, the flexural rigidity method states that the total force acting on the plate is equal to mass times acceleration of the plate, but now the bending force is included in this total force.

The assumption of uniform displacement (Section 3.2.1), which assumes that the baseplate behaves as a rigid body, is now compared with the theory that includes baseplate bending. The conventional assumption of uniform displacement leads to the rigid-body equation of motion of the baseplate, given by equation (3.21). An approximate version of the flexural rigidity method is derived, in which the displacement is also distributed uniformly over the plate. It is shown that the resulting expression differs from the expression obtained using the conventional assumption of uniform displacement.

At first sight, it seems that that the first term in the differential equation (3.85), representing the bending force, vanishes when the displacement is assumed uniform since the spatial derivatives of U3 are zero. This, however, is not the case, as can be seen from the solution to the differentia] equation (3.85):

^3(*i. * 2 ) = f f ( ParP'iCd(x\ ,x2) + T3 (x\, x2) ) Gf'cx (A-, - x\ ,x2-x2) dx\dx2 . JJS

(3.90)

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in which

flex 1 <» O G' e\r, a» = - J _ [H0 (or)- H0 (iar)]. (3.91)

8V0D ico

From this equation, it follows that the assumption of a uniform displacement is equivalent to assuming that the Green's function G/*" is constant over the plate. The variation of the Green's function over the baseplate area is contained entirely in the Hankel functions. Therefore, for the Green's function to be approximately constant, the argument of the Hankel functions must be constant. Thus,

ar-

I 2\

D

1/4 r = constant. (3.92)

The value r = 0 is always attained in the spatial convolution integral, giving the condition

1/

-fr\ r = 0- (3-93)

This assumption is true for low frequencies and low mass densities; however, more interesting is the case where the flexural rigidity of the plate, D, becomes very iarge:

2 4 D»aco r . (3.94)

For values of D which satisfy condition (3.94), the Green's function G^ is approximately constant, so that

V,Ui , * 2 ) = ui = r=— \\ I W , x'2) + PappUc\x\ ,xi)} dx\dx2 .

(3.95)

Thus, for the flexural rigidity method, the assumption of uniform displacement does not lead to the integrated equation of motion pertaining to the rigid-body assumption:

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f W % ) ,x2)+ff r3(*i - *2>^id j c2 = -co2 MpU3. (3.96)

Instead, equation (3.95) should be used to describe the baseplate deflection if the displacement is distributed approximately uniformly over the plate. The apparent contradiction of this statement with the vanishing spatial derivatives in differential equation (3.85) can be explained by realizing that a uniform displacement corresponds to an infinitely high flexural rigidity D. Therefore, when the displacement is distributed uniformly, the first term of equation (3.85):

4 4 4 3 t/3 3 l/3 d U3 , D{—±+2-^L + —±}, ( 3 9 7 )

dx1 dXidx2 dx2

is equal to infinity times zero, and thus undetermined. The outcome of this cannot readily be obtained, but is not necessarily zero.

In conclusion, uniformity of displacement is not physically realizable, since it corresponds to an infinitely high value for the flexural rigidity/). For all physically realizable values of D (that is, a finite, positive value), the baseplate behaviour is described by equation (3.90). If the plate is very stiff, the displacement is approximately constant. To describe the baseplate behaviour in this case, equation (3.95) should be used rather than the integrated equation of motion of the baseplate.

3.3 Conclusions

In this chapter, a model is developed to describe the mechanical behaviour of a seismic vibrator. The model contains two elements.

First, a description is given of the force applied by the drive mechanism of the seismic vibrator to the baseplate. Expressions for the applied force are obtained using Lerwill's (1981) mechanical model of a hydraulic P wave vibrator. More complicated vibrator models or models for different vibrator types can easily be incorporated in the theory since they only alter the expressions for the applied force.

Second, four different models describing the behaviour of the baseplate of a seismic vibrator are discussed. Two conventional models, assuming uniform displacement or uniform traction under the plate, are not in agreement with field measurements, which show that neither of these quantities is uniform. Two new models which yield non-uniform distributions of both traction and displacement are presented. The first new model, referred to as the mass-loaded boundary

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condition, assumes that the applied force is distributed uniformly over the baseplate. The formulation of this model then follows from the condition that the total force acting on the plate is identical with the inertial force related to the acceleration of the mass of the baseplate. The second new model contains a more rigorous treatment of the plate stiffness; the analysis of the plate behaviour in this model is referred to as the flexural rigidity method.

The differential equation that governs the baseplate deflection for the flexural rigidity method is derived, and the formulation of the boundary conditions at the plate edges is given. The structure of the baseplate is accounted for by using the concept of elastic equivalence, which. allows the magnitude of the components of the flexural rigidity to be determined for structurally orthotropic plates. An approximate solution to the governing differential equation is obtained, in which the baseplate dimensions are neglected. From this solution, it follows that the displacement of the baseplate is given by a spatial convolution of the input force acting on the plate, and a Green's function that describes the deflection of the baseplate due to the action of a point force. For the Vibroseis configuration, the input force consists of the force exerted by the drive mechanism of the vibrator to the plate, and the reaction force exerted by the earth on the baseplate. The Green's function is causal and mathematically correct, but does not rigorously satisfy the boundary conditions at the edges of the plate.

It is shown that for a very stiff plate, for which the displacement is approximately uniform, the flexural rigidity method predicts a different baseplate behaviour than the model using the conventional assumption of uniform displacement.

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Chapter 4 The combined earth-vibrator model

In this chapter, the distribution of traction and displacement directly underneath the baseplate of a seismic vibrator is discussed. The distributions of traction and displacement depend on the elastic parameters of the earth, as well as on the properties of the seismic vibrator.

It follows from the earth model developed in Chapter 2 that the displacement at the surface of the medium is a spatial convolution of the surface tractions and a Green's function. The Green's function depends on the earth model; in this chapter, only the homogeneous halfspace model is adopted in the calculations.

In Chapter 3, four different vibrator models are developed. The four models differ in their description of the behaviour of the baseplate of the seismic vibrator. All models yield an additional relation between traction and displacement direcdy underneath the baseplate.

The earth model and the vibrator model are coupled by assuming that traction and displacement are continuous across the baseplate - earth interface. Using the combined earth-vibrator model, the distributions of traction and displacement directly underneath the baseplate are calculated for all four baseplate models, and for different frequencies and earth models. Using these distribution functions, the radiation impedance and power output of the vibrator are calculated. It is shown that the four different baseplate models yield substantially different distribution functions, but that the average radiation impedance and the average power factor are similar for all models.

Finally, the distribution of the acoustic power flow in the far field of the medium in compressional, shear and Rayleigh waves is investigated.

4.1 The earth model

4.1.1 Theory

The earth is modelled as a homogeneous, isotropic, perfectly elastic halfspace. It is assumed that no horizontal components of the traction are present at the earth's surface:

7"a(*i,x2) = 0 . (4.1)

Since only the normal components of the particle displacement, particle velocity and traction are considered in this chapter, these quantities are denoted simply as displacement, velocity and traction.

For the calculation of the traction - and displacement distributions directly underneath the baseplate of a seismic vibrator, an expression relating the two quantities at the surface of the

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'jf earth is needed from the earth model. If equation_(2.93) is applied to the depth level ;t3 = 0, the following expression for the displacement at the surface is obtained:

U3(xl,x2) = [f T3,(x\,x'2)G(xl-x\,x2-x2)dx\dx2 , (4.2) JJs

in which 5 denotes the part of the surface which is occupied by the seismic vibrator(s) (the remainder of the surface is assumed to be traction-free), and where the Green's function G is given by (equations (A.62), (A.68) and (A.72))

o) r -]

G(xx-x\,x2-x2) = G(r) = - - pypA (p)J0(copr)dp . (4.3) . 2mpcs ■'o

In this equation, r represents the distance between observation point xa and integration point

r=[(xa-x'a)(xa-x'a)]/2 . (4.4)

The Rayleigh-denominator A( p) is given by

4( P)= [ — - 2p2 f + 4p\pys , (4.5)

in which the vertical P and S slownesses y„ and y. are given by p 3

yp^i-^--p2)A > (Re.Im) ypsZ0 (4.6) cP.s

1 The Rayleigh denominator has simple zeros at p = ±pR =±—, in which cR denotes the CR

Rayleigh wave velocity. These poles have their physical expression as surface waves. These waves are attenuated exponentially in depth, but it is shown later that they have a major influence on the behaviouT of the Green's function at the surface of the earth. If the earth is assumed to be a layered elastic halfspace, an expression identical to equation (4.3) is obtained from the earth model. In this'case, the Green's function is given by equation (2.82). Thus, the general theory developed in this chapter is not restricted to a homogeneous halfspace. Also, the

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theory is valid if non-vanishing horizontal components of the surface traction are present. For computational convenience, however, results are shown for the homogeneous halfspace model with only vertical traction components acting at the surface.

The numerical evaluation of the Green's function is discussed in Appendix C. To avoid the evaluation of integral (4.3) for all possible values of co and r, the modified Green's function (-'mod is introduced, which is defined as

< w ( / 2 ) = ^ > . (4.7)

co

In this equation, Q is given by

Q = cor . (4.8)

Therefore, the evaluation of integral (4.3) for all co and r can be restricted to the calculation of Gmod for all Q. -values of interest.

4.7.2 Amplitude and phase of the earth's Green's function

In this section, the dependence of the earth's Green's function on the elastic parameters of the medium is discussed. The elastic parameters of the medium are expressed in terms of the density, the compressional (P) wave velocity and the shear (S) wave velocity. It is observed from equation (4.3) that the density only scales the amplitude of the Green's function. Thus, in all calculations the density is taken constant with magnitude 1700 (kgm-3). The effect of the compressional and shear wave velocity of the medium is investigated by calculating the Green's function for different earth models, in which only the compressional or the shear wave velocity is varied. Figure 4.1 shows amplitude and phase of the modified Green's function for three earth models with different P wave velocities. The S wave velocity is equal to 150 ms-1. The three models have P wave velocities of 400, 325 and 250 ms-1, respectively. Figure 4.2 shows modulus and phase of the modified Green's function for three earth models with different S wave velocities, and a constant P wave velocity of 400 ms-1 . The S wave velocity has values of 150, 100 and 50 ms-1, respectively. The scale for the amplitude of the Green's function is different in Figures 4.1 and 4.2. The amplitude variations of the models where the shear wave velocity is varying (Figure 4.2) are much more pronounced than the variations in the models where the P wave velocity is varying (Figure 4.1). Variation of the P wave velocity results predominantly in significant amplitude variations at low Q -values. This behaviour is dictaied by the value of the Green's function when Q equals zero (see Appendix C):

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Omega Omega (a) W

Figure 4.1 The modified Green's function Cmod for three earth models with different P wave velocities, and with a constant S wave velocity of 150 ms"1.

: cp = 250, : cp = 325, — : cp = 400 ms"1. (a) amplitude, and (6) phase in degrees.

3 5e-10l-

Omega (<•)

Omega m

Figure 4.2 The modified Green's function Gm<ld for three earth models with different S wave velocities, and with a constant P wave velocity of 400 ms" .

: c.= 50, : cs= 100, — : c t = 150 ms"' . (a) amplitude, and (b) phase in degrees.

A0 G(a>, r=0) = — , (4.9)

in which the constant A0 is given by

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, 1 °~ 4/1 1\ ' (4.10)

\CP Csl

Therefore, the amplitude of the Green's function at low il -values, as a function of the P wave velocity, will increase as the P wave velocity approaches the S wave velocity. The singularity for r=0 in the Green's function (equation 4.9) vanishes when the surface integration given by equation (4.2) is performed. This is shown in Appendix C.

4.1.3 The asymptotic behaviour of the earth's Green's function

The behaviour of the Green's function for large Q - values is mainly determined by the contribution of the Rayleigh pole. This contribution is given by (equation A. 102):

JKF{PR) H«XnPR), (4.1D A'(PR)

in which pR denotes the Rayleigh wave slowness, A ' denotes the derivative of the Rayleigh-denominator with respect top, and where F(p) is given by

F(P)* *-• (4.12) 2mpcs

Figures 4.3a and 4.3b show the real and imaginary part, respectively, of the calculated modified Green's function, and the approximated modified Green's function, using only the contribution of the Rayleigh pole. The correspondence between the calculated and approximated modified Green's function is excellent, even at Q -values as low as 50 (e.g., a value of r of 0.5 m and a frequency of ±16 Hz). The Rayleigh wave velocity is mainly determined by the S wave velocity. It can be shown that (Ewing et al, 1957)

0.8743 cs < cR< 0.9554 cs . (4.13)

Therefore, the oscillations of the Green's function due to the behaviour of the Hankel function (equation (4.11)), are mainly governed by the S-wave velocity.

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"6ï""o 2000 «oo " 6000 -'"-"i j5oö Jöof Omega Omega

(o) W

Figure 4.3 The modified Green's function (dashed line) and its asymptotic approximation by the Rayleigh pole contribution (equation (4.11); solid line), as a function of O. In the calculations, a P wave velocity of 400 ms~' and an S wave velocity of 150 ms"1 are used, (a) real pan, and (b) imaginary part

4.2 The combined earth-vibrator model

In this section, the expressions from the earth model that relate particle displacement and traction at the surface of the earth are combined with a vibrator model. Four vibrator models are investigated, each of them assuming different baseplate properties. The baseplate models have been discussed in the previous chapter. The vibrator model and the earth model are coupled by assuming perfect contact between baseplate and earth surface. The mechanism by which the baseplate is kept in contact with the earth is, in practice, the use of the weight of the truck as a holddown mass.

The earth model yields a relation between traction and displacement. The vibrator model yields an additional relation between those two quantities. The model is completed by specifying the source strength. For example, one could require that the acceleration in the centre of the baseplate is prescribed. Also, the assumption of a known weighted sum signal can be used to complete the model. However, the response of the earth-vibrator system to a constant force applied on top of the baseplate gives more physical insight in the problem. This force is taken to be of unit magnitude and zero phase, and is applied at the centre of the baseplate. In practice, the force applied on top of the baseplate, Fapp 'c , can be obtained from measurements of the reaction mass acceleration Ar (equation (3.19)):

Fapplied=-MrAr. (4.14)

Since the response of the earth to an array of vibrators is also investigated, the complete model pertaining to the array configuration is given. The total number of vibrators is denoted by

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K. The part of the surface occupied by the fcth vibrator is denoted by 5*; displacement, traction and mechanical parameters referring to this vibrator are also indicated by the superscript k. The relation (4.2) from the earth model can thus be written as

£/*(*!,x2) = X JJ .T'-i(x\,x'2)G'\x]-x\,x2-x2)dx\dx2 , (4.15) i = l S

where the suffices i and k for the Green's function indicate that x'a and xa are situated on the ;'th and klh vibrator, respectively. The notation is illustrated in Figure 4.4.

Baseplate/

Figure 4.4 The notation used in the calculation of the array response.

The earth-vibrator model can now be formulated for the four different baseplate models: the assumption of uniform displacement, the assumption of uniform traction, the mass-loaded boundary condition and the flexural rigidity method.

4.2.1 Uniform displacement

The assumption of a uniformly distributed displacement can be formulated as (Section 3.2.1)

V\{xl,x2) = u\ . (4.16)

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The assumption of uniform displacement implies that the baseplate is treated as a rigid body, whose equation of motion is given by

F ^ * + \\j\{x\,x2)dx\dx2 = - co2MkpUk

3 . (4.17)

Combining this equation with the relation (4.15) from the earth model yields

a MPYJ JJ p' (X\~X\ 'X2~X2)T^X\ > x'2)dx\dx'2 +

+ ff Tk3(x\,x2)dx\dx2 = - F w W , ( , foralU1 , .r2on5*, k = \,2 ,K .

s' (4.18)

The resulting integral equation can be solved numerically for the traction distribution. The (constant) displacement can be obtained by substituting the traction values in the integrated equation of motion (4.17). A numerical check can be performed by calculating the displacement at all vibrator points using the earth model (equation (4.15)); the displacement values should be constant over the plate, with the same magnitude as the previously calculated displacement value.

4.2.2 Uniform traction

The assumption of a uniformly distributed traction can be formulated as (Section 3.2.2)

T\{X^X2) = T\. (4.19)

Combining the earth model with the integrated equation of motion of the baseplate,

r.applicd.k . k -.k I I , , , . r.k, , , . , , . , F" +A 7/3 = - co JJ to (x],x1)U3(x1,x2)dx]dx2 , (4.20)

where Ak denotes the magnitude of Sk, yields

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IJ dx\dx'2 o {x\, x'2) X T'3 JJ . G' (X\ -x\, x'2- x~2) dx\dx2 + 2

co 1=1

, , k _,* „applied.k „ + A T 3 = - F " , k=\,2,...,K. (4.21)

Equation (4.21) represents K equations with the K traction values as unknowns. The displacement can be calculated using equation (4.15) from the earth model. A numerical check on the solution can be obtained by using the integrated equation of motion (4.20) of the baseplate; the calculated traction and displacement distributions should also satisfy this equation.

4.2.3 The mass-loaded boundary condition

The mass-loaded boundary condition can be formulated as (Section 3.2.3):

1 k -.applied.k, . „ ,* , . , . vrk, . P FF (*,, x2) + T3(x1, x2) =-co a (x1,x2)Ui(xl, x2) . (4.22)

The applied pressure p<vplitdk is now assumed to be distributed uniformly over the plate, with magnitude

~applied,k napphed,k. . nappüed,k t P (xltx2)=PFy = . (4.23)

A

Combination of equations (4.15) and (4.22) yields the following relation

* * 2 ■& ff ik i A a{.xx.x2)o) ZJ j) P U]-x\,x2-x'2)Tj(x\,x'2)dxldx'2 +

1=1 s' , k —A, . _applied,k - _ k , - _ . ,

+ A Ti(xl,x2) = - FH , for all A'1,^2 ° n ^ - k= 1>2. K- (4.24)

The integral equation (4.24) can be solved numerically. A numerical check on the validity of this solution can again be obtained by calculating the displacement distribution by substituting the obtained traction distribution in equations (4.22) and (4.15), respectively.

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4.2.4 The flexural rigidity metliod

The flexural rigidity method gives the following relation between traction and displacement (Section 3.2.4):

u\{xl,x1) = jJkGflex,\x1-x\,x2-x'2)x

x { Tj(x\ ,x'2) + papphed' {X\, x2) }dx\dx'2 , (4.25)

where G ' is given by

„flex.k 1 I F ( 1 ) / * N u t 1 ) / - * M G' = — — [H0 (a r)-H0 (ia r)] . ( 4 2 6 )

8V e Dk ico

In these equations, the applied pressure P ' is defined as

paPplicd,k

P =< - (4.27) , elsewhere

Ft in which 5 ' denotes the area on top of the hh baseplate where the applied force is exerted, F k flex k

and AF-k denotes the magnitude of 5 ' . The suffix k for the Green's function G ' indicates

that x a and x'a are situated on the kih vibrator, r denotes the distance between observation point

x a and integration point xa (Figure 4.4):

r = \ixa-x'a)(xa-x'a)}A , (4.28)

k D denotes the flexural rigidity of the /hh baseplate, o denotes the (constant) surface density of mass of the £th baseplate and a is given by

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. »\* * CT ft)

(4.29)

F k.

Since the pressure on top of the kxh baseplate is applied at the centre region 5 ' of the £th baseplate, and is assumed constant in this region, the convolution of G a' with the applied pressure can be performed analytically. This yields

Uk3(x1,x2)= HtGfltX'k0cl-x\,x2-x'2) Tk^x\,x'2)dx\dx'2 + C

kF<"""ied'k,

(4.30)

in which

C * « - ' (4.31) SVoDkico

Combining equations (4.15) and (4.30) results in the following equation:

r l / / , £ ' * ( * ! - x \,x2- x'2)T'3(x\,x'2)dx'ldx2 + C t l JJs

+ - 7 J I . G (xl-xl,x2-x2)Tj,{xl,x2)dxxdx2 =-F HH

C JJs

forallATj, x 2 o n S \ it = 1,2, ,K . (4.32)

Since in this equation only the traction appears as unknown, the problem is now fully specified and can be solved numerically. The displacement distribution can be obtained from the traction distribution by substituting the latter values in equation (4.30), or in equation (4.15). As both equations should yield the same displacement distribution, this duality can again be used as a numerical check on the solution.

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4.3 The numerical procedure

Table 4.1 lists the parameters used in the modelling program.

Applied force, FaPPlicd

Length of baseplate, / Width of baseplate, w Mass of baseplate, M Reaction mass, Mr

Surface mass density of baseplate, o Flexural rigidity of baseplate, D Length and width of sample areas on baseplate, Ax Number of samples in Xj direction, NXl

Number of samples in x2 direction, NX2

Table 4.1 Parameters used in the modelling program.

The force F°PPlied is applied at the centre of the baseplate over 1 sample area (0.092 m x 0.092 m).The traction and displacement are assumed to be constant within each sample area. Using this assumption, the integral equations (4.18), (4.24) and (4.32) are solved for the traction distribution underneath the baseplate by a matrix inversion operation. Three numerical checks on the results are performed:

1. The accuracy of the numerical evaluation of the earth's Green's function is checked using Hankel's theorem (see Appendix C).

2. The displacement distribution underneath the baseplate is calculated from the traction distribution by using the earth model, and by using the vibrator model. The two displacement distributions are compared.

3. The complex power balance for lossless elastic media is applied to the numerical results (see Appendix C).

All three checks gave excellent results regarding the correctness of the solutions.

1 [Ns] 1.84 [m] 1.104 [m] 1400 [kg] 1540 [kg] 692.7 [kg m-2] 2.2 x 107 [Nm] 0.092 [m] 20 12

-98-

4.4 Numerical results

In the following, two different earth models are used, whose properties are listed in Table 4.2.

Model I Model n

Cptms-1) 400 1000 c,(ms-') 150 400 p (kgnr3) 1700 1700

Table 4.2. The two earth models.

First, the distribution of traction and displacement over the baseplate is discussed. Results are shown for different baseplate models and frequencies, and for two different earth models. Then, the average radiation impedance and the average power factor are calculated for the earth-vibrator system. The power distribution in the far field of the medium is shown for different model parameters. The behaviour of synthetic vibrator signals that are of practical interest (the ground force and the baseplate acceleration) is discussed. Finally, a brief discussion is presented in which the effect of the use of a vibrator array on the calculated distribution functions is shown.

4.4.1 The distribution of traction and displacement over the baseplate

Figures 4.5-4.8 show contour plots of amplitude and phase of the traction and displacement distribution directly underneath the baseplate at 80 Hz, using four different baseplate models. Earth model I (c_ = 400 ms_1, cs=150 ms-1 ) is used in the calculations. Figures 4.5a and 4.5b show contour plots of the amplitude and phase of the traction distribution directly underneath the baseplate, using the flexural rigidity method. The amplitude of the traction, shown in Figure 4.5fl, is non-uniform, ranging from 0.2 to 0.5 [Ns m-2 ]. The traction has a maximum value in the centre of the plate, and a minimum towards the side of the plate. The distance between the two minima corresponds to a Rayleigh wavelength (1.66 m at 80 Hz). The phase of the traction, shown in Figure 4.5ft, lies in the range from 166° to 269°. The phase distribution has a minimum at the centre, and monotonically increases towards the sides of the plate.

Of the four baseplate models to be discussed, only the mass-loaded boundary condition and the flexural rigidity method yield non-uniform distributions of both traction and displacement.

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(b)

Figure 4.5 The distribution of traction and displacement at 80 Hz for earth model I, using the flexural rigidity method, (a) amplitude of the traction, contour interval 0.05 (Ns nr* ), (i) phase of the traction, contour interval 10 degrees.

The non-uniform displacement distribution using the flexural rigidity method is illustrated in Figures 4.5c and 4.5d, which show the modulus and phase of the baseplate displacement. The modulus of the baseplate displacement has a maximum of 14 [x 10~10 ms] in the centre, and decreases very regularly to a value of 8 [x 10_,° ms] at the edges of the plate. The phase plot of the displacement distribution has a structure which is similar to the amplitude distribution, but has a minimum of 72° at the centre and increases to a value of 134° towards the edges of the plate.

The distributions of traction and displacement when the mass-loaded boundary condition is used are shown in Figures A.ba-d. The traction distribution shown in Figures 4.6a and 4.6/; shows a highly irregular behaviour, with amplitudes varying from 0.1 [Ns nr 2 ] at the baseplate centre to 0.75 [Ns nr 2 ] towards the edges, and phase variations of 145° to 280°. A highly non-uniform traction distribution is present in the centre part of the plate. Also, artifacts

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«o

Figure 4.5 (continued) The distribution of traction and displacement at 80 Hz for earth model I, using the flexural rigidity method, (c) amplitude of the displacement, contour interval 1 (x 10-10 ms), and (d) phase of the displacement, contour interval 10 degrees.

associated with the Rayleigh waves can be observed in the amplitude plots. The same irregular behaviour is present in the displacement distribution (Figures 4.6c and 4.6d). The displacement has its maximum value in the plate centre, which is expected given the small traction in this area. The range in which the displacement lies for the mass-loaded boundary condition is 1.5 -35 [x 10"10 ms] for the amplitude, and 70°-235° for the phase.

If the displacement is assumed to be distributed uniformly over the plate, the traction is distributed as shown in Figures 4.7a and A.lb. The amplitude of the traction, shown in Figure 4.7a, increases very rapidly towards the plate edges, which is a well-known artifact of this baseplate model: the traction tends to infinity at the baseplate edges (van Onselen 1982). The phase shows a more regular behaviour, increasing from 180° at the plate centre to 260" at the edges of the plate. The (constant) displacement of the baseplate has a modulus of 16 [x 10" !0

ms] and a phase of 113°.

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Figure 4.6 The distribution of ttacuon and displacement at 80 Hz for earth model I, using ihe mass-loaded boundary condition, (a) amplitude of the traction, contour interval 0.05 (Ns m-2), (fc) phase of the traction, contour interval 10 degrees.

The distribution of the displacement if the traction is assumed to be distributed uniformly over the plate is shown in Figures 4.8a and 4.8b. The traction has a (constant) amplitude of 0.42 [Ns n r 2 ], and a phase of 213°. The modulus of the displacement, shown in Figure 4.8a, has its maximum value of 32 [x 10"10 ms] at the plate centre, and monotonically decreases to a value of 2.5 [x 10-1 0 ms] at the edge of the plate. The phase curve has a more irregular shape, with values in the range of 100°-150°.

From these figures, the following observations and conclusions can be made:

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<e)

(d)

Figure 4.6 (continued) The distribution of traction and displacement at 80 Hz for earth model I, using the mass-loaded boundary condition. (c) amplitude of the displacement, contour interval 5 (x 1(T10

ms), and (d) phase of the displacement, contour interval 20 degrees.

1. The mass-loaded boundary condition leads to highly irregular traction - and displacement distributions. This is explained by realizing that the mass-loaded boundary condition describes the relation between traction and displacement independently at each individual sample point, resulting in a non-coherent baseplate behaviour. This is in contrast to, for example, the flexural rigidity method, where the displacement at any point is influenced by the traction values at all other baseplate points. The mass-loaded boundary condition includes the plate stiffness by assuming a uniformly distributed applied force. Given the highly non-uniform displacement distribution, this is not sufficient to describe the plate stiffness for this particular earth model, and at this particular frequency of operation of 80 Hz.

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(a)

Figure 4.7 The distribution of the traction at 80 Hz for earth model I, using the assumption of uniform displacement, (a) amplitude of the traction, contour interval 0.05 (Ns m~2 ), and (fc) phase of the traction, contour interval 10 degrees.

2. Comparing the displacement distributions corresponding to the mass-loaded boundary condition and the assumption of uniform traction, it is observed that the range in which the amplitiude varies is identical, and that the geometrical distributions of the two methods look similar, apart from the Rayleigh wave artifacts which are absent in the uniform traction plots. The phase plots also have similar structures, although the non-uniformity is more pronounced for the mass-loaded boundary condition.

The assumption of uniform traction assumes a flexible plate. It was just shown that the plate stiffness is not included properly in the mass-loaded boundary condition, which explains the global correspondence between the two methods.

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Figure 4.8 The distribution of the displacement at 80 Hz for earth model I, using the assumption of uniform traction, (a) amplitude of the displacement, contour interval 5 (x 10"10 ras), and (b) phase of the displacement, contour interval 10 degrees.

3. The assumption of uniform displacement and the flexural rigidity method both explicitly take into account the plate stiffness. In the case of uniform displacement, the plate is treated as a rigid body, whereas in the flexural rigidity method a stiff plate is modelled by attaching a large value to the flexural rigidity D of the plate. The flexural rigidity method yields moderate variations in the displacement over the plate, which is to be expected for a stiff plate. Still, large differences are observed in the traction distribution. Also, particularly the phase of the displacement differs for both models: the centre value of the phase of the displacement is, for example, 72° for the flexural rigidity method, whereas the assumption of uniform displacement leads to a phase angle of 113°. The reason for the different results from these two methods is given in Chapter 3, where it is shown that a plate with a high modulus of rigidity under these loading conditions does not behave like a rigid body.

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(a)

Figure 4.9 The distribution of traction and displacement at 80 Hz for earth model II, using the mass-loaded boundary condition, (a) amplitude of the traction, contour interval 0.05 (Ns m-* ), (6) phase of the traction, contour interval 1 degree.

4. Of all four baseplate models, the flexural rigidity method yields, on average, the lowest values for the traction underneath the plate. Only the mass-loaded boundary condition has considerably smaller traction values concentrated in the centre region of the plate. The reason for the low traction values when the flexural rigidity method is applied is, that this method includes the bending forces acting in the baseplate in the equation of motion of the plate (see Section 3.2.4/).

5. The Rayleigh waves have a pronounced effect on most distributions. This is explained by Figure 4.3, which shows that the Rayleigh pole contribution is the main constituent of the response of the earth to an impulsive point force, especially at higher frequencies.

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(d)

Figure AS (continued) The distribution of traction and displacement at 80 Hz for earth model II, using the mass-loaded boundary condition. (c) amplitude of the displacement, contour interval 1 (x 10"10

ms), and (d) phase of the displacement, contour interval 5 degrees.

Next, the influence of the earth model on the traction - and displacement distributions is investigated. The distributions are calculated for earth model II, which has a P-wave velocity of 1000 ms - 1 and an S-wave velocity of 400 ms - 1 . Results are only shown for the mass-loaded boundar)' condition, since these results most clearly demonstrate the rigorous effects a different earth model has on the distribution of traction and displacement.

The modulus and phase of the traction and displacement distributions are shown in Figures 4.9a-d. Comparing these distributions with the distributions calculated with earth model I (Figures 4.6a -rf), it is observed that the distributions from earth model II are much more uniform. Also, a decrease in displacement, as well as the absence of Rayleigh wave interferences is visible in these figures.

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Figure 4.10 The distribution of traction and displacement at 40 Hz for earth model I, using the mass-loaded boundary condition, (a) amplitude of the traction, contour interval 0.05 (Ns m-2), (6) phase of the traction, contour interval 2 degrees.

The major differences of the results obtained with earth model II, and the previously calculated distribution functions of earth model I are summarized as follows:

1. the non-uniformity of both traction and displacement underneath the baseplate is less pronounced for earth model II, which has the higher P and S wave velocities.

2. the modulus of the displacement is decreased significantly for earth model II, whereas the traction attains higher values for this earth model.

3. the artifacts visible in the distribution functions of earth model I, associated with the Rayleigh waves, are absent in the results of model II.

The reasons for these differences are as follows. First, the behaviour of the amplitude of the displacement and the traction for the two earth models can be explained by the shape of the

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V)

Figure 4.10 (continued) The distribution of traction and displacement at 40 Hz for earth model I, using (he mass-loaded boundary condition. (c) amplitude of the displacement, contour interval 5 (x 10~10

ms), and (d) phase of the displacement, contour interval 5 degrees.

average radiation impedance curves that are presented in the next section. It is shown there that at a frequency of 80 Hz the average radiation impedance (which is defined as minus the ratio of the integrated traction and the integrated velocity) is significantly higher for earth model II than it is for earth model I. Second, the variation in the earth's Green's function is mainly determined by the Rayleigh pole contribution, as is shown in the previous section. The period of the Rayleigh pole oscillations is much smaller for earth model I than it is for earth model II, yielding a pronounced Rayleigh-wave interference pattern as well as a stronger non-uniform behaviour for model 1. For earth model II, the Rayleigh wavelength is much larger than the plate dimensions.

Another factor which affects the shape of the traction - and displacement distributions is the frequency of operation. From the phase curves of the earth's Green's function, it is obvious

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Figure 4.11 The distribution of traction and displacement at 40 Hz for earth model I, using the flexural rigidity method, (a) amplitude of the traction, contour interval 0.1 (Ns m-2), (b) phase of the traction, contour interval 10 degrees.

that due to the decreased Q -range the variation of the Green's function over the plate is less at lower frequencies (longer wavelengths) than it is at higher frequencies (shorter wavelengths).

The distributions of traction and displacement are calculated for earth model I, using a frequency of 40 Hz. The results using the mass-loaded boundary condition are shown in Figures A.\0a-d. Comparing these results with the distributions at 80 Hz (shown in Figures 4.6a-d), it is concluded that a lower frequency indeed yields a less pronounced non-uniform behaviour. If the flexural rigidity method is used, however, the decreasing nonuniformity as frequency decreases is less obvious, as shown by Figures 4.1 \a-d. The frequency dependence of the baseplate behaviour for the different baseplate models is discussed further in the next section.

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M

, /

1 1 1

\ \

^,-f.

/

1 f l \ \ . \

\ \ ^

\ . \

\

) .& I

1 /

. /

-, \

\

,

/

(d)

Figurt 4.11 (continued) The distribution of traction and displacement at 40 Hz for earth model I, using the flcxural rigidity method, (c) amplitude of the displacement, contour interval 1 (x 10"10 ms), and (d) phase of the displacement, contour interval 10 degrees.

In conclusion, the four different baseplate models used in the modelling all yield significant differences in the distributions of traction and displacement. These distributions also depend on the earth model and the frequency of operation.

To get more insight into the meaning of these different distributions, the average radiation impedance of the baseplate and the average power factor of the seismic vibrator, representing measures for the overall baseplate behaviour, are discussed as a function of frequency, and for different earth models.

- I l l -

4.4.2 The radiation impedance function

An important practical question is the shape of the radiation impedance function. The radiation impedance Imp is defined as the ratio of the pressure and the normal component of the particle velocity, or, since the traction equals minus the pressure,

T3(xltx2,<ti) Imp(.x1,x2,co) = • (4.33)

V-i(xi,x2,co)

From the non-uniform distributions shown in the previous section, it follows that "the" radiation impedance of the earth - vibrator system does not exist, since the radiation impedance varies over the plate for all four different baseplate models. Therefore, an average radiation impedance Imp^o) is defined as

1 ff 1 f f Ts(x1,x2,(o) lmpx((D) = T lmp(.x1,x2,0))dx1dx2 = ~T \\ dxxdx2 ,

A JJs A J}sV3(Xl,x2,<o) (4.34)

in which S denotes the baseplate area, and A denotes the magnitude of S. This definition is rather arbitrary. One could, for example, also define an average radiation

impedance Imp2{co) as minus the average traction divided by the average velocity:

- j j j T3(.x1,x2,co)dx1dx2

lmp2(co)= } S • (4.35)

j )) V3(x1,x2,co)dxidx2

The introduction of an average radiation impedance is meaningless if the two definitions (4.34) and (4.35) yield different results. Obviously, the two definitions yield identical results when assuming either uniform traction or uniform displacement. It is shown in the next figures that this is also true if either the flexural rigidity method or the mass-loaded boundary condition is used in the calculations.

The integration of traction and displacement over the baseplate area was numerically unstable for data of earth model I obtained with the mass-loaded boundary condition. This is due to the extreme variations of traction and displacement this baseplate model yields at high frequencies (see Figures 4.6a-d). Although not quite as severe, the same holds for the results obtained with the assumption of uniform traction. Therefore, results from earth model I will only be shown for the flexural rigidity method and the assumption of uniform displacement.

-112-

% !e*06

50 100 150

Frequency (Hz) (")

100 150

Frequency (Hz) (b)

Figure 4.12 The average radiation impedances Impi(co) (dashedline)andlmp2(co) (solid line) as a function of frequency for earth model I, using the flexural rigidity method, (a) amplitude (Nsm-3 ), ani\b) phase in degrees.

B

cu u c to

S»»07

OJ en to 03 > to ^_ ie+07 o

•o n

a ** n I i t i

50 100 150

Frequency (Hz) (a)

50 100 150

Frequency (Hz) (b)

200 » 0

Figure 4.13 The average radiation impedances Imp] (a)) (dashed line) and Impi(có) (solid line) as a function of frequency for earth model II, using the mass-loaded boundary condition, (a) amplitude (Nsm-3), and (b) phase in degrees.

Figures 4.12a and 4.12ft show modulus and phase, respectively, of the average radiation impedances Impi(co) and lmp2(oS), as defined by equations (4.34) and (4.35). The data are calculated for earth model I, using the flexural rigidity method. It is observed that the agreement between the two different radiation impedance definitions is excellent. The same is true for the average radiation impedance calculated with the mass-loaded boundary condition (earth model

-113-

II; Figure 4.13). It is thus conjectured that the introduction of an average radiation impedance is unambigious, and that this average radiation impedance can be calculated either by using equation (4.34) or by using equation (4.35).

It can already be observed from the previous figures that the two different baseplate models (the flexural rigidity method and the mass-loaded boundary condition) yield very similar radiation impedance curves, in spite of the significant differences between the individual traction - and displacement distributions as predicted by these different baseplate models. This is shown more explicitly in Figure 4.14, where modulus and phase of the average radiation impedance are shown for earth model n, using four different baseplate models.

«_ ie«J7 -

50 100 150

Frequency (Hz) (a)

50 100 150

Frequency (Hz) <b)

Figure 4.14 The average radiation impedance as a function of frequency for earth model II, using four different baseplate models: : flexural rigidity method, : mass-loaded boundary condition, — : uniform displacement, and - - - -: uniform traction, (a) amplitude (Nsirr3 ), and (b) phase in degrees.

The baseplate models all yield a similar radiation impedance function. Close agreement exists both in amplitude and in phase. This point is also illustrated in Figure 4.15, which shows the modulus and phase of the average radiation impedance for earth model I, using the flexural rigidity method and the assumption of uniform displacement. The high radiation impedance earth model II and the low radiation impedance model I show a similar behaviour. The modulus has its maximum at low frequencies, then rapidly decays and becomes more or less constant at higher frequencies. The modulus of the radiation impedance of model I even shows a slight increase at high frequencies. The phase of the radiation impedance is 90° at zero frequency, and then decays towards the higher frequencies. For earth model I, this decay is slightly more rapid (phase range more than 90°) than it is for earth model II (phase range less than 90°). The behaviour of the radiation impedance curve can be explained by considering a simple lump

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model for the earth containing a spring and a dashpot (Lerwill 1981). At low frequencies, the vibrator sees an elastic load (spring), and no energy is radiated. At baseplate-earth resonance (dashpot), pressure and particle velocity are in phase, and the baseplate sees a purely resistive load: all energy is used for the radiation of seismic waves. It must be noted that, due to the complexity of the earth's Green's function, this simple lump model can only serve as a qualitative illustration of the earth-vibrator interaction.

nl 1 1 t L_ 0 50 100 150 JOO

Frequency IHz] (a)

Figure 4.15 The average radiation impedance as a function of frequency for earth model I, using two different baseplate models: : flexural rigidity method, and — —: uniform displacement, (a) amplitude (Nsnr3 ), and (b) phase in degrees.

In conclusion, in spite of significant differences in the distributions of traction and displacement obtained with the four different baseplate models, all four models predict identical average radiation impedance curves. The average radiation impedance is highly reactive at low frequencies, but becomes purely resistive at higher frequencies. The frequency at which the radiation impedance becomes purely real depends on the earth model: the higher the velocities, . the higher this frequency.

4.4.3 The average power factor

Another issue that is investigated is the power output of a seismic vibrator. Following Tan (1975), the time-average of the outward acoustic power flow through the earth surface, Pm"'bi\ is defined as (see Appendix C)

Re {P0"'-bp(co) ) = - 1 Re { ƒ ƒ 73(.x,, x2 ,©) VjCx,, x2 ,ft>) dx]d.x2 } , (4.36)

50 100 150

Frequency (Hz) (b)

-115-

where S denotes the contact area between the baseplate and the earth's surface. In equation (4.36), only the vertical components of traction and particle velocity are taken into account.

Equation (4.36) states that the time-averaged outward power flow is determined by the phase difference of traction and velocity. If traction and velocity are 90° out of phase, no acoustic power is delivered to the earth. To illustrate this point, equation (4.36) is written as

Re [P°u'-bp (a>) ) = - 1 Re { ƒ ƒ T3(Xl ,x2,a» v\(Xï, x2 ,co)dx1dx2 }

= - 1 Re {\A | exp(i0„) } = - - \A | cos(flA) . (4.37)

The term cos(Ö/1) in equation (4.37) is an average measure for the phase difference between

traction and velocity (and thus for the time-averaged, radiated power), and is denoted as the average power factor.

50 100 150 Frequency (Hz)

-J ^ ^

S/ .//

s/ / / •V

ƒ / / /

f/ /

* \ V

50 100 150 Frequency 1Hz)

Figure 4.16 The average power factor as a function of frequency for earth model I, using two different baseplate models: : flexural rigidity method, and : uniform displacement.

Figure 4.17 The average power factor as a function of frequency for earth model II, using four different baseplate models: : flexural rigidity method, : mass-loaded boundary condition, — --: uniform displacement, and - - - -: uniform

Figure 4.16 shows the average power factor for earth model I, using two different baseplate models: the flexural rigidity method and the assumption of uniform displacement. The average power factor is small at low frequencies: the earth is moving up and down, but no acoustic

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energy is radiated into the earth. At 50 Hz, however, the power factor is close to 1, and stays more or less constant up to 200 Hz. The average power factor for earth model II, using all four different baseplate models, is shown in Figure 4.17. For this earth model, the frequency at which the average power factor (nearly) equals 1 is 200 Hz, which is much higher than it was for earth model I. Note that the average power factor is basically independent of the baseplate model: it only depends on the earth model and the size of the plate.

4.4.4 The radiated power distribution in the far field

100 150 200 Frequency (Hz)

1 / '" ~^>

100 150

Frequency (Hzl (b)

100 150

frequency (Hz) (c)

100 150

Frequency (Hz) (d)

Figure 4.18 The power flow (Nms) in the far field of the medium for eanh model 11, using four different baseplate models. : total power flow, — : P wave contribution, - - - -: S wave contribution, and : Rayleigh wave coniribution. (a) flcxural rigidity method, (f>) mass-loaded boundary condition, (c) uniform displacement and (d) uniform traction.

-117-

Next, the distribution of the outward power flow in the far field of the medium is investigated. Van Onselen (1982) derived expressions for the acoustic power flow through a surface composed of the earth surface, supplemented by either the surface of a hemisphere (for the P and S wave contributions) or by a cylinder (for the contribution of the Rayleigh waves).The configuration is shown in Appendix C. The relations in the far field of the medium are obtained by taking the radius of the hemisphere.and the cylinder to infinity. The power flow through the earth surface is zero except at the baseplate area; thus, this term represents the complex power radiated by the baseplate into the underlying halfspace. The outward power flow in the far field of the medium is composed of three terms, corresponding to the contributions of the P, S and Rayleigh waves, respectively. The expressions derived by van Onselen for these contributions are given in Appendix C; here, only the resulting distributions are shown.

100 150

Frequency (Hz)

(a)

100 150

Frequency (Hz)

0»

Figure 4.19 The power flow (Nms) in the far field of ihe medium for eanh model I, using two different baseplate models. : total power flow, — : P wave contribution, - - - -: S wave contribution, and : Rayleigh wave contribution, (a) flexural rigidity method, (b) uniform displacement.

Figure 4.18 shows the distribution of the far field power flow for eanh model II, and for four different baseplate models. Figure 4.19 shows similar curves for earth model I. The following observations can be made:

"1. The radiated power flow mainly consists of the contributions of the S waves and the Rayleigh waves. The P wave contribution is small, but increases with increasing frequency.

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Earth model I (Figure 4.19) shows a significantly higher P wave contribution than earth model II.

2. The Rayleigh wave contribution and the S wave contribution are of the same order of magnitude, but the Rayleigh wave contribution decreases more rapidly towards the higher frequencies.

3. The trend in the distributions for the four different baseplate models is identical, although in detail differences exist. This is illustrated in Figure 4.20, where the P wave contribution to the far field power flow is shown as a percentage of the total power flow using earth model II, for four different baseplate models.

50 100 150 Frequency (Hz)

200 250

Figure 4.20 The P wave contribution as a percentage of the total far field power flow. This percentage is shown as a function of frequency for earth model II, using four different baseplate models: : flcxural rigidity method, : mass-loaded boundary condition, : uniform displacement. and - — : uniform traction.

Figure 4.20 illustrates that for earth model II the contribution of the P waves to the total power flow is less than 15 % for all four baseplate models.

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4.45 Synthetic vibrator measurements

The results from the four previous sections seem to indicate that the different baseplate models result in detailed differences in the distribution functions of traction and displacement, but that the overall vibrator output is hardly affected by the choice of the baseplate model. In this section, three vibrator signals that are of practical interest are discussed: 1. the traction directly underneath the baseplate, integrated over the plate area (referred to as "integrated traction"); 2. the baseplate acceleration; and 3. the acceleration of the reaction mass. It is shown that the synthetic results for these quantities are affected by the choice of the baseplate model.

1. The integrated traction is of interest because its time derivative represents the far field wavelet shape (Chapter 2). Figure 4.21 shows amplitude and phase of the integrated traction as a function of frequency for earth model II, using four different baseplate models.

S \.i

'P. o->

- / / s \

-

1

\N \

\ \ \ '" \s 1 . 1 p

SO 100 ISO Frequency (Hz]

50 100 150 Frequency (Hz)

(a) (b)

Figure 4.21 The integrated traction underneath the baseplate as a function of frequency for earth model II, using four different baseplate models: : flexural rigidity method, : mass-loaded boundary condition, — : uniform displacement, and - - - -: uniform traction, (a) amplitude (Ns), and (b) phase in degrees.

All four baseplate models give different results, although the differences between the results obtained with the assumption of uniform displacement, the assumption of uniform traction and the mass-loaded boundary condition are small. Calculations for a different earth model (earth model I; Figure 4.22) demonstrate that the effect of the choice of baseplate model is even more pronounced for this medium.

-120-

60 100 150

Frequency (Hz)

(a)

50 100 ISO

Frequency (Hz) (b)

Figure 4.22 The integrated traction underneath the baseplate as a function of frequency for earth model I, using two different baseplate models: : flexural rigidity method, — ~ : uniform displacement (a) amplitude (Ns), and (6) phase in degrees.

2,3. In practice, either the reaction mass acceleration, or the baseplate acceleration, or both, are measured to serve as a feedback signal on the seismic vibrator. Since the response to a constant applied force is calculated, the reaction mass acceleration is identical for all four baseplate models (equation (3.19)). The baseplate acceleration in the centre of the plate is shown in Figure 4.23.

£ 0.0005

50 100 150 200

Frequency (Hz) (a)

cn <u L. en

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a. in

a

a ->50

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-

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// ^-— yy^^"^ ' ^ ^ .^-^

■ i

/ / /

/''--" £..--•

, , 100 150

Frequency (Hz) (h)

Figure 4.23 The acceleration in the centre of the baseplate as a function of frequency for earth model II, using four different baseplate models: : flexural rigidity method, : mass-loaded boundary condition, — : uniform displacement, and - - - -: uniform traction, (a) amplitude (ms-1), and (b) phase in degrees.

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The results are obtained using earth model II, using four different baseplate models. Significant differences are observed in both the amplitude and the phase behaviour of the baseplate acceleration for the different baseplate models. The global correspondence between the results obtained with the assumption of uniform displacement, the assumption of uniform traction and the mass-loaded boundary condition is only observed in the phase of the baseplate acceleration; the amplitude of the acceleration calculated using the assumption of uniform displacement is well below the amplitude values calculated with the other baseplate models. Again, a different earth model results in more pronounced differences between the results obtained with different baseplate models (Figure 4.24).

/ /

j i i i 50 100 150 200 J50

Frequency (Hz)

(b)

Figure 4.24 The acceleration at the centre of the baseplate as a function of frequency for earth model I, using two different baseplate models: : flexural rigidity method, and : uniform displacement, (a) amplitude (ms_1), and (b) phase in degrees.

It is concluded that the determination of the integrated traction (or ground force) is non-trivial, since the amplitude and phase behaviour of the ground force depends on the adopted baseplate model. The ground force calculated with the flexural rigidity method has much lower amplitudes than the ground force obtained with the other three baseplate models; also, phase differences exist between the results obtained with the flexural rigidity method and the results from the other three baseplate models. Synthetic baseplate acceleration curves confirm that the vibrator motion is different for different baseplate models. The validity of the different baseplate models using measurements of the baseplate acceleration and the ground force is discussed in Chapter 5.

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Frequency [Hz) (a)

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4.4.6 Vibrator arrays

In this section, the interaction between two adjacent vibrators is investigated. In practice, an array of vibrators is used (a) in order to emit a sufficient amount of energy into the ground, and (b) to attenuate surface waves. The results shown in the previous section are calculated using a single vibrator model. The effect of interaction is illustrated by comparing the response of a single vibrator to the response of an array of two vibrators.

Results are shown for earth model I, using the flexural rigidity method and the assumption of uniform displacement. The array configuration is shown in Figure 4.25.

■//

/

f Fapplied | /

/

M /*

/ fe

/ /

5

L

, /

7 ^

" ~ ) !

I

/ 7 Figure 4.25 The array of two (identical) vibrators, and the notation used in this section.

The array consists of two identical vibrators, with the same force FaPPlied of magnitude 1 (Ns) exerted on top of the baseplate of each of the vibrators. The numerical values used for the different parameters are listed in Table 4.3. The distributions of traction and displacement are calculated for 64 different frequencies. The frequency increment corresponds to an increment in the ratio R of centerline to centerline distance L and Rayleigh wavelength XR of 1/16. The ratio R = LIXR is referred to as scaled frequency. Thus, distributions are calculated for scaled frequencies R in the range 1/16 - 4. The calculated distribution functions are compared with the response of a single vibrator with the same dimensions, to the same applied force FaPPUcd. Results are shown as a function of scaled frequency R, along the centre line AB shown in Figure 4.26.

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Figure 4.26 The lines AB along which the traction distribution is shown, (a) vibrator array, and (b) single vibrator.

Length of baseplate, / 1.84 [m] Width of baseplate, w 1.104 [m] Centre line to centre line distance, L 5.52 [m] Spacing between vibrators, s (= 2 x / ) 3.68 [m] P wave velocity, cp 400 [ms-1] S wave velocity, cs 150 [ms-1] Rayleigh wave velocity, CR 141.7 [ms-1] Applied force, FaPP,ied 1 [Ns]

Table 4.3 Parameters used in the vibrator array calculations.

The modulus of the traction is presented as the percentage difference Perc between the array response and the single vibrator response:

Perc (x i, x 2, co )= T3{Xi,x2, co)

array T3(xj,x2, (0)

single

I iSingle x 100 % , (4.38)

in which the subscripts "array" and "single" refer to the array response and the response to a single vibrator, respectively. The phase of the traction, Phase, is presented as the phase difference between the array response and the single vibrator response:

array single Pliase (Xi ,x2, co)= ®T} (xltx2, co) - 0 7 j (*! ,x2, co) , (4.39)

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in which <PT denotes the phase of the traction. Figure 4.27 shows the differences in the traction

along line AB, as a function of scaled frequency R. The flexural rigidity method is used in the calculations.

(b) Figure 4.27 The differences in amplitude and phase of the traction between the array response and

the single vibrator response. X,-axis: scaled frequency, -Y2-axis: distance along line AB. (a) percentage difference in modulus of the traction, and (b) phase difference in degrees.

The percentage difference Pcrc in the modulus of the traction, shown in Figure 4.27a, has a maximum value of 38 % at R = 53/16. The phase difference Phase between array response and

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single vibrator response, shown in Figure 4.27ft, contains values up to 20° (at R = 59/16). Similar calculations for the distribution of the displacement on the vibrator show that the the displacement is less affected by these interaction effects: the maximum variation of the modulus of the displacement is 10 %, and the maximum phase difference is equal to 10°. The period of the oscillations occurring in the direction of the scaled frequency axes cannot be readily correlated with the ratio of some measure of the array dimension and the Rayleigh wavelength. At low frequencies, the period of the oscillations is equal to 1, but the period increases with increasing frequency.

These results do not change drastically for a different baseplate model. This is illustrated in Figure 4.28, which shows the difference in the modulus and phase of the traction at point B (shown in Figure 4.26) as a function of scaled freqency R, for two baseplate models: using the flexural rigidity method, and using the assumption of uniform displacement.

Although differences in absolute values occur, it is observed that the general shape of the curves is very similar for both baseplate models.

Finally, the effect of interaction on the integrated traction underneath the baseplate is calculated. Figure 4.29 shows the modulus and phase of the integrated traction underneath the baseplate, for two different baseplate models: the flexural rigidity method and the assumption of uniform displacement. The modulus of the integrated traction is presented as the percentage difference ?ercint between the array response and the single vibrator response, and is defined as:

Scaled frequency

Figure 4.28 The percentage difference in the modulus of the traction at point B on the baseplate, as a function of scaled frequency R. : flexural rigidity method, — : uniform displacement.

Perc (&))=

array

[ j r3(X], A-2, co)dxldx2 - jj TjiXi, x2, o))dx}dx2

single

j j T£xx,x2, co)dx]dx2

single x 100%.

(4.40)

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The phase of the integrated traction, Ptiase'"1, is presented as the phase difference between the array response and the single vibrator response, and is defined as:

int ""^ singU

Phase (CD) = fpj.* (co) - 0T^ (co) (4.41)

in which 0T^denotes the phase of the integrated traction. 1 3

1 2 Scaled frequency

1 2 Scaled frequency

(b)

Figure 4.29 The differences in amplitude and phase of the integrated traction between the array response and the single vibrator response, for two different baseplate models:. :: flexural rigidity method, —

: uniform displacement (a) percentage difference in the modulus of the integrated traction, and (b) phase difference in degrees.

Figure 4.29 demonstrates that the effect of interaction on the integrated traction is small (the maximum amplitude difference is only 5 %, and the maximum phase difference is only 3° ).

As a last remark, it should be noted that the effect that interaction has on the traction distribution underneath the vibrator is automatically included in the feedback system of the seismic vibrator. Unlike for example the signature of an air gun array, the effect of interaction on the ground force need not be taken into account explicitly in the determination of the far field wavelet. The formulation given in Chapter 2 of the wavefield in the medium, and in the far field of the medium, as a spatial convolution between a Green's function and the surface tractions remains valid when a vibrator array is used. However, the vibrator array introduces a frequency dependent directivity pattern if the array dimensions are not small compared with a wavelength. Then, the far field wavelet must be calculated by adding the contributions of the individual vibrators.

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4.5 Conclusions

The Green's function, which describes the displacement at the surface of an elastic halfspace due to the action of a point force acting at the surface, depends on the elastic parameters of the earth, the distance from the source and the frequency of operation. The density of the elastic medium is a simple amplitude scaling factor, and the compressional wave velocity only has a significant effect on the amplitude of the Green's function at low frequencies and/or small distances from the source. The shear wave velocity is the determining parameter for both the amplitude and phase of the earth's Green's function. The Rayleigh waves, whose propagation is also mainly determined by the shear wave velocity, dominate the behaviour of the Green's function for a large range of frequencies and source distances.

The coupling of the earth's Green's function with different vibrator models yields significant differences in the distributions of traction and displacement underneath the baseplate. Only the flexural rigidity method and the mass-loaded boundary condition yield non-uniform distributions of both traction and displacement. The mass-loaded boundary condition, however, does not yield moderate variations in the displacement distribution which are to be expected for a stiff plate.

The overall plate behaviour as defined by the average radiation impedance and the average power output is very similar for all baseplate models. The different models do, however, affect the amplitude and phase behaviour of the ground force and the baseplate acceleration at the centre of the plate. Therefore, the validity of the baseplate model is crucial for the determination of the ground force from measurements on the vibrator, and a comparison of the different model results with field measurements is essential.

Finally, it is shown that only a small part of the radiated power is contained in the far field compressional wave. The P wave contribution increases with frequency and is larger for a low-impedance medium (earth model I) than it is for a high-impedance medium (earth model II).

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Chapter 5 Test of the theory by experiment

In this chapter, the theory developed in the previous chapters is compared with measurements from a land vibrator experiment.

The land vibrator experiment is described in Section 5.1. In Section 5.2, the validity of the earth model is tested by modelling the reponse of a downhole geophone array and comparing the synthetic response with downhole measurements. Section 5.3 investigates the correctness of the four baseplate models that were introduced in Chapter 3: uniform traction, uniform displacement, the mass-loaded boundary condition and the flexural rigidity method. For the latter three baseplate models, a comparison is made between the measured baseplate acceleration, and the predicted baseplate acceleration using measurements of the traction underneath the baseplate and the acceleration of the reaction mass.

In Section 5.4, the question of which feedback signal to use on a seismic vibrator is addressed. From the earth model, it follows that the far field wavelet essentially equals the time derivative of the ground force. The determination of the ground force is non-trivial because the traction underneath the baseplate is not uniform. The ground force predicted by three conventional feedback signals (baseplate acceleration, reaction mass acceleration and the weighted sum method) and one new alternative feedback signal (the flexural rigidity feedback signal) is compared with the exact ground force. The exact ground force is obtained by summing the contributions of the individual pressure transducers over the baseplate area.

For the frequency-domain results shown in this section, the following convention for the phase curves is adopted. If two measurements A and B, with phase spectra <pAand <t>B, respectively, are compared, the phase difference Atf> = <pA - (j>B is denoted as the phase difference between A and B (in this order). All amplitude plots show true amplitudes, unless stated otherwise.

5.7 The data set

In this section, the setup for a series of experiments with a single land vibrator, performed on August 25-26 1987 by the Koninklijke Shell Exploration and Production Laboratory, Rijswijk, the Netherlands, is described.

In this experiment, a single vibrator was located above a borehole which contained an array of downhole geophones. The geophone spacing was 2.5 m, and the deepest phone was clamped at a depth of 176.5 m. The vibrator used in the experiments was a Prakla Seismos VVDA vibrator with a baseplate mass Mp of 1,400 kg, a reaction mass Mr of 1,540 kg and a holddown mass Mh of 15,000 kg. The dimensions of the baseplate were 1.84 m x 1.1 m. The peak driving force was 80 kN.

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In all experiments, a linear, 14 sec, 8-200 Hz sweep was emitted with a taper length of 250 msec. Three different feedback signals were used: baseplate acceleration, reaction mass acceleration and the weighted sum method. During each sweep emission, the following data were recorded (Figure 5.1):

'A. ■ti T

Reaction mass M

MDS 10 recording instrument

pressure transducers

Figure 5.1 Measurements made on the seismic vibrator during the land vibrator experiment.

1. The baseplate acceleration a3, measured with a Brüel &Kjaer type 4321 accelerometer of known sensitivity, and positioned on the top cross of the baseplate/stilt assembly.

2. The reaction mass acceleration an also measured with a Briiel «feKjeer type 4321 accelerometer.

3. The distribution of the pressure directly underneath the baseplate, using a crystal bed of known sensitivity. Measurements of the pressure underneath the baseplate were made using pressure transducers at 40 different positions on the baseplate. By summing the contributions of the individual pressure transducer responses, the true force exerted by the baseplate on the earth could be obtained. The individual pressure responses were also recorded. The pressure exerted by the baseplate on the pressure transducers has the same magnitude and polarity as the traction at the surface of the earth. Pressure transducer

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measurements were only recorded when the weighted sum signal was used as a feedback signal.

4. The vertical component of the downhole particle velocity at 24 positions down the borehole, using Geosource SM-6B geophones of known sensitivity s, and with a known damping factor d of 0.67 and natural frequency/0 of 4.5 Hz. These measurements were corrected for the geophone response using the expression for the impulse response 1(f) of the geophone (Pieuchot 1984):

I(f)=s^—f ' (5-D f-ft-lidffs

in which ƒ denotes frequency. 5. Other measurements (for example, the vibrator sweep produced by the Texas Instruments

VCS-5 vibrator controller, and the weighted sum of baseplate and reaction mass acceleration).

The recording instrument was a 96-channel MDS 10 (no. 108) system. The sampling rate was 2 msec, with a total recording time of 16 s. A high-cut filter of 72 dB per octave was applied at 250 Hz. All signals were recorded using the same recording instrument. In the data shown in this chapter, the weighted sum method was used as a feedback signal, and both amplitude and phase were controlled.

Originally, all traces consisted of 8000 samples per trace, with a sampling interval of 2 msec. To obtain the frequency domain data, the traces were padded out with zeros to 16384 samples, a Fourier transform was performed, and the frequency domain data were resampled at 1 Hz sampling intervals to reduce computation times. No further pre-processing was applied to the data. Since the phase spectra of the recorded signals oscillate very rapidly, only phase differences between signals will be shown. All amplitude spectra represent true amplitudes, that is, the sensitivities of accelerometers, geophones and pressure transducers are taken into account throughout this chapter.

5.2 The earth model

During the land vibrator experiment described in Section 5.1, measurements were recorded of the response of a downhole geophone array to a sweep emitted by a seismic vibrator located near the borehole. In this section, a comparison is made between these measured signals and the modelled downhole response using the theorj' described in Chapter 2. Since the vibrator was positioned only a few metres from the borehole, the theory for the acoustic layered medium

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Legend **9 r , n ,

m

I . g r

I . k

y e n

f.b g

zand(z)

klci (It)

veen (v)

schelpen (s)

grim (gr)

groen (g)

bruin (b)

= sand

= clay

«= peal

= shells

= gravel

= green

» brown

Figure 5.2 The geology at the location

of the experiment (top 50 m).

(Section 2.11) was used in all calculations. The 24 downhole geophones were located at 2.5 m intervals, between depth levels 119 m and 176.5 m. During the experiment, a linear, 14 s, 8-200 Hz sweep was emitted using the weighted sum signal as a feedback signal.

A log indicating the geological boundaries was available for the location of the experiment. The log for the top 50 m is shown in Figure 5,2.

5.2.1 The velocity profile

The log shown in Figure 5.2 was combined with a velocity distribution obtained from first arrival-times of a dynamite survey. Figure 5.3 shows the raw velocity profile; the dotted lines in this figure indicate the geological boundaries derived from the log, and the solid dots indicate the positions where travel time data were available (every 2.5 m).

Figure 5.3 The raw velocity profile for the top 50 m. Dots indicate geophone positions where travel lime data were available; dotled lines indicate geological boundaries from the log.

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Figure 5.4 shows the velocity profile that is obtained by combining the log and the traveltime data.

Figure 5.4 The velocity profile for the top 50 m (bold line). Dots indicate geophone positions where travel lime data were available; these measurements are connected by the solid lines. Dotted lines indicate geological boundaries from the log.

Below 50 m, the velocity distribution indicated that the geology is very homogeneous, with interval velocities of 1720 (ms_1 ) between depth levels 43.7 m and 130 m, and 1840 (rns-1 ) for the deeper layers. From dynamite uphole times and land airgun downhole arrival times, an anomalous velocity of 270 (ms_1 ) was found for the top layer (first 3 m).

5.2.2 The density profile

I. Clay Because the clay layers are located at shallow depth, the clay is expected to be uncompacted with a porosity between 0.3 and 0.5; a value of 0.4 is assumed. The density of the clay formation can then be calculated using the equation

Pb=(l-<P)P™+Pf, . (5.2)

in which p denotes the density (kg m - 3 ), (j)denotes the porosity and where the subscripts b,

ma and// denote the rock, the matrix and the fluid, respectively. For p m , the density of

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kaolinite (2650 kg rrr3 ) is assumed. The density of the clay formation is thus found to be 2000 (kg rrr3 ).

2. Sand The density of the sand formations is also calculated using equation (5.2). The matrix density is now equal to 2650 (kg n r 3 ), which is the density of sandstone. The porosity is estimated using the Wyllie equation (Wyllie et al 1956):

At-Atma 1

in which At denotes the interval transit time (pts/h), and Bcp is a correction factor for unconsolidated sands. An estimate for Bcp is obtained from the equation (van Baaren 1987)

**=%r • . (5.4)

Combining the velocity profile with the log indicating the geological boundaries, the velocities of the sand layers and the clay layers are estimated to be 2000 (ms-1 ) and 1700 (ms-1 ), respectively. From these velocities, the interval transit times are calculated. Then, the porosity of the sand formations follows from equation (5.3), and is equal to 0.41. The density of the sand formations is calculated using equation (5.2), yielding a value of 2000 (kgm-3).

3. Peat An estimate of the density of formations containing clay and peat is obtained by weighting the densities of clay and peat with the thicknesses of these layers. For the layers containing peat, the density of lignite (1200 kg n r 3 ) is assumed. The density of the clay-peat formation between depth levels 10.4 m and 14.5 m is then found to be 1720 (kg n r 3 )'. The same procedure is applied to the sand-peat formation between 18.2 m and 22.7 m, yielding a density of 1560 (kg rrr3) .

The resulting density profile for the top 50 m is shown in Figure 5.5. Since the velocity profile at larger depth levels was more or less constant (approximately 1800 ms-1 ), a constant density of 2000 (kg n r 3 ) is assumed in the deeper layers.

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t

MO — 1: : : t 1:..: t r—~—; - r - :—-

1900-

i«oo- :

170O-

IfiOO- ;

15H—i—i—i—i—i—i—i—i—i—i—r—i—i—i—i—i—i ' i ' i '~i—r~i—r~i—r (I 10 20 .VI 40 Si

► depth imj

Figure 5.5 Density profile for the top 50 m. Dotted lines indicate geological boundaries from the log.

5.2.3 Modelled and measured downhole response

The following steps are performed to obtain the modelled downhole response.

1. Using the (known) acoustic parameters of the earth as input, the response in the co- p domain is calculated by means of a recursive scheme (equations (2.102)-(2.105) and (A.94)). The calculations were performed for 200/?-values; an increase in the number of p-values did not yield any significant changes. The denominator in the recursive expressions yields an infinite number of internal multiples. To diminish wrap-around effects caused by the finite time interval that is considered in the calculations, this denominator is rewritten as a numerator in its Taylor expansion so that any integer number of internal multiples can be taken into account. At post-critical incidence, however, this expansion does not converge, and no multiply reflected energy is included for these p values to avoid numerical instabilities. The highest p-valuep,^ is chosen to be

1.2 Pmax = — > (5.5)

where cmi-n denotes the lowest velocity in the medium, which ensures that the contribution of the evanescent wavefield is included for all practical purposes (Spaans, 1988).

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TRRCE-NUMBERS

0. 10. 15. 20. 25. 3D. 35. to. «5.

50.

100.

ï 150.

200.

250.

Figure 5.6 Real and synthetic downhole geophone response plotted alternately. Distance between traces is 2.5 m. The first trace corresponds to the measured geophone response at 176.5 m depth, trace 2 to the modelled response at 176.5 m depth, trace 3 to the measured response at 174 m depth, etc. One internal multiple is included in the calculation of the synthetic seismogram.

2. The co-p response is transformed to the space domain according to equation (2.108), and spatially convolved with the recorded surface tractions.

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50.

£ 100.

150.

200.

?50.

Figure 5.7 Real and synthetic downhole geophone response plotted alternately. Distance between traces is 2.5 m. The first trace corresponds to the measured geophone response at 176.5 m depth, trace 2 to the modelled response at 176.5 m depth, trace 3 to the measured response at 174 m depth, etc. Three internal multiples are included in the calculation of the synthetic seismogram.

3. The resulting seismogram is multiplied with the complex conjugate of the weighted sum signal (corresponding to a cross-correlation with the weighted sum signal in the time domain).

4. The result is then transformed to the time domain.

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The downhole measurements are processed in the following way. First, the measurements are transformed to the frequency domain. The measurements are then corrected for the sensitivity and frequency-dependence of the geophpne response. Finally, the measurements are multiplied with the complex conjugate of the weighted sum signal, and transformed to the time domain.

Figure 5JS Detailed view of real and synthetic downhole geophone response at 176.5 m depth. (A) measured response. (B) modelled response, 0 internal multiples included, (C) modelled response, 1 internal multiple included, (D) modelled response, 2 internal multiples included, (E) modelled response, 3 internal multiples included.

The processing sequences described above show that not only the shape of the modelled and measured downhole signals can be compared, but that also the amplitudes are preserved. Figure 5.6, showing both modelled and measured seismograms, is plotted with a constant scaling factor which is equal for all traces. Measured and modelled signals are plotted alternately: trace 1 corresponds to the measured geophone response at 176.5 m depth, trace 2 to the modelled response at 176.5 m depth, trace 3 to the measured response at 174 m depth, etc. From Figure 5.6, it is observed that the modelled seismogram is very close to the measurements for the first arrivals and the multiples immediately after this first arrival. The multiple train at later arrival times (starting at 150 msec for the first trace) is not modelled correctly. In Figure 5.6, one

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internal multiple was included in the calculation of the synthetics; Figure 5.7 shows the synthetic result with three internal multiples. Again, the first arrival and the multiples immediately after this first arrival are modelled correctly, but the multiple train at later arrival times, which is not present in the measured data, is even more pronounced in this figure than it is in Figure 5.6. Also, some wrap-around effects due to the finite time interval that is considered in the calculations are visible at early arrival times.

TRRCE-NUfBEfi.S TRUCE-NUMBERS

0. 10. 5. 10. 15. 20.

50.

100.

150.

(a) (b) Figure 5.9 As Figure 5.6, but with a density - and velocity distortion in the model, (a) density distortion;

densities between 1500 and 2500, and (b) velocity distortion of 10 %.

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Figure 5.8 shows the part of the modelled and measured data that contains the first arrival, for the depth level 176.5 m. The modelled result is shown with 0, 1, 2 and 3 internal multiples included. Again, the same constant scaling factor is applied to all traces. The synthetic results which include one or more internal multiples show a good match with the measured response.

Part of the differences at the later arrival times between modelled and measured response can be explained by the extreme inhomogeneity in the top 50 m of the earth, where very thin layers containing peat and clay or sand and clay are embedded in sand and clay layers, which at certain depth levels also contain gravel and shells. These layers are described as plane, homogeneous, isotropic, acoustic layers in the modelling. Second, differences will exist because different types of errors are present in the synthetics (numerical errors, errors in the transducer measurements) and the real data (noise, errors in the geophone sensitivities). The noise in the field data is small because the experiment was repeated several times, and no significant differences between a single trace and a stack of four traces could be observed. Third, the field measurements will be distorted by the imperfect coupling of the geophones in the borehole.

Next, the effect of inaccuracies in the density and velocity profiles is investigated. Figure 5.9a shows the effect of a distortion of the density profile on the synthetic seismogram. Densities of 1500 and 2500 [kg m -3 ] were used alternately for the different layers in the model; thus, layer 1 has a density of 1500, layer 2 of 2500, layer 3 of 1500, etc. The main effects of the perturbation of the density contrast are that the amplitude of the first arrival is decreased, and that the multiple starting at 180 msec also has a lower amplitude. There is still a good correspondence between measurements and synthetics. Figure 5.9ft shows the effect of a distortion of 10 % in the velocity profile. The velocity of the first layer is increased by 10 %, the velocity of the second layer is decreased by 10 %, the velocity of the third layer is increased by 10 %, etc. The main effect of the distortion in the velocities is that the multiple at 135 msec has a higher amplitude.

It is concluded that the acoustic earth model described in Chapter 2 accurately predicts the downhole geophone reponse. Using the acoustic parameters of the earth and the pressure distribution directly underneath the baseplate as input, both the shape and absolute amplitudes of the first arrival and first multiples are modelled correctly.

5.3 The vibrator model

In Chapter 4, it was shown thai different baseplate models result in differences in the synthetic baseplate acceleration and ground force. Results from the land vibrator experiment are used to investigate the validity, or otherwise, of the different baseplate models. In the measurements used in this chapter, the weighted sum signal is used as a feedback signal. The amplitude and phase of the baseplate acceleration, reaction mass acceleration, the weighted sum

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signal and the traction under the baseplate integrated over the plate area (the measured ground force) are shown in Figures 5.10 and 5.11.

™ 20000

50 100 150

Frequency (Hz)

(a)

Z 20000 -

50 100 150

Frequency (Hz)

50 100 150

Frequency (Hz)

(c)

50 100 150

Frequency (Hz)

(d)

Figure 5.10 Amplitude of measurements made on the seismic vibrator during the land vibrator experiment, (a) baseplate acceleration (ms-1 ), (fc) reaction mass acceleration (ms-> ), (c) the weighted sum signal (Ns) and (d) the traction under the plate, integrated over the baseplate area (Ns).

The phase plots show the phase difference with minus the acceleration of the reaction mass. It was shown in Chapter 3 that the reaction mass acceleration is a measure of the force applied to the plate:

Farp'ial=-M,A, (5.6)

As a test on the accuracy of the different baseplate models, the acceleration of the plate is predicted using measurements of the traction underneath the plate and of the acceleration of the

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50 100

Frequency (Hz)

(a)

50 100

Frequency (Hz)

(b)

50 ' 100

Frequency (Hz)

(c)

Figure 5.11 Phase difference in degrees between measurements made on the seismic vibrator during the land vibrator experiment and minus the measured reaction mass acceleration, (a) baseplate acceleration, (b) the weighted sum signal and (c) the traction under the plate integrated over the baseplate area.

reaction mass. This prediction of the baseplate acceleration is compared with the true (measured) baseplate acceleration. The reason for this choice of comparison is as follows. First, no measurements of the baseplate acceleration were available outside the baseplate centre, but the traction underneath the plate was measured and recorded at 40 different positions on the plate. Second, the flexural rigidity method is obtained by deriving an expression for the baseplate acceleration in terms of the traction and the reaction mass acceleration.

The measurements of the baseplate acceleration were not made on the baseplate itself, but on the top cross of the baseplate/stilt assembly. Measurements made by Sallas et al (1985; Figure

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5.12) show that accelerometers on the top cross and on the baseplate give different responses, with phase differences between top cross and pad average up to 50° at 100 Hz.

TOP CROSS

PAD AVERAGE

0 20 40 60 80 100

FREQUENCY (HZ)

Figure 5.12 The phase of the baseplate acceleration: on the lop c ross of the baseplate/stilt assembly, the pad average and overall assembly, (from: Sallas et al, 1985).

The vector representation of the different vibrator signals at 70 Hz (Figure 5.13) shows that the effect of the position of the accelerometer on the amplitude of the baseplate acceleration is small.

The effect of the positioning of the accelerometer on the phase of the baseplate acceleration is thus expected to be significant, but in the land vibrator experiment no measurements were available of the acceleration of the baseplate except at the top cross of the baseplate/stilt assembly.

Finally, one must bear in mind that a Prakla Seismos VVDA vibrator was used in the experiment. This is a different vibrator from the one used by Sallas et al (1985). The construction of the baseplate can affect the vibrator behaviour.

-90

-143-

Figure 5.13 Vector representation of signals measured on the vibrator, at a frequency of 70 Hz (from: Sallas et al, 1985).

5.3.1 Uniform displacement

The assumption of uniform displacement (see equations (3.20) and (3.21)) yields the following prediction of the baseplate acceleration:

A3 =■

Fa"pUe-\\\ T,{x„x2)dxldx2

M„ (5.7)

The measured and predicted baseplate acceleration are shown in Figure 5.14. The phase of the predicted baseplate acceleration is in good agreement with the measurements, but the amplitude behaviour shows discrepancies. The measurements show an increase in amplitude with increasing frequency, whereas the predicted acceleration remains more or less constant, and even slightly drops towards the high-frequency end.

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50 100 150 200 Frequency (HzI Frequency (Hz)

(b)

Figure 5.14 Predicted baseplate acceleration using the assumption of uniform displacement, and measured baseplate acceleration, (a) amplitude (ms-> ), predicted acceleration is solid line, measured acceleration is dashed line, (b) phase difference in degrees between predicted and measured acceleration.

The phase of the predicted baseplate acceleration is in good agreement with the measurements, but the amplitude behaviour shows discrepancies. The measurements show an increase in amplitude with increasing frequency, whereas the predicted acceleration remains more or less constant, and even slightly drops towards the high-frequency end.

Another estimate for the accuracy of the assumption of uniform displacement are Sallas's (1985) measurements of the baseplate acceleration over the baseplate area (Figure 3.3). These measurements show that the displacement is non-uniform, although the non-uniformity is moderate compared with the traction distribution. It was explained in Section 3.2.4/ that moderate variations in the baseplate acceleration can still have an appreciable effect on the baseplate behaviour.

5.3.2 Uniform traction

The assumption of uniform traction cannot be tested in the way described above. Sallas's (1985) measurements (Figures 3.3 and 3.4) indicate that the traction is varying much more over the baseplate area than the baseplate acceleration does. The non-uniformity of the traction in the land vibrator experiment can also be observed in Figure 5.15, which shows the traction distribuiion at 80 Hz for the land vibrator experiment.

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(o)

\ \ V \ \

w

Figure 5.15 The distribuüon of the traction underneath the baseplate at a frequency of 80 Hz. Data are from the land vibrator experiment, (a) amplitude,

6 contour interval 0.2 (x 10 Nsm-* ), and (b) phase, contour interval 5 degrees.

5.3.3 The mass-loaded boundary condition

The mass-loaded boundary condition (equation (3.31)) can be rewritten to yield the following estimate for the acceleration at the baseplate centre:

■^applied

■ U ( * , , * , ) = -■+ T3(x} ,x2)

C r ( A ] , X2) (5.8)

in which xadenotes the coordinates of the centre point on the baseplate, and A denotes the

magnitude of the baseplate area. Although the surface mass density o"is allowed to vary over

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the baseplate for this baseplate model, and is expected to be largest at the plate centre because of the presence of the hydraulic piston in this area, o is assumed constant over the plate:

M o=- p _ 1400 - 2 ,

1.84x1.1 = 692.7 (kgm ) (5.9)

The magnitude of crat the centre of the plate only scales the amplitude of the predicted baseplate acceleration. The measured baseplate acceleration, and the predicted baseplate acceleration using the mass-loaded boundary condition are shown in Figure 5.16.

50 100 150 200

Frequency (Hz)

(o)

Frequency \m) (b)

Figure 5.16 Predicted baseplate acceleration using the mass-loaded boundary condition, and measured baseplate acceleration, (a) amplitude (ms-1 ), predicted acceleration is solid line, measured acceleration is dashed line, (b) phase difference in degrees between predicted and measured acceleration.

It is observed from this figure that the amplitude and phase behaviour of the mass-loaded boundary condition is similar to the results obtained with the assumption of uniform displacement (Figure 5.14). This is not surprising since both models make use of the same equation of motion of the baseplate; the only difference is that the mass-loaded boundary condition uses this equation in its local rather than in its integrated form.

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5.3.4 The flexural rigidity method

The flexural rigidity method yields the following estimate for the plate acceleration (see equation (3.84)):

A 3 (x , , j r 2 )= -ö> ƒƒ Gf'"(xl-x\,x2-x2){Papp'ied(x\,x2) + T3(x\,x2)}dx\dx2 ,

(5.10)

in which the applied pressure PaPPlie^ is acting in the region SF (with area AF), which is assumed to consist of one sample in the centre of the baseplate:

PaPPUe\x\,X2):

~MrAr , , F

0 , elsewhere (5.11)

100 ISO

Frequency 1Hz) SO 100

Frequency (Hz) m

Figure 5.17 Predicted baseplate acceleration using the flexural rigidity method, and measured baseplate acceleration, (a) amplitude (ms-] ), predicted acceleration is solid line, measured acceleration is dashed line, (b) phase difference in degrees between predicted and measured acceleration.

Application of these equations to the centre part of the baseplate yields

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A-i(x\,xc2) = -(a jj Gfl'x(xc

l-x\,xc2-x'2) T3{x\, x'2)dx\dx'2 + m MrA.

8V0D (5.12)

In the calculations, D equals 2.2 x 107 [Nm] (equation (3.64)), and crequals 692.7 [kgnr2 ] (equation (5.9)). The measured baseplate acceleration, and the predicted baseplate acceleration using equation (5.12), are shown in Figure 5.17. The amplitude match is excellent, but a phase difference of approximately 90° exists between the two signals over the entire frequency band. Although the phase shift is not exactly 90°, it will be referred to as a 90° phase shift for brevity. The measured and predicted accelerations in the time domain are shown in Figure 5.18, for the time interval between 6 and 6.08 sec. This time interval corresponds to an instantaneous frequency of approximately 90 Hz.

rvj in

o

CD U

D . CU

in m

400000

200000 ->

0 -

-200000

-400000

I\P'

— 1 , | * 1

i j \\'i \ \\i

'ill

1

n*

<? \^

\\ ' i W ' v 1 \y' vi';

1

V- IV- t^\ Mi \\\\ ! '

i j i j 11 Ij \\\\ \ \\J i \\ i \

1

h \ j ili '\j

6.02 Time

6.04 (sec)

6.06 6.OB

Figure 5.18 The predicted baseplate acceleration in the lime domain using the flexural rigidity method (solid line), and the measured baseplate acceleration in the time domain (dashed line), in the time interval 6-6.08 sec.

In the time domain, the phase shift causes the maxima and minima in the predicted acceleration to coincide with the zero crossings of the measured signal. A discussion on the possible causes of the 90° phase shift is presented in the next section.

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The predicted baseplate acceleration is composed of two terms: one term represents the spatial

convolution of the Green's function with the traction measurements (shown in Figure 5.19),

and the second term contains the contribution of the applied force (shown in Figure 5.20).

% «000 -

150 Frequency (Hz)

(a)

50 100 Frequency (Hz)

(b)

Figure 5.19 Predicted baseplate acceleration using the contribution of the traction measurements to the flexural rigidity method, and measured baseplate acceleration, (a) amplitude (ms-1), predicted acceleration is solid line, measured acceleration is dashed line, (b) phase difference in degrees between predicted and measured acceleration.

£ 20000

50 100 150 Frequency (Hz)

(o)

SO 100 Frequency (Hz)

(b)

Figure 5.20 Predicted baseplate acceleration using the contribution of the applied force to the flexural rigidity method, and measured baseplate acceleration, (a) amplitude (ms -1), predicted acceleration is solid line, measured acceleration is dashed line, (b) phase difference in degrees between predicted and measured acceleration.

At low frequencies the major contribution to the predicted acceleration is obtained from the

traction undernea th the plate, whereas at high frequencies the predicted acceleration is

determined by the magnitude of the applied force. The phase shift of 90° is very pronounced in

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the contribution of the traction, but cannot be readily recognized in the term corresponding to the applied force. This could indicate that a phase error is present in the traction measurements, but no evidence was found to support this.

An interesting simplification to the expression (5.12) is obtained by assuming that the baseplate is stiff, so that the Green's function is approximately constant over the baseplate area. This simplification will be referred to as the simplified flexural rigidity method. In this case, the acceleration at the plate centre is given by (see equation (3.95))

c c ÏCO f [ f -. A-i(x1,x2) = = = Tz{X],x1)dxldx2 - MrAr \ .

8V0D s (5.13)

The baseplate acceleration predicted with this expression is compared with the results obtained with the exact flexural rigidity method, in which the spatial convolution of the traction under the plate and the Green's function is performed over the entire baseplate area. Figure 5.21 shows that the two methods agree very closely.

% <0000 -

100 150

Frequency (H2I (a)

so 100

Frequency (Hz)

Figure 5.21 Predicted baseplate acceleration using the simplified flexural rigidity method, and predicted baseplate acceleration using the exact flexural rigidity method, (a) amplitude (ms-1 ), simplified flexural rigidity method is solid line, exact flexural rigidity method is dashed line, (b) phase difference in degrees between simplified and exact flexural rigidity method.

The simplified flexural rigidity method is of practical importance for the estimation of the ground force from measurements of the accelerations of baseplate and reaction mass. This is discussed in the next section.

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5.4 The feedback signal on the seismic vibrator

From the earth model, it follows that the far field particle velocity is essentially equal to the time derivative of the ground force. Only the vertical component of the ground force, denoted by F 0 , is considered. F° is given by minus the integrated traction underneath the plate:

FG= - f f 7 3 (x, , x2)dxldx2 ■ (5.14)

In practice, it may be more appropriate to control the ground force itself instead of its time derivative, for the folllowing reasons. First, vibrators are operated as close to the maximum holddown weight as possible without decoupling. Second, the time derivative can be handled in processing. Third, controlling the time derivative leads to a reduced output at higher frequencies. Because of absorption, we need all the high frequency energy we can get. Therefore, the question which feedback signal best represents the ground force is discussed in this section. Four methods are discussed: baseplate acceleration (Edelmann 1982), reaction mass acceleration (Lerwill 1981), the weighted sum method (Castanet et al 1965; Sallas and Weber 1982; Sallas 1984), and a new method derived from the simplified flexural rigidity method. The latter method is referred to as the flexural rigidity feedback signal. For all four possible choices of the feedback signal, the predicted ground force is compared with the exact ground force. The exact ground force is obtained by summing the contributions of the 40 pressure transducers that were positioned underneath the baseplate during the experiment. The comparison between the different feedback signals and the directly measured ground force is performed in the frequency domain.

The effect of the errors present in the different feedback signals on the shape of the far field wavelet is investigated. From the predicted ground force, an estimate for the shape of the far field particle velocity is obtained by a simple differentiation with respect to time. This predicted far field wavelet is cross-correlated with the true (measured) far field wavelet, which is the time derivative of the measured ground force. The exact ground force is again obtained by summing the contributions of the 40 individual pressure transducers. Time-domain results of this far field cross-correlation function are compared with the autocorrelation of the true far field wavelet for all possible choices of the feedback signal. A comparison is made between both normalized cross-correlation and autocorrelation functions, and true amplitude cross-correlation and autocorrelation functions.

The baseplate acceleration feedback signal and the reaction mass acceleration feedback signal are not justified by any of the four models for the baseplate discussed in the previous section. The baseplate models assuming uniform traction, uniform displacement or the mass-loaded boundary condition all make use of the equation of motion of the baseplate without including

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the bending of the plate. The weighted sum method is derived from this equation of motion. The assumption of uniform traction and the mass-loaded boundary condition, however, require measurements of the baseplate acceleration over the entire baseplate to obtain an estimate for the ground force; these measurements were not available. The flexural rigidity feedback signal is based on the baseplate model described by the flexural rigidity method.

All results shown here use measurements of the baseplate acceleration with the accelerometer positioned on the top cross of the baseplate/stilt assembly. It was mentioned in the previous section that this will introduce phase errors, but that no measurements of the baseplate acceleration at different positions on the baseplate were available.

To avoid confusion, it should be noted that the phrase "measured ground force" refers in this chapter to the summation of the output of the 40 pressure transducers underneath the baseplate, and not to the weighted sum approximation of the ground force.

5.4.1 Baseplate acceleration

The method of using the baseplate acceleration as a feedback signal is based upon the assumed relation between the baseplate velocity and the far field particle velocity. Therefore, in all results that are shown for this feedback signal, the measurements of the baseplate acceleration have been transformed to baseplate velocity data by a simple division by - ico.

Using baseplate acceleration as a feedback signal, it is assumed that

1. The far field particle velocity is in phase with the near field particle velocity. 2. The near field particle velocity is equal to the baseplate velocity.

However, the radiation impedance functions calculated in Chapter 4 for different baseplate models indicate that a complex relation exists between the ground force and the baseplate velocity:

1. Significant amplitude and phase differences exist between the ground force and the baseplate velocity.

2. These amplitude and phase differences depend on the frequency of operation, and 3. These amplitude and phase differences are different for different earth models.

Thus, no theoretical justification can be given for the use of the baseplate acceleration as a feedback signal. This conclusion is supported by measurements of the baseplate velocity and the ground force, shown in Figure 5.22. Because of the large amplitude differences, both measurements are normalized to have a maximum value of 1.

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SO 100 150 Frequency (Hz)

(a)

SO 100

Frequency (Hz)

m Kiyure 5.22 Baseplate velocity and measured ground force, (a) normalized amplitude, baseplate velocity is solid

line, ground force is dashed line, (i) phase difference in degrees between baseplate velocity and measured ground force.

From this figure, it is obvious that large errors in both amplitude and phase exist when the baseplate acceleration is used as a feedback signal. The effect of these errors on the estimation of the far field wavelet is illustrated in Figure 5.23, which shows the normalized far field cross-correlation and autocorrelation functions. It is concluded that the use of the baseplate acceleration as a feedback signal yields large errors in the estimation of the far field wavelet.

The other three feedback signals (reaction mass acceleration, weighted sum signal and the flexural rigidity feedback signal) all have in common that they explicitly claim to give a measure of the ground force. In the discussion of these three feedbacks signals, the equation of motion of the plate, including the plate bending, plays an important role. This equation of motion is given by

4 4 4 9 t / 3 ( x i , s 2 ) | 2 3 U3(x},x2) | d U3(x},x2)

dx 2 2

3jTjÖ.Ï2 dx~

<7(A, , X2 )CO t / 3 (JC, , Jt2 ) = P"PPlied{x, , X2 ) + 7-3 (*1 ,X2), (5.15)

or, integrating both sides over the baseplate area 5,

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- 2 0 O

Time (msec) (c)

Figure 5.23 (a) normalized far field cross-correlation function using baseplate acceleration as a feedback signal, (fc) normalized far field autocorrelation function and (c) difference between the cross-correlation and the autocorrelation.

ƒƒ.[»< 4 4 4

dU-i(xl,x2) 3 [ / 3 ( A - , , i 2 ) 9 t / 3 (A , , J : 2 ) +2 ;—; + ) \ax irf.x'2 .+

ox, 2 2

ÖAJÖA'J dx*

fjj f f cr(.Vi , .v,) t /3 (.v, , .v,) rfr ,</*-, = F w M + ( [ 7 \ (.v, . .v2 ) dxrfx-, JJS - - - .i./^. (5.16)

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The equation of motion consists of four terms: the first term represents the bending force, the second term the inertial force, the third term the force applied on top of the baseplate and the fourth term (minus) the ground force.

5.4.2 Reaction mass acceleration

Lerwill (1981) first proposed to use the acceleration of the reaction mass as a feedback signal. Using the reaction mass acceleration as a feedback signal, it is assumed that

1. The far field pressure or particle velocity is in phase with the near field pressure. 2. The near field pressure is proportional to the reaction mass acceleration.

The use of the reaction mass acceleration as a feedback signal yields a measure for the force applied to the baseplate (equation (5.6)):

FaPPlieö = _M (5.17)

The following errors are introduced when the reaction mass acceleration is used to predict the ground force. First, the far field pressure or particle velocity is not in phase with the near field pressure, but with the time derivative of the near field pressure. However, this error, although not recognized by Lerwill, is not essential. A more fundamental error is introduced because the bending forces and inertial forces that are present in equation (5.16) are neglected when the reaction mass acceleration is used as a feedback signal.

Pi ML

100 IK

requency IHzl (a)

~ ^

frequency (Nz! (b)

Figure 5.24 The ground force predicted using the reaction mass acceleration, and ihe measured ground force, (a) amplitude (Ns), applied force is solid line, measured ground force is dashed line, (b) phase difference in degrees between predicted and measured ground force.

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The measured ground force, and the predicted ground force using the reaction mass acceleration as a feedback signal (the force FaPP,ied, given by equation (5.17)) are shown in Figure 5.24. From this figure, it is observed that the amplitude behaviour of the predicted ground force differs from the measured ground force amplitude spectrum, and that a large phase error is present, increasing from 0° to 180° towards the high frequencies. The effect of these errors on the far field cross-correlation function is shown in Figures 5.25 and 5.26.

Figure 5.25 (a) normalized far field cross-correlation function using the reaction mass acceleration as a feedback signal, (b) normalized far field autocorrelation function and (c) difference between cross-correlation and autocorrelation.

ïme (msec! «■>

The correlation functions show that the use of the reaction mass acceleration as a feedback signal results in a non-zero phase cross-correlation function. The main peak is not at time zero, and large differences in amplitude exist.

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-20 o

Time (msec) (b)

-20 0

Time (msec) (c)

Figure 5.26 (a) true-amplitude far field cross-correlation function using the reaction mass acceleration as a feedback signal, (6) true-amplitude far field autocorrelation function and (c) difference between cross-coiTelation and autocorrelation.

5.4.3 The weighted sum method

Using a weighted sum of baseplate acceleration and reaction mass acceleration, it is assumed that

1. The far field displacement is in phase with the ground force. 2. The ground force is equal to a weighted sum of baseplate and reaction mass acceleration.

The weighted sum method is based upon the integrated equation of motion of the baseplate, assuming uniform displacement (rigid body assumption):

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p.appUed^jj 7 3 ( A . ] ) A . 2 ) ^ l r f A - 2 =-Mp(l) U3 , (5.18)

or, using equations (5.6) and (5.14),

F G = - Ij T3(x1,x2)dx1dx2 = -[MpA3 + MrAr) . (5.19)

The weighted sum method neglects the bending forces in the plate, and neglects the effect of the non-uniformity of the displacement on the calculation of the inertial force. The effect of these approximations on the estimation of the ground force is shown in Figure 5.27.

n bi -J 1 1 1= ' -?oo ' 1 . i l— 0 50 100 150 SOC S O "o 50 100 ISO

Frequency (Hzl Frequency 'Hz)

Figure 5.27 The ground force predicted using the weighted sum method, and the measured ground force, (a) amplitude (Ns), weighted sum is solid line, measured ground force is dashed line, (b) phase difference in degrees between predicted and measured ground force.

The amplitude spectrum of the weighted sum signal is nearly flat because the weighted sum signal is used as a feedback for these measurements. The phase error between predicted and measured ground force is small, but the amplitude prediction of the weighted sum method is less accurate.

Normalized and true-amplitude far field cross-correlation functions are shown in Figures 5.28 and 5.29. Although some differences exist between the far field cross-correlation and autocorrelation functions, both normalized and true-amplitude results indicate that the use of the weighted sum method as a feedback signal yields an accurate estimate of the far field wavelet. It must be remembered, however, that in this experiment the weighted sum signal has a (nearly) flat amplitude spectrum, but the exact ground force (and consequently the far field wavelet) contains little high frequency energy (Figure 5.27a). Since the errors in the weighted sum method increase as frequency increases, the time domain cross-correlation result will be

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-20 O

Time (msec) (a)

-20 0

Time (msec) (b)

Time (msec) (c)

Figure 5.28 (a) normalized far field cross-correlation function using the weighted sum signal as a feedback signal, (b) far field autocorrelation function and (c) difference between cross-correlation and autocorrelation.

degraded if more high frequency energy is present in the exact ground force. Since it was shown in Chapter 4 that the plate bending and the non-uniform behaviour of the baseplate displacement depend not only on frequency, but also on the mechanical properties of the baseplate and the elastic parameters of the earth, the performance of the weighted sum method is also expected to be different at different locations, and for different vibrator types. The effect of these factors is not investigated here.

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-20 0

Time (msec) (a)

2e+aa

/ I * A.

-20 0

Time (msec) (c)

Figure 5.29 (a) irue-ampliiudc far field cross-correlation function using the weighted sum signal as a feedback signal, (b) true-amplitude far field autocorrelation function and (c) difference between cross-correlation and autocorrelation.

In practice, the measured response is not cross-correlated with the feedback signal f(t) (which is, for this experiment, the weighted sum signal), but with the pre-determined sweep input q(r) (see Chapter 1). The normalized far field cross-correlation function using the pre-determined sweep input is shown in Figure 5.30. Comparison of Figure 5.30 with Figure 5.28 demonstrates that the feedback signal f(t) and the pre-determined sweep q(i) yield similar normalized far field cross-correlation functions.

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0.5

0

-0.5

-

— A A A

■

ATV/V-V^- -^ -

I 1 -20 0

Time (msec) (a)

-20 . 0

Time (msec) (b)

-20 0

Time (msec) (e)

Figure 5.30 (a) normalized far field cross-correlation function using the pre-determincd sweep input, (i) far field autocorrelation function and (c) difference between cross-correlation and autocorrelation.

5.4.4 The flexural rigidity feedback signal

It was shown in Section 5.2 that the simplified flexural rigidity method, in which the Green's function G^7" is assumed constant over the baseplate area, gives results which are in close agreement with the results obtained with the exact flexural rigidity method. The simplified flexural rigidity method (equation (5.13)) allows the ground force to be determined from the equation

r n sVoD c c F = - JJ T3(x 1,x2)dxldx2 =-[ : A 3 ( J C , , X 2 ) + MrAr]. (5.20)

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Figure 5.31 shows amplitude and phase of the predicted ground force using the flexural rigidity feedback signal (equation (5.20)), and the measured ground force.

? 5 f07

50 100 150 200

Frequency (Hz) (a)

Frequency (Hz) 0»

Figure 5.31 The ground force predicted using the flexural rigidity feedback signal and the measured ground force, (a) amplitude (Ns), flexural rigidity signal is solid line, measured ground force is dashed line. (p) phase difference in degrees between predicted and measured ground force.

Despite the good performance of the flexural rigidity method when predicting the amplitude of the baseplate acceleration (Figure 5.17), the 90° phase shift present in this prediction introduces large differences in both amplitude and phase of the measured and predicted ground force. The

o

90 phase shift that is observed in the prediction of the baseplate acceleration has two possible causes.'

1. errors in the measurements It is unlikely that the measurements of the baseplate and reaction mass accelerations contain significant errors. Accelerometers have been used in the Vibroseis method for more than twenty years. Also, in the land vibrator experiment, high-quality calibrated accelerometers were used. The measurement of the traction underneath the baseplate is less trivial. A phase

o shift of 90 in the traction measurements alters the measured ground force in Figure 5.31.

o but not the predicted ground force. In the case of a 90 phase error in the traction

o

measurements, the weighted sum method will contain an additional 90 phase error if compared with the exact ground force. Sallas et al (1985) also measured the ground force using pressure transducers underneath the baseplate, and compared this ground force with the weighted sum signal. Their results, shown in Figure 5.32, are similar to the results

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obtained with the data from the land vibrator experiment used in this chapter. Therefore, an error in the measurements is considered unlikely.

30

DB 2D

10

22.5

S o

-22.5 !

.GROUND FORCE

LOAD CELLS

20 40 60 80 100 FREQUENCY ( H Z I

(a)

GROUND FORCE

LOAD CELLS

20 40 60 100

(b) Figure 5.32 The ground force predicted using the weighted sum method (denoted by "ground force"), and

the ground force measured using load cells underneath the baseplate (denoted by "load cells"). Measurements are from Sallas et al (1985). (a) amplitude (dB), and (b) phase in degrees.

2. errors in the tlicory If it is assumed that the Green's function derived for the flexural rigidity method is out by

o

90 (or a factor i), the predicted ground force using the flexural rigidity feedback signal is altered accordingly. This is shown in Figure 5.33, in which the ground force is predicted using the expression

F C = - ff 73(A1,A-2)d.v1dA2 =-[i*—^LA3(x'itXc2) + MrAr]. (5.2i)

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so too ISO Frequency (Hz)

100

Frequency (Hz) (b)

Figure 5.33 The ground force predicted using the phase-corrected flexural rigidity feedback signal (equation (5.21)) and the measured ground force, (a) amplitude (Ns), phase-corrected flexural rigidity feedback signal is solid line, measured ground force is dashed line, (fc) phase difference in degrees between predicted and measured ground force.

200 i

100

CD CD C_

cn

CD to

-100

- 2 0 0 ' 50 100 Frequency (Hz)

150 20C-

Figure 5.34 The phase errors (in degrees) in the predicted ground force. solid line: phasc-correcied flexural rigidity feedback signal (equation (5.21)), dashed line: weighted sum signal.

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The prediction of the ground force as expressed by equation (5.21) is referred to as the phase-corrected flexural rigidity feedback signal. The amplitude of the predicted ground force using the phase-corrected flexural rigidity feedback signal is now in almost perfect agreement with the measured ground force. The phase error is very similar to the phase error introduced by using the weighted sum method. This is illustrated in Figure 5.34, which shows the phase errors in the ground force prediction for the weighted sum signal and the phase-corrected flexural rigidity feedback signal.

The normalized and true-amplitude far field cross-correlation functions using the phase-corrected flexural rigidity feedback signal are shown in Figures 5.35 and 5.36.

-20 0

Time (msec) (b)

Figure 5.35 (a) normalized far field cross-eorrelalion function using the phase-corrected flexural rigidity feedback signal, (è) normalized far field autocorrelation function and (c) difference between cross-correlation and autocorrelation.

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-20 0

Time (msec)

Figure 5.36 (a) true-amplitude far field cross-correlation function using the phase-corrected flexural rigidity feedback signal, (b) true-amplitude far field autocorrelation function and (c) difference between cross-correlation and autocorrelation.

The far field cross-correlation functions demonstrate a perfect timing and an amplitude

behaviour which is slightly better than the results obtained with the weighted sum method.

Returning to the discussion of the 90 phase error, the derivation of the flexural rigidity

method consists of two steps:

a. derivation of the differential equation governing the baseplate deflection. This part contains many assumptions, but most of these assumptions have been well established in plate analysis for more than 100 years. Other assumptions, such as the assumption of

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infinite stiffness of the core in the direction perpendicular to the stiffeners, can, in my opinion, not account for a phase error in the Green's function.

b. solution of the differential equation. It was shown in Section 3.2 that the solution to the differential equation is causal and mathematically correct. If the phase correction is exactly 90°, the resulting phase-corrected Green's function is non-causal. An error is made because the finite plate dimensions are not included in the derivation, so that the boundary conditions at the plate edges are not rigorously satisfied. Again, no connection could be made between the 90° phase error and the mismatch in the boundary conditions.

As a last remark, it must be stated that if the measurements are correct, the phase error in the theory can never be exactly 90°, since this violates the condition of causality: a multiplication in the frequency domain with / sgn(ü)) corresponds to a convolution in time with \l(n t). In these expressions, the signum-function sgn(ü)),

sgn(ö)) = 1 , co>0

-1 , co<0 (5.22)

is introduced because the time domain signal is real. Since the Green's function &a is causal, a phase shift of 90° in the frequency domain results in a non-causal function in the time domain. Nevertheless, the results obtained with this phase-corrected Green's function are in good agreement with the field measurements. A possible explanation for this phenomenon can be found in the time domain behaviour of the Green's functions.

-

i

<

{ ^

^

Time (msec) -100 0

Ti^ie (msec)

(b)

Figure 5.37 Time domain versions of the Green's function for the baseplate velocity, C"'", and its phase-corrected version i G/1"", using a frequency band of 0-250 Hz. (a) time domain version of &"", and (fc) time domain version of i G^1"-"

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Since the transformation of the Green's function to the time domain cannot be done analytically, the Green's functions were calculated for a number of frequencies, and transformed to the time domain by a Fourier transform algorithm. Figures 5.37 and 5.38 show the time domain versions of the Green's function and the phase-corrected Green's function, using two different frequency bandwidths.

„„„„ . JOOOO i .

i I . 1 I I

"-} - i 0 1 ! - 2 - 1 4 1 2

Time (msec) Time (msec) (o) (b)

Figure 5.38 Time domain versions of the Green's function for the baseplate velocity, GIUxy, and its phase-corrected version i GP"-*, using a frequency band of 0-400 kHz. (a) time domain version of G "'v, and (b) time domain version of / &"■*,

For reasons explained in Section 3.2.4e, the Green's functions for the baseplate velocity are shown. The constant c (equation (3.79)) was chosen equal to 0.1, which is a realistic value for this problem. Figure 5.37a and 5.31b show the time domain versions using frequencies up to 250 Hz (a simple cosine taper was applied between 225 and 250 Hz). The non-causal behaviour of the phase-corrected Green's function is very pronounced in this figure. However, if the Green's functions are calculated for frequencies up to 400 kHz the non-causal behaviour of the phase-corrected Green's function is hardly visible because of the short duration of the signal (Figures 5.38a and 5.38fc). If the calculations are performed in the frequency domain, the pronounced non-causal behaviour induced by the bandlimitation will not be present, and effectively the Green's functions shown in Figures 5.38a and 5.38ft are used.

5.5 Conclusions

In this chapter, the theory developed in the previous chapters is compared with measurements from the land vibrator experiment.

The earth model developed in Chapter 2 is used to predict the downhole response to a seismic vibrator acting at the surface, using the traction measurements underneath the baseplate and a

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5000

-5000

10000

J

-

I / Y ^ -j l / ^ ^ ^

given geology as input. The predicted seismogram agrees closely with the measured downhole signals.

The accuracy of die different models describing the baseplate behaviour is investigated by using measurements of the traction underneath the baseplate and the acceleration of the reaction mass to predict the acceleration at the centre of the baseplate. .This predicted baseplate acceleration is then compared with the measured baseplate acceleration. This procedure cannot be applied to the baseplate model assuming uniform traction, but measurements of the traction distribution underneath the baseplate show that this model is not realistic. The assumption of uniform displacement yields a small phase error, but does not predict the correct amplitude behaviour. The mass-loaded boundary condition yields results that are very similar to the results obtained with the assumption of uniform displacement. The flexural rigidity method accurately predicts the amplitude of the baseplate acceleration, but a phase error of approximately 90° is introduced.

From the earth model, it follows that the time derivative of the ground force represents the shape of the far field wavelet. The time derivative is best handled in processing; the determination of the ground force from measurements on the seismic vibrator is non-trivial because the traction is not distributed uniformly underneath the baseplate.

Four approximate methods to predict the ground force have been compared to the exact, directly measured ground force. Using the baseplate acceleration as a feedback signal is not based on a theoretical foundation, and results in large amplitude and phase errors. The use of the reaction mass acceleration as a feedback signal predicts the force applied on the top of the baseplate, but this force differs from the measured ground force because bending forces and inertial forces are acting in the baseplate. The weighted sum method assumes that the baseplate behaves as a rigid body, and therefore neglects the bending forces in the plate and the effect of the non-uniformity of the displacement on the calculation of the inertial force. The errors that are introduced by these assumptions increase with increasing frequency, and also depend on the elastic parameters of the earth and the mechanical properties of the seismic vibrator. The amplitude behaviour of the ground force is not predicted correctly by the weighted sum method, but the phase errors are small. The effect of these errors on the estimation of the far field wavelet is calculated. A good match is obtained between the far field cross-correlation function using the weighted sum method as a feedback signal, and the far field autocorrelation function obtained from direct measurements of the ground force. The flexural rigidity method includes all forces acting in or at the baseplate. Due to the abovementioned phase shift of 90° in the prediction of the baseplate acceleration, the flexural rigidity feedback signal does not yield an accurate estimate of the ground force. If the phase shift is accounted for in the Green's function pertaining to the flexural rigidity method, both amplitude and phase of the ground force are matched very closely. However, no theoretical justification has been found for applying this phase correction.

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Chapter 6 The marine vibrator source

6.1 Introduction

Since the introduction of the Vibroseis principle in seismic exploration in the early 1960's, considerable effort has been put into the development of a marine version of the land vibrator. Figure 6.1 shows one of the latest versions of the marine vibrator.

Figure 6.1 The marine vibrator developed by I.V.I, (courtesy I.V.I.).

The incentive for the development of the marine vibrator was the feeling that further refinement of available marine sources was insufficient for future exploration needs. The marine vibrator's ability to fill this gap lies mainly in the perfect control that one has, at least in principle, over the emitted source signal. Funher, the marine vibrator can be used in shallow water areas, where conventional seismic sources are difficult to deploy. Figure 6.2 shows the setup for a marine Vibroseis survey.

The aim of this chapter is to investigate some geophysical issues related to the performance of the marine vibrator. Since the Vibroseis method has been and is being used extensively on land, problems and important aspects of the marine vibrator performance can be deduced from more than 20 years of experience with land vibrators. Also, differences in performance between land and marine vibrators are discussed.

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seismic vessel

X r n Q_Q_ °2

fixture

marinevibraior

Figure 6.2 Sel-up for a marine Vibroscis survey.

A particular issue that deserves attention is the choice of the signal that should be monitored on the vibrator for phase and amplitude control. This choice is intimitely related to to the way the wavefield behaves in the far field of the medium. Also, the power output of the vibrator is of great practical importance and is investigated here.

To analyse these problems a model of the marine vibrator and its surrounding medium is developed. The model contains two elements: (1) a description of the propagation of seismic waves in the acoustic medium, and (2) a description of the mechanical properties of the marine vibrator. The earth model and the vibrator model are coupled by assuming continuity of pressure and normal component of particle displacement at the boundary between the fluid and the marine vibrator.

6.2 Configuration

The configuration to be investigated is illustrated in Figure 6.3. The marine vibrator is modelled as a hemisphere, which is a good approximation to its actual shape. The radius of the sphere is denoted by a. The surface of the vibrator is denoted by 5, and the surrounding acoustic medium is denoted by V. In reality, the vibrator is attached to a seismic

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vessel. This will be taken into account in the mechanical part of the marine vibrator model. In accordance with the terminology used for land vibrators, the cylindrical plate at the bottom of the marine vibrator is called the baseplate, and the hemisphere is referred to as the reaction mass.

Orthogonal Cartesian coordinates are used with the origin at the centre of the baseplate, and the x-x, axis pointing downwards.

Figure 6.3 The configuration.

6.3 Vibroseis: land and marine

The land vibrator possesses the following characteristics:

1. the far field pressure or particle velocity is essentially equal to the time derivative of the integrated traction directly underneath the baseplate (Chapter 2).

2. the traction and displacement directly underneath the baseplate are distributed non-üniformly over the baseplate. This non-uniformity increases with increasing frequency (Chapter 4)

3. the far field compressional wave power flow is small, and in the order of 10 % of the total power flow (see Chapter 4).

These results may not be valid for the marine case, because some distinct differences exist between land and marine Vibroseis applications. First, the marine vibrator is immersed in water, whereas the land vibrator vibrates on the surface of the earth. Second, only the baseplate of a land vibrator exerts a force on the earth. In the marine case, the reaction mass is also emitting seismic waves. Third, a large part of the energy of the land vibrator is converted into Rayleigh waves. In the marine case, which vibrates in an acoustic environment, these Rayleigh waves cannot exist. Fourth, the land vibrator is vibrating on a near-surface zone, which in general is irregular, generates shear waves and has a low velocity. The marine vibrator is operating under much more stable conditions, in a very homogeneous medium (water). Fifth, monitoring the signal on the land vibrator is only possible by making measurements on the vibrator itself. In marine applications, the near field signal can be measured directly by putting a

n reaction mass

Surface S

"baseplate"

Surrounding medium V

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hydrophone in the water. Sixth, the marine vibrator is moving through the water while emitting its sweep. Dragoset (1988) proposed a solution to the problem of the resulting smearing of the data.

From these differences, it is concluded that the previously mentioned land vibrator characteristics do not necessarily apply to the marine environment. The following questions related to the performance of the marine vibrator arc therefore formulated.

1. How can the far field pressure or particle velocity be related to measurements at or near the marine vibrator?

2. Is non-uniformity of pressure and displacement on. the surface of the marine vibrator important in the marine case?

3. What is the power output of the marine vibrator?

To answer these questions, a model of the marine vibrator and its surrounding medium is developed.

6.4 The marine vibrator model

The modelling of the response of the acoustic medium to the pressure exerted by the marine vibrator on the medium is performed in a way that is very similar to the approach followed in the modelling of the land vibrator. The model again consists of two parts: a description of the wave propagation in the fluid, and a description of the properties of the marine vibrator.

It is assumed that the acoustic medium is of infinite extent. The formulation of the wave propagation in an infinite acoustic medium is well-known (see for example de Hoop 1978, Herman 1981).

The vibrator model consists of two parts. The first is a mechanical model, which includes effects of the fixture of the vibrator to the seismic vessel, and of the connection between baseplate and reaction mass. The second is a model which describes the behaviour of the vibrator casing. For the latter model, the mass-loaded boundary condition, introduced in Section 3.2.3, is used. The flexural rigidity method proved to be impractical for reasons given in Section 6.4.2ft.

The fluid model and the marine vibrator model are combined by assuming continuity of pressure and normal component of displacement at the surface of the vibrator.

6.4.1 The earth model

In this section, the wave propagation in a homogeneous, isotropic, acoustic medium is formulated. The analysis is performed in the frequency domain. Due to the low energy density

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of the emitted signal, the wavefield emitted by the marine vibrator is assumed to behave linearly even in the vicinity of the source. Then, the wavefield in the water can be described by means of the two linear equations:

1. Hooke's law, which is the linear relationship between pressure and volumetric strain, and 2. Newton's second law of motion.

Since the acoustic medium outside S is source-free, these two equations become, respectively

~KP = djUj, (6.D

2 diP = pcoUi, (6.2)

_2

in which P = pressure in the medium [Ns m ], Ui = particle displacement [ms],

_3 p = mass density of the acoustic medium [kg m ],

2 -1

K = compressibility [m N ], <£ = angular frequency [s ],

and where 9t- denotes the spatial derivative with respect to*,. Starting from these equations, it can be shown (e.g. Herman 1981) that the pressure in the medium or at the surface of the medium can be represented by the equation

ƒ ƒ {paG(x,x')Uj(x-)nj,(x') + djG(x,x')nj(Lx')P(.x')}dx-=-{j,l)P(ix),

for x e {S,V} . (6.3)

In the remainder of this chapter, only the component of the particle displacement which is oriented normal to the surface of the vibrator, Un, is considered:

U„(x) = nj(x)Uj(.x) , (6.4)

where rij denotes the normal to S. The orientation of n; is illustrated in Figure 6.3. For a medium of infinite extent, or for an infinite halfspace bounded at the top by a pressure-free surface, the Green's function G can be readily obtained (e.g. Ones 1987). The Green's function pertaining to an infinite medium is given by

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r , ,, _ exp(/&?) G ( x , x ) = , ( 6 5x

in which k = acoustic wavenumber [m ] = 7",

c = acoustic wavespeed [ms ]=(p>c)

and R denotes the distance between observation point x and integration point x ' ,

K = [(*,-*;•) ( x , - * ; - ) / 2 . (6.6)

0

If the configuration is bounded at the top by a pressure-free surface at depth level x3 = x3, the Green's function is given by

r i v v.s exp(ikR) expQfcft") 4nR AnR (pJ'

in which R" is given by

R°=[ (xa-x'a)(xa-xa) + (x3+X'i-2x°3)2y2. (6.8)

The second term in equation (6.7) can be interpreted as the response to a mirror-imaged, virtual source. The interaction of the source with the sea-surface, which is expressed by this virtual source term, is also known as the ghost interaction. The Green's function pertaining to the infinite halfspace is used in all calculations for two reasons. First, the field data that were available were measured at such a depth that ghost interactions were negligible. Second, the effect of the free surface is well-known (see, for example, Ziolkowski 1986).

6.4.2 The marine vibrator model

6.4.2a The mechanical model

In the first part of the marine vibrator model, the relation is defined between the input force supplied by the hydraulic drive system, and the force which, as a result of this input force, acts on the inside of the surface of the marine vibrator. For this purpose, a mechanical model of the interior of the marine vibrator is used, which takes into account the effects of the connection

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Rigid fixture between baseplate and reaction mass, and the fixture of the marine vibrator to the ship (Christensen 1987, personal communication). The model is shown in Figure 6.4.

The expressions given in this section for the mechanical model of the marine vibrator are different from the equations pertaining to the land vibrator for three reasons. First, the marine vibrator is attached to a fixture which is assumed to be rigid. Second, in the marine case the reaction mass also exerts a pressure on the surrounding fluid. Third, field measurements were available in which the driving force F was constant. It is therefore desirable to obtain model results which

represent the response of the marine vibrator to a constant driving force F, rather than the response to a constant applied force Fapp 'e .

From the mechanical model, the forces acting on the baseplate and the reaction mass can be defined. The forces acting on the reaction mass are the driving force/, and the forces f\, f2, ƒ3, and/4, which are given by

Figure 6.4 The mechanical model of the marine vibrator (after:

Christensen, I.V.I.).

lop , fi = s1[un(x )-«„(x )] (6.9)

/ 2 = A T , - [ M „ ( X , 0 / , ) - M „ ( X C ) ] , dt

(6.10)

/3 = - A r 2 - « B ( x ' 0 ' ) , dt (6.11)

top , fA = -S2 M„(X ) . (6.12)

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Here, X'°P denotes the coordinates of the top of the reaction mass (that is, X'°P = (0,0,-a)), and xc denotes the coordinates of the centre of the baseplate (that is, xc = (0,0,0) ). These coordinates denote the points at which the individual forces are exerted. In equations (6.11) and (6.12), it is assumed that the fixture is rigid. On the baseplate, only the forces//] and/2are acting.

Since it is advantageous to work in the frequency domain, a Fourier transform is performed to equations (6.9)-(6.12). This yields:

F 1 = 5 1 [ t / „ ( x ' o p ) - ( / n ( x c ) ] , (6.13)

F2 = - icoK! [ Un(x '°p) - Un(x c)J, (6.14)

Fi = icok2Un(x'op) , (6.15)

F4 = -s2 Un{x'op). (6.16)

The total force applied to the baseplate and reaction mass, denoted by Fabpphcd and Fap£hed,

respectively, is then given by the expressions

Flppp!icd= F+C2[Un(x,op)-Un<ixc)l, (6.17)

FZp!iC"=-[F+ClUn(x,op)-C2Un(xc)], (6.18)

in which Cj and C2 are given by

Cï = (s1-s2)-ico(K1-K2), (6.19)

C2= S]-ico K] . (6.20)

From measurements on the marine vibrator, the spring constants 5] and s2 were estimated to

be 1.36 x 109 |Nm - ] and 7.1 x 103 [Nm~ ], respectively. Since no reliable estimates were available for the damping constants Kt and K2, these were set to zero. The masses Mp and Mr

of baseplate and reaction mass equal 435 kg and 1,425 kg, respectively; due to the presence of the hydraulic piston in the centre part of the baseplate, the mass density in this area was taken to be. 1.4 times the mass density outside the centre region of the baseplate. The radius a of baseplate and hemisphere was set to 1 m.

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6.42b The marine vibrator casing

The description of the behaviour of the marine vibrator casing when a force is exerted on its inner surface is a complicated problem due to the complex structure of the casing, shown in Figure 6.5.

Figure 6.5 The structure of the marine vibrator casing (from: I.V.I.).

Application of the flexural rigidity method to the marine vibrator problem proved to be impractical, partly because of the entirely different shape of the marine vibrator (that is, a hemisphere instead of a simple plate), and partly because the mechanical structure of the marine vibrator is different from the baseplate structure of the land vibrator. However, application of the mass-loaded boundary condition (see Section 3.2.3), which yielded highly non-uniform traction and displacement distributions in the land vibrator case, indicated that non-uniformity of pressure and displacement was small in the marine case. Consideration is therefore restricted to the application of this mass-loaded boundary condition.

The formulation of the mass-loaded boundary condition is very similar for the marine vibrator problem and the land vibrator problem. In the marine case, however, the reaction mass is also exerting a pressure on the surrounding medium, so that the mass-loaded boundary condition also has to be applied to the reaction mass. The mass-loaded boundary condition is given by

papphcd^ )_P{x) = _a{x)(0 Un{x ) t x e s . (6.21)

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Since different expressions were obtained for the force applied to the baseplate and the force applied to the reaction mass, this expression is divided into the two equations:

Pabphed- P{x) = - o(x )co t/„(x ) , x e S bp, (6.22)

-, applied P/m

p -P(x) = - CT(X )<o t/„(x ) , x e S rm , (6.23)

where the surfaces of baseplate and reaction mass are denoted by Sbp and Srm, respectively, and the pressure applied to the interior of baseplate and reaction mass by Pa

bp

pp " and P°% 'e ,

respectively. The plate stiffness is taken into account by assuming that the applied force is distributed uniformly over the surface of the marine vibrator; this yields the following

_ „applied , „applied expressions for Pbp and Pr„ :

-.applied -applied r bp PbP =—L

T- . (6.24) m

and

^.applied

/C^^V . (6.25) 27E3

where the forces Fabp 'e and F°j£'" are given by equations (6.17) and (6.18), respectively.

6.4.2c Combined mechanical model and vibrator casing model

As mentioned earlier, the response of the marine vibrator to a known driving force F is calculated, since then the model results can be related to the field measurements. This, together with the radiation of waves by the reaction mass itself, results in a pressure - displacement relation on the marine vibrator which is more complicated than in the land vibrator case. Therefore, this section is devoted to establishing relations in which the displacement U„ on the vibrator is expressed in terms of the driving force F and the pressure on the vibrator P.

Combining equations (6.17)-(6.18) and (6.22)-(6.25), the following expressions are obtained for the pressure on the baseplate and the reaction mass, respectively:

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F+C2UJx'°p)-C2UJxc) 2 P(x)= — . L.2 + a(x)wUn(x),xeSbp, (6.26)

702

F + C1Un(x'°p)-C2UJxc) 2 f ( * ) = — ^ 2-^--^+o{x)coUn(x),xeSrm. (6.27)

2na

Bearing in mind the formulation of the wave propagation in the fluid (see equation (6.3)), it is convenient to express the displacement in terms of the pressure. For this purpose, first equation (6.26) is applied to the centre point of the baseplate (with coordinates xc ), and equation (6.27) to the top part of the reaction mass (with coordinates x'°P). The displacements at these points can then be expressed by the equations:

2 c -C1[2na2P(x'°p)+F} + [2nalo{xop)w - Cj] [ r a V c x ^ - F ]

</„(* ) = Wa

= al + a2P(.x'op) + a3P(xc), (6.28)

, -C2[7m2P(xc)-F] + [na2a(xc)a) -C2]\27ta2P(x'op) + F) Un{x ) = _

= P] + p2P(x'op) + p3P(.xc), (6.29)

in which the determinant Det is given by

Det = [2xa2c(x'op)(o -Cl][na2c{xc)- C2] + c\ . (6.30)

Substitution of these expressions for the displacement in the top and centre of the vibrator in equations (6.26)-(6.27) yields the two equations:

x e Sbp:

^m2P(x) +\C$2- C2a2) P(x'°p) + [ C ^ - C2a^ P(x') + \C2{5i - C2«, - F) 2 Z

na o(x )co (6.31)

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and

f^_ 2nalP(x ) HC^-CrfJ P(.x°p) + [C^-CrfJ P(xc) + [Cxax -C^^ + F] U n{\ ) - .

2 -2m a(x )o)

(6.32)

6.5 The combined earth-vibrator model

From the earth model, the following relation was obtained between the normal component of the displacement and the pressure on the surface of the vibrator:

ƒ ƒ {paG{x,x')Un(x') + djG(\,\-)nJ(x')P(x'))dx-=-^P(x), xeS. (6.33)

An additional equation describing the relation between the two quantities was obtained from the marine vibrator model (equations (6.31)-(6.32)):

x e Sbp :

_ m2P(x ) +[C2/32- C2a2] P(x'op) + [ C ^ 3 - C2a3] P(xc) + [C2/31 - C2a^ - F] " « ( x ) Ö >

2 TUX 0(X )C0

(6.34)

2m2P(x ) + [C ,o 2 - Cj/JJ P(x'op) + [C,03- C^3] P(xe) + [C.a^ - C2J3] + F) U„(x ) =

2 -27ia CT(X )w

(6.35)

If it is assumed that the input force F is known, the problem is fully determined, and the distributions of pressure and normal component of displacement over the plate can be calculated. Once these distributions are known, the pressure anywhere in the acoustic medium

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can be obtained using equation (6.3). Before the results of these calculations are presented, the numerical techniques employed to solve the set of equations (6.33) - (6.35) are discussed.

6.6. The numerical procedure

The expressions (6.34) and (6.35) for the displacement are substituted in equation (6.33), so that an integral equation is obtained in which the pressure is the only unknown quantity. Polar and spherical coordinates for points lying on the baseplate and the reaction mass, respectively, are introduced:

Polar coordinates: (6.36)

*i = acos(ff) sin(a) Spherical coordinates: { x2= asin(ö) sin(a)

ix3 = - acos(a) (6.37)

Due to the cylindrical symmetry present in the configuration, the wavefield on the surface or outside the marine vibrator does not depend on the polar angle 6 . For convenience, the polar angle 6 corresponding to the observation point is made.equal to zero.

Figure 6.6 Discretization of the marine, vibrator surface.

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The polar angle 9' for the integration point was also set to zero in connection with the pressure at a certain position, because the pressure on the surface of the vibrator does not depend on this polar angle. Analytic removal of the artificial 9' dependence of the Green's function could not be achieved; instead, these integrals are calculated numerically using Simpson's rule. The lines along which the pressure is calculated are shown in Figure 6.6. The increments Ar and A a were chosen to be 0.1 (m) and 0.15 (radians), respectively. The singularities in the Green's function are removed analytically in the surface integration using the method described previously in connection with the elastic earth model (see Appendix C).

The integral equation formed by equations (6.33) - (6.35) is discretized as shown in Figure 6.6. It is assumed that the pressure is constant in each r or a interval shown in this figure. If the total number of intervals is denoted by N, A' equations are obtained with N unknowns (the latter being the pressure at the N intervals). This set of equations is solved by a matrix inversion routine.

6.7 The distribution of pressure and displacement at the surface of the vibrator

Using the model described in the previous section, the distributions of pressure and displacement on the vibrator can be calculated. In this section, some results of the distributions are presented. In all figures, phase angles are given in degrees. The force F is chosen to be of zero phase, with amplitude 50,000 (Ns). The profiles along which the distributions are shown are illustrated in Figure 6.7. Figure 6.8 shows the distribution of the

Figure 6.7 The profiles along which the different distributions pressure on the baseplate, for two different frequencies: 10 and 200 Hz.

From the distribution of the modulus of the pressure, shown in Figure 6.8a, two things can be observed: the variations of the pressure along the plate are less than 10 %, and there is no dramatic difference between the distributions at 10 and 200 Hz. The phase of the pressure, shown in Figure 6.8fc, is almost uniform and does not exceed 5°.

In Figure 6.9, the displacement distribution along the profile BC is shown.

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14400

14200

14000

13800

13600

f / 1

1 1

1 1

"

0 0.2 Q.4 0 6 0.1

i Position on plate I».]

4 (ai Position on plate in .

Figure 6.8 The distribution of the pressure on the baseplate along profile AB, for frequencies 10 and 200 Hz. : 10 Hz , - - - : 200 Hz (a) amplitude (Nsm-2), and (ft) phase in degrees.

Position on plate (».l

fa)

-

/ y

\ i i . ,i,.

Position on plate (■.)

<b)

Figure 6.9 The distribution of the displacement along the profile BC, for frequencies 10 and 200 Hz. 10 Hz , — : 200 Hz. (a) amplitude (ms), and (ft) phase in degrees.

Again, the modulus shown in Figure 6.9a is very constant over the upper shell, but decreases with frequency. This can be explained by the formulation of the mass-loaded boundary condition (equations (6.22)-(6.23)), from which it follows that, if the pressure does not vary very much with frequency, the same must hold for particle acceleration, not for the displacement. The increase of the modulus near the top of the vibrator is caused by the higher value of the mass density in this area (the value of CTin the centre of the baseplate is 1.4 times

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the mass density of the remainder of the vibrator surface, because of the presence of the hydraulic piston in this region). The phase plot in Figure 6.9b shows a more pronounced phase difference of about 30° at 200 Hz. Since all these distributions show a very gradual increase of pressure and displacement with frequency, a fixed point D (shown in Figure 6.7) is chosen on the baseplate, and the distributions as a function of frequency instead of position are calculated.

Figure 6.10 shows the displacement at D as a function of frequency.

Figure 6.10 The distribution of the displacement at point D on the baseplate, as a function of frequency, (a) amplitude (ms), and (fc) phase in degrees.

As already discussed, the modulus shown in Figure 6.10a indicates a decrease with increasing frequency. Close inspection of this curve reveals that the modulus is inversely proportional to the frequency squared, which means that the acceleration is more or less constant. The phase as a function of frequency, shown in Figure 6.10ft, is a very smooth, almost linear curve, with a maximum phase angle of 20° at 200 Hz.

Note that nonuniformity of displacement and pressure is a serious problem in land Vibroseis. However, in the marine case surface waves and low velocity weathering zones are absent, which partly explains the less distorted distribution functions.

Now that the displacement and pressure distributions are calculated, two practical questions will be addressed: the power factor of the marine vibrator, and the far field characteristics of the emitted wavefield.

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6.8 The average power factor

From the calculated distribution functions of pressure and displacement, the average power factor of the marine vibrator can be calculated. For the marine vibrator, the applied force is used not only to radiate acoustic energy, but also to move the mass of the vibrator and the surrounding water. The time-average of the acoustic power flow P00 is defined as (see Chapter 4):

Re{ />"«») ) = I Re { ƒ ƒ P(x ,co) V* (x ,a>) dx }

= j Re {\A\txp(ieA)} = I|A|costf,,). (6.38)

50 100

Frequency IHz.l

Figure 6.11 The power factor as a function of frequency : : mean power factor , - - -: maximum of power factor, : minimum of power factor.

The cosine-term in equation (6.38) represents the average phase difference between pressure and velocity, and is commonly referred to as the average power factor. In Figure 6.11, the average power factor on the vibrator as a function of frequency, as well as the minimum and maximum value the power factor attains on the marine vibrator surface are shown. As could be expected, the power factor is zero at 0 Hz: the vibrator is moving the mass and the water, but no acoustic energy is radiated. At higher frequencies, however, the power factor increases until it reaches a value of 30 % at 200 Hertz. For lower frequencies, under 100 Hertz, the power factor is of the order of 10 %.

6.9 Far field relations

The second question investigated is the shape of the far field wavelet. This question is important because it is the far field wavelet that should be used in the cross correlation process.

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Since only the near field wavefield can be measured, the relation between the near and far field wavelet must be known in order to be able to control the cross correlation process.

The observation point in the far field, x / is defined by its coordinates (see Figure 6.12)

A A 4

= Rf

= 0, = /?'

sin(<D,

cos(£) (6.39)

where the x2 coordinate is set to zero. This can be done without loss of generality since cylindrical symmetry is present in the configuration. In the far field analysis, the phase factor zxp(ikRf), as well as the l//?/-scaling, will be omitted troughout for clarity.

Figure 6.12 The notation for the far field relations.

The distance between the integration point and the observation point is denoted by /?rf; the Green's function corresponding to this distance Rd is denoted by Gf ■ When the integration point lies on the reaction mass, the following three equations are obtained as Rf —» °° :

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Rd=Rf-a[ sin(Öcos(e')sin(a')-cos(^)cos(a') } + 0(1/R1) , (6.40)

Gf=c\p[-ika { sin(<£)cos(e')sin(a')-cos(£)cos(a') }]/47i + 0(1///) , (6.41)

djGfnj(\') = ikGf{ sin($cos(0)sin(a')-cos(£)cos(a')} + 0(1/Rf) . (6.42)

When the integration point is situated on the baseplate, the following equations are obtained as Rf ->«>:

Rd = Rf- f sin(§ cos(d') + 0(Wf) , (6.43)

G /=exp[-i'/trsin(^)cos(0')]/4n: + 0(11R?) , (6.44)

BjGf nj(.x') = ikcf cos(& + OWR*) . (6.45)

The far field expressions are obtained by neglecting terms of order one over the distance (note that, because the trivial 1/Rf- scaling is omitted in all expressions for the Green's function in this section, in reality terms of order one over the distance squared are neglected). If the l/Rf terms are neglected in the Green's function, and expressions (6.40)-(6.45) are substituted into equation (6.3), this yields

2 2 71

Pf(Rf,^) = - ^ ^ - f2 da\m(a)exp[ikacos(Z)cos(a')]J0[kasin(£)sm(a)]Un(a) + • ' n

+ -=— | 2 da sin(a ) P(a ) exp[/tocos(£) cos(a )] x 2

x {(' sin(£) sin(a ) J:[ka sin(<jj) sin(a ) ] + cos(|) cos(a ) J0\ka sin(f) sin(a ) ] } +

I dr'rJ^kr'sm(^){ikcos(£,)P(r') + pcoUn(r')} . (6.46) 2 ,

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In deriving these equations, the integration over the polar angle 0' is performed analytically using the relations (Abramowitz and Stegun, 9.1.21)

I COS(K) exp[z'zcos(K)] dK=2mJl(z) , (6.47)

(2n

J exp[/zcos(*r)] dK=2xJ0(z) . ( 6 4 8 )

To distinguish the far field relations (6.40)-(6.46) from other far field relations that will be derived making additional approximations, these expressions are referred to as the exact relations. The exact far field relations are used in all the calculations. These expressions are not very transparent; therefore, to obtain more insight into the physical aspects, further approximations are made to make the equations simpler.

The first approximation is that the dimensions of the vibrator are small compared with a wavelength:

ka«l, (6.49)

or, using a value for a of 1 m, and a value for c of 1500 m/s,

ƒ « 2 4 0 . (6.50)

The exponential terms and the Bessel function of order zero can then be approximated to the. value 1, and the integral with the Bessel function of order one can be neglected. The second approximation is that pressure and displacement do not vary over the vibrator. It has already been seen in Section 6.7 that this approximation is a very reasonable one. The (constant) pressure and displacement of baseplate and reaction mass are denoted as Pbp, Prm, Ubp and Urm, respectively, where Ubp and Urm denote the normal displacement components of the baseplate and the reaction mass.

Using these assumptions, the following expression is obtained for the far field pressure ƒ> at distance/?/, and at an angle £ from the vertical (see Figure 6.12):

Pf(Rf,Z)=-\PCD2a2[Ubp+2Urm] -'COa™i®{Pbp-Pm~l, (6.51)

or, in the time domain,

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ƒ,„ƒ. . , 1 2 d

dt \.ubp+2urm\

a cos(£) d ~4c~ J^Pbp-Prm\-

(6.52)

From this equation, it can be seen that the far field pressure is composed of two terms. The first term shows that the particle acceleration at the surface propagates undistorted to the far field. In the second term, the time derivative of the pressure difference between the top and bottom of the vibrator in the water can be recognized, as well as a directivity pattern, which is similar to the land vibrator case. The first term, containing the mass acceleration, can be written in a different form by applying the mixed boundary condition. This yields

applied applied

pf(RU)=TPa2 [Pbp ~Pbp +2Prm ~Prm ] + a cos(|) d

Ac iPbp-Prmh (6.53) ■jjlfbp-Prm JbP

180 180

270 270

in which it is assumed that the mass densities of baseplate and reaction mass are constant over the plate, with mass densities ob and am, respectively. From equation (6.53), it follows that the pressure difference between the inside and the outside of the vibrator propagates undistorted to the far field. This term corresponds to a monopole term, and is related to an injection of volume. The undistorted propagation of the pressure difference between the inside and the outside of the vibrator, or the driving pressure, is very similar to the behaviour of, for example, a bubble emitted by an air gun, where it is the pressure difference across the bubble wall that

o o propagates undistorted. The second term in equation (6.53), which can be recognized as a dipole part, contains the pressure difference between top and bottom. This term is related to the resulting body force. If baseplate and reaction mass are moving out of phase, or, in other words, if the baseplate moves downwards and the reaction mass moves upwards, the dipole term will become small. The directivity patterns of the monopole and the dipole part are shown in

Figure 6.13. In fact, the 'amplitude of the dipole part also depends on the frequency: it increases linearly with increasing frequency.

The far field characteristics of the marine vibrator have now been established. Although it may not be very surprising that the marine vibrator source has both monopole and dipole

DIPOLE H

Figure 6.13 The dipole

pa items.

MONOPOLE

and monopole directivity

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characteristics, the question now is which pan determines the far field behaviour. From the polar diagrams shown in Figure 6.13, it is obvious that when the angle £ equals 90°, the dipole term will vanish. At 0°, it will have its maximum value. The directivity pattern, together with the frequency-dependent amplitude of the dipole pan, makes it difficult to predict which term dominates. To answer this question, the far field pressure is calculated as a function of frequency, using equation (6.46). Figure 6.14 shows the amplitude and the phase of the ratio of the far field pressure and the pressure at the centre of the baseplate (Figures 6.14a-b), and of the ratio of the far field pressure and the acceleration at the centre of the baseplate (Figure 6.14c-d).

•5. 3

£ 5 £ c. Q.

^ V

"-i~ 10

. o <u

« to -C CL

u

-10

-20

»!•»

. - . ^ "-— \ N ~~~"~ — —

\ v \ \ \^ N->v X .

>v ^ v \ "s .

^ v "V. > v **»■

\ . ""•' ^ ^ - s ^ ■—- _ _

^ ^ ^ ^ ~~~~———_

1 1 1 ,

Figure 6.14 The ratio of ihc far field pressure, and the baseplate pressure - and acceleration, for three different angles t,: :£, = 0°, : £, = 45°, : £, = 90°. (a) ratio of the modulus of the far field pressure and the modulus of the baseplate pressure, (6) phase difference in degrees between the far field pressure and the baseplate pressure, (c) ratio of the modulus of the far field pressure and the modulus of the baseplate acceleration, id) phase difference in degrees between the far field pressure and the baseplate acceleration.

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At 90°, the amplitude ratio of the far field pressure and the baseplate pressure, shown in Figure 6.14a, is constant, and the phase difference is very small. At 0°, however, the dipole term is playing a role, and an increase in the amplitude ratio is visible, as could be expected. Also, the phase difference shown in Figure 6.14b is becoming more important. The values at an angle of 45° are in between the values for the horizontal and vertical direction, as could be expected from the directivity patterns. In Figures 6.14c and 6.14d, again the modulus and phase of the far field pressure are shown, but now the far field pressure is shown relative to the baseplate acceleration. The same characteristics are present, but the deviations from the desired flat zero phase spectrum are slightly less pronounced.

^1 \r~

Tut (■set.1

(a) lb)

Figure 6.15 (a) autocorrelation of the sweep, (b) far field cross-correlation function ai 0° of the far field pressure and the near field pressure, and (c) far field cross-correlation function ai 0C of the far field pressure, and the baseplate acceleration.

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The corresponding far field cross correlation functions can also be calculated. Since the worst case obviously occurs vertically beneath the vibrator (at 0°), where the dipole term is largest, results are shown corresponding to this direction. A sweep ranging from 10 to 100 Hz is used. The autocorrelation of the sweep is shown in Figure 6.15a, whereas the cross-correlation function at 0° of the far field pressure and the near field pressure is shown in Figure 6.15ft. The cross-correlation function looks very stable, almost zero phase and has short side lobes. This can be explained by realizing that a phase difference of a few degrees per 10 Hz, as was present in the far field pressure field, yields a time shift of the order of a millisecond. The amplitude distortion of the wavelet is hardly noticeable. The same pattern is present when the far field pressure is cross-correlated with the baseplate acceleration. The cross-correlation at 0°, shown in Figure 6.15c, looks very similar to the autocorrelation of the sweep shown in Figure 6.15a. It even looks slightly better than the cross-correlation function of the near- and far field pressure.

In conclusion, the pressure in the far field of the medium contains both monopole and dipole terms. Model results show that the dipole term is small; therefore, either the mass acceleration or the near field pressure can be used as a feedback signal on the vibrator.

The assumptions (6.49) and (6.50) are valid for a single vibrator. In practice, an array of marine vibrators is used. For the array configuration, the wavefield in the far field of the medium can be obtained by summing the contributions of the individual vibrators (the individual contributions are given by equation (6.52)). Similar to the land vibrator case, the use of an array of vibrators introduces a frequency-dependent directivity pattern which makes application of a directional deconvolution algorithm necessary.

6.10 Test of the theory by experiment

In this section, the synthetic results from the marine vibrator model are compared with field measurements. An experiment was carried out using the set-up shown in Figure 6.16. A typical vibrator depth for seismic surveys is 6 m. Although the depth of 75 m at which the vibrator was located during this experiment does not conform to practical applications, this depth was convenient since sea-surface reflections do not then interfere with the measurements (these ghost effects were not included in our model calculations). Furthermore, the depth of the vibrator will only affect the magnitude of the hydrostatic pressure, but not the emitted wavefield (apart, of course, from the ghost interaction). Unfortunately, no far-field measurements were made; the (calibrated) hydrophone was positioned only 6 m away from the vibrator. The hydrophone was located vertically below the vibrator, corresponding to an angle £ in Figure 6.12 of 0°. Since in seismic reflection surveys the wavefield at near-vertical incidence is of primary importance, this hydrophone position gives an adequate measurement of the outgoing

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water surface

^ .. .,

marinevibrolor / V W X N X X X ^

/S^\\S\VVV^ 1 i

75 m

6m

hydrophone

water bottom

'mmÉÊMB. 1 nn

180 m

' wfflt m.

Figure 6.16 The experimental set-up.

vibrator signal. A linear, 10 s sweep ranging from 35 to 220 Hz was transmitted. No phase control was applied; it is assumed that the input sweep equals the force F applied by the drive system to the vibrator. The hydrophone measurement was corrected for spherical divergence and travel time delay. The hydrophones were of the type F37 Transducer Serial A68, and a Hewlett-Packard recording instrument was used. Second and higher order harmonics were removed from the data.

Using the marine vjbrator model described in this chapter (calculating the pressure and displacement distribution on the vibrator for a constant applied force F, and subsequent application of equation (6.46) to obtain the far field pressure), a synthetic far field response is obtained. To allow for a comparison of this"synthetic response with the field measurements, the synthetic far field pressure is calculated vertically below the marine vibrator, and the spherical divergence and travel time delay are omitted in the model results.

The amplitude and phase of the field measurements and the synthetic results are shown in Figures 6.17 and 6.18, respectively.

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100 150

Frequency (Hz

(a)

50 100 150

Frequency IH2.1

lb)

Figure 6.17 The measured hydrophone signal at 6 m depth, (a) amplitude (Nsm-2), and (6) phase in degrees.

100 150

Frequency (Hz) (o)

100 150 rrnquency (H7)

Figure 6.18 The modelled far field hydrophone signal, (a) amplitude (Nsm-2), and (b) phase in degrees.

Figure 6.17a shows the amplitude spectrum of the hydrophone signal. Comparing this signal with the corresponding modelled measurement shown in Figure 6.18a, the following are observed:

(1) An increase in the amplitude of the pressure as frequency increases is present in the' measurements as well as in the model results.

(2) The increase in amplitude is slightly more pronounced in the field measurements than it is in

the model results.

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Figure 6.176 shows the phase difference between the hydrophone signal and the feedback signal. Comparing this figure with the corresponding modelled phase difference shown in Figure 6.1 &b, the following are observed:

(1) The decrease in phase difference towards the higher frequencies is present in both the field data and the model results (although the field measurements show a less monotonically decreasing behaviour).

(2) The range in which the phase varies (0° - 50°) is also identical for field measurements and model results.

It is concluded that field measurements and model results show a good match. The remaining deviations between the two are caused by

(1) errors in the model; (2) near-field effects present in the field data; (3) discrepancies between measurement set-up and model configuration (the instrument

response of the recording system, sea-bottom and sea-surface reflections, etc.)

6.11 Conclusions

The marine vibrator is modelled as a hemisphere vibrating in an acoustic medium of infinite extent. From Hooke's law and Newton's second law of motion, a relation is obtained between the pressure and the normal component of the displacement at the surface of the vibrator. An additional relation between the pressure and the displacement of the vibrator is obtained from the mass-loaded boundary condition, in which the thickness of the marine vibrator casing is neglected, but not its mass. The stiffness of the casing is included in this boundary condition by assuming that the pressure applied at the inside of the marine vibrator is distributed uniformly over the vibrator's surface. From the combined earth-vibrator model, the distributions of pressure and displacement over the vibator's surface can becalculated.

From the numerical results of the marine vibrator model, the following conclusions are drawn. First, the nonuniform behaviour of pressure and displacement at the vibrator's surface. which is a problem with the land vibrator, is very moderate in the marine case. Second, the power factor of the marine vibrator is on average 10 %, increasing to 30 % at 200 Hz. Third, the marine vibrator has both dipole and monopole characteristics. The dipole effect, however, is hardly noticeable in the far field cross correlation functions. Therefore, either the mass acceleration or the near field pressure can be be used as a feedback signal on the marine vibrator.

The model results agree closely with field measurements.

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Concluding remarks

The aim of the present study is to determine the wavelet that is emitted into the earth by a seismic vibrator. If this wavelet is cross-correlated with the measured data, the resulting seismogram will contain the desired zero phase wavelet. Normally, the measured data is cross-correlated with the pre-determined sweep, which is assumed to be equal to the feedback signal chosen on the vibrator. Therefore, the desired zero phase wavelet after cross-correlation can be obtained by choosing the right feedback signal on the vibrator. To determine which choice of feedback signal is correct from a geophysical point of view, an analysis of the wavefield emitted by a seismic vibrator is presented. The analysis contains three sequential steps: 1. the development of a model for the wave propagation in the medium, 2. the development of a model for the seismic vibrator, and 3. combination of die earth model and the vibrator model.

The earth model gives a description of the wavefield in or at the surface of the medium. The response of a layered medium is of interest for forward modelling purposes. The wavefield in the medium can be modelled if the tractions at the surface of the earth and the elastic parameters of the earth are known. Comparison between measured and modelled downhole responses for one experiment show that the method for modelling wave propagation in a layered medium from measurements of the surface tractions accurately describes the wavefield in the medium. For the determination of the outgoing vibrator signal, the response of an elastic, homogeneous halfspace is of interest. From this model, the shape of the far field wavelet is obtained. If the effect of the far field wavelet on the measured earth response is removed, the desired impulse reponse of the geology is obtained. For the elastic halfspace model, the far field wavelet is equal to the time derivative of a weighted sum of the components of the ground force, in which the weighting factors are frequency-independent directivity functions, provided that the dimensions of the source are small compared with a wavelength. Therefore, the time derivative of the surface tractions should be removed to obtain the impulsive point force response of the earth. If the source dimensions are not small compared with the wavelength (for instance, when a vibrator array is used, or at high frequencies), the wavefield becomes directive, with a frequency-dependent directivity pattern. For the array configuration, the far field wavelet at a given position in the earth can be modelled by summing the contributions of the individual vibrators.

The vibrator model consists of two parts. The first part is a description of the mechanism by which a force is exerted by the vibrator on the top of the baseplate. For a simple hydraulic P wave vibrator model, this applied force is equal to minus the mass of the reaction mass multiplied by its acceleration. Other vibrator types or vibrator models may result in different expressions for the applied force, but these can easily be incorporated in the remainder of the

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vibrator model. The important thing is that the applied force can be obtained from measurements on the vibrator.

The second new part of the vibrator model describes the behaviour of the baseplate when a given force is applied to the plate. This force is composed of the force applied by the drive system of the vibrator on the top of the baseplate, and the reaction force exerted by the earth on the plate. The force on the top of the plate is acting in the centre region of the plate.whereas the reaction force of the earth is distributed over the plate area at the bottom of the plate. The bending forces and inertial forces in the plate result in a non-uniform distribution of the displacement and the traction directly underneath the baseplate. Several baseplate models have been discussed, each model assuming different baseplate properties. Only the two new methods developed in this thesis, the mass-loaded boundary condition and the flexural rigidity method, yield the non-uniform distributions of both traction and displacement that are present in field measurements.

Combining the earth model and the vibrator model, the problem is to determine the time derivative of the ground force (representing the far field wavelet) from measurements on the vibrator. Before discussing the choice of the feedback signal on the vibrator that results in a zero phase wavelet after cross-correlation, two observations are of practical importance. First, measurements indicate that the phase of the baseplate acceleration varies significantly with the position of the accelerometer on the plate. Consequently the determination of the'ground force from measurements on the vibrator is a function of the position of this accelerometer. Second, the interaction between vibrators need not be accounted for explicitly in the determination of the ground force, since the effect of interaction on the ground force underneath each individual vibrator is automatically included in the estimate for the ground force by means of the feedback system.

The determination of the ground force from measurements on the vibrator is non-trivial because the displacement and traction are distributed non-uniformly over the baseplate. The ground force is equal to the traction underneath the baseplate integrated over the baseplate area; this ground force must be determined from measurements of the accelerations of baseplate and reaction mass. In practice, three different feedback signals are currently being used: the baseplate acceleration, the reaction mass acceleration and the weighted sum of the baseplate acceleration and the reaction mass acceleration. No theoretical justification can be given for the use of the baseplate acceleration as a feedback signal. Measurements show that the use of the baseplate acceleration to predict the ground force leads to large errors, both in amplitude and in phase. The use of the reaction mass acceleration neglects the mass of the baseplate and the bending forces in the plate, and therefore yields an unreliable estimate of the ground force. The weighted sum method assumes uniform displacement, and therefore neglects the bending of the baseplate and the effect of the non-uniform displacement on the magnitude on the inertial force.

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The effect of these errors on the far field cross-correlation function is small; for the experiment described in this thesis, the weighted sum method yields a reliable estimate of the ground force. However, this may not be generally valid because the bending of the baseplate, and the effect this has on the far field wavelet, depend on the vibrator type and the elastic parameters of the earth. Also, the force exerted by the baseplate on the ground decreased towards the high frequencies in this experiment; if the amplitude spectrum of this force is kept constant, a degraded performance is expected for the weighted sum method. The only method which includes all forces in the plate (including the bending forces) is the flexural rigidity method developed in this thesis, but an, as yet, unexplained phase shift of approximately 90° exists. between the measured baseplate acceleration, and the predicted baseplate acceleration using this method. This method yields a reliable estimate of the ground force only if this phase shift is corrected for, but no theoretical justification for applying this correction has been found.

Only the weighted sum method and the (phase-correcetd) flexural rigidity method yield an accurate estimate of the integrated surface traction. Both methods use a weighted sum of the accelerations of the baseplate and the reaction mass to obtain an estimate of the integrated surface traction; the only difference between the two methods is the shape of the weighting factors.

For the marine case, a model has been developed which contains the same elements as the hind vibrator model. Some distinct differences exist between the marine vibrator model and the land vibrator model.

The earth model for the marine case is chosen to be an acoustic medium of infinite extent. The far field wavelet for a single vibrator consists of two terms: a monopole term, which is related to an injection of volume, and contains the acceleration of the vibrator, and a dipole term, which is related to the resulting body force, and equals the time derivative of the pressure difference between top and bottom of the vibrator.

The marine vibrator model includes the waves radiated by the reaction mass. The marine vibrator differs from the land vibrator in its mechanical structure, and because the marine vibrator is attached to a seismic vessel by a fixture. Only the mass-loaded boundary condition is considered.

Combining the earth model and the marine vibrator model shows that the non-uniformity of pressure and displacement is very moderate in the marine case. The combined model is used to establish the effects of the monopole and dipole parts on the shape of the far field wavelet. The effect of the dipole part is small, so that a single marine vibrator can be regarded as a monopole source. The far field wavelet then essentially equals the acceleration of the vibrator. Therefore, the acceleration of the vibrator should be used as a feedback signal. The model results demonstrate that the use of the near field pressure as a feedback signal also yields a reliable estimate of the far field wavelet.

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The use of an array of vibrators leads to a directivity pattern in the emitted wavefield, and the wavefield at a given position in the medium can be modelled by summing the contributions of the individual vibrators. This is similar to the land vibrator case.

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Appendix A Determination of Green's matrices for acoustic and elastic media

In this appendix, explicit expressions are derived for die components of the Green's matrices for the elastic layered medium and the elastic halfspace, and the Green's functions for the acoustic layered medium. First, the components of the Green's matrices in the wavenumber domain are derived from the recursive scheme given in Chapter 2. Then, the transformation to the space domain is described, taking into account the cylindrical symmetry present in the configuration.

Finally, the shape of the far field wavelet is determined using the relations pertaining to the elastic halfspace.

A.1 The horizontally layered elastic medium

A.].] Expressions for the Green's matrices in the wavenumber domain

In Chapter 2, it is shown that the components of particle velocity and traction at any point in a layered elastic medium can be obtained by first decomposing the wavefield in upgoing and downgoing waves. The desired decomposition is achieved by application of the transformation

F, = P,-\V; , (A.1)

in which Pj denotes the composition matrix (see equation (2.20)-(2.22)). F contains the

components of particle velocity and traction:

F ; = [ F i < / - f 2 j l T . ' ( A- 2 )

F i j =[V3j,fUj,T2j]T . (A3)

?2j ^ITlj'Vijftjf , (A.4)

and W • denotes the components of the upgoing and downgoing wavefield, respectively:

\V, = [ UP;. , D, ] T , (A.5)

UP. =| ÜPlJ,ÜP2j,UP3j]J , (A.6)

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Vj=lDuj,D2J ,D3Jf- (A.7)

The upgoing and downgoing wave components in layer k, at depdi level x$k, can be expressed as (see equations (2.65) and (2.66)):

D * ( * # ) = txplicoQkhlbs]RY A, ST x

x [l-exp[icoQkh°kbs]Rk

yymexp[icoQkhk] Rk expfuuQ* (hk- h°bs)] ] ,

(A.8)

for the downgoing wavefield, and

UPtU3,*ï) = exp[i'£öQt(/ii-//t s))Rk zxp[icoQk(hk-h°ks)] x

[l-txp[icoQklils}R,. exp[ia>Qkhk]Rk exp\icoQk (hk- hfs)} ] -l

x

x exp[iö)Qfc/jf"]Rf A,ST, (A.9)

for the upgoing wavefield. The notation associated with the depth level in layer/: is explained in Figure 2.4. In equations (A.8) and (A.9), Q* is a diagonal matrix containing the vertical P and S wavenumbers,

Qi = diag(rp,ys, ys) , (A.10)

in which

I 1 rP.s = \—-p

7 (Re, l n ^ , > 0 (A. i l l

A] is a matrix which depends on the medium parameters of layer 1 (see equation (2.54)), and ST is a matrix containing the components of the surface traction:

ST =[ST3,STl,ST2]T= [f,,f l,f2Z^o ■ (A.12)

UP DOWN ST

The reflection matrices Rk , R* and Rk that appear in equations (A.8) and (A.9) can be calculated by means of a recursive scheme. Symbolically, these equations can be written as

D t ( 4 6 / ) = R D t ( ^ ) A 1 S T , (A.13)

and

UPi(AT^) = R U , ( ^ ) A 1 S T . (A.14)

Now the wavefield in terms of its particle velocity and traction components can be determined by applying the composition matrix P to equations (A.13) and (A.14). This yields

F , C*&> = [ V3 C*3°J) • ? l C*£) . ?2 (X») ] T

= P ) i t [ R U , ( ^ 6 ; ) + R D t ( ^ ) ] A 1 S T , (A.15)

p . obs. r C. . obs. f) . ofcs. C, , obs. -7

F 2 (A3,i ) = [ T3 ( x 3 i t ) , V, (x3*) ,V2 (x3i4 ) ]

= ^ P 2 . * [ R U * ( X M ) - R D * (*?!?)] A, ST . (A.16)

From equations (A.15) and <A.16), it follows that the components of particle velocity and traction at any depth level are, in the wavenumber domain, given by the multiplication of the vector ST, containing the components of the surface traction, and a matrix that will be referred to as the Green's matrix G a (a = 1, 2):

p . obs p? obs. e^r F«(^3,t) = G a ( j c 3 t t )ST , (A.17)

in which the Green's matrices are given by

V2 G ]U3l*) = 7WP >•* l RU* (A '3.t) + RD/t(*3.* ) ] A , , (A. 18)

and

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obs, 1 obs. ofcs. G2(*3.*) = y jP2 . t tRUi(*3 .*) -RD*^3.*) ]A, . (A.19)

Thus, for the layered elastic medium two Green's matrices have to be calculated, as expressed by equations (A. 18) and (A.19). Beginning with Gj, the reflection matrix R+is defined as

R = RU t + RD*, (A.20)

in which the subscript k for the reflection matrix R+ is omitted for brevity, and the different components of R+ are denoted by

R =

*pp

,+ 'SVP

0

R

R

+ PSV + svsv 0 R SHSH

(A.21)

~ ^.PSV The Green's matrix G j is split up in two parts: a matrix G j corresponding to the

^SHSH compressional and vertically polarized shear waves, and a matrix G j corresponding to the horizontally polarized shear waves:

G i = G j + G ] . (A.22)

If the matrix multiplications are performed as given by equation (A. 18), it is found that the ^PSV

Green's matrix for the compressional and vertically polarized shear waves, G j , is given by

^PSV G , = CO

(In) nlAl

K

2 1 K2cos(x) /if4 cos {%) tf4Tsin(2#)

, ATjCOsCt) A'3sin(^) 1 2S

1 2 A'2sin(2) A'4jsin(2^) /f4sin (x)

(A.23)

^SIISII and the Green's matrix for the horizontally polarized shear waves. G j . is given by

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G, SUSH &

(2a)J

O

O

O

O O

A-5sin2Cr). -A-5isin(2^)

1 2 - /^jsintf*) /r5cos (x)

(A.24)

where the substitutions

I Pi=pcos(x) \p2 = psin(z) (A.25)

,£ are made, in which p = [p] +p2 J . Equation (A.24) shows, as expected, that the horizontally polarized shear waves do not contribute to the vertical component of the particle velocity, and are independent of the vertical component of the surface traction.

Using the notation

■U=4- -2p 2 , cs,k

(A.26)

the elements in these matrices are found to be

Ki = . [ 2 * * R+PP + p{pk7s/,R+

SVP]psl(p1rp/ +

v* . - * „ + *

[ F ^ Rlsv + P(Pkrs,>) 'Ksvsv] 2P7P.i(PJs,y> (A.27)

+ [2pkp - M RPSV- PkH(Pkys/V:Rsvsv] 2p2YpA(PirfJ' • Pkl

(A. 2 8)

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\Pk

Rpp+p(pkys,k) 'Rsvr) ip2y,,,(Pir^.i)' +

\Pkl Rpsv+P(Pk7s,k) '*RSVSV]P M P i ^ . i ) ' . (A.29)

*4=O tp f-^J ^ - P ^ t ( P ^ / / ! / ? w ] 2/?27,.1(P1rp,1)y3 +

\PJ Rpsv+Mksk (pkysk)~%Rsvsv]p s}(p:rs^ , (A.30)

and

-'/, K5=P (Pkïs,k)' (MiYsj) 'RSHSH- (A.31)

In the same way in connection with the expression for Gi (equation (A.19)), the reflection

matrix R is introduced as

R = R U t - R D t , '(A.32)

and the different components of R are denoted by

R

RpP Rpsv 0 Rsvp R svsv 0

0 0 RSHSH

(A.33)

^PSV Again, the Green's matrix G 2 is splitted up in two parts: a matrix G 2 corresponding to the

compressional and vertically polarized shear waves, and a matrix G 2 corresponding to the horizontally polarized shear waves:

G 2 = (> 2 + G 2 (A.34)

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If the matrix multiplications are now performed according to equation (A.19), the Green's ^PSV

matrix for the compressional and vertically polarized shear waves, G 2 , is given by

G2 = CO

(2a)/i ,4 ,

# ! H3cos(x) H3sin(x)

H2cos(x) H4cos2(x) HAjsin(2x) 1 2

H2sin(x) H4jsinQ.x) //4sin (x)

(A.35)

^SHSII and the Green's matrix for the horizontally polarized shear waves, G 2 , is equal to

G2 = ;

0 0

H5sin2(x) -Hsjsmdx)

-H5jSm(2X) H5cos2(x)

(A.36)

The components of the Green's matrix are given by

i \ * '

Hi = -[l*kSk(PtYp,i)'*K7t> + 2/i*/> — R~svp]psi(P17P/' + \Pkl

-[vkSk(Pkrp,kYVlRp'sv+2nkp — ^5vsv]2p2yp.1(p,n,/1 . (A.37)

»z= [p(PkYP.k)~%R~Pr- — R~SVP]P*I (P1rp,1)y,+ \Pkl

+ [p(PtYp,tyViR'psx--\^A K'SVSVWYP.^PIYJ1 . (A.38)

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w3= - O*** (Pkïp^Rpp + 2^kP tyj \Pki

R~svp]2p27s.APirPJ2 +

+ [ / v * (pkrP,kY%Rpsv + 2p-kP \— Pk

RSvsv\ps\(PJ^)' (A.39)

" 4 = [ p ( P * 7 ^ * f f - { — j R~svp]2p\](piïpJ1 +

[p(PkrP,k)y^psv-\—I / W ] P * I (P,rs,,)y (A.40)

and

-y -y Hs = PQ*tfs,k) ' Oi}Ys,i) 'RSHSII

A.l.2 Transformation to the space domain (A.41)

In the previous section, it is shown that the components of traction and particle velocity in the wavenumber domain are given by a multiplication of the surface traction components and a Green's matrix:

F~ . obs. r w , obs. C; , obs. c; . obs. _T 1 (*3.t ) = f V3 C*3.t ) , Ti (X3ik ) ,T2 (*3iJt ) ]

= ip),,[RL't(.r^) + R D , a - f ; ) ] A 1 S T

F 2 (*3,A J = I ^3 (*3,t ) . V 1 (*3.* ) . ^ 2 (*3.* ) J

= jfPi.k I RU t CTJ'T) - RD t Cvf/) 1 A , ST

(A.42)

(A.43)

Since a multiplication in the wavenumber domain conesponds to a convolution in the space domain, the components of traction and displacement in the space domain, at position x ', are given by

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_ , obs. (°° f°° ,_ . obs obs obs F a ( x * ) = J I Ga(x3,k'Xiik-x1,x2k-X2)ST(xi,x2)dx1dx2 , (A.44)

where the Green's matrices in the space domain follow from

„ , obs obs obs-—•"~t~-" a ^ 3 , t > xl,k~ xl'X2,k~x2) ~

= (2*)~ J J G a (x3 i / ,* i ,* 2 )exp[ i^(^ i t5 - jcp)]d* 1 d* 2 . (A.45)

The components of the Green's matrices in the wavenumber domain are given in Section A. 1.1. To appreciate the symmetry present in the configuration, and to exploit this symmetry to arrive at a more elegant wavenumber-space transformation, the following substitutions are made. First, since the Green's matrices are derived in terms of slownesses pa, the substitution

ka=copa (A.46)

is made, and the analysis is restricted to positive frequencies. The time domain expressions can be obtained by the transformation

- Re { f F(co) expi-ico t) da> } for t Z 0

0 forr<0 m = l n (A.47)

Then, the cylindrical symmetry in the earth model is accounted for by applying the substitution (A.25) and

A " * - * ] = rcos(0) obs . to. - (A.48)

in which

2 i/ obs . i ' 2 r = [ U u " * i ) + U # - * 2 > ] • (A.49)

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The Green's matrices in the space domain are now given by the expression

n r -obs -««1*3.* > r

2

. ö ) = — pdp I Ga(xfj,p,z)z*Vl icopr cos(6 - %)] d% .

(A.50)

The notation employed for the space coordinates is illustrated in Chapter 2, Figure 2.5. As the desired integration over X cannot be performed if the recursive scheme depends on %,

it is first noticed that, as expected for a cylindrically symmetric earth model, the reflection matrices RUt and RD^ are only dependent on p, and hence independent of %. These matrices can be written in the general form

RU, RUpp RUpsv 0 R V svp R Usvsv 0

0 0 RUSHSH

(A.51)

and

RD,= RDpp RDPSV 0 RDsvp RDsvsv 0

0 0 ' RDSHSH (A.52)

Following the notation used in the previous section, two Green's matrices are defined, which satisfy

(ja(x3.k<r>(')-ija (xi.k ' r > ö ) + b a \xi.k SW (A.53)

As the reflection matrices do not depend on X> the integration over X can be performed analytically, using the following relations (Abramowitz and Stegun, 9.1.21):

[In I cos(^) expf/rcosOlf-0)] dx = 2m' cos(0) ./^z),

■'e. (A.541

[In cos(2*) exp|i'2cosOr-0)] dx = - Incos(20) ./2(z), (A.55)

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{2n j sin&) exp[/zcosCC-e)] dx = 2ia sin(0) J,(z), (A.56)

{ sin(2*) exp[/zcosCi:-0)] dx= - 2ii sin(20) y2(z). In (A.57)

and

/ • 2 * I exp[/zcos(*-0)] dx = 2/r J0(z). (A.58)

Writing

J (copr,6) = 70(öpr) ± cos(26) J2(Ctipr), (A.59)

the space domain Green's matrices follow from

„ PSV obs f ^PSVI obs . . .

G „ (jc3Jk,r,0) = I G „ (j :3 i t ,p ,r,0)dp J0

(A.60)

and

Ga (*3,* . r . 0 ) = I G a (A3jt ,p ,r ,0 )dp , (A.61)

in which

*i PSV G, =

2 CD

2>r//]4)

A: , J^copf)

iK2cos{6)J}((üpr)

iKï%\n{eyJi{wpr)

iK3 cos(0) J}(apr) iK3 sin(0) Ji(copr)

■K4j'(0)pr,9)

1

■A:4sin(20)y2(ftP')

iA-4sin(20) y2(C(^r) jKAj\(opr,6)

(A.62)

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G SHSIl 0)

2n jK5j\opr,8) jKisin(26)J2(apr)

^K5sin(29)J2(opr) ^K5J~(o)pr,6)

(A.63)

^psv G 2 =

co 2n\i,A 1 " !

H^Joiopr)

iH2cos(8)Ji(ccpr)

iH2sin(d)Jl(0f>r)

iH3 cos(0) J^capr) i H3 sin(0) J^apr)

^HA] (topr,G) ■jH4sm(26)J2(ecpr)

- \HA sin(20) J2(opr) jH 4 Acopr,®

and

(A.64)

^SIISH a G 2 = — 2n

jH*,J+(0)pr,8) y Hs siM26)J2(opr)

jH<, sin(20) J2(copr) }rH5 J'{(üpr,d)

A.2 The elastic halfspace

(A.65)

Since no upgoing waves occur when no contrast is present in the medium, the. global reflection matrices for an elastic halfspace are given by

K ™ = 0

R

R? \ =° (A.66)

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Thus, the reflection matrices RUj and RD, at depth level x3* are given by

/ R U I ( ^ ) = O

U D j C x ^ - e x p t f a Q ^ S * ' ) ' ( A"6 7 )

Substitution of these values in the expressions for the Green's matrices for the layered elastic medium , and omitting all layer indices, yields

T-, , obs. i f°° „ . obs obs obs F « ( x ) = J J Ga(x3 , * , -xY,x2 -x2)ST(.x1,x2)dxldx2, (A.68)

where, applying the substitutions (A.25) and (A.48), the Green's matrices follow from

G«(x3 ,r,6) = Ga (x3 ,r,B) + Ga (x3 ,r,6) , ( A 6 9 )

with

G f V'(.vf \r,e) = f~ GPa\?S,PS,e) dp , (A.70)

•'o

and

G™SH(x°3"s,r,e) = f" GT'\xfs,p,r,e) dp . (A.71)

Using the same notation as for the layered elastic medium, it is found that the Green's matrices have the same form as equations (A.62)-(A.65), but now the Green's matrix constituents are found to be:

Ki=-psypexp(icoYpx3 )-2p ypcxp{iwysx3 ) , (A.72)

K2 = 2fip s yptxp(icoyrx3 s)- 2 p p s yp zxp(i(oysx° *), (A.73)

K3 = -2 p2yp7sexp(icoYpx3bs)+p J e x p O o ^ x f ' ) , (A.74)

KA = *H /53r„rsexp(/co7 x3 s) + nps txp(iajysx3

bs), (A.75)

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Ks = pexp(iarysx3 s), (A.76)

H] = VPS2exp(ioxypxfs)+ 4/J p yp ysexp(iarysx3bs), (A.77)

H2 = - p s txp(iü}ypx03 "')+ 2/7 yp ^expO'tyy^j * ) , (A.78)

W 3 =2/ ip j^expO'öj^jTj ' ) - 2 ^ p sy, expt/ory^j *) , (A.79)

W4 = -2/537jexp(/ft)V j r f ' ) - p . r ^ e x p O ü ^ . x f ' ) , (A.80)

and

,, p ,. obs H5 = --i—txp(icj}ysx3 ) . (A.gn

^4.5 77ie layered acoustic medium

A short review of the wavefield in the acoustic medium is given to indicate the changes to be made in the vectors and matrices used in the derivation of the wavefield in the elastic medium.

First of all, the dimensions of all matrices and vectors used in the elastic case are reduced in the acoustic case: (3x3) matrices become scalars, (6x6) matrices become (2x2) matrices, (6>; 1) vectors become (2x1) vectors and (3x1) vectors become scalars. For the acoustic case, the vector F is defined as

F = [ F , , F 2 ] T , (A.82)

where now

\F2=T3 '

The elements of the system matrix E are found to be

(A.sr>

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£'"7 (A.84)

£ 2 = p

The eigenvalue matrix is in the acoustic case a scalar containing the vertical P-wavenumber,

S = V (A.85)

and the elements of the decomposition matrix P, containing the eigenvectors of the system matrix, are scaled in such a way that the symmetry properties valid in the elastic case also hold in the acoustic case:

p (A.86)

P2=( — ) A

Performing the steps performed when calculating the response for the elastic medium, the global reflection coefficients for the acoustic case are given by the following expressions:

DOWN „UP .«. , . UP_ 0 > i +^;+iexp(2ia)ypJ+1^+i) j up up ' (A.87)

1 - rjj+iRfr\txvQ-iafYpj+xhj+x)

UP „DOWN . . . , . DOWN _ rjj+1+ Rj exp(2it»yp j hp j + l , DOWN „DOWN . . . ' (A.88)

1-OV+i RJ expQuoypJl>P

and

DOWN ...... , , „ST ST tjJ+{ txp{iar/„jhpR) j + 1 DOWN „DOWN . . . . . " (A.89)

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These recursions are initialized by the conditions

/?/v+i - 0

R™™=1 (A .90)

,R ST = 1

The components of the wavefield in the medium in the space domain, contained in the vector F, are now given by

„ . obs. (°° f°° ~AC obs obs obs Fa(*k ) = l 1 ,Ga(x3<k,x1Jc-x1,x2Ji-x2)ST(xl,x2)dxidx2 , (A.91)

where ST is a scalar containing the vertical component of the traction at the surface:

ST = 7-3(*3 = 0 ) . (A.92)

The superscript "AC" is used to indicate that the Green's functions refer to an acoustic rather than an elastic medium. The Green's functions, after the introduction of polar coordinaie?. again follow from an integration over the radial wavenumberp:

„AC obs ~ f- ~ A C . obs . . . Ga (*3,k S,8) = Ga (x3k ,p,r,6 ) dp . (A.93)

The components of the Green's functions are now given by

^AC CO G] = p Yp,kyp,\ l n \ Pkp\ j

% ,obs., „UP obs. txp[icoy (2hk-hk )]Rk + exp(icoylik ' ) ST

, „UP „DOWN . . . , . }-Rk Rk cxp{2icoyhk)

Rk JQ{03pr) ,

(A .94)

and

C-, .AC 0)

X obs Vn.iPk\ expfzïuy■ ( 2hk-h t ) ) R .UP

\?p.tPl , „UP „DOWN . . . , . ]-Rk Rk exp(2icoyphk)

e\p(ior/hk ) ST '-■ fit J^ocpr)

(A.95)

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A.4 Far field relations

The starting point for the derivation of the far field properties is the integral representation for the particle velocity components in an elastic halfspace, in cylindrical coordinates (equations (A.68)-(A.71)). The Green's matrix is separated into a Rayleigh wave contribution and a body wave contribution. The Green's matrix of the Rayleigh wave contribution is given by a simple analytic expression, whereas the Green's matrix pertaining to the body wave contribution is given by an integral whose integrand contains a first order Hankel function. The latter integral is approximated by performing the integration over all radial slownesses over the so-called path of-steepest descent. Finally, the traction and particle velocity components are expressed in their cylindrical or spherical coordinates. The procedure outlined above is essentially the same as van Onselen's (1982) far field approximation for the particle displacement components caused by a vertical load applied at the surface.

The notation used in this section is shown in Figure 2.6. The coordinate transformations are performed according to the formulae:

sin(<£) cos(ö) sin(<ü)sin(0) cos(£)

cos(£) cos(0) cos(i?) sin(0) —sin(^)

-sin(0) cos(0) 0 (A.96)

and

cos(0) sin(0) 0

-sin(0) cos(0) 0

0 0 1 (A.97)

The distance r is approximated as follows. The use of

2 2 2 r = r 0 + r, - 2 r0r , cos(y) ;

(A.98)

yields

/• = rQ- r, cos(i//) + Gtrö ) as r0 -» . (A.99)

Until now, the position of the origin of the coordinate system was not specified. Choosing the origin in the centre of the source leads to the smallest error, since then the maximum value of r,

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is minimized. Furthermore, the error in the far field approximation for the Vibroseis configuration can be minimized by treating each vibrator separately; then the total wavefield can be obtained by vector summation of all individual contributions, and the far field relations are independent of the array dimensions.

Next the evaluation of the Green's matrix is discussed. It is seen that all Green's matrix constituents can be written in the general form

1=C K{p)Jm(apr)dp (A. 100)

and

K(p) = F(p) 4(p)

(A.101)

where A( p) is known as the so-called Rayleigh denominator.

lm(p)

k\mf\ws

complex p - plane

'^'"" ©" 1 1

'. — ~r

Re(p)

f igure A .1 . Deformed palli of integration used in connection with integral representation (A.)O(D.

Van Onselen (1982) showed that by deforming the integration path along, the contours C and C.x

shown in Figure A.1, this integral can be written as

-219-

/ = i7[ f ( P j ? ) H £> (wp^) + 1 ƒ ff( p) tf L V p r ) dp , (A.102) 4 ' (p*) 2 J c

in which A ' (p) denotes the derivative of the Rayleigh denominator with respect top, where the Rayleigh slowness pR is given by

1 PR~7R~' (A.103)

and c^ denotes the Rayleigh-wave velocity. In equation (A. 102), the first term is the contribution of the Rayleigh-pole, corresponding to the surface wave propagation, which follows from the integration over the contour C\ shown in Figure A.1. The second term is the body wave contribution, which follows from the integration over the contour C shown in Figure A.1. The surface wave contribution is not present for the Green's matrix pertaining to

^SHSH the contribution of the horizontally polarized shear waves; in the expressions for G a , the Rayleigh denominator is absent

The far field expansion of the surface wave contribution can now be obtained by using approximation (A.99), and by^applying the asymptotic expansion of the Hankel function (AbrarrlGwitz and Stegun, 9.2.3):

H%\z) - (—\ *exp[i(z - my - - ) ] . (A.104) \nz] z 4

This leads to the following expressions for the surface wave contributions:

\' = A

txp[i(cor0pR-—)] 4

r " rr JA }}s 6)%JJ Trtxp[-icopKr,cos(\i/)]dA +

cxp[i(cor0pR )] + Ar3 wA\\ T3cxp[-io)pRr,cos(y/)]dA ,

Jï S

exp[i(cor0pR )] Vg = Agg a)/*)) Tgexp[-i(opRr,cos(\l/)]dA ,

(A. 105)

'A } J s (A. 106)

-220-

and

exp[i(ar0pR )]

V3 = A 3r 0)% j [ Trtxp[-io)pR r, cos(y) ]dA +

txp\i(cor0pR )]

+ A3 3 o)/i\\ T3exp[-icopRrleos(\fi]dA , (A.107) _/2 S

in which Arr, Ar3, Aed,A3r andA33 are simple amplitude scaling factors. The amplitude

scaling factors are given by

, . „ 3 * , » obs. j , * , * obs. , /„ YA 2pRystxp(-coy x3 )+pRs*ystxp(-coysx3 )

Arr=[2nPft) . (A. 108)

,, 2 j . , » obs. _ 2 * * . » obs. /„ VK PR'f*exp(-W7x3 ) + 2pR yy sexp(-&>yvx3 )

Ar3 = {2npR) £ ILL -1 , (A.109) P-A\PR)

/ y'A P«exp(-co ysx°3bs)

A8G=-\2-XPR) . (A. 110)

Vis

.. - 2 » * , * obs. 2 j , , * 06.5 „ / , \-J4 2 p „ y y , e x p ( - © y j r 3 )+ />*J*exp(-o>y,* 3 )

^ 3 r = - ( 2 ^ / f ) £ £ , ( A . l l l )

PA\PR)

and . • , . ... * , * obs. 3 * , » o i s

/- r/2 ^ * * 7 p e x p ( - ü ) v , X 3 ) + 2p f fy exp(-ojy v x 3 ) A33 = \2npR) f- £ '- ■ . (A.112)

P-A'(PR)

In these expressions, the following notation is used

1 T 2

2~ZPR < (A. 113) y* = —~2pR ,

<\

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and

yp,s - \ P K ~ — \ ■ (A.114)

What remains, then, is to establish the contribution of the body waves, given by

'A= \lcK{p)H(iXa>pr)dp . (A.115)

As can be inferred from equations (A.72)-(A.81) and (A.100)-(A.102), the integrand has the general form

K(p) H^Xcopr) = M(p) cxtficoyxi1") H„\copr), ( A .116)

where y denotes either yp or y s. The calculation of the integral IA involves three steps:

1. Application of the asymptotic expansion of the Hankel function.

2. Taking the approximation (A.99)

r = r0-r,cos(y/) =/?0sin(0 -r,cos(y/) + 0(/?ö ) as/?0-»«> , (A.117)

and the geometrical identity (see Figure 2.6) xf '=/?0cos(£) . (A.118)

3. Application of the method of steepest descent (Bath and Berkhout 1984) to the resulting expressions for large values of RQ, leading to the following asymptotic expression for IA:

M [ s i n ( ^ ] cos(£) exp[/(*/?0- ( /w+l£) ] exp[-/*r, cos(y/) ]

' A = „ . , . , • (A.119) coR0sm(c,)

co co w h e r e * = • * = — if y = y , and* = *, = — i f y=ys.

*• p C S

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The body wave contribution in the far field, expressed in its spherical particle velocity components, then follows from

expuKr,K„) rr VR= -icoARr y \ ° p' I 7>xpHV,cos(v/)]<M +

K0 JJS

exp((Rnkn) rr -icoAR3 . " p T3exp[-ik r,cosW]dA , (A.120)

«o JJs

e\p(iR0ks) rr V,c= -iaAfy -= 11 TrexpH'/^cosCY/)]^ +

exp(//?0fcj rr -icoAg ^ j j r 3 exp[ - i^ r , cos ( i / ) ]^ , (A.121)

and

exp(iRnk^) rr Ve=-icoAge j — reexpH^^^cosCv^)]^ . (A.122)

" 0 JJS

The components of the particle velocity in the radial (R) and tangential (£) directions are determined by the Green's matrices for the compressional (P) and vertically polarized shear (SV) waves only; the particle velocity in the tangential (6 ) direction is determined entirely by the Green's matrix for the horizontally polarized shear (SH) waves.

The functions A Rr, AR3 , A , A £3 and A ee are frequency-independent directivity functions.

They are given by the following expressions:

^5/«(2£)(±--Lsin2(£)'2

CP \cs CP &Rr= : . (A.123)

2nn A (— sin(£)) cp

- — cos(£)| — - — sin2(^) CP \Cs CP

ARI = : . (A.124) 2«/z4(^-sin(ö)

-223-

- -Uos (£ ) (1 - 2 sin2(£))

*fy = £ j . (A.125) 2ir/J A (-L sin(Ö )

^ s i n ( 2 £ ) ( - ^ - ^ s i n 2 ( £ ) W

^ 3 = i . (A.126) 2>r/M(-i-sin(Ê))

and

'4ee= ~ - — • (A.127) 2^/i

The same procedure can be applied to the acoustic halfspace. The vertical component of the far field particle velocity is, in the acoustic case, given by

Vi = -ico ^ i ! ^** t*o* , ) n T e x p H , cosW]dA m Inpc] R« JJ*

The exponential factors in the integrals (A.120)-(A.122) can be neglected provided that

£p j r, c o s ( y ) « 1 , (A.129)

or

2xr,«\ps, (A. 130)

where X„, denotes the P and S wavelength , respectively:

). =f££ (A.131) V ƒ ' in which ƒ denotes frequency. Condition (A.130) holds if the far-field evaluation is performed for each vibrator separately, since the maximum value of r, for present industrial applications is about 1 m, whereas the wavelengths of seismic interest are in the range of 10-100 m. If the vibrator array is considered as a whole, an additional phase shift is no longer negligible when

-224-

the array dimensions become large compared to the wavelength. If the influence of the exponential factor can be neglected, it can be seen from equations (A.120)-(A.122) that the far field particle velocity is essentially the time derivative of a weighted sum of the components of the ground force, where the weighting factors are frequency-independent directivity functions.

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Appendix B Solution of the differential equation describing the plate bending

This appendix contains the derivation of the solution of the differential equation (3.67) describing the baseplate deflection under the action of an input pressure P"1:

4 4 4 3 U3(xltx2) | 2d U£cltx2) [ d U3(xltx2) |

3jT] dx$X2 d*2

- o t t ! , x2)ea l/3(x, ,x2) = / ' ' V i ,x2). (B.l)

In this derivation, the dimensions of the plate are neglected; thus, it is assumed that the plate is of infinite extent. Also, the surface mass density cof the plate is assumed constant. It is shown that the plate deflection is equal to a spatial convolution of a Green's function and the input pressure. The plate dimensions enter the analysis in this convolution. Two methods are discussed to perform the spatial convolution describing the plate deflection.

The analysis is performed entirely in the frequency domain; for brevity, the index co for the frequency-dependent variables is omitted troughout the appendix.

B.l The response to a point force

In order to solve the differential equation (B.l) governing the baseplate deflection due to an applied pressure P'm, the baseplate deflection under the action of a point force at position

(x\, x'2) is first considered. The displacement of the baseplate caused by this point force is denoted as the spatial Green's function Gfla. Assuming that the surface density of mass a of the plate is constant, Gfla satisfies the equation:

D{ d4Gf'ex^x2) ^yGfUx{xhx2) | dAGf,ex(xhx2)] |

dX) dx}dx2 3*2

2 - aoi C' "(A-,, X2) = 8(XX-X\ , x2-x'2) ■ (B-2)

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B.l.1 Derivation of the integral relation describing the plate deflection

In order to solve this equation, a two dimensional spatial Fourier transform is first applied in the Xj andx2 direction. The derivatives with respect to x am the space domain become multiplications with ika'm the wavenumber domain. Assuming that the plate is of infinite extent, the following expression is obtained for the Green's function in the wavenumber domain:

C/?«(M-2) = ■ 2 — r - . (B.3) 4 2 2 4 D[ ki + 2kxk2 + k2 } - aco

The Green's function in the space domain follows from the inverse Fourier transform

-2f°° f°° ~flcx

Gf,cHx],x7) = (2K) I I G (k],k2)txpU(k1x1 + k2x2))dk]dk2. (B.4)

To arrive at an analytic expression for the space-domain Green's function, polar coordinates are introduced:

A'i = krcos(a)

k2 = krsin(a) (B.5)

and

(B.6) J xl = rcos(d) )x2=rs'm(6)

where X:r=(/t]' + k2j ■ and r= (A'I+JT^) • Using (Abramowitz and Stegun. 9.1.21)

1 f -/o(z) = - exp[/zcos(a)]da , (B.7) K Jr.

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in which J0 denotes the Bessel function of the first kind of order zero, the integration with respect to a can be performed analytically. This results in the following expression for the Green's function:

Gfie*(r) = — ( ' litDh

KHrK) 2nD)0 k*_a* " (B.8)

in which the constant a is introduced, given by

a =

2nl/4

oa> (B.9)

If the substitution kr = ap is made, and if, for brevity, the notation £2 = a r is introduced, this becomes

_1 f~ pJdfll rDa2 h P

A. l „flex, . 1 (°° pJtfflp) , GJ (r) = r | —-2 dp . (B.10)

2nDa

B.l.2 Calculation of the integral relation describing the plate deflection

The problem of determining the space-domain Green's function is now reduced to the evaluation of the integral /, given by:

I y <* v> dp ( B n )

The relationship between the zero order Bessel function and the Hankel function is given by (Abramowitz and Stegun, 9.1.3 and 9.1.4)

J^i2p)=jHlo\Hp) + ^H{i\aP) , (B.12)

in which H0 and H0 denote the Hankel functions of order zero of the first and the second kind, respectively. This can be used to convert the integrals to a form which can be solved by contour integration in the complex plane. Substitution of the Bessel function by Hankel functions yields

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/ = 2 ^ + / B ^ ' (B.13)

in which

•'n

and

•'n

P#o \i2p) j -4 dp (B.14)

P - 1

3 * • (B.15) p - 1

These integrals can be evaluated by deforming the contour of integration in the complex p-plane. In this approach, the integral IA with the Hankel function of the first kind in the integrand is closed over the positive imaginary axis, and the integral IB with the Hankel function of the second kind in the integrand is closed over the negative imaginary axis (van den Berg 1981). This ensures convergence of the integrands on the entire integration path. The real and imaginary parts of the (complex) integration variable p are denoted by u and v, respectively:

p = u + iv. (B.16)

The deformed integration contours are illustrated in Figure B.l . The integrand of each remaining integral contains simple poles at p= 1 , -1, i and - / . To determine in which quadrant the poles are situated, it is observed that, since a time factor e\p(-icot) is used in the calculations, Im(cu) is greater than zero. This means that the poles are located as illustrated in Figure B.l. For negative frequencies the poles are located just on the other side of the real and imaginary axes; it can be shown that the final result for negative frequencies is the complex conjugate of the result now derived, which is to be expected for real functions of time.

Integration along the contours defined above yields

[ pH^{Qp) , I dp = 2m Res , (B.17) L l p - 1

and thus

-229-

■du - I - idv = 2m Res ; r- uHlxnu) _ f- IV/VQ\ia A J 4 , " J 4 ,

"'O U ~ 1 J0 V - 1

(B.18)

Figure B.1 Location of the poles in the complex p-plane, and the closed contours for the complex integration.

and

pH%\&P)

/ - I dp - - 2m' Res' (B.19)

and thus

/„= | : du » ' - i -ƒ'

-/v//ó2)(-/ï3v) 4

V -

H)dv =

= -l « 4 - i " " 4 v4-ivH(

0l\inv) a

u (-i)rfv = - 2 m Res (B.20)

-230-

where the relation

H%\z) = -H$\-z) (B.21)

is used, and where a minus sign is introduced for the second residue because the contour C2 is not running counter-clockwise. Res' and Res" denote the residues of the poles located at p= 1 and p= -i, respectively. These residues are found to be

R e s ^ i / t f t o , (B.22)

and

Rcs"=ljHil)(iQ) . ( B 2 3 )

Adding up the contributions of the Hankel functions yields

/ = " * dp = i H0 (£2) - H0 (iÜ)} . (B.24) Jr\ n _ 1 H

Thus, the space-domain Green's function is given by

(lex 1 W (D G,le\r) = - _ L f H0 (or) - HQ (iar) ] , (B.25)

8Vc£> ico

or (from Abramowitz and Stegun, 9.6.4)

GPc\r) = j=L— [ Ho\ar) + - KQ(ar) ] . (B.26) 8VoD ia) 7T

When 12equals zero, the contour is closed over the positive imaginary axis, yielding

I -f—dP-\ ~r—idv = 2jti R e s > (B.27) Jo p -\ Jo v - 1

-231-

where the residue Res is now given by

R e s " 4 • (B.28)

This yields

G e\r = 0) = 7 = — . (B.29) 8V0D ieo .

The same result is obtained by letting Q tend to zero in equation (B.26).

B.2 The response to a pressure distribution

Now that expressions have been obtained for the baseplate deflection under the action of a point force, the response to a pressure distribution Pin can be obtained by a spatial convolution of this applied force with the impulse response C " :

Ufc!, x2) = J \ s Gfle\x1 - x \ , x2-x'2)P'\x\, x2) dx\ dx'2, (B.30)

where 5 denotes the baseplate area. The spatial convolution between this input pressure and the Green's function can be

performed as follows. The baseplate is divided into NX x NY evenly spaced discrete samples with side lengths Ax j and Ax 2, respectively. The centre point of the sample area (j,k), x a (j,k), has coordinates:

*i(M) = 0 ' - 2 ) 4 * i 1 • (B.31)

x2{j,k) = {k-^)Ax2

The notation is illustrated in Figure B.2. The area of this sample is denoted by S(j,k). Pressure and displacement are assumed to be constant within each sample area. These quantities will be denoted as Pin(j,k) and U$(j,k), respectively.

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X

1

sample area S(m,n)

0

2 k centrepoint x'a(j,k) integration point x'^ -k

m \ m j s

sample area S(j,k)

\ »■ •* l

observation point xa(m,n)

Figure B.2 The discretization of the baseplate area.

Two approximate methods to calculate the spatial convolution integral are briefly discussed. The first method is a straightforward summation; the second method is more sophisticated since it allows for variations in the Green's function within the sample areas. In the processing of the field measurements, the second method did not result in any significant changes in the convolution result. For these field measurements, a satisfactory result is already obtained when the Green's function is assumed constant over the entire baseplate area. The second convolution method is used in all synthetic results shown in Chapter 4. Also, the results from this method are used in the discussion of the boundary conditions in Section 3.2.4e.

B.2.1 Summation method

Since the Green's function is slowly varying as a function of distance (see Figure 3.13), it is assumed that both G^* and P'n are constant within each sample area. This yields:

NX NY U3(m,n) = £ X P'nU^) & (r (m-ln-k)) Ax,Ax2,

7=1 k=\ (B.32)

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in which r°bs is the distance between the centre points of two sample areas (m,n) and (j,k) (Figure B.4). When observation point and integration point coincide, equation (B.29) should be used to determine the Green's function.

B2.2 Circle approximation

In the circle approximation (van den Berg, personal communication, 1988), it is assumed that the input pressure /*'" is constant in each sample area, but the variation of the Green's function within the individual sample areas is accounted for. To arrive at analytic expressions for the contributions of the individual sample areas to the convolution integral, the rectangular sample area is approximated by a circular sample area with radius R, which has the same area as the original sample area. This is illustrated in Figure B.3. Thus,

R = A*iAx2\

:A (B.33)

equivalent circular integration area

integration area S(j,k)

observation point xjm,n)

Figure B.3 The integration and observation point in the convolution integral, and the rectangular and circular integration areas.

The input pressure in the rectangular element S(j,k), which is replaced by an equivalent circular element with radius R, is assumed to have a constant value Pm(j,k) in this area. The contribution of this element to the total displacement in the observation point (m,n) is denoted by U3jk(m,n). Thus,

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NX NY

(B.34)

The notation used in the following is illustrated in Figure B.4.

jk,mn

observation point

integration area S(j,k)

Figure B.4 The circular integration area and the notation used

Assuming that the input pressure is constant in the integration area under consideration,

t /3 jk(m,n) = - Pin(j,k) J— f( I H{0])[arjk(m,n)] - //^[Jar-rfm./i)] ] dx' .

MoDia) JJS(.j,k)

(B.35)

This equation is written as

U3jk(m,n) = p'n(j,k) Gfj"(m,n), (B.36)

-235-

flex where the discrete Green's function Gjk (m,n) is given by

G%x(m,n) = - J _ ( f [ //^[«•■rfm.n)] - W J / V ^ m , " ) ] ] dxa jk . (B.37). WaDico JJS{j,k) -J

Deformation of the rectangular integration element to a circular one yields

G$'(m,n)= - J- i "de'nk' f drj,k'r^[H^[a-jlf.m,n)]-Hio)liarji.m,n)] ] .

8V0D ico Jo Jo (B.38)

To evaluate the contribution, the following integral is investigated:

ljk(m,n) = ( dejk' ( dr% $ HUhzTjAmril , (B.39)

from which the results for z = a or 2 = ia can easily be derived. In this derivation, use is made of Grafs addition theorem (Abramowitz and Stegun, 9.1.79), which yields

Ho)[zrjk(m,n)]= £ tf(j [zrobs{m-j,n-k)}Jv{zr"k)co%{vejkmn) ,

. int obs, . , . for rjk<r (m-j,n-k ) , (B.40)

and

H 0 } [z0 *("».«)] = X W u* (2 Ó *' > y «tz r°b\m-j,n-k)] cos( t>0;jt m n)

for rjk>r (m-],n-k), (B.41)

respectively. The angle 6:k m n is given by

ota' in 1 e > t . m - = » + e ; t . m - - ^ • (B.42)

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It can be verified that the condition

zr^'exp[±i'0 i ( .mn] <\zr°b\m-j,n-k)\ , (B.43)

which is necessary for the addition theorem to be valid, is fulfilled. When the integration over 6kmnis performed, a non-zero contribution to the integral is obtained only when in the

summation v equals 0. It can easily be verified that this contribution equals 2n. Bearing in mind the different summation theorems applied depending on whether rJ/t is smaller than or

obs. . . . greater than r (m—j,n—k), the following two integrals are considered:

IA=2nH0 [zr (m-j,n-k)]) I drjk rjk Ja(zrjk) • f o r rjk - r (rn-],n-k), (B 4 4)

and

, _ . r obs. . , , . / int int . . ( 1 ) . int int obs, . , . h = 2n J0[zr (m-j,n-k)]) I drjk rjk HQ (zrjk) , for rjk <r (m-j,n-k) , ( B 4 5 )

•'o

Using the relations (Abramowitz and Stegun, 11.3.20 and 11.3.24)

J tJff.t)tt=aJï(a) , ( B 4 6 ) ■'o

and

( tH(o\t)dt=aH\%) +-, ( B 4 7 ) Jr. 7T

these integrals can be performed, obtaining

I A = —Ho)[zr° \m-j,n-k)] J}(za), (B.4SÏ

and

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; ^n<X t r obs, INT , r ü ) / \ 4j' r Ofc*. . . . , /B = - 7 " - / o [ z r (w-j.n-*)] # i (za) + — 70[zr (m-j.n-/:)] . (B.49)

z

The desired expression for Ijk(m,n) can now be obtained by combining equations (B.39), (B.44), (B.45), (B.48) and (B.49). If the observation point lies within the integration area, the integration result is obtained by a proper combination of equations (B.44) and (B.45). This yields:

ljk(m,n) =^ + ~J0[zr°bs(m-j,n~k)]H%R) , z

for 0<rob\m-j,n-k) <R , (B.50)

and

/M(m,n) = ^H$)[zrob\m-jji-k)]JlizR) ,

for rob\m-j,n-k) > R , (B.51)

where use has been made of Abramowitz and Stegun, 9.1.16. Explicit expressions for the displacement can now be written down:

NX NY I/3(m,n) = X X GTjk\m,n) P'n(j,k) , (B.52)

j=\ *=i

in which

G%\m,n) = - - 1 - + - ^ {j0{arob\m-j,n-m < W ) + i J0[iarob\m-j,n-k)] H{?\iaR) } Da 4Da

= --^l + ^ T {-'o \ar°bs(m-j,n-k)]H^\aR) - -/„ \ar0%i-j,n-k)]K,(<,/?)), Da ADa- K

for 0 < rob\m-j,n-k) < R , (B.53)

-238-

and

Gfkx{m,n) = —A H^[ar°bs(m-j,n-k)) J^aR) + i H^[iar°b\m-j,n-k)] J^iaR) }

ADa

- mR { H$)[arob\m-j,nr-k)] J^aR) + ^-K0 \ar°b\m-j,n-k)} I^aR) } , 4Da3

for r° \m-j,n-k) > R , ( B ^

respectively.

-239-

Appendix C The numerical procedure for the combined earth-vibrator model

In this appendix, the numerical calculation of the Green's function relating displacement and traction at the surface of the earth is described. A check on the accuracy of the numerical scheme is derived using Hankel's theorem.

In Chapter 4, the earth Green's function is coupled to a model of the seismic vibrator, and the distributions of traction and displacement at the surface of the earth are calculated. A numerical test on the accuracy of the calculated distribution functions is obtained by application of the. power balance to a closed surface tf. The power balance is applied to two configurations: the baseplate of the seismic vibrator, and a surface comprising the surface of the earth, and extending to the far field of the medium.

C.l The earth's Green's function

C.l.1 Numerical calculation of the earth's Greens function

In the development of a numerical scheme for the calculation of the earth's Green's function at the surface of the earth, use has been made of the numerical scheme van Onselen (1982) developed. The Green's function at the surface of the earth is given by

a f°° -i C(r,o)) = pypA (p)J0(o>pr)dp. • (C.l)

2mpcs Jo

Therefore, the evaluation of the integral /, given by /(r,<u)= f pypA (p)J0(copr)dp , (C.2)

•'o

is investigated. First, for sake of brevity, the function X(p) is introduced:

-1 X(P)= PYpA (P) • (C-3>

Furthermore, the variable Q is introduced which is given by

Q=cor. (C.4)

-240-

The integral I can now be written as

I(il)= rX(p)J0(ï2p)dp . (C.5)

To evaluate the behaviour of the integrand for large p-values, the function X(p) is written as a Taylor expansion in descending powers of p. This yields

A'(p) = £ c2kP . (C.6)

in which the first three terms of the expansion are found to be

\cl cll

c -r2l 2

C2 Co 2 2 i C C

t

3 3

" 2c4

(C.7)

(C.8)

and

C4 = C 1 - + - i »_ + »

. 8 - 6 2 - 4 4 - 2 6 . 8 ^ 2cpc5 2cpcs 2cpcs Ac SJ

(C.9)

It follows from equations (C.5) and (C.6) that the decay of the integrand is of the order _ i/

0 ( p 2) asp —> oo. To accelerate the convergence of the integrand, the first term of the expansion is subtracted from the integrand. Using the relation

f •/()(')<"= 1, (c.10) •'o

this yields

l(Q)= ( {X(p)-C0)J0(np)dp + - J . ( d j )

-241-

Due to the singularity atp=0, the second term of the expansion cannot be subtracted over the whole integration interval, but only from a certain limit a up to infinity. This yields

I(Ü) = ƒ"{X(P)-c0}j0(np)dp +ƒ {X(p)-c0-c2P2)j0(np)dp +

-2 p J0{Qp)dp . (C.12)

The second integral on the right-hand side of (C.12) can be evaluated by realizing that

-2 -4 -6 X{p)-C0-C2p =C4p + 0 ( p ) a s p - > ° ° . (C.13)

Therefore, neglecting orders p and higher, this integral can be written as

ƒ> {X(p)- c 0 - c2p~2} j0(Qp) dp=\ c 4 P " 4 y 0 ( ^ )dp . a

(C.14)

a can be determined such that the contribution of integral (C.14) is negligible:

ƒ" CAp J0(Gp)dp <e , (C.15)

where £ is small. A rough estimate for a can be obtained using the inequality

J ' C4p~4JQ(Qp)dp < J \CAp'4 f°° \C I \ja{Qp)\dp < I \C4\P'4dp = L-^ < c ,

« 3a (C.16)

where use has been made of the knowledge that the value of the Bessel function is always smaller than one, so that

'A a> 3e

(C.17)

-242-

The third integral on the right-hand side of (C.12) is evaluated by performing an integration by parts, and by subsequent application of Abramowitz and Stegun, 11.4.16 and 9.1.27. The final result is that (Q *■ 0)

{X( p) - c0)j0(Qp) dp + ic2n- -A J0(t)dt +

C0 C2J0(Ga) + _2 +_L_ei—L -c2ü[\+J1(üa)] , Q a

(C.18)

where a is given by (C.17). The second integral on the right hand side is tabulated using Abramowitz and Stegun, equation 11.1.16, thus allowing for a very fast evaluation of this integral.

If ^2=0, the result is

/(J2=0) = n

(C.19)

The singularity in (C.19) is removed in the subsequent surface integration. For this purpose, it is recalled that the substitution

n=ar (C.20)

is made. Herman (1981) showed that

ƒ/. dXr

= Xi In r2+x+

2 + x2\n r 3 - x ,

r2-x\_ - x , In

rA-x2

+ r 3 ~ x2

-x2\n r i+ .x |

r/l + x~l. (C.21)

The notation used in this expression is explained in Figure C.l. The first integral on the right hand side of (C.18) is evaluated using Simpson's rule.

Singularities in the derivative of the integrand are removed by substituting

2 2 2 2 2 p =a cos (6) + b sin (6) , (C.22)

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U l . * 2 >

(.X\,X2)

(*i,X2)

Figure C.1 The notation used in Herman's (1981) solution of the presence of the l/r-singularity in the numerical evaluation of the surface integral.

in the integration intervals

[a,b] = [0,\/cp],

[a,b] = [\lcp,Vcs)

(C.23)

(C.24)

and

[a,b] = [l/cs,a] , (C.25)

respectively.

The contribution of the Rayleigh pole is extracted by subtracting from the integrand its Taylor expansion around the singular point:

-244-

fx{p)J^üp)dp = C^É-dp = f •'„ Ja A( n) J-? 4(p)

JV(p) _ H(pR) Mp) A'{pR){p-pR)

dp +

MPR) f dp r_dp }„ T>-\ &\PR) Jq P-PR

(C.26)

In this equation, the Rayleigh wave slowness/?^ is given by

Pr= — , CR

(C.27)

in which cR denotes the Rayleigh wave velocity, and the interval [q,r] contains the singular point pp. A'(p) denotes the derivative of the Rayleigh denominator with respect to p. The first integral on the right-hand side of equation (C.26) no longer suffers from a singularity; the second integral is solved using Cauchy's theorem. This yields

f'-ÊL-.iK + to Jq P~PR

r-PR PR-Q

(C.28)

C.l.2 Check on the numerical calculation of the earth's Green's function

The integration results can be checked in the following way. The expression for /,

KQ)= f X(p)J0(i2p)dp , (C.29)

is first written as

/(/2)= I X(p)J0(Up)dp + in N(PR) = Ic(Ll) + lRay(0), J0 A\PR)

(C.30)

where ƒ denotes the Cauchy principal value integral. The reason for the separation of the

Rayleigh pole contribution will become clear later; in what follows, the pole contribution lKay(Q) is excluded from the analysis. Then, both sides are multiplied with DJG(Qa), and the resulting equation is integrated over Q. This yields

-245-

f QIc(ü)J0(£h)dQ = ( \ ^-J0{Ch)J^Qp)QpdQdp. (c.31) •'o •'o ■'o

Application of Hankel's theorem (see Koefoed 1979),

I I f(x)J(fyx)Jo(yu)yxdydx = f{u) , (C.32) Jo •'o

to equation (C.31) yields the relation

^§-=\ mc(i2)J^ih)di2 . (C.33) ■'o

Thus, the integration result I(Q) can, after subtraction of the pole contribution, be inserted in equation (C.33) to check if the integration expressed by this equation indeed yields the (known) kernel function X(a)/a. The reason for subtracting the pole contribution now becomes clear: the pole contribution, containing the term J0 (iipK), converges very slow, and the integral (C.33)

cannot be evaluated numerically if this term is included. Since the asymptotic behaviour of the integral is given by (see equation (A.102))

KO)~C H$\nP/i) , (C.34)

the integral (C.33) can only be evaluated for the real part, since the imaginary part still contains a slowly converging y0-term. The integral (C.33) is calculated for 100 a-values, with an a-

increment of 10 . The integral Ic is calculated for il -values up to 7000. Since the scaling factor 5, given by (see equation (C.l))

5 = — ^ 7 - (C35) 2mpcs

is included in the calculations, Figure C.2 shows the imaginary part of SX(a)/a , using earth model I (cp = 400 ms_I , cs = 150 rns-1 , p = 1700 kgirr3 ). It is observed from this figure that the analytic and calculated kernel function match very closely.

-246-

Ze-O

-2B-07

-6e-07

Figure C.2 Imaginary part of the kernel function S X(a)/a (dotted line), and the inverted kernel function calculated using equation (C.33) (solid line).

C.2 The complex power balance

In this section, the complex power balance for a lossless elastic medium is discussed. The power balance serves as a check on the accuracy of the numerical calculation of the distributions of traction and displacement underneath the baseplate. Also, expressions are given for the outward power flow in the far field of the elastic medium, expressed in terms of its compressional, shear and Rayleigh wave contributions.

C.2.1. General theory

The complex power balance for an elastic medium in a closed volume V with surface »9 is, in the absence of body forces, given in the frequency-domain by (Tan 1975):

P(co) = -Ho ( Wdef(o» - Wkin(co) } .

In this equation, P is the complex outward acoustic power flow through ó ,

P(a») = - I ƒ ƒ V*(x,ü>) 7"0.(x,ö>) nj (x) dA ,

(C.36)

(C.37)

where T,j denotes the stress tensor, V.the components of the particle velocity and where "*" stands for complex conjugate. WdeA$ the complex elastic deformation energy stored in V',

-247-

Wdcf(có)= I ƒƒƒ r y (x ,o))£y(x,a))dV , (C.38)

in which the strain tensor Etj is given by equation (2.4). WHn is the time-averaged kinetic energy stored in V,

«'tm(<a)= j ƒƒƒ pVi(xfffl) V* (x,Q»dV . (C.39)

In the remainder of this section, the index "at" is omitted for brevity. Since the elastic medium is assumed to be lossless, the condition

Re(/>) = 0

must be satisfied.

(C.40)

C.22 Application of the power balance to the elastic halfspace

First, the power balance (C.40) is applied to a closed surface in the elastic halfspace. The surface i? is chosen to be the surface of the earth Se, supplemented by a hemisphere H with (large) radius R for the compressional and shear wave contributions to the acoustic power flow, or a cylinder C with (large) radius R for the contribution of the Rayleigh waves. The earth's surface Se consists of a traction-free surface 5', and the baseplate area S. The configuration is shown in Figure C.3.

surface

C \H

baseplate 7 ' ^ > K^***r

— . — _ _ i

s Earth J

R I

' _ ^ - ^ ^

o « O

Figure C.3 The configuration used in the calculation of ihc power balance.

-248-

The acoustic power flow is divided into two parts. The first part is the power flow through the eanh's surface. The contribution of Se to the power flow is zero everywhere except at the baseplate area S, since the remainder of the earth's surface, S', is assumed to be traction-free. The complex power flow radiated by the baseplate of the seismic vibrator is denoted as the input power flow P'n. The second part is the power flow through the hemisphere or cylinder. This contribution is denoted as the outward power flow P°m. Equation (C.40) can thus be written as

Re(/5"1) = Re(/>°"') . (C.41)

The outward power flow can be obtained by taking the radius R of the hemisphere and cylinder to infinity. Then, the far field expressions for the particle velocity and traction are substituted in the expression (C.37) for the power flow. Van Onselen (1982) showed that in the far field of the elastic medium the compressional, shear and Rayleigh wave contributions decouple, so that the outward power flow can be decomposed according to

Rc(Pou') = Pou'= PP+PS+PR , (C.42)

in which the superscripts P, S and R denote the compressional, shear and Rayleigh wave contribution, respectively, and where it is taken into account that the far field power flow is purely real. If, as is the case for the numerical examples presented in Chapter 4, only normal traction components are present at the earth's surface, the P 'wave contribution Pp is given by

PP =jpcp t * dB f2^s in ( | ) cos (ö |v^$ ,Ö) | 2 . (C.43)

Similarly, the S wave contribution equals

PS = i pcs ( "dB hdSsm(S)cos(&\v^,6)\ . (C.44) "'o h

The surface wave contribution is given by

R (o R tïn ft P = IV' I d6 I I r 3 txp[-icopR r, cos(i//) 1 dA

16s- •'n JJS (C.45)

-249-

in which WR is given by

*rP[\-V*\p\

P- \A'(PR)\ iyp+ys) *P2R-\\YP +

c, % A + - 7 7, ) +

+ CO

p \A'(PR)\ r„

1 + U^J-L] U» o 4 / 1 _ 2

PR \~2~ PR

H\A'(P*)\ YS

A 2 3 *>PR - ^ (C.46)

Frequency (Hz) 150

Frequency (Hz)

frequency (Hz) equency (Hz) id)

Figure C.4 The real pan of the input power flow (solid line) and the far field power flow (dashed line) for eanh model II, using four diffcrcnl baseplate models: (a) flexural rigidity method, (i) mass-loaded boundary condition, (c) uniform displacement and (d) uniform traction.

-250-

For the notation used in these equations, the reader is referred to Appendix A. The distribution of the far field power flow over the compressional, shear and Rayleigh waves is discussed in Chapter 4; in this appendix, results are shown of the comparison between the input power flow Pin and the outward power flow P0"', calculated from the distributions of traction and displacement obtained with the coupled earth-vibrator model (Chapter 4). This comparison .serves as a check on the correctness of the numerical solution of the coupled earth-vibrator problem. Figure C.4 shows the real parts of the input and output power flow for earth model II (cp = 1000 ms-1 , cs = 400 ms_1 , p = 1700 kgm-3 ), using four different baseplate models. The results illustrate that the accuracy of the numerical scheme developed to calculate the traction and displacement distributions is good.

C2.3 Application of the power balance to the baseplate of the vibrator

Next, the power balance is applied to the baseplate o£.the seismic vibrator. The surface ■& is now chosen to be the baseplate surface. Although the assumption of a lossless elastic medium does not enter explicitly into the derivation of the different baseplate models, equation (C.40) applies to results obtained with the mass-loaded boundary condition and the assumptions of uniform displacement and uniform traction. This can be seen by simply multiplying the equation of motion of the baseplate with the complex conjugate of the particle velocity, and subsequent integration over the baseplate area. The validity of the power balance (C.40) cannot be shown for the flexural rigidity method. The reason for this is that the flexural rigidity method is derived using the fundamental equations of elastic wave propagation, but in the derivation several terms which are assumed small compared to other terms are dropped. As a result, a difference between input power and output power occurs.

The total outward acoustic power flow through the baseplate surface, PbP, is again divided into an input power flow and an output power flow, according to

Pbp(C0) = Pin-bp^) + Pou'-h!,{co), (C.47)

and the-power balance can be formulated as

Re{P,'"'"'(fl» }+Re{P°u'-bp(w)) = 0 . (C.48)

The input power flow is defined as

Rt{Pin-bp(o}) ) = - j R e { ƒ ƒ Papp,ied(xx, x2M v\{xx, x2,co) dx} dx2 } , (C.49)

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in which S denotes the baseplate area on which the driving force is exerted, and the output power flow is given by

Rc[Pou'-bp (co)} = - l R e { ƒ ƒ T3(x],x2,a»vlui,x2,(o)dxidx2 } , (C.50)

where 5 denotes the contact area between baseplate and earth surface. In equations (C.49) and (C.50), only the vertical components of traction and particle velocity are taken into account. Also, use has been made of the knowledge that the surface 5 is flat, with the normal pointing outwards S. The input power delivered by the force applied on top of the baseplate, and the output power radiated by the baseplate into the earth, are shown in Figure C.5 for earth model II, using four different baseplate models.

SO 100 150 »0

Frequency (Hzl

(a)

1.5t-07

ie-07

5e-0S

\ r^ r /

y Frequency (Hz)

(c)

100 ISO

Frequency (Hz]

(W

100 150 200

Frequency (Hz)

(d)

Figure C.5 The input power delivered by the applied force on the baseplate (solid line) and the output power radiated by the baseplate into the earth (dashed line) for earth model II, using four different baseplate models: (a) flcxural rigidity method, (b) mass-loaded boundary condition, (c) uniform displacement and (d) uniform traction.

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All baseplate models using the equation of motion of the baseplate show a perfect agreement between input power and output power. As expected, the flexural rigidity method does not rigorously satisfy the power balance, although the differences between input power and output power are small.

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References

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Company, San Fransisco Allen, H.G. 1969. Analysis and design of structural sandwich panels. Pergamön Press Ltd., Oxford. Anstey, N.A. 1980. Resolution, bandwidth and money. Paper presented at the 50th meeting of the Society of

Exploration Geophysicists, Houston, Texas. Awojobi, A.O. and Grootenhuis, P. 1965. Vibration of rigid bodies on semi-infinite elastic media. Proceedings

of the Royal Society of London, Series A, 287, 27-63. Baaren, van, J.P. 1987. Lecture notes Petrophysics. Delft University of Technology. Bath, M. and Berkhout, A.J. 1984. Mathematical Aspects of Seismology, Vol. 17, in the Seismic Exploration

series, section 1,484 pp., 2nd enlarged edition. Geophysical Press Ltd., London. Berg, van den, P.M. 1981. Transition matrix in acoustic scattering by a strip. Journal of the Acoustical Society

of America 70, 615-619. Bracewell, R.N. 1978. The Fourier Transform and Its Applications. 2nd edition, McGraw-Hill Book Company,

New York. Bycroft, G .N. 1956. Forced vibrations of a rigid circular plate on a semi-infinite elastic space and on an elastic

stratum. Philosophical Transactions of the Royal Society, Series A, 248, 327-368. Castanet, A., Ruell-Malmaison and Lavergne, M. 1965. Vibrator controlling systems. US patent 3,208,550. Cloux, du, R. 1986. Symmetry properties of elastodynamic wave fields and their application to space-time

scattering theory. Ph.D. Thesis, Delft University of Technology. Dragoset, W.H. 1988. Marine vibrators and the Doppler effect. Geophysics 53,1388-1398. Edelmann, H.A.K. 1982. A contribution to the investigation of amplitude characteristics of vibrator signals.

Geophysical Prospecting 30,774-785. Ewing, W.M., Jardetzky, W.S. and Press, F. 1957. Elastic Waves in Layered Media. McGraw-Hill Book

Company, Inc., New York. Gorman, DJ. 1982. The free vibration analysis of rectangular plates. Elsevier North Holland, Inc., New York. Hargrove, K.L., Bonacoris, D.G. and Andrew, J.A. 1984. The use and misuse of "nonlinear" Vibroseis

techniques in reflection seismic acquisition. 37th Annual Exploration Meeting of The Midwestern Sections of the Society of Exploration Geophysicists, Midland, Texas.

Herman, G.C. 1981. Scattering of transient acoustic waves in fluids and solids. Ph. D. Thesis, Delft University of Technology.

Hoop, de, A.T. 1970. The surface line source problem in elastodynamics. Nederlands Elehronika en Radiogenootschap 35,19-21.

Hoop, de, A.T. 1978. Diffraction of acoustic waves. Lecture notes, Delft University of Technology, Department of Electrical Engineering.

Hoppmann, W.H. 1955. Bending of Orthogonally Stiffened Plates. Journal of Applied Mechanics 77,267-271". Huffington, NJ. 1955. Theoretical Determination of Rigidity Properties of Orthogonally Stiffened Plates.

Journal of Applied Mechanics 78,15-20. Huber, M.T. 1914. Die Grundlagen einer Rationellen Berechnung der Kreuzweise bewehrten Eisenbetonplatten.

Zeitschrift des Osterreichischen Ingenieur-u. Architehen-Vereines 66, no. 30. Janak, P.M. 1982. A Comparison and Analysis of Seismic Land Source Energy Relationships and Radiation

Patterns. 347 pp. in: Colorado School of Mines Land Seismic Source Study, final report, vol I, pan II. Kelvin, Lord and Tait, P.G. 1883. Treatise of Natural Philosophy, vol. 1, p. 188. Kennett, B.L.N. 1974. Reflection, rays and reverberations. Bulletin of the Seismological Society of America 64,

1685-1696. Kennett, B.L.N. 1979. Theoretical reflection seismograms for elastic media. Geophysical Prospecting 27, 301-

321. Kennett, B.L.N. 1983. Seismic Wave Propagation in Stratified media. Cambridge University Press, Cambridge. Koefoed, O. 1979. Geosounding Principles I, Resistivity Sounding Measurements, in Methods in Geochemistry

and Geophysics. Elsevier Scientific Publishing Co., Amsterdam Lamb, H.L. 1904. On the propagation of tremors over the surface of an elastic solid. Philosophical Transactions

of the Royal Society, Series A, 203, 1-42. Lerwill, W.E. 1981. The amplitude and phase response of a seismic vibrator. Geophysical Prospecting 29, 503-

528. Lerwill, W.E. 1982. Reply to comments by Sallas and Weber on "The amplitude and phase response of a

seismic vibrator". Geophysical Prospecting 30, 939-941. Levy, M. 1899. Sur 1'équilibre élastique d'une plaque rectangular. Compt. Rend. 129. Libove, C. and Hubka, RJE. 1951. Elastic Constants for Corrugated Core Sandwich Plates. NACA TN 2289.

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Love, A.E.H. 1927. A treatise on the mathematical theory of elasticity. 4th ed. Cambridge University Press, Cambridge.

McFarland, D., Smith, B.L. and Bcrnhart, W.D. 1972. Analysis of plates. Spartan Books, New York. Miller, G.F., and Pursey, H. 1954. The field and radiation impedance of mechanical radiators on the free surface

of a semi-infinite isotropic solid. Proceedings of the Royal Society, Series A, 223, 521-541. Navier, C.L.M.H. 1823. Bulletin des Sciences de la Sociélé Philomathique de Paris. Onselen, van, C. 1980. Analysis of the elastic wave motion in a semi-infinite solid excited by a vibrating disk

on its stress-free surface ("Vibroseis problem"). Internal report no. 1980-22, Laboratory of Electromagnetic Research, Department of Electrical Engineering, Delft University of Technology.

Onselen, van, C. 1982. Vibroseismic excitation of elastic waves in a semi-infinite solid. Internal report no. 1982-10, Laboratory of Electromagnetic Research, Department of Electrical Engineering, Delft University of Technology.

Ottes, W. 1987. Calculation of the acoustic wavefield emitted by a point source of volume injection in a cylindrical configuration. M. Sc. Thesis, Delft University of Technology, Department of Mining and Petroleum Engineering.

Pieuchot, M. 1984. Seismic Instrumentation, Vol. 2, in the Seismic Exploration Series, section 1. Geophysical Press Ltd., London.

Robertson, I.A. 1966. Forced vertical vibration of a rigid circular disk on a semi infinite elastic solid. Proceedings of the Cambridge Philosophical Society 62,547-553.

Rietsch, E. 1977a. Computerized analysis of Vibroseis signal similarity. Geophysical Prospecting 25, 541-552. Rietsch, E. 1977b. Vibroseis Signals with Prescribed Power Spectrum. Geophysical Prospecting 25, 613-620. Sallas, J.J., and Weber, R.M. 1982. Comments on "The amplitude and phase response of a seismic vibrator" by

W.E. Lerwill. Geophysical Prospecting 30, 935-938. Sallas, J.J. 1984. Seismic vibrator control and the downgoing P-wave. Geophysics 49, 732-740. Sallas, J.J., Amiot, E.A. and Alvi, H.I. 1985. Ground force control of a P-wave vibrator. SEG seismic field

techniques workshop, August 13-16 1985, Monterey, California. Spaans, P. 1988. The effect of the evanescent wavefield on the calculation of synthetic seismograms. M. Sc.

Thesis, Delft University of Technology, Department of Mining and Petroleum Engineering. Tan, T.H. 1975. Diffraction theory for time-harmonic elastic waves. Ph. D. Thesis, Delft University of

Technology. Tan, T.H. 1985. The elastodynamic field of N interacting vibrators (two-dimensional theory)- Geophysics 50,

1229-1252. Timoshenko, S. and Woinowsky-Krieger, S. 1959. Theory of Plates and Shells. 2nd edition. McGraw-Hill, New

York. Troitsky, M.S. 1967. Orthotropic Bridges Theory and Design. The James F. Lincoln Arc Welding Foundation,

Cleveland. Ursin, B. 1983. Review of elastic and electromagnetic wave propagation in horizontally layered media.

Geophysics 48, 1063-1081. Wyllie, M.R., Gregory, A.R. and Gardner, L.W. 1956. Elastic wave velocities in heterogeneous and porous

media. Geophysics 21,41-70. Ziolkowski, A.M. 1986. The scaling of airgun arrays, including depth dependence and interactions. Geophysical

Prospecting 34, 383-408. Ziolkowski, A.M. and Lerwill, W.E. 1979. A simple approach to high resolution seismic profiling for coal.

Geophysical Prospecting 27,360-393.

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Summary

In the exploration of oil and gas using the seismic method, seismic vibrators are frequently used to generate a source signal. In this thesis, the wavefield emitted by one or more seismic vibrators acting at the surface of the earth is investigated. In theory, the shape of the far field wavelet can be determined from measurements on the vibrator. If this wavelet is cross-correlated with the measured data, the resulting seismogram will contain the desired zero phase wavelet. Normally, the measured data is cross-correlated with the pre-determined sweep, which. is assumed to be equal to the feedback signal chosen on the vibrator. Therefore, the desired zero phase wavelet after cross-correlation can be obtained by choosing the right feedback signal on the vibrator. To determine which choice of feedback signal is correct from a geophysical point of view, an analysis of the wavefield emitted by the seismic vibrator is presented. The analysis contains three sequential steps:

1. a description of the wave propagation in the earth, 2. a description of the behaviour of the seismic vibrator, and 3. combination of the earth model and the vibrator model.

In Chapter 2, the wave motion in a layered medium that consists of plane horizontal, homogeneous, isotropic and perfectly elastic layers is investigated. The source is characterized by the presence of a distribution of traction components at the surface. The analysis of the wavefield emitted by this source is based on the reflectivity method of Kennett. Modification of this method to account for the presence of traction components at the surface shows that the components of the wavefield in the medium are given by a spatial convolution of the surface traction components and a Green's matrix. The Green's matrix in the wavenumber domain is given by a recursive scheme in which the elastic parameters of the individual layers are included. Because cylindrical symmetry is present in the configuration, the transformation of this Green's matrix to the space domain can be achieved in an efficient way.

For the determination of the far field wavelet shape, the response of an elastic halfspace is of interest. From the expressions describing the wave propagation in a layered elastic medium, the shape of the far field wavefield is obtained. It is shown that, if the source dimensions are small compared with a wavelength, the far field wavelet essentially equals a weighted sum of the ground force components, in which the weighting factors are frequency-independent directivity functions.

Chapter 3 describes a model of the seismic vibrator. First, expressions are derived for the force exerted by the vibrator's drive system on the baseplate of the vibrator. Then, the behaviour of the baseplate under the combined action of this applied force and the reaction force

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of the earth is described. Four different baseplate models are considered. Two conventional models, assuming uniform displacement or uniform traction under the plate, are not in agreement with field measurements. Two new models which yield non-uniform distributions of both traction and displacement are presented. -The first new model assumes that the applied force is distributed uniformly over the baseplate. The formulation of this model then follows from the condition that the total force acting on the plate is identical with the inertial force related to the acceleration of the mass of the baseplate. The second model, the flexural rigidity method, contains a more rigorous treatment of the plate stiffness.

The earth model and the vibrator model are coupled in Chapter 4, where the distributions of traction and displacement underneath the baseplate are calculated. The coupling of the earth's Green's function with the different baseplate models yields significant differences in the distributions of both traction and displacement. Also, the integrated traction underneath the baseplate (equal to minus the ground force), which determines the far field wavelet shape, is different for the different baseplate models. However, the average radiation impedance and the average power output are very similar for all four models.

In Chapter 5, the theoretical results from the previous chapters are compared with experimental data. Using measurements of the pressure distribution underneath the baseplate and the known acoustic parameters at the location of the experiment as input, the response of a downhole geophone array is modelled using the earth model developed in Chapter 2. Comparison of model results with the measured downhole response shows that the earth model accurately describes the wave propagation in the medium. The performance of the four different baseplate models is investigated by predicting the baseplate acceleration for each model from measurements of the pressure distribution underneath the baseplate and the reaction mass acceleration. Comparison of the measured baseplate acceleration with the predicted baseplate acceleration shows that only the flexural rigidity method results in the correct amplitude behaviour, but this method suffers from an, as yet, unexplained phase difference of 90° between measured and predicted baseplate acceleration.

The integrated traction underneath the baseplate is of practical interest for the determination of the far field wavelet shape. In practice, the integrated traction underneath the baseplate (which is equal to minus the ground force) must be determined from measurements of the accelerations of baseplate and reaction mass. Because the traction is not distributed uniformly underneath the baseplate, the determination of the ground force from measurements of the baseplate and reaction mass acceleration is non-trivial. Three different feedback signals that are currently in use (baseplate acceleration, reaction mass acceleration and a weighted sum of the baseplate acceleration and the reaction mass acceleration), and one new feedback signal based on the baseplate model that includes the flexural rigidity of the baseplate, are used to estimate the ground force. Comparison of the predicted ground force with the true ground force obtained from measurements of the traction distribution underneath the baseplate show that the use of the

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baseplate acceleration or the reaction mass acceleration as a feedback signal yields a poor estimate of the ground force. The phase difference between measured and predicted ground force is small when the weighted sum method is used as a feedback signal, but the amplitude behaviour shows larger discrepancies for this signal. The prediction of the ground force using the feedback signal based on the flexural rigidity method yields a reliable estimate if the phase error that is observed for this method in the prediction of the baseplate acceleration is corrected for.

In Chapter 6, the wavefield emitted by a marine vibrator is investigated. A model is developed which contains the same elements as the land vibrator model. It is shown in this chapter that the distribution of pressure and displacement on the marine vibrator is almost uniform. Far field expressions for the emitted wavefield show that the marine vibrator has both monopole and dipole characteristics. In practical applications, the dipole effect is small, and the pressure in the far field of the medium essentially equals the acceleration of the marine vibrator.

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Samenvatting

(summary in Dutch)

Theoretische en praktische aspekten van de Vibroseis methode

Bij de opsporing van olie en gas met behulp van de seismische methode wordt voor de generatie van een bronsignaal veelvuldig gebruik gemaakt van seismische vibratoren. In dit proefschrift wordt het golfveld voortgebracht door één of meerdere seismische vibratoren aan het oppervlak van de aarde onderzocht. Door metingen op de seismische vibrator is het in theorie mogelijk de vorm van de uitgezonden golf in het verre veld van de aarde te bepalen. Indien deze golf wordt gecorreleerd met de gemeten seismische data zal het resulterende seismogram de gewenste golf met fase nul bevatten. Normaliter wordt de gemeten data gecorreleerd met de vooraf bepaalde sweep invoer, welke gelijk wordt gesteld aan het feedback signaal op de vibrator. Hieruit volgt dat de gewenste golf met fase nul na kruiscorrelatie verkregen kan worden door het juiste feedback signaal op de seismische vibrator te kiezen. Om te bepalen welke keuze voor het feedback signaal correct is vanuit een geofysisch oogpunt wordt een analyse gepresenteerd van het golfveld voortgebracht door de seismische vibrator. De analyse bevat drie opeenvolgende stappen:

1. een beschrijving van de golfvoortplanting in de aarde, 2. een beschrijving van het gedrag van de seismische vibrator, en 3. combinatie van het aardmodel en het seismische vibrator model.

In Hoofdstuk 2 wordt de golfbeweging beschouwd in een gelaagd medium dat bestaat uit een aantal vlakke horizontale, homogene, isotrope en volkomen elastische lagen. De bron wordt gekarakteriseerd door de aanwezigheid van een verdeling van tractie-componenten aan het oppervlak. De analyse van het door de bron gegenereerde golfveld in het medium is gebaseerd op de reflectiviteitsmethode van Kennett. Modificatie van deze methode teneinde de invloed van de aanwezigheid van componenten van de oppervlakte-tractie te beschrijven leidt tot het resultaat dat de componenten van het golfveld in het medium worden beschreven door een spatiele convolutie van de componenten van de tractie aan de oppervlakte en een Greense matrix. De Greense matrix in het golfgetal domein volgt uit een recursief schema, waarin de elastische eigenschappen van de verschillende lagen een rol spelen. De transformatie van deze Greense matrix naar het spatiele domein kan door de aanwezigheid van cylindrische symmetrie in de configuratie op efficiënte wijze worden bereikt.

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De responsie van een elastische halfruimte is van belang voor de bepaling van de golfvorm in het verre veld van het medium. Uitdrukkingen voor het golfveld in het verre veld van de halfruimte worden verkregen uit de beschrijving van de golfvoortplanting in een gelaagd elastisch medium. Aangetoond wordt dat indien de afmetingen van de bron klein zijn ten opzichte van de uitgezonden golflengte, het golfveld in het verre veld in essentie gelijk is aan de tijdsafgeleide van een gewogen som van de componenten van de "ground force" (de kracht die door de aarde op het oppervlak wordt uitgeoefend). De weegfactoren zijn frequentie-onafhankelijke functies die de richtingsafhankelijkheid van het golfveld beschrijven.

In Hoofdstuk 3 wordt een model beschreven van de seismische vibrator. Allereerst worden. uitdrukkingen afgeleid voor de kracht die door het aandrijfsysteem van de seismische vibrator op de bodemplaat wordt uitgeoefend. Vervolgens wordt het gedrag van de bodemplaat onder de gecombineerde werking van deze dynamische aandrijfkracht en de reactiekracht van de aarde beschreven. Vier verschillende modellen voor de bodemplaat worden beschouwd. Twee modellen, die uitgaan van de conventionele aannames van constante verplaatsing en constante tractie onder de plaat, respectievelijk, geven geen realistische beschrijving van het gedrag van de bodemplaat. Twee nieuwe modellen worden ontwikkeld waarin zowel de tractie als de verplaatsing variëren over het oppervlak van de bodemplaat. De eerste van deze modellen neemt aan dat de dynamische aandrijfkracht gelijkmatig is verdeeld over het oppervlak van de plaat. De formulering van dit model volgt dan uit de conditie dat de totale kracht die op de plaat werkt gelijk is aan de kracht gerelateerd aan de versnelling van de massa van de bodemplaat. Het tweede model, de buigstijfheidsmethode, bevat een meer rigoreuze behandeling van de stijfheid van de plaat.

De modellen voor de golfvoortplanting in het medium en het gedrag van de seismische vibrator worden gekoppeld in Hoofdstuk 4, waar de verdelingen van tractie en verplaatsing onder de bodemplaat worden berekend. De koppeling van de Greense functie voor de aarde met de verschillende modellen voor de bodemplaat leidt tot significante verschillen in de verdeling van tractie en verplaatsing onder de plaat. Ook de geïntegreerde tractie onder de bodemplaat (gelijk aan minus de "ground force"), die bepalend is voor de verre veld golfvorm, is verschillend voor de verschillende modellen voor de seismische vibrator. De gemiddelde stralingsimpedantie van de bodemplaat en het gemiddelde geleverde vermogen zijn echter voor alle vier modellen nagenoeg gelijk.

In Hoofdstuk 5 worden de theoretische resultaten uit de voorgaande hoofdstukken vergeleken met experimentele gegevens. Gebruik makend van metingen van de drukverdeling onder de bodemplaat en bekende gegevens over de akoestische parameters van de aarde ter plaatse, wordt de responsie van een array van geofoons in een boorgat gemodelleerd met behulp van het aardmodel uit Hoofdstuk 2. Vergelijking van de modelresultaten met de gemeten responsie in het boorgat toont aan dat het aardmodel een nauwkeurige beschrijving van de golfvoortplanting in het medium geeft. De vier verschillende modellen voor het gedrag van de bodemplaat van de

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seismische vibrator worden getoetst door, gebruik makend van metingen van de drukverdeling onder de plaat en de versnelling van de reaktie massa van de vibrator, de versnelling van de bodemplaat te voorspellen met behulp van elk van deze modellen. Vergelijking van de voorspelde versnelling van de plaat met de gemeten versnelling leert dat alleen de buigstijfheidsmethode het juiste amplitude gedrag oplevert, maar dat deze methode resulteert in een, tot op heden onverklaard, faseverschil van ongeveer 90° tussen gemeten en voorspelde versnelling.

Voor praktische toepassingen van de Vibroseis methode dient de geïntegreerde tractie onder de plaat (gelijk aan minus de "ground force") bepaald te worden uit metingen van de versnellingen van bodemplaat en reaktie massa. Door de niet-uniforme verdeling van de tractie onder de bodemplaat is de bepaling van de "ground force" uit metingen van de versnellingen van bodemplaat en reaktie massa niet triviaal. Drie bestaande feedback signalen (bodemplaat versnelling, reaktie massa versnelling en een gewogen som van de bodemplaat versnelling en de reaktie massa versnelling), en één nieuw feedback signaal gebaseerd op het buigstijfheidsmodel van de plaat, worden toegepast om de "ground force" te bepalen. Vergelijking van de voorspelde "ground force" met de werkelijke "ground force", berekend uit metingen van de drukverdeling onder de bodemplaat, leert dat van de drie bestaande feedback signalen alleen de gewogen som methode een redelijke schatting geeft van de "ground force". Het faseverschil tussen gemeten en voorspelde "ground force" is klein voor dit feedback signaal, maar het amplitudegedrag vertoont grotere verschillen. De voorspelling van de "ground force" met behulp van de buigstijfheidsmethode levert een betrouwbare schatting indien voor het faseverschil waargenomen in de voorspelling van de plaatversnelling wordt gecorrigeerd.

In Hoofdstuk 6 wordt het golfveld uitgezonden door een marine vibrator onderzocht. Een model wordt ontwikkeld dat dezelfde elementen bevat als het model voor de land vibrator. Aangetoond wordt dat de verdeling van de druk en de verplaatsing op de marine vibrator vrijwel uniform is. Verre veld uitdrukkingen voor de marine vibrator tonen aan dat het uitgezonden golfveld zowel dipool als monopool eigenschappen heeft. Voor praktische toepassingen is het dipool effect klein, en is de druk in het verre veld van het medium in essentie gelijk aan de versnelling van de marine vibrator.

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Acknowledgements

The research described in this thesis could only be carried out with the help and support of others.

First of all, I like to thank Jil for all her help, moral support and patience over the years. The Vibroseis project originated from a cooperation of our group in Delft with Industrial

Vehicles International, Inc. I gready appreciate the financial, moral and technical support I was given by I.V.I, over the past four years, and would like to thank mr. J. Bird sr., mr. J. Bird jr. and Elmo Christensen for all their help.

I am also greatly obliged to the Koninklijke Shell Exploration and Production Laboratory in Rijswijk for supplying me with the land vibrator data, and for assistance during the processing of the data set.

I wish to express my special gratitude to my promotor, prof. A.M. Ziolkowski, and to dr. ir. J.T. Fokkema, for their continuous support and stimulating ideas.

The critical reading of the manuscript by John Sallas, Bob Pearce and Wietze Eckhardt is greatly appreciated. Also I must mention prof. dr. ir. P.M. van den Berg, prof. dr. ir. A.T. de Hoop and ir. J.P. van Baaren for their helpful suggestions.

I am very grateful to drs. P.H. van der Kleyn and Ties Steenhuis for their assistance in making the figures.

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Curriculum vitae

2 sep. 1961 geboren te Ens, de Noordoostelijke Polder

1978 diploma Atheneum B

Newman Scholengemeenschap te Breda

1978 -1984 Technische Universiteit te Delft

29 nov. 1984 diploma mijningenieur

1 jan. 1985 - 31 okt. 1986 wetenschappelijk assistent, werkzaam aan een onderzoeksproject over de Vibroseis methode bij de vakgroep Petroleumwinning en Technische Geofysica van de afdeling Mijnbouwkunde en Petroleumwinning aan de TU Delft

1 nov. 1986 - heden universitair docent bij bovengenoemde vakgroep

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