physics of acoustic wave propagation

Chapter 2

Physics of Acoustic WavePropagation

To expedite the data processing and simplify the theory explorationistsoften invoke the acoustic approximation, i.e., shear effects in the dataare negligible. This is an acceptable approximation for somewhat lay-ered media, near-offset traces recorded by vertical component phones,and surface-wave filtered data. To deepen our understanding of thisacoustic approximation we now present an overview of the physics ofthe acoustic wave equation.

2.1 Acoustic Media and Acoustic Waves

Assume a compressible, non-viscous (i.e., no attenuation) fluid with noshear strength and in equilibrium (i.e., no inertial forces). Small local-ized displacements of the fluid will propagate as an acoustic wave, alsoknown as a compressional wave. Due to the lack of shear strength, lo-calized deformations of the medium do not result in shear deformationsbut instead cause changes in the fluid elements volume, as shown inFigure 2.1.

The equilibrium force/unit area on the face of a volume elementwill be called the time-independent equilibrium pressure Peq(r), whilethe change in pressure due to a localized compressional wave will bedenoted as P (r, t). For example, the atmospheric pressure decreases

1

2CHAPTER 2. PHYSICS OF ACOUSTICWAVE PROPAGATION

Equilibrium

CompressionRarefraction

Figure 2.1: Cube of air in (top) equilibrium and (bottom) disturbedfrom equilibrium. The lower left diagram depicts a rarefraction wherethe surrounding medium pulls the cube into a larger volume so thatthe air density is less than the surrounding medium. The lower rightdiagram is similar, except the surrounding medium compresses the cubeinto a smaller volume resulting in denser air.

with elevation increases and can be considered to be independent oftime. If I begin talking, however, I excite transient acoustic wavesP (r, t) that disturb the equilibrium pressure.

Snapshots of the particle distribution for a condensation wave anda rarefraction wave are shown in Figure 2.1. In the compressionalcase the element volume is filled with denser (shaded) air while in therarefraction case the element volume has lighter (unshaded) air thanthe surrounding medium. We can physically create the condensationwave by injecting air from our lungs into the medium (HELLLL!), andthe rarefraction wave by sucking air into our lungs (UHHHHH!). Usinga spring-mass model, rarefractions are created by pulling a spring (i.e.,tension) and condensations by compressing a spring (compressions).

2.2. ACOUSTIC HOOKES LAW: P = U 3

2.2 Acoustic Hookes Law: P = uHookes law for an acoustic medium says stress is linearly proportionalto strain for small enough strains. A simple 1-D example will firstbe given to demonstrate Hookes law, and then we will apply it to thecase of an acoustic medium.

1. 1-D Spring: The force on a mass connected to a spring disturbedfrom equilibrium (see Figure 2.2) is given by

F = k(du/l)k, (2.1)where du is the displacement from equilibrium, k is the springconstant, l is the length of the spring in equilibrium, and k isthe downward point unit vector in Figure 2.2. The ratio du/l isalso known as the compressional strain of the spring model. Notethat in the equilibrium position there is no motion because thegravitational force balances the elastic force. When disturbed,the net force on the mass is given by the above equation.

2. 3-D Acoustic Springy Cube: Hookes Law says that the sur-rounding disturbed medium exerts a pressure on the face of thecube that is linearly proportional to the corresponding volumechange of the cube. That is,

P = dV/V, (2.2)where dV is the change in volume after disturbance, V is theundisturbed cube volume, and is the bulk modulus. Note that avolume change is the 3-D equivalent of a 1-D displacement change.

Figure 2.2 shows that the relative volume change dV/V is givenby:

dV

V= [dxdydz (dx+ u)(dy + v)(dz + w)]/(dxdydz),

=udydz + vdxdz + wdxdy

dxdydz+O(2nd order terms)

ux

+v

y+w

z, (2.3)


dx

dx dy dzdu dv dw+ +du dv dw

1-D Spring

du {equilibrium disturbed state

3-D Acoustic Cube

dz

equilibrium disturbed state

dy

{dw

du

{

dv

du dv dw

disturbance=

} }=dVV dx dy dz

du dydz + dv dxdz + dw dxdy}

Figure 2.2: (Top) Spring in (left) equilibrium and (right) disturbed fromequilibrium. (Bottom) Elemental cube of air in (left) equilibrium and(right) disturbed from equilibrium. In this case the cube has expandedso net tensional forces of surrounding medium must be expanding cube.Note that the 1st-order volume change is outlined by the solid heavylines, where the higher-order terms are associated with the corner parts.

2.3. NEWTONS LAW: P/X = (X, Y, Z)2U/T 2 5

where V = dxdydz is the cubes volume before the disturbance.Substituting equation 2.3 into equation 2.2 yields:

P = (x, y, z) u+ S(x, y, z, t), (2.4)where u = (u, v, w) are the cartesian components of displacementalong the x, y, and z axes and S(x, y, z, t) is a time dependentsource term that is independent of the displacement field. Forexample, S(x, y, z, t) can be a source that injects material intothe medium such as an air gun used for marine seismic surveys.Often the spatial and temporal variables are suppressed for theP and u field variables, but the spatial coordinates are explicitlyexpressed for the bulk modulus to remind the reader that thephysical properties of the medium can vary with location.

Note:

Neglecting second-order terms is the small displacement approxi-mation, valid for V/V < 104 or sound quieter than a jet engine(Kinsler and Frey, 1961, Fundamentals of Acoustics).

Sign convention: P is the force/area that the surrounding mediumexerts on the face of the elemental cube, where tensions (forcepointing away from cube into external medium) are negative andcompressions (force pointing into cube) are positive. For example,the divergence of u is positive if the volume expands by tensionalforces, which is consistent with the sign in equation 2.4.

2.3 Newtons Law: P/x = (x, y, z)2u/t2The external force on an acoustic cube is given in Figure 2.3. Theseforces have a non-zero gradient along the x-axis, and so there is a netelastic force imposed upon the cube by the external medium. This netforce must be balanced by an inertial force (i.e., it accelerates) so thatNewtons law says:

[P (x+ dx, y, z, t) P (x, y, z, t)]dydz [(x, y, z)dxdydz]u(x, y, z, t),(2.5)


P(x, y, z, t) P(x+dx, y, z, t)

....

TENSIONS ARE PULLS (-)

COMPRESSIONS ARE PUSHES (+)

.

Figure 2.3: External (top) tension and (bottom) compressional forceson an elemental cube. In the lower case there exists a spatial gradientof the disturbed pressure field along the x-axis.

where the double dot corresponds to two time derivatives. Expandingthe LHS in a Taylor series about the point (x, y, z) we get

[P (x, y, z, t) P (x, y, z, t) + P/xdx+HighOrder Terms]dydz [dxdydz]u(x, y, z, t), (2.6)

and dividing by dxdydz and neglecting higher-order terms we get New-tons Law:

P (x, y, z, t)/x = (x, y, z)u(x, y, z, t). (2.7)

For an arbitrary force distribution the general form of Newtons law is:

P = (x, y, z)u(x, y, z, t), (2.8)

where u(x, y, z, t) is the particle acceleration vector.Note:

2.4. ACOUSTIC WAVE EQUATION 7

The minus sign is used so that we are consistent with the notationfor pressure. If P (x + dx, y, z, t) is positive and greater thanP (x, y, z, t) in Figure 2.3 then the cube should accelerate to theleft, which it will according to the above form of Newtons Law.

2.4 Acoustic Wave Equation

Applying 2/t2 to equation 2.4 and applying to equation 2.8 afterdividing by /(x, y, z) gives:

P = (x, y, z) u+ S(x, y, z, t), (2.9)

[1/(x, y, z)P ] = u. (2.10)The above two equations are the first-order equations of motion in anacoustic medium. Substituting equation 2.9 into equation 2.10 yieldsthe 2nd-order acoustic wave equation:

(1/(x, y, z)P ) 1/(x, y, z)P = S(x, y, z, t)/(x, y, z).(2.11)This equation is valid for arbitrary velocity and density distributions.

Assuming negligible density gradients the above equation reducesto the oft-used wave equation:

2P c22P/t2 = (x, y, z)S(x, y, z, t)/(x, y, z), (2.12)where

c =(x, y, z)/(x, y, z) , (2.13)

and c is the compressional wave propagation velocity.The above equation is known as the inhomogeneous acoustic wave

equation for negligible density variations, expressed as

2P c22P/t2 = F , (2.14)where F = S(x, y, z, t)/(x, y, z) is the inhomogeneous source termthat specifies the location and time history of the source. For example,F specifies the strength of the hammer blow and its location in theLiberty Park experiment.


2.5 Solutions of the Wave Equation

The physics of wave propagation will now be explained using some spe-cial solutions to the wave equation. A harmonic plane wave propagatingin a homogeneous medium will be first examined, and then the case ofa 2-layered medium will be studied.

2.5.1 1-D Wave Propagation in an HomogeneousMedium

A harmonic wave oscillates with period T and has a temporal depen-dence usually given by eit, where = 2pi/T is the angular frequencyinversely proportional to the period T . A plane wave is one in whichthe wavefronts line up along straight lines, and can be described by thefollowing function:

P (x, t) = A0ei(kxt) (2.15)

which also solves the homogeneous wave equation 2.12. This can beshown by plugging equation 2.15 into equation 2.12 to get

(k2 (/c)2)P (x, z, t) = 0, (2.16)

which admits non-trivial solutions if

k2 = (/c)2. (2.17)

This equation is known as the dispersion equation and imposes a con-straint on the temporal and spatial variables in the Fourier domain.It says that the wavefront will move a distance of one wavelength during one period T of elapsed time, and the propagation velocity of

this movement is equal to c =/. See Figure 2.40 for an illusration

of a plane wave propagating plotted as Offset vs Time.The real part of equation 2.15, i.e., cos(kxt), plots out as slanted

lines in x t space, and these lines move to the right as t increases. Inan xy z volume, these lines (or wavefronts) become planes that areperpendicular to the x-axis. Therefore this function represents a planewave propagating along the x-axis. The shortest distance between 2

2.5. SOLUTIONS OF THE WAVE EQUATION 9

00.05

0.10.15

0.2

0

0.05

0.1

0.15

0.21

0.5

0

0.5

1

Offset (m)Time (s)Offset (m)

Time (s

)

0 0.05 0.1 0.15

0

0.02

0.04

0.06

0.08

0.1

0.12

Wavelength

Period

Figure 2.4: 2-D wavefront propagating along the Offset coordinate.

adjacent peaks of the wavefront is defined to be the wavelength and its reciprocal is given by k = 2pi/, where k is known as thewavenumber. Using this definition of k and that for = 2pi/T thedispersion relation can be re-expressed as:

c = /T. (2.18)

The constant c is also defined in terms of the material properties of the

rock as c =/, otherwise known as the compressional wave velocity.

Note: The function ei(kxt) describes a right-going plane waveand ei(kx+t) describes a left-going wave. To see this, note that wefollow a wavefront if the phase = (kx t) stays constant for anincrease in both t and simultaneous increase in x (such that x/t = /k).Because x increases this means that the wave is moving to right with thecompressional velocity given in equation 2.18. Conversely, = (kx+t)is a constant if t increases and x decreases; thus, the wave is moving toleft.


2.5.2 2-D Wave Propagation in an HomogeneousMedium

The following function

P (x, z, t) = A0ei(krt), (2.19)

solves the wave equation 2.12, where the wavenumber vector is given byk = (kx, kz) and the observation vector is given by r = (x, z). Similar tothe 1-D dispersion equation, the 2-D dispersion equation can be derivedby plugging 2.19 into equation 2.12 to get

k2x + k2z (/c)2 = 0. (2.20)

The real part of equation 2.19, i.e., cos(kxx+kzzt), plots out asstraight lines perpendicular to the wavenumber vector k, and these linespropagate in a direction parallel to k as t increases. This is easy to provebecause the general equation for a straight line is given by k r = cnst,where k is a fixed vector perpendicular to the straight line. The locus ofpoints that satisfy this equation defines the wavefront where the phase(i.e., = kxx+ kzzt) is a constant. Thus as the time increases, i.e.as cnst increases, the straight line also moves such that the directionof movement is parallel to the fixed k vector, as shown in Figure 2.5.

Therefore equation 2.19 represents a harmonic plane wave propagat-ing along the direction parallel to k. Similar to the discussion for a 1-Dplane wave, the shortest distance between two adjacent peaks of thewavefront is defined to be the wavelength and is given by = 2pi/k,

where k =k2x + k

2z is known as the wavenumber. Using this defini-

tion of k and that for the 2-D dispersion relation takes the same formas equation 2.18. A good illustration of the relationship between thewavenumber vector and the direction of wave propagation is given inthe movie. Here the left figure is a plot of points in (kx,kz) space, andthe right figure is the corresponding snapshot of the harmonic planewave. Note that as the length of the wavenumber vector increases thewavelength decreases, and as the wavenumber direction changes so doesthe direction of the propagating wave.


k

T

x = /cos( )

APPARENT WAVELENGTHS

z

x

z = /sin( )

Figure 2.5: 2-D wavefront propagating along the direction parallel tothe wavenumber k vector.

00.05

0.10.15

0.2

0

0.05

0.1

0.15

0.21

0.5

0

0.5

1

Offset (m)Time (s)Offset (m)

Time (s

)

0 0.05 0.1 0.15

0

0.02

0.04

0.06

0.08

0.1

0.12

Figure 2.6: (Left) 3D seismograms and (right) 2D seismograms.


T e

R ee

ikz

ik z -ik zc

c

xz

Incident Wave Reflected Wave

Transmitted Wave

Figure 2.7: Plane wavefront normally incident on a flat interface thatseparates two homogeneous media. The unprimed medium indicatesthat of the incident wave.

2.5.3 Plane Wave Propagation in a 2-Half SpaceMedium

Figure 2.7 depicts a plane harmonic wave normally incident on an in-terface separating two half-spaces of unequal stiffness. The functionsare those for the up and downgoing solutions of the wave equation,but it is understood that the geophones record the sum of the up- anddown-going wavefields, the total wavefield. That is, the total pressurefields in the upper (+) and lower (-) media are expressed as

P+(z) = eikz +Rpeikz, (2.21)

P(z) = Tpeikz, (2.22)

where Rp and Tp denote the pressure reflection and transmission coef-ficients, respectively. The harmonic function eit has been harmlesslydropped because it cancels out in the final expressions for Pp and Tp.

The two unknowns in these linear equations, Rp and Tp, can bedetermined by imposing two equations of constraints at the interface


at z = 0: continuity of pressure across the interface

P+(z = 0) = P(z = 0) [eikz +Rpeikz]|z=0 = Tpeikz|z=0,or

1 +Rp = Tp, (2.23)

and continuity of particle velocity (recall Newtons Law in equation 2.8)across the interface

1/P+/z|z=0 = 1/P/z|z=0 1//z(eikz +Rpeikz)|z=0= 1//z(Tpeikz)|z=0,

or

(k/)(1Rp) = (k/)Tp, (2.24)

Setting k = /c and k = /c, and solving for Rp in equations 2.23and 2.24 yields the pressure reflection and transmission coefficients fora normally incident plane wave on a flat interface:

Rp = (c c)/(c + c), (2.25)

Tp = 2c/(c + c). (2.26)

Here c is known as the impedance of the medium, and roughly indi-cates the stiffness of a medium. For example, a plane harmonic planewave in a homogeneous medium exerts a pressure denoted by P =eikxit and has a particle velocity denoted by u = k/()P= 1/(c)P .Therefore, the ratio P/u becomes

P/u = c. (2.27)

This says that decreasing impedances imply larger particle velocities fora fixed elastic pressure on a cubes face. This is exactly what one wouldexpect in a really soft medium: larger displacements for springier-softerrocks, which is one of the reasons that earthquakes shake sediment-filledvalleys more than the surrounding bedrock. Conversely, stiffer medialead to smaller displacements for a given elastic pressure.

Note:


e-ikz e-ikzeikz- eikz-

P = 0 at the Free Surface

xz

Node

Upgoing Downgoing Total Field

Figure 2.8: The pressure at the free surface is always zero because theair has no stiffness to resist motion. Mathematically, the downgoingwave has equal and opposite amplitude to the upgoing wave at the freesurface.

The pressure reflection coefficient is negative if the impedance ofthe incident layer is greater than that of the refracting layer, i.e.,c > c. Thus gas sands (which typically have lower velocitythan the overlying brine sand) have negative polarity reflections.

The free-surface reflection coefficient due to an upcoming waveis Rp = 1 because the rock impedance of the incident layer isgreater than that of air (impedance of air 0 so c > c = 0).Equation 2.21 says that the total pressure field value on the freesurface is P = 1 +R= 1 1 = 0! See Figure 2.8.

In a land experiment geophones record particle velocity of theground while a marine experiment records pressure with hydrophones.If P = 0 on the free surface then we must lower the hydrophonesseveral feet beneath the water surface, otherwise we record noth-ing.

The pressure transmission coefficient Tp is larger than 1 if the inci-dent medium has a lower impedance than the refracting medium,


i.e. c < c. For a given displacement, a really stiff mediumexerts a greater elastic pressure than a soft medium. Check outthe a href=fd1mov.mpgmovie/a of a 1-D propagating waveand see if the transmission amplitude changes and the polarity ofthe reflection changes according to the reflection and transmissionformulae derived above.

2.5.4 Reflection Coefficient for Particle Velocity

Marine experiments measure the pressure field, so this is why the hy-drophones must be sufficiently below the sea surface in order to measurea non-zero pressure. On the other hand, land experiments use geo-phones that measure the particle velocity. Typically, only the verticalcomponent of particle velocity is measured. The reflection coefficientfor particle velocity has a different form than that for pressure. To seethis, assume that the up and downgoing vertical particle-velocity fieldsin the top and bottom layers are given by

w+(z) = eikz +Rweikz,

w(z) = Tweikz. (2.28)

The boundary conditions at the interface are continuity of vertical-particle velocity w+|z=0 = w|z=0 and pressure w+/z|z=0 = w/z|z=0,where Hookes law is used w = P/z = c2P/z.

These two continuity conditions yield the following boundary con-ditions:

1 +Rw = Tw,

c(1Rw) = (1 + Rw)c, (2.29)

which can be solved for the particle velocity reflection and transmissioncoefficients:

Rw = (c c)/(c + c), (2.30)

Tw = 2c/(c + c), (2.31)


where the unprimed variables again refer to the medium of the incidentwave, and the subscript w denotes vertical-particle velocity. Note thatthe reflection coefficient above will have opposite polarity comparedto the pressure reflection coefficients. Also, note that in some casesthe transmitted amplitude can be greater than the amplitude of theincident wave! Does this violate conservation of energy? No, energy isthe squared modulus of amplitude scaled by the impedance (see latersection). Thus, a weaker medium with weak rocks (small impedance)can have transmit larger amplitude waves than the incident waves in amuch stronger (larger impedance) medium. It takes much more energyto rapidly displace strong rock 1 mm than it does in a weak rock.

Free-Surface Reflection Coefficent. The particle velocity reflectioncoefficient is equal to +1 at the free surface, so the total particle velocityfield at the free surface is 1 + Rw = 2. Thus the free surface, becauseit straddles a zero stiffness medium, can oscillate with great vigor andhas the largest amplitude compared to the underlying rock motion ina half-space. When an earthquake hits, dig a hole, jump in, and coveryourself with dirt! See Figure 2.9.

2.6 Oblique Incidence Angles, Reflection

Coefficients and Snells Law

If the incidence angle of the plane wave is non-zero (measured w/r tothe normal) then the normal derivatives in equation 2.24 bring downkz rather than k. Therefore, equation 2.24 becomes

(kz/)(1Rp) = (kz/)Tp,(2.32)

or kz = 2picos()/ and kz = 2picos

/

(cos/())(1Rp) = (cos/())Tp, (2.33)and because = c/f we have

(cos/(c))(1Rp) = (cos/(c))Tp. (2.34)

2.6. OBLIQUE INCIDENCE ANGLES, REFLECTION COEFFICIENTS AND SNELLS LAW17

Solving for the reflection coefficient yields

Ru = (cosc cosc)/(cosc + cosc), (2.35)

If the horizontal wavenumbers in the upper (kx = sin/c) andlower medium (kx = sin/c

) are equated, kx = kx, then this impliesSnells law:

sin /c = sin/c. (2.36)

This means that transmission rays bend across an interface, bending to-wards the vertical when entering a slower velocity medium and bendingtowards the horizontal when entering a faster medium (see Figure 2.10).At the critical incidence angle crit the refraction angle of the transmit-ted ray is 90 degrees so that Snells law says crit = arcsin(c/c) ifc > c. This gives rise to refraction head waves that propagate parallelto the interface at the velocity c of the underlying medium.

A consequence of Snells law is that a medium with a velocity thatincreases linearly with depth always turns a downgoing ray back to-wards the surface, as shown in Figure 2.10. This can easily be shownby approximating the linear velocity gradient medium with a stack ofthinly-spaced layers, each with a homogeneous velocity that slightly in-creases with depth. The velocity increase is the same across each layer.Applying Snells law to a downgoing ray shows that each ray trans-mitted across an interface bends a little bit more towards the horizontluntil it goes back up. As the thickness of eash layer decreases, the raytrajectory will be the arc of a circle.

Why did we equate the upper and lower medium wavenumbers? Wedid this because the apparent horizontal velocities along the interfacemust be equal to one another (i.e., vx = /kx = vx = /kx). The hor-izontal velocities of the incident and transmitted waves must be equalat the interface otherwise they would break away from one another andnot be coupled at the interface.

How does the formula for transmission coefficient in equation 2.31change for an oblique incidence angle?


e-ikz e-ikz

xz

Upgoing Downgoing Total Field

Anti-Node

eikz eikz+ +

Vertical Displacement at the Free Surface

Figure 2.9: The vertical displacement at the free surface is maximumbecause no elastic resistance at z = 0 means vigorous ground shak-ing. Mathematically, the downgoing wave has equal amplitude to theupgoing wave at the free surface.

V2

1V

V3

V4

Linear Increasing Velocity V(z) MediumV(z+dz) > V(z)

dz

sin = sin V3 V4

Figure 2.10: Downgoing rays bend across an interface towards thehorizontal if the velocity increases with depth. For a medium wherec(z) = a + bz, all downgoing rays eventually bend back towards thesurface.

2.7. ENERGY OF PROPAGATING ACOUSTIC WAVES 19

2.7 Energy of Propagating Acoustic Waves

Elastic energy is stored in a cube of acoustic material as it is deformedfrom equilibrium. That is, squeeze a cube of acoustic material, releaseit, and then the cube moves to perform work on the medium. In thedeformed state the potential to perform work takes the form of elasticpotential energy. Figure 2.11 shows that the work (i.e., area force/area distance cube moves) performed by the surrounding medium on acube along the z axis is given by (Pdxdy)dz, where the limits ofintegration are from the undeformed volume to the deformed volumeat some given time. This figure shows that, using equation 2.3 and 2.2and = c2, the expression for instantaneous potential energy densityis given by

PE = P 2/(2c2). (2.37)

Therefore the total instantaneous energy density of an acoustic wavepropagating along the x-axis in a homogeneous medium is given by

= E/V = 1/2[u2 + P 2/(c2)], (2.38)

where the kinetic energy is given by the second to the last term andthe potential energy is given by the last term. For a harmonic planewave c = P/u this equation becomes:

= u2. (2.39)

As one might expect, it takes more energy to move denser rock withthe same particle velocity as moving lighter rock.

2.8 Elastic Wave Equation

If the medium has non-zero shear strength then there can be shearstrains supported by the rock. This means that the shape of a cubecan be distorted into a, e.g., trapezoidal-like shape after applicationof a shear stress on the cube. Although most of our treatment ofexploration seismology will assume the acoustic approximation, we will


dxdy

dz c2

c2 c2

Instantaneous Potential Energy Density

dV = - V dP

Hookes Law

P dx dy ]dz = - P dV = V P dP = 1/2 P V-undeformed

deformed

[

undeformed

deformed

2

Figure 2.11: The potential energy of a cube of deformed material alongthe z-axis.

need to acknowledge the underling physics of elastic wave propagationwhen dealing with surface waves and shear waves in our data.

The notation for the shear strain is ij = 1/2(ui/xj + uj/xi),which says that a shear strain exists if there is a non-zero gradientof displacement that is perpendicular to the direction of displacement.We can also use Einstein index notation so that ij = 1/2(ui,j + uj,i),where the index follwoing a comma indicates a partial derivative withrespect to that indiexs coordinate. Figure ?? illustrates the idea of ashear strain that distorts the shape of a cube.

Hookes Law for elastic strain in an isotropic medium Aki and Rci-hards, 1980) is given by

ij = ijkk + 2ij, (2.40)

where ij is the stress tensor thgat denotes the force/area imposed bythe outside medium in the ith direction along the face with normal j.Also, is the shear modulus and is Lames constant. A consequenceof a medium supporting shear strain is that the shear stresses can benon-zero.

Newtons law is given by

ui = i1/x1 + i2/x2 + i3/x3 = ij,j, (2.41)

2.8. ELASTIC WAVE EQUATION 21

where Eintein notation says that repeated indicies indicate summationover all three components. Inserting equation 2.40 into equation 2.41gives the elastic wave equation in terms of components only:

ui = (+ )jj,i + ui,jj (2.42)

or in vector notation:

u = c2p( u) c2s ( u), (2.43)where the compressional velocity is given by cp =

+ 2 and the

shear velocity is given by cs =.

For a homogeneous medium we can decompose u into a sum ofcompressional and shear potentials u = + , and plug itinto equation 2.43 to get the wave equation for a compressional wavepropagating at velocity cp:

= c2p2 (2.44)and a shear wave propagating at velocity cs:

= c2s2 (2.45)Here we use the identities = 0, phi = 0, and 2 =( ) ( ).

2.8.1 Reflection Coefficients

Figure 2.12 depicts the plane wave impinging upon a horizontal elasticinterface. Here, there are 5 different wave types to consider becausea shear wave can be generated at the interface. The shear wave hasparticle motion that is perpedicular to the propagation direction whilethe compressional components is parallel to the direction of propaga-tion. Imposing continuity of vertical and horizontal particle velocityand normal and shear stress tensors provide four equations of con-straint. Similar to the acoustic case, we can solve these four equationsfor the unknown amplitudes PP, PS, PS , andPP . The PP reflectioncoeficcient at the free surface is given in Aki and Richards (1980):

PP =44p2cos(i)cos(j)/() (1 22p2)244p2cos(i)cos(j)/() + (1 22p2)2 (2.46)


PPP

PP

PS

PS

Plane Wave Reflection Coefficients (4 unknowns)

Figure 2.12: Plane wave impinging upon a horizontal interface.

where , , and indicate P-wave velocity, S-wave velocity and densityrespectively.

2.8.2 Small Angle Approximation to Reflection Co-efficients

For a two layered medium separated by a horizontal interface the PPreflection coefficient for small jumps in the medium parameters is givenby (Aki and Richards, 1980, p. 153):

PP = .5(1 42p2)/+ 2 cos2 i

42p2

, (2.47)

where = 2 1, = 2 1, and = 2 1. here the1 subscript indicates the upper medium where the plane P-wave isincident and 2 indicates the lower medium.

Further manipulation of this equation by defining the Poisson fac-tors

= 1 2 ; 2 = 2(1 2)/(2[1 ]), = (1 + 2)/2 ; = (1 + 2)/2; (2.48)

2.8. ELASTIC WAVE EQUATION 23

gives

R() = A+B sin2 + C(tan2 sin2 ) (2.49)where

A = R0;R0 1/2(/ +/);B = B0 + (1 2)R0 ; (2.50)

B0 = C 2(1 + C)1 21 ; C =

/

/ +/; (2.51)

The first angle-dependent term in equation 2.49 significantly contrubutesfor 0 < < 30 degrees, while the second starts to significantly con-tribute for > 30 degrees.

For < 30 degrees, geophysicists will use the small angle approxi-mation to equation 2.49:

R() A+B sin2 , (2.52)and plot up crossplot curves (Foster et al., 1997) to assess geology. Forexample, Figure 2.13 depicts the crossplot of A and B pairs taken froma well log (i.e., estimate density and P- and S-wave velocities from soniclog). It shows a linear trend, and the idea is that any deviations fromthis trend represent a significant change of geology such as oil or gasbearing rocks. The departures can be estimated by finding A and Bpairs from the R() vs curves estimated from the seismic reflectionamplitudes along a horizon of interest.

Another measure of hydrocarbon anomalies can be estimated byfinding the linearization of the fluid factor (Shuey, 1985):

Rf () = R0 + 1/2/tan2, (2.53)

where tan2 sin2 for small angles. This factor is used in the casehistory described in the next chapter.

Details for implementing this AVO (i.e., Amplitude vs Offset) pro-cedure are non-trivial because much data processing must be performedbefore the A and B pairs can be picked. Nevertheless, significant oiland gas deposits have been discovered by the AVO method.


Figure 2.13: Crossplot of slope (B) and intercept (A) pairs using welllog data from the North Sea. The dashed line corresponds to the fluidline for / = 1.9. The A and B pairs tend to lie on a trend that isconsistent with the fluid line. Note, the fluid line assumes a constantdensity, the actual density profile is used to compute A and B.

Bibliography

[1] Aki, K., and Richards, P., 1980, Quantitative seismology: Theoryand Methods: Freeman Co., NY, NY.

[2] Foster, D., Keys, R., and Reilly, J., 1997, Another perspective onAVO crossplotting: The Leading Edge, September issue.

[3] Kinsler, L. and Frey, A., 1961, Fundamentals of Acoustics, . ley andSons, NY, NY.

[4] Shuey, R.T., 1985, Application of the Zoeppritz equations: Geo-physics, 609-614.

2.9 Problems

1. Identify the direct arrival, air wave, surface waves, refraction ar-rivals, and reflection arrivals in the CSG shown in Figure 2.14.Estimate the apparent velocity in the x-direction Vx and the as-sociated period for each event. From these calculations determinethe wavelengths. Show work.

2. Which arrivals have the same apparent velocity as the actualpropagation velocity of that event? Why?

3. The 1-D SH wave equation is the same form as the 1-D acousticwave equation, except c becomes the shear wave velocity, P be-comes the y-component of displacement v, c=sqrt(mu/rho) wheremu is the shear modulus, and the SH wave equation is

1/c22v/t2 2v/z2 = 0 (2.54)

25

26 BIBLIOGRAPHY

Figure 2.14: Shot gather from salt lake valley. The trace interval is 5feet along the horizontal axis and the time units along the vertical axisare seconds.

SH (or shear horizontal) refers to the fact that the shear waveparticle motion is perpendicular to the direction of particle mo-tion, and is along the horizontal direction (in and out of planeof paper). The SH continuity conditions at the interface at z=0are a). Continuity of y-displacement v+ = v., b). Continuityof shear traction: v/z+ = v/z, where is the shearmodulus.

Derive the y-displacement reflection and transmission coefficientsfor a plane SH wave normally incident on a planar interface in anelastic medium.

physics of acoustic wave propagation

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