Download - The influence of surface inhomogeneities on deep electromagnetic soundings of the Earth

Geophys. J. R. astr. SOC. (1987) 90,61-73

The influence of surface inhomogeneities on deep electromagnetic soundings of the Earth

E. B . Fainberg and B . Sh. Singer Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation (IZMIRAN), USSR Academy of Sciences, 142092 Troitsk, Moscow region, USSR

Accepted 1986 November 17. Received 1986 November 17; in original form 1985 April 3

Summary. Asymptotic expressions for components of the electromagnetic field of a grounded electric dipole are considered for the model consisting of a thin surface-layer overlapping a stratified medium with a highly resistive screen on the roof. It is shown that the method of spatial derivatives makes it possible to obtain proper estimates of the impedance at distances of r , lhol from the nearest edge of the surface anomaly ( IXo I being the effective depth of the field penetration in the underlying section). The magnetotelluric methods allow one to obtain the true values of impedance, provided r s max {lhol, lF/(S-' + Z, 11'2} where S is the integrated conductivity of the surface layer, F i s the transverse resistance of the screen, and& is the Tikhonov-Cagniard impedance for the medium underlying the surface layer.

Key words: magnetotellurics, electrical dipole, subsurface inhomogeneities, distortions

Introduction

Difficulties of interpretation of findings yielded by deep electromagnetic-exploration, associated with inhomogeneities of the surface layer, are well known. The difficulties are of a common nature both for exploration with use of electromagnetic fields of natural originf' and for exploration with fields excited by controlled sources of current. To overcome such difficulties, use should be made of electromagnetic field modelling. Still, it should be noted that despite a high level of development of physical (Moroz, Kobzeva & Timoshin 1975) and numerical (Vasseur & Weidelt 1977; McKirdy, Weaver & Dawson 1985; Singer e t af. 1984) modelling, it is not readily accessible and flexible instrument of research. In con- nection with the above, the question of how far one should go away from any inhomogeneity in order to obtain information on the Earth's deep structure, by any method based on a simple laterally uniform model, gains essential significance. In order t o 'answer this question, the law governing attenuation of anomalous electromagnetic field components with distance from a surface inhomogeneity should be elucidated. One should bear in mind that the distance for which the anomaly distorts the electromagnetic field is dependent on

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both the law of change of the anomalous field and the magnitude of the anomaly. In cases when the anomalous field is small compared to the normal one (weak inhomogeneity), sounding may be performed even above the anomaly itself without essential errors. On the other hand, such vast anomalies as seas, oceans, mountain massifs of regional extension, large active faults, etc., cause a high distortion of electromagnetic fields, but at sufficiently great distances, the anomalous fields created by them have a structure that allows a proper determination of parameters of the medium. Thus, the question posed should be formulated in the following way: under what conditions will a reliable estimate of the impedance be obtained by the magnetotelluric method, the method of spatial derivatives, and other methods, if sounding is performed in the total field created by both the external source and the anomaly of surface conductance. As regards the problem of determination of the anomalous field values, it can, with a few exceptions, be solved only with the help of modelling.

Determination of the law of decrease of the anomalous electromagnetic field is also important for the methods of numerical modelling, as the numerical grid may be restricted at distances greater than a typical scale of an anomalous field decrease.

An integral equation (Vasseur & Weidelt 1977) for determining tangential components of electric field is:

E. B. Fainberg and B. Sh. Singer

E,(r,O)=Er(r,O)+ 6(r -r')SaET(r')ds'. .!4

This equation is derived for the model consisting of a thin surface layer with conductance S, which differs from the constant value S, only in an anomalous area A ; S, = S - S, is the value of the anomalous conductance. The inhomogeneous surface layer is underlain by a laterally uniform section; C(r) is the Green's tensor of the normal section. Once the electric field is known, components of the magnetic field may be found with the help of the integral relation

H(r,O -)=H"(r,O-)+ dH(r -r'))S,E,(r')ds'.

The term E; of equation ( 1 ) represents the electric field excited in the normal model, i.e. in the model that does not contain lateral inhomogeneities; Hf is the associated normal magnetic field, the components of the Green's tensor functions d(r) and d~(r ) represent components of the field created by a grounded electric dipole. Therefore, in order lo reveal the law of anomalous field decay beyond the inhomogeneity it is necessary to study the asymptotic behaviour of a field of such a dipole at a great distance. A review on 'the adjustment distance' problem has recently been published by Jones (1983). The analytic solution for the B-polarization case has been found by Dawson, Weaver & Raval(l982).

1 Electromagnetic fields and impedances in the stratified medium containing a resistive screen

The problem of computation of a field component excited by a horizontal electric dipole with a moment p exd- ju t ) , lying on the surface of a laterally uniform section is one of the classical problems of electrical prospecting. Thus, Vanyan (1965) has shown that in the absence of intermediate non-conducting screens in the section, tangential components of the electric and magnetic fields decrease at a great distance from the source according to the law l/?, and the vertical component of the magnetic field according to the law l/r4. As revealed elsewhere (Vanyan 1959; Dmitriev, Skugarevskaya & Fedorova 1970): the presence

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of non-conducting screens in the section leads to another law E,- l/r2. The question of which of the laws (namely l / r z or l/r3) operates in practice is of essential significance for geoelectrics. Assuming that the medium is such that at great distances from the dipole, the electric field will decrease as E,- I / ? , whereas H , - l /r3, it is likely that the MTS results a t considerable distances from inhomogeneities will be distorted. However, in practice, geoelectric sections do not include layers, being ideal insulators. Therefore, the asymptotic behaviour of the dipole field in a medium with a resistive screen is of central importance. The model to be considered below consists of a uniform surface layer underlain by a resistive layer of thickness h and a conductivity uo(z) that is dependent (generally) upon depth. At depths where z > h, conductivity varies in accordance with the law u(z), where u% uo, referring to the typical values of conductivity at depths z > h and 0 < z < h, respectively (Fig. 1).

Figure 1. The lateraily uniform model excited by a grounded electrical dipole.

The electromagnetic field created by a grounded electric dipole at the surface of a stratified medium (i.e. z = 0 -) may be expressed in the form (Appendix A):

I d d

r dr dr r

Equations ( 3 ) and (4) determine the horizontal components of electric and magnetic fields, the last equations determining the vertical component of a magnetic field and the density of leakage currents. Here r is the vector from the dipole to the observation point, e , = r/r, e9 = n x e,, where n is unit vector of normal to the surface of the Earth being directed upwards, p = prer i- p9eq, p r = p * cos cp, pq = -p * sin 9, cp is the angle between electrical

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dipole and observation point directions, S is the conductance of the thin surface layer. Functions Q, are defined by

Q,(r> = 1- qv(k)J1 (kr) dk / (2n) , v = 1 , 2 , 3 , 4 ,

E. B. Fainberg and B. Sh Singer

0

where J1 ( z ) is the first order Bessel function,

and

q 4 ( k ) = [ l t SZg(O)]-’ Here

2kXk(0)

1 + khk(0) ’ a k =

(9)

the value &(o) determines the spectral impedance of the underlying medium for the inductive (toroidal) mode:

ZL(0) = -iapOhk(o). (10)

The absolute value IXk(0)J is an effective penetration depth of inductive mode for the spatial harmonic exp (- ikr), ZZ(0) is the spectral impedance of underlying section for the galvanic (poloidal) mode of the field.

A conventional method of calculation of asymptotic expressions for functions (7) consists of the following. In integrals (7) values of k < l/r are essential. Accordingly, if r is great it is possible to substitute factors before Bessel’s function in the integrals with a power series in k . It is well known that field behaviour within a uniform layer is controlled by the value (k2 - iwpoa)”’. Hence, the above expansion is legitimate for such great distances as k - l / r < ( ~ p ~ a ) ” ~ is of interest for. We shall assume that this condition is satisfied’ for all layers of the section except for the screen, since for the latter the parameter ( W ~ ~ O ~ ) ’ / ~ is small. Accordingly, we shall suggest that

lh~(h) l (a) i.e. the distance from the observation point to the source is much greater than the depth of field penetration into the section underlying the screen (for k = 0). The presence of the screen affects the behaviour of the inductive and galvanic modes of the field quite differently. Thus, with the assumptions that

r > h (b) and that the screen conductivity is limited by the condition I ~ ~ l h e 1 (where K ~ =

(- i ~ p ~ u ~ ) ~ ’ ’ ) , it is shown in Appendix B that for the inductive mode

hk(0) X ho = h t- ho(h) ( 1 1)

[ho will be used below instead of hO(0)]. Thereby, both limitations (a) and (b) for the required distance from the source could be replaced by r * I hol, where I hdl is the effective

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depth of penetration of the inductive mode into that part of the section lying beneath the thin surface layer. With the same restrictions, the following expression for the spectral impedance of the galvanic mode is obtained:

Zg,(O) =z; +Yk2, (12) where Z , is Tikhonov-Cagniard's impedance of the underlying section, while

.F= [ o ; ' ( z ) d z i - ~ * (13)

is the effective transverse resistance of the underlying section. F* designates the contribution to the transverse resistance due to the section lying beneath the resistive layer ( z > h). As a result, the following expressions for integrand in (7) are obtained

ak 2kXo( 1 - kho), (14)

X O AJ -1

2k

where Xi1 = -iwpoS and A;' =hi' i- A;'. The value lXs l determines a characteristic size associated with the horizontal skin-effect; !Ao I is an effective depth of field penetration into the section including the surface thin layer. This value is closely connected with Tikhonov-Cagniard's impedance 2, = -iwpoAo on the Earth's surface. Expansion (14) is valid, provided that kl Xo I e 1, and expansion ( 1 5 ) required the condition that IkXoAo/XsI< 1 , which, as it is evident, is less strict than klhol< 1. Indeed, in the high- frequency range (the S-interval) l X o l % lhsl and, accordingly, A. = As. So the restriction klhoAo/XsIe 1 leads to klXol< 1. In the low-frequency range (the h-interval) Ihol< IXsl, A. = A o andsoklhoAo/XSI =klhoI- Iho/Xsle 1 iscertainly fulfilled ifklXoIe 1 .

Integrals appearing after substitution of (14), (15) in (7), (8) can be calculated if use is made of the known expressions

[Jl(x)dx = 1,

X2m+i J , ( x ) dx = (-1)"(2tn + 1)

Integrals included therein should be considered as the limit at E + Oi- of integrals containing an additional factor exp(-ek) in the integrand. As a result, asymptotic expressions for Q , , Q2 and Q3 are as follows

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which are valid at r , Iho I. Substitution of ( 1 2 ) into (9) leads to an integral which can be calculated with the help of the known expression


where r ( z ) is Euler's gamma function, K,(z) is McDonald's function of order v. Thus function Q4 has the asymptotic form

where

hL = (S.EAo/Xo)l/z = [ .E/(S-' + 2;)

designates a parameter that controls the effects of current leakage via the resistive layer. Within the h-interval A. = ho, and hL = (,§a1/' (Ranganayaki & Madden 1980). Within the S-interval hL = (F/Z;) ' /2 - ~;'(h/h,,)'/* and X L doesn't depend upon S. Substitution of (18)-(20) and (22) into (3)-(6) makes it possible to obtain asymptotic expressions for the field components of a horizontal electric dipole, which are valid at distances r > I ho I from the source:

Thus, at distances r , Ihol from the source, tangential and vertical components of the magnetic field decrease in accordance with the laws l / r 3 and l/r4, respectively. At the same time the electric field behaviour at the distance r s lhol is additionally dependent on the ratio between r and hL. In the case of r lhLl contributions of terms containing McDonald's function are exponentially small and expression (23) leads to a l / r 3 dependence of the tangential component of the electrical field upon distance. This is evident if the 'function K 1 ( 2 ) is replaced by its asymptotic representation at large values of the argument:

At small values of z, K l ( z ) = z-'; hence, within the distance range I hol Q r Q I XL I , the behaviour of electric field components is substantially changed to the form

A0 2nE, = S-' - (prer - p,e,) P .

A0

Expressions ( 2 3 ) and (24 ) make it possible to evaluate relationships between mutually orthogonal components of the electric and magnetic fields on the Earth's surface

EJH, = -Zo [ l - (SZoXoh~) - ' r 3 K ; ( r / X L ) ] , (28 )

E,/Hr = Z o [l -t. (2SZohohL)-' r z K l ( r / l q , ) ] .

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Surjace inhomogeneities and deep electromagnetic soundings 67

It is helpful t o obtain similar ratios for field components on the t o p of the section underlying the surface layer. The latter may be useful, for instance, for ocean bot tom MT- soundings or soundings with the help of vertical gradient methods. The magnetic field H; beneath the surface layer is calculated from (23) and (24) using ratios (A7) and (A8), so

An estimate o f the impedance can also be made with the help o f the spatial derivatives method (Berdichevsky, Vanyan & Fainberg 1969). Making use of (24), (25) we obtain

Zo = -iwpoHz/ V. H,. (30)

Thus, the impedance may be determined by the spatial derivative method at distances r s lhol from the source in case the field is excited by the dipole source, whereas the magnetotelluric method requires the additional restriction r B I XL I t o be satisfied.

2 Impact of surface inhomogeneities

The magnetotelluric method is based upon determination of the impedance Zo from the ratio E, =Zo * n x H,, which is valid for the laterally uniform medium excited by a plane wave. Similar t o the magnetotelluric method are vertical gradient methods with vertically separated measurements o f tangential components of the magnetic field. The method of spatial derivatives is based on the use of the vertical component of the magnetic field and its vertical derivative. The impedance is in practice calculated from the ratio -iwpoHz = 2, V. H,. This method is valid as long as the spectral impedance remains practically unchanged in the spatial frequencies domain covering the major part of the magnetic field energy. Since the magnetic field observed on the Earth's surface belongs to the inductive mode, whereas both modes make contributions t o the telluric field, it is likely that the method of spatial derivatives will be less sensitive t o distorting effects of geoelectric inhomogeneities than the magnetotelluric one.

The electromagnetic field in the presence o f surface inhomogeneities may, be determined from equations ( I ) , (2). The Green's functions included in these equations 6(r - r') and GH(r - r?, as was already mentioned, have a simple meaning: if the source of a field is a surface electric-dipole jse(r) = ep(r - r'), i = 1 , 2 located a t point r ' , then the fields created by it at the point r will be equal t o ,

E, = e l GIi(r - r ') + e2 C2'(r - r ')

and

H=elG:(r -r ' )+e2G$(r -r')+e3G$(r -r ' ) ,

where e l , e2 (horizontal) and e 3 = -n (vertical) are the unit vectors o f the Cartesian coordinate system.

In the case the distance o f the observation point r from the anomalous area substantially exceeds its geometric size, the Green's function can be taken out of the integrals in ( I ) , (2). In this case the contribution of the anomaly t o the field observed at point r coincides with

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the field of an equivalent electric dipole having a moment

E. B, Fainberg and B. Sh. Singer

P = s, SaE, ds

and located within the anomalous area. If the size of the anomalous area is comparable with the distance to the observation point, or exceeds it, then the areaA can be subdivided into parts Ai(A = UiAi), which are small compared with the distance to the observation point. Consequently, the anomalous field at the point r may be considered as the field of the set of electric dipoles

Pi =S,: E, ds

located in points ri within Ai, i.e.

E,W = EXr) + 1 Ef(r),

H(r) = H"(r) t. 1 Hi(r),

(31)

(32)

i

i where subscript i denotes the contribution of the dipole (pi, ri}. We assume that the fields E:(r) and H"(r) allow the determination of the impedance Z , in accordance with the formulas to be used in magnetotelluric method and the method of spatial derivatives. Now, we shall try to answer the question of how far one should get away from the edge of the anomaly to calculate properly Zo basing on total fields (31), (32). It is not difficult to answer this question bearing in mind that the ith member of the sum is determined by formulas differing from (23)-(25) by the substitution of pi for p and r - ri for r. Therefore, if the distance from the nearest edge of the anomaly is rmin > Ihol, the formula -iwpoHz = Z , V. H, is valid for each member of the sum, hence, the whole sum satisfies this formula. Analogous considerations show that at a distance of r,in > max { I ho I, I hLI } from the nearest edge of the anomaly, the impedance is correctly evaluated with the help of the magnetotelluric method.

Thus, the method of derivatives poses, generally speaking, less strict constraints upon the geoelectric structure of the region in which it can be applied, since it yields the true results at distances exceeding lXo l from the nearest edge of anomaly, whereas the use of the magnetotelluric method requires, not only rmin > Ihol, but also rmin s I h ~ l . Expressions (29) show that the same statement is valid for determinations of the impedance bkneath the surface layer (bottom of the ocean) as well.

The ratio between scales IXo I and lhLl is determined by the geoelectric section. Thus, for crystalline foundations of shields it is feasible to adopt 9- 10"Ohm m2 (Jamaletdinov 1982). At S - 100--1000 S and in h-frequenty interal hL - 1000-3000 km, IhLJ greatly exceeds the value Iho 1 5 400 km. This result makes clear the wide spreading of impedance distortions due to the surface inhomogeneities' influence. For oceanic areas hL - 450 km, assuming that 9- 1 O7 Ohm - m2 (Cox 1980) and S - 2 - 1 O4 S.

If the values of transverse resistance of the Earth's crust assumed herein do not differ much from the actual ones, then magnetotelluric observations at the ocean bottom seem to be more promising compared with those made on continents.

Another advantage of the method of derivatives is that impedance determinations are independent of both intensity of the source and its geometry and orientation. This advantage seems to be quite essential for deep electromagnetic soundings with controlled sources, such as the experiment 'Khibini' in the Kola peninsula, where the geometry of the effective source is unknown.

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Surface inhomogeneities and deep electromagnetic soundings

Acknowledgments

We are grateful to Drs T. W. Dawson and P. Weidelt for help in preparing this paper.

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References

Berdichevsky, M. N., Vanyan, L. L. & Fainberg, E. B., 1969. Magnetovariational sounding according to spatial derivatives of a field, Geomagn. Aeron., 9, 369-371 (in Russian).

Cox, C. S . , 1980. Electromagnetic induction in the oceans and induction inference on the constitution of the Earth, Geophys. Surveys, 4,137-156.

Dawson, T. W., Weaver, J. T. & Raval, V., 1982. B-polarization induction in two generalized thin sheets at the surface of a conducting half-space, Geophys. J. R. astr. Soc., 69, 209-234.

Dmitriev, V. I., Skugarevskaya, 0. A. & Fedorova, E. I., 1970. On high frequency asymptotics of electromagnetic field in the stratified medium. I zv . Akad. Nauk. SSSR, Physics of the Earth, 2,44-51.

Jamaletdinov, A. A., 1982. Normal electrical section of crystalline foundation and its geothermal interpretation according to the data of MHD-soundings in the Kola peninsula, in Deep electromagnetic soundings with the use of pulse MHD-generators, pp. 35-48, Apatites, (in Russian).

Jones, A. G., 1983. The problem of current channelling: a critical review, Geophys. Surveys, 6,79-122. McKirdy, D. McA., Weaver, J . T. & Dawson, T. W., 1985. Induction in a thin sheet of variable conduct-

ance at the surface of a stratified earth - 11. Three-dimensional theory, Geophys. J. R. astr. Soc.,

Moroz, I. P., Kobzeva, V. M. & Timoshin, B. V., 1975. Modelling of electrodynamic processes in the inhomogeneous conducting medium. Kiev, (in Russian).

Price, A. T., 1949. The induction of electric currents in nonuniform thin sheets and shells, Q. J. Mech. appl. Math., 2, 283-318.

Ranganayaki, R. P. & Madden, T. R., 1980. Generalized thin sheet analysis in magnetotellurics: an extension of Price's analysis, Geophys. J. R . astr. Soc., 60,445-457.

Singer, B.Sh. & Fainberg, E. B., 1985. Electromagnetic Induction in Non-uniform Thin Sheet and Shells, (Chapter 2, 3) Moscow, (in Russian).

Singer, B.Sh., Dubrovsky, V. G., Fainberg, E. B., Berdichevsky, M. N. & Ilamanov, K., 1984. Quasi three- dimensional modelling of magnetotelluric fields in the South-Turanian platform and in the South- Caspian magadepression, I zv . Akad. Nauk. SSSR, Physics of the Earth, 1,69-81.

Vanyan, L. L., 1959. Some problems of the theory on the frequency soundings of horizontal stratifi- cations, Prikladnaja Geophysika, 23, p. 3-45, (in Russian).

Vanyan, L. L., 1965. Basics of Electromagnetic Soundings, Moscow, (in Russian). Vasseur, G . & Weidelt, P., 1977. Bimodal electromagnetic induction in non-uniform thin sheets with an

application to the northern Pyrenean induction anomaly, Geophys. J. R. astr. Soc., 51,669-690.

80,177-194.

Appendix A

T H E F I E L D O F A S U R F A C E E L E C T R I C A L D I P O L E

The electromagnetic field in a laterally homogeneous medium is expressed via two scalar functions V(r, z ) and W(r, z) using the following relationships (Vasseur & Weidelt 1977)

E, = i w p , , n x ~ , ~ - V, u-l - , ( 3 av az

H, = V, -- n xV,W,

Ez = a-'V W,

Hz = - V t V ,

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where E,, H, are the tangential components of the electric and magnetic fields; E,, H, are the corresponding vertical components; a(z) is the conductivity of the medium which depends only upon the depth z ; n is the unit vector of external normal to the surface of the Earth, and V, is the operator of spatial differentiation wrt horizontal radius-vector r; time factor is exp (-iot). The potential V(r, z ) of the inductive (toroidal) mode and the potential W(r, z) of the galvanic (poloidal) mode satisfies the equations

E. B. FainbergandB. Sh. Singer

V' V + iwl.coo(z) V = 0,

V * [a' (Z)VW ] + iwpo w = 0

(A51

(A6)

and zero boundary conditions at t ++m. In a non-conducting atmosphere (z<O) W(r, z) = 0. As a consequence, the magnetic field is expressed only via the potential V(r, z).

The tangential component E, of the electrical field and the vertical component of the magnetic field H, are continuous at the surface thin sheet (Fig. 1) while the tangential component of the magnetic field suffers a discontinuity

n X[HT(r, 0-) - H,(r, O+) ] = js, (A71

where the surface current density is

j S = S E T + j s e .

Here S(r) is the conductance of the surface layer, jse(r) is the density of the external surface current. From (A7) and (A2) follows the expression for the surface current density

js = - n xV,$ - V, W(r, O+), ('49)

where

aw aw +(r) = - (r, 0+) - - ( r, 0-1

a Z az

is a current function. (Al), (A8) and (A9) lead to the vector equation for functions $(r) and W(r, O+):

S-'(-nxV,J/ - V,W)= iwponxVTV- V, ( a-' - z) + S-ljse .

Simple operations reduce this to a system of two scalar equations (Singer & Fainberg 1985)

V . [~- ' (v ,+-nxV,W)] =-iol.coV:c/+ V; (S-'nxjse), ('41 1)

The first equation can be reduced to the well-known Price's equation (Price 1949) if one omits the second term in the rhs and the term with W(r, 0+) in the lhs. The latter corresponds to the absence of leakage currents. For a uniform surface layer, (A1 1 ) leads to two independent equations

V+($ + iopoSv) = V, (nx j*), (A12)

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It is not difficult to solve these equations, if one makes use of the relationships

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d - V(k, z ) = k V(k, z ) dz

d - V(k, z ) = - hi1 ( z ) V(k, z ) dz

if z < 0,

if z > 0

and d

dz u-l - W(k, z ) =- ZB,(z)W(k, z) if z > 0

between Fourier components V(k, z ) , W(k, z) of the potentials V(r, z ) , W(r, z ) and their derivatives with respect t o z. The value hk(z) determines the spectral impedance of the inductive mode

ZL(Z) = -iwpoh&) (A 16)

for the medium lying at depths greater than z, Z:(z) is the spectral impedance of the galvanic mode. The solutions of(A12), (A13) are

where integration is carried out along the Earth surface, R = r - TI, eR = R/R. Equations (AIO), (A14) leads also to expressions

V(r , 0) = LQ, (R ) (n x eR ) - jse (r ’) ds ’,

a Q3(R)(n xeR) * jse(r’) ds

Functions Q,(r), Y = 1 , 2 , 3 , 4 are defined via(7)-(9). Expressions (A17)-(A20) together with (Al)-(A4) leads to the expressions for the electromagnetic field components at the Earth surface. Replacement of the external current density jse with p6(r’) results in expressions (3)-(6) for the field of grounded electrical dipole with a moment equals to p.

Appendix B

I M P E D A N C E F U N C T I O N S O F A L A T E R A L L Y U N I F O R M SECTION C O N T A I N I N G A

R E S I S T I V E L A Y E R

We shall obtain an approximate expression for hk(0) under the assumption that the upper part of the underlying section contains a resistive screen of thickness h and conductivity uo(z) , beneath which lies a more conducting part of the section u(z), { u % uo }. Changes of the value Xk(z) within the range of depths 0 < z < h are described by the equation

d

dz - hk - [k2 -k K$(Z)] h2 = - 1,

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the solution of which should coincide with the value hk(h) at z = h. The modulus of the latter determines the effective penetration depth of the inductive mode into that part of the section lying beneath a resistive layer. At rather low conductivities of the layer, such as


I K o I h < 1, U o * U (B2)

and for such small values of k that

kh< 1 , klh,(h))* 1,

it is possible to omit the second term in the lhs of (BI). Indeed, Ihk(h)l < Iho(h)l - ( w l . ~ ~ u ) - ~ ’ ~ and, consequently I ~ ~ h ~ ( h ) l < ( U ~ / U ) ” ~ . It

should be noted that we do not suggest homogeneity of a resistive layer. The value uo or K~

included in the estimates determines only the order of the relevant values within the boundaries of the resistive layer. Thus, equation (BI) after having been simplified is reduced

.

to dhk/dZ = - 1 , hence

hk(0) = hk(h) h + 0 [K:(k, 2) A:],

Ki(k, Z ) = kZ + K i ( Z ) .

Bearing in mind that the dependence of hk(h) upon k is determined by the terms - [kh0(h>l2 one concludes that

hk(0) = ho(0) + 0 [K:(k, Z ) h i ] ,

h,(O) = h f A@).

(B4)

where

The impedance of the galvanic mode within a resistive layer satisfies the equation

d

dz - 2; - U o ( Z ) (Z;),>’ = -K:(k, Z ) / U o ( Z ) . (B5)

In order to determine the dependence of 2; upon k the following expansion should be substituted in (B5):

Z;(Z> = Z 0 ( z ) + f l ( z )k2 -I- f 2 ( z ) k 4 + . . . (B6)

Here the equality Zk(z) = 2 { ( z ) = Zo(z ) is used (Vasseur & Weidelt 1977). Comparison of the terms with the same power of k leads to a chain of equations

- 2U020(Z)fi = - U i ’ ( Z ) , d f 1

dz -

and so on. The solution of equation (B7) has the form

f l ( Z ) = [fm+ ~ ( S ) u ; l ( S ) ~ ~ ] / F ( z ) ,

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The exponent may be estimated as

in accordance with (A16). This value is small compared to unity since K~ complies with constrains (B2). Thus, F(z) = 1 and formula (B9) can be simplified to

fi ( z ) = fl ( h ) + 00' ( 5 ) d5'.

The value fi(h) represents the coefficient of k2 in the power series expansion of Zi (h) . Below it is designated a s F * . Thus,

Z8,(h) = Zo(h) + F * k Z t- . . . (B11)

It is not difficult to evaluate that .T* - Ao(h)/a. Thus, the coefficient of k2 in the series for Zi(0) is

yEfl(0)=.T* + a i ' ( z ) d z + O(K;hi).

The solution of (B8) has the form

I f 2 (h) - f 2 (0) = I," atf: ( z ) dz - h/ao - [h + Ao(h) a. /uI2.

Taking into account that f 2 ( h ) - Al(h)/a we obtain the series

Zi (0 ) =Z,(O) + Y k 2 + . . . The first omitted term is small compared withFk'.

by guest on Decem


Dow

nloaded from


As shown by Weidelt (2007), the Green's function of the TM-mode has an infinite number of polesin the complex k - plane.

By restricting our consideration by the range of small wavenumbers, we essentially considered thepole, which is closest to the real k - axis. This procedure is justified only if this pole issignificantly closer to the k - axis the the other poles. In terms of the attached publication, thisimposes a restriction ∣λL∣≫∣λ0∣ . Considering that at low frequencies ∣λL∣∼∣ω∣

−1 /4 growsslower than ∣λ0∣∼∣ω∣

−1/2 , the frequency range where the results are valid is limited at lowfrequencies as well as at high ones.

This probably explains the discrepancy pointed out by Weaver and Dawson (1992).

Weaver, J. T. & T. W. Dawson, 1992, Adjustment distance in TM mode electromagnetic induction, Geophys. J. Int., 108, 293 - 300.

Weidelt, P., 2007, Guided waves in marine CSEM, Geophys. J. Int., 171, 153 - 176.

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