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GEOPHYSICS, VOL. 67, NO. 2 (MARCH-APRIL 2002); P. 610617, 8 FIGS.10.1190/1.1468622

The boundary element method for 3-D dc resistivitymodeling in layered earth

Qinzhong Ma

ABSTRACT

The integral equation for dc resistivity modeling of

3-D inhomogeneous bodies buried in a layered earth isderived by using Greens theorem. The main features ofthis method are (1) the layers above and below the 3-Dobject can be included, (2) multiple subsurface inhomo-geneous bodies can be embedded in the different layers,and (3) the boundary element method (BEM) is used tosolve the integral equation using triangular surface ele-ments. Linear variation of the electrical properties is as-sumed withineach element. The potentialon thegroundsurface is obtained by solving thelinear equation systemwith Gaussian elimination.

Model calculations demonstrate that the results ob-tained by this method compare well with the analyti-cal solution of a sphere in a uniform half-space and theasymptotic behaviorfor thesolutionof a buriedbodybe-

neath a surficial layer as the layer resistivity approachesthat of the half-space. A comparison of responses overelongate 3-D bodies with responses over 2-D bod-ies of identical cross-section also shows satisfactoryagreement.

INTRODUCTION

In the dc resistivity method, the development of numeri-cal methods and computer techniques allow forward modelingof subsurface 3-D resistivity structures. Usually the problemsrelyonly on numericalmethods, whichincludefinite-differencemethods (Dey and Morrison, 1979; Zhao and Yedlin, 1996);finite-element methods (Coggon, 1971; Fox et al., 1980; Zhouand Zhong, 1986); integral equation methods (Alfano, 1959;

Dieter et al., 1969); and boundary element methods (BEM)(Okabe, 1981; Xu et al., 1984,1988; Schulz, 1985; Xu andZhao,1985). Generallyspeaking,BEM isa special case of theintegralmethod,wherethenumberofunknownsinthesetofsimultane-

Manuscript received by the Editor August 10, 1998; revised manuscript received June 18, 2001.Formerly Lanzhou University, Geography Department, Lanzhou 730000, China; presently Seismological Bureau of Shanghai, No. 87 Lanxi Road,Putuo, Shanghai 200062, China. E-mail: mqz1234@sina.com.cn.c 2002 Society of Exploration Geophysicists. All rights reserved.

ousequations is usually very large. In BEM, thesolutioncan bereduced to a 2-D problem, reducing the number of gridpointsdramatically. Thus, BEM is simpler in element division and ini-

tial data preparation and requires less memoryparticularlyappropriate fordomains extendingto infinity (Adey andNiku,1985; Brebbia et al., 1985; Brebbia, 1988). BEM is especiallysuitable in the case of 3-D problems.

Many distributions of 3-D anomalous bodies under a flatsurface have been investigated with boundary integral tech-niques. Lee (1975) presents a surface integral equation suitedfor the model of a heterogeneity embedded in a two-layeredmedium; the solutions were found using the Galerkin method.Okabe(1981) gives the generalized integral equations describ-ingthe arbitraryinhomogeneitiesproblemwith weightedresid-ualscheme,whichcan becalculated byBEM. Schulz(1985)de-rives the integral equations for a body buried in a horizontallystratified half-space according to Okabes generalized formu-las, where in the numerical treatment the unknown is assumedto be constant on every boundary element.

This paper presents a derivation of the 3-D potential inte-gral equations in which multiple 3-D bodies can be embeddedin arbitrary layers within a layered earth. The derivation isstraightforward and comprehensive, being based on Greenstheorem. BEM (Brebbia, 1978) is used to solve the integralequation. The boundary of a 3-D body is divided into trian-gular elements. Linear variation of the potential is assumedwithin each element, and a Gaussian quadrature formula anddigital linear filtering techniques are used to calculate the in-tegral. Thus, the integral equation is converted into a set oflinear algebraic equations that can be solved by Gaussianelimination.

The apparent resistivity curves obtained by this methodagree with that obtained with approximate analytic solutions.The agreement is quite good. I also examined the asymptotic

behavior of the apparent resistivity curves for a buried spherebeneath a layer over a half-space; I find good agreement. Fi-nally, I compare the responses over elongate 3-D bodies withthose over 2-D bodies of identical cross-section.

610

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BEM for 3-D dc Modeling in Layered Earth 611

BOUNDARY INTEGRAL EQUATION FORMULATION

Boundary value problem

Consider the configuration shown in Figure 1. The x- andy-axes are horizontal, and the z-axis is downward with the ori-gin on the ground surface. An I-A current source is impressed

at point A (the originof thecoordinate system). The earthis ann-layeredhostinwhichthereisa3-Dbodyinthe mth layer. Thevalues 1, 2, . . . , n are the resistivities of the layers, and b isthe 3-D inhomogeneity. Let 1, 2, . . . , n be the boundariesof the layers and b be the boundary of the 3-D body. We drawa hemisphere with a very large radius r

= under theground surface 1. In this way, the total volume consists of re-gions 1, . . . , m, . . . , n , b. Let H1, . . . , Hm1, Hm, . . . , Hn1represent the depth corresponding to each boundary. Also letU1, . . . , Um, . . . , Un, Ub represent the electric potentials in re-gions 1, . . . , m, . . . , n, b, respectively. The normal vectoron each boundary is n.

The governing equations for Uj can be written as

2Uj = 21I3(A), j = 1, . . . , m, . . . , n, b, (1a)

where 2 is the 3-D Laplace operator and 3(A) is a 3-D Diracdelta function at point A on the ground surface. Because thecurrent does not cross the ground surface 1, we have

U1

n

1

= 0. (1b)

From the continuity of the electric potential, we have

Uj1|j = Uj |j; Um|b = Ub|b, j = 2, . . . , m, . . . , n.

(1c)

From the continuity of normal components of current density,we have

FIG. 1. Principal view of a 3-D body in an arbitrarily lay-ered earth. Dashes outline an infinite boundary. The x- andy-axes are horizontal, and the z-axis is downward with theorigin on the ground surface. A current source of I A is im-pressed at point A on the ground surface:1, 2, . . . , m, . . . , n,and b are the resistivities of the layers and the 3-D inho-mogeneity, respectively: 1, 2, . . . , m, m+1, . . . , n, b arethe boundaries of the layers and the 3-D body, respectively.Thesubsurface consists of regions1, . . . , m, . . . , n, b.Thedepths corresponding to each boundary are represented byH1, . . . , Hm1, Hm, . . . , Hn1, respectively: n is the normal vec-tor on each boundary.

1

j1

Uj1

n

j

=1

j

Uj

n

j

;1

m

Um

n

b

=1

b

Ub

n

b

,

j = 2, . . . , m, . . . , n. (1d)

Since is far from the anomalous body, the electric potential

on is the same as that in a half-space excited by a pointsource of current on 1:

Uj | =C

r;

Uj

r

=C

r 2; j = 1, 2, . . . , n, (1e)

where C is a constant of proportionality.

Greens theorem and fundamental solution

The basic idea behind BEM is to transform the partial dif-ferentialequations in thespacedomain into integral equationson the boundaries using Greens second identity:

(U2 2U) d =

U

n

U

n

d, (2)

where is the boundary of the domain , n is the outsidenormal vector, U is a potential to be determined, and is thefundamental solution of the differential equation (1a).

Suppose in a horizontally stratified medium that a pointsource of unit current is located at point p in the mth layer.Then the potential = j (p, q) at point q in the jth layersatisfies

2j = 3(p), (3a)

1

n

1

= 0,

j | =C

r;

j

r

=C

r 2,

j = 1, 2, . . . , m, . . . , n, (3b)

j |j+ 1 = j+1|j+ 1 , (3c)

1

j

j

n

j+ 1

=1

j+1

j+1

n

j+ 1

,

j = 1, 2, . . . , m, . . . , n 1. (3d)

Here, = 21 when p 1 and = j when p / 1. The value3(p) is the 3-D Dirac delta function centered at p. The dif-ferential equation (3) is solved by separation into cylindricalcoordinates (e.g.,Daniels, 1978). Thecomplete solutionis givenby

j (p, q) = Q

0

G j (z,zp, )J0(r) d, (4a)

Q =

1

4, zp > 0

1

2, zp < 0

, (4b)

where r=

(x xp)2 + (y yp)2; (xp, yp,zp) and (x, y,z) arethe coordinates of points p and q, respectively; and J0 is

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the Bessel function of order zero. Function G j can be simpli-fied into the computational convenient forms shown inAppendix A.

The normal derivatives (m/n) and therefore the par-tial spatial derivatives ofm are needed for the integral rep-resentation of the potential of buried bodies. Considering

J0(r) = J1(r), we obtain

m

x= Q

x xp

r

0

Gm(z,zp, )J1(r) d,

m

y= Q

y yp

r

0

Gm(z,zp, )J1(r) d,

m

z= Q

0

zGm(z,zp, )J0(r) d.

All of the integrals above andthe integral of equation (4)havethe form

H(r) =

0

f()J (r) d; = 0, 1,

the Hankel transform of order , and can be evaluated bymeans of linear digital filters (Anderson, 1982, 1984).

Boundary integral equation

Using Greens theorem, we can finally obtain the boundaryintegral equations as follows (see Appendix B):

U1(ps) = U0s

b

mbUbm(ps, q)

nd, (5)

CpUb(pI) = U0I

b

mbUbm(pI, q)

nd, (6)

where U0s = I1(ps, A); U0I =I1(pI, A); ps 1; pI b;q b; and Cp is a coefficient.

Formulas (5) and (6) are the boundary integral equationsthat U1 and Ub must satisfy. Equation (6)can besolved by BEMto find the potential Ub on the boundary b of the body. Then,substituting the result into integral equation (5), the potentialU1 on the ground surface can be calculated.

If there exist d1 anomalous bodies with the correspondingresistivities bj (j = 1, 2, . . . , d1) in the m1th layer whose resis-tivity is m1 and d2 anomalous bodies with the correspondingresistivitiesbj (j = d1 + 1, d1 + 2, . . . , d1 + d2)inthem2thlayerwhose resistivity is m2 , the integral equations correspondingto equations (5) and (6) are

U1(ps) = U0s d1+d2j=1

bj

mbjUbm(ps, q)

nd, (7)

1

m

4

d1+d2j=1

bjpmbj

Ub(pI) = U0I

d1 +d2j=1

bj

mbjUb m(pI, q)

nd, (8)

where mbj = (1/m) (1/bj); m = m1 when j d1 and m = m2when j > d1; bj p isthesolidangleofpoint pI against the region

bj ; and bj p = 0 if pI is outside ofbj whose boundary is bj(Figure 2).

BOUNDARY ELEMENT METHOD

The following discussions are mainly for a single 3-D body,and the index b at the lower right corner ofUb is neglected. Wedivide b into triangular elements and approximate each ele-ment as a triangular plane. The vertices of the triangular planeare nodes. There are N nodes on b. The boundary integral,equation (6), is divided into a sumof integralson each elemente. For node i we have

CiUi = U0i

b

e

mbUm

nd, (9)

where Ci = (4 bi mmb)/(4) and U0i =I1(pi , A). Thenode numbers of the three vertices of an element are takenas j , k, and l with coordinates (xj , yj ,zj ), (xk, yk,zk), (xl , yl,zl)(Figure 3). We assume that U is composed of linear functionsin each element. Using Uj , Uk, Ul to represent U at j , k, and l,Ucan be written as

FIG. 2. Principal view of multiple bodies in the different lay-ers. The valued 1, . . . , m1 , . . . , m2 , . . . , n represent the resi-stivities of the first layer, . . . , the m1th layer, . . . , the m2ndlayer, . . . , the nth layer; b1 , . . . , bd1

, bd1+1, . . . , bd1+d2

rep-resent the resistivities of these inhomogeneous bodies,respectively.

FIG. 3. Boundary of a 3-D body divided into triangular ele-ments. Point i is the ith node on the boundary: n is the normalvector on each triangular element: j , k, l represent the nodenumbers of the three vectices of an element.

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BEM for 3-D dc Modeling in Layered Earth 613

U = jUj + kUk + lUl = [j , k, l]{Uj , Uk, Ul }T,

where j , k, l are linear interpolation functions defined as

x = xj j + xkk + xl l ,

y = yj

j

+ yk

k

+ yl

l,

z = zj j + zkk + zl l .

It is obvious that at point j, j = 1, k = l = 0; at pointk, j = l = 0, k = 1; and at point l, j = k = 0, l = 1. The in-tegral over an element can be derived as follows:

e

mbU m

nd =

e

[j , k, l]mbm

nd{Uj , Uk, Ul }

T

= [fi j , fik, fil ]{Uj , Uk, Ul }T,

where

fi j = mb

e

jm

nd; fik = mb

e

k m

nd;

fil = mb

el

m

nd. (10)

All these integrals are evaluated using a Gaussian formula(Brebbia, 1978) and a linear digital filtertechnique(Anderson,1984) for the Hankel transform. In this paper a four-pointGaussian quadrature formula for the triangular elements isused. For example, we can calculate fi j as follows:

fi j = mb

e

j m

nd = mb

4t=1

(t)j

m(pi , qt)

nWt,

where is thearea of thetriangular elements, Wt is theweight,

(t)j isthevalueofj at point qt (Brebbia, 1978). Thecalculation

of fik and fil is the same as that of fi j .The sum of all integrals over each element on b is obtained

asb

e

mbUm

nd = [Fi1, . . . , Fi j , . . . , FiN]

{U1, . . . , Uj , . . . , UN}T = Fi u,

(11)

where u = {U1, . . . , Uj , . . . , UN}T is the column vector formed

by U at each node on b, Fi j is the sum of fi j at the elementsaround node j , and Fi = [Fi1, . . . , Fi j , . . . , Fi N]. Substitutingequation (11) into equation (9), we obtain

CiUi = U0i Fi u. (12)

For each node we can derive an equation as above. From all

the nodes i (1 i N) on b, we derive an equation system

Cu = u0 Fu. (13)

where C = diag[C1, . . . , Cj , . . . , CN], u0 = [U01, . . . , U0j , . . . ,U0N]

T, F = [F1, . . . , Fj , . . . , FN]T.

We can obtain a linear equation system from equation (13)as follows:

[C + F]u = u0. (14)

The system of linear equations (14) contains N equations withN unknowns that can be solved to obtain the values of u atnodes on b. Substituting these values into equation (5), weobtain

U1(ps) = I1(ps, A) Fsu. (15)

We can calculate U1 at each point on the ground surface.

NUMERICAL EXAMPLES

To test the theory and program for the most general case,we first compare the resultsof a sphere under a flat surface ob-tained by this method with that of approximate analytic solu-tion (solution of half-space as the double of the whole spacesolution). Second, we compute the solution of a buried spherebeneath an overburden and examine the asymptotic behavioras the layer resistivity approaches that of the half-space. Thenwe can compare computations for elongate 3-D bodies withthose for 2-D structures of identical cross-section. The pro-gram described in this paper can accommodate an arbitrarynumber of layers, but for simplicity we consider an inhomo-geneity in thelowerlayer of a two-layerearth (Figures5 and6).

Thefour-electrodesystem is arrangedso thetwo potentialelec-trodes move in the middle segment between the two fixedcurrent electrodes. The current electrode spacing is 600 min Figures 4 and 5 and 1200 m in Figures 6 and 7, and the

FIG. 4. Model curvesof apparent resistivity for a sphereburiedin a half-space earth. Solid curvesand dotted curvesare there-sults obtained by BEM and the approximate analytic solution,respectively; x represents the midpoint of the two potentialelectrodes. (a) When r/h0 = 1/2, the solid curve is comparedwith the dotted curve. (b) When r/h0 = 5/7, the solid curveis compared with the dotted curve. (c) The cross-section ofthe model. 1 = 1 ohm-m.b = 0.05 ohm-m. Thespheres radiusr= 5 m; h0 is the depth of a sphere center.

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FIG. 5. Model curvesof apparent resistivity for a sphereburiedin a two-layer earth. (a) The curves are plotted for differ-ent overburden resistivities 1 (in ohm-m), while the substra-tum and the sphere have the resistivities 2 = 100ohm-m andb = 1 ohm-m, respectively. The dotted curve is obtained with

the approximate analytic solution of a sphere in a uniformhalf-space (1 = 2 = 100 ohm-m). (b) The cross-section of themodel. Thethickness of the overburden ish = 5 m. Thespheresradius andthe depth of its center are10 and 20 m, respectively.A and B represent the two current electrodes, M and N repre-sentthe two potentialelectrodes, andx represents themidpointof the two potential electrodes.

FIG. 6. Model used to compare the response of this elongate3-Dbody to theresults of a 2-Dbody of identicalcross-section.The L represents the strike length of the 3-D body, and h isthe depth of the 3-D body beneath the lower layer. The val-ues 1, 2, b are the intrinsic resistivities of the upper layer,lower layer, and 3-D body, respectively. A four-electrode sys-tem is arranged along the x-axis. The two current electrodesare located at A and B, and the two potential electrodes are atM and N.

potential electrode spacing is 5 m in Figures 4 and 5 and 30 min Figures 6 and 7.

Figure 4 shows the apparent resistivity a curves of asphere buried in a half-space medium in which the results ofBEM (solid curves) are compared with that of an approxi-mate analytic solution (dots). In Figure 4a, r/ h0 = 1/2; the

results agree well. Here, r and h0 represent the radius of asphere and the depth of the sphere center, respectively. InFigure 4b, r/ h0 = 5/7. The maximum error appears at thepointover the sphere and can reach 10%. This error is mainlyin the approximate analytic solution. When r/h0 1/2, theapproximate analytic solution is convincing.

Figure 5 shows the results of a sphere buried in a two-layermedium. The spherical surface is divided into triangular ele-ments with 29 nodes. The curve computed with this methodfor 1 = 2 = 100 ohm-m is compared with the dotted curveobtained by approximate analytic solution of a half-space con-taining a sphere . The results agree well. On the other hand, asthe overburden resistivity1 approaches thatof the substratum2 = 100ohm-m, the asymptotic value of apparent resistivity isclose to that of the half-space containing the sphere. For the

calculations illustrated in Figure5, the computer time is severaltens of seconds on a Pentium II microcomputer.

Theapparent resistivitycurves fromelongate3-D bodies arecompared with those from the corresponding 2-D structurecalculated using a boundary element routine (Qian and Ma,1992) in Figures 7 and 8. The body is a tetragonal prism.

In Figure 6, the upper layer is 50 m thick, and the cross-section of the 3-D body in the lower layer is 100 100 m. Itsstrike length is L, and its depth beneath the lower layer is h.The resistivity of the bottom layer is 2 = 10 ohm-m.

In Figure 7, the shape of apparent resistivity curves for the3-D body is similar to that of the 2-D body, although the 2-Dcase shows a stronger influence of the inhomogeneity. WhenL = 100 m, h = 5 m, apparent resistivity curve 2 is similar tothat of a 2-D body. The cubic surface is divided into triangular

FIG. 7. Profile curves of apparent resistivity along thex-axis ofthe 3-D body of Figure 6 and across a 2-D body of identicalcross-section for 1 = 100 ohm-m, 2 = 10 ohm-m, b = 0.1ohm-m, and h = 5 m. Solid curves are the responses of the 3-Dbody. The dashed curve is the response of the 2-D structure.The L is the strike length of the 3-D body, and x represents themidpoint of the two potential electrodes.

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BEM for 3-D dc Modeling in Layered Earth 615

FIG. 8. Profile curves of apparent resistivity along the x-axisof the 3-D body of Figure 6 and across a 2-D bodyof identical cross-section for1 = 1.0 ohm-m, 2 = 10 ohm-m,b = 10 000 ohm-m, and h = 10 m. The L is the strike length.The results of the3-D body areplotted usingsolidcurves, whilethat of the 2-D structure is shown as dashed curve. The x rep-resents the midpoint of the two potential electrodes.

elements with 98 nodes. When L = 480m, h = 5 m, apparent re-sistivitycurve 3 approaches that of the2-D body, andthe agree-ment between them is satisfactory. The surface of the elongate3-D body is divided into triangular elements with 260 nodes.The results with the high-resistivity body (Figure 8) are alsoqualitative agreement between the two models. On the otherhand, the extent of the agreement between the 2-D and 3-Dresults may be influenced by the nature of the layering (seeFigures 7 and 8). In Figures 7 and 8, L = 0 represents the caseof no body in the layered earth.

CONCLUSIONS

The method used in this paper can be used to model the 3-Dbodies located in earths with layer interfaces both above andbelow the body. Furthermore, it can be used to model multiplebodiesembedded withinthe differentlayers in a layered earth.The approach outlined here is straightforward and easy to im-plement. The numerical solution can be run on a microcom-puter (Pentium II) in a matter of minutes. The formulationspresented in this paper can become an applicable techniquefor simulating the dc resistivity responses of 3-D anomalousbodies buried in a layered earth.

ACKNOWLEDGMENTS

I express my sincere thanks to Colin Farquharson; DavidBoerner,the associateeditorof Geophysics; andan anonymous

reviewer for their comments and suggestions. Thanks are alsoextended to Larry Liner, the editor of Geophysics, for his en-couragement. In addition, I thank Xu Shi-Zhi for his usefuladvice at an early stage.

REFERENCES

Adey, R. A., and Niku, S. M., 1985, Comparison between boundaryelements method and finite elements for CAD of cathodic protec-tion system, in Brebbia, C. A., and Noye, B. J., Eds., BETECH 85:Computational Mechanics Publications, 269298.

Alfano, L., 1959, Introduction to the interpretation of resistivity mea-surements for complicated structural conditions: Geophys. Prosp.,7, 311366.

Anderson, W. L., 1982, Fast Hankel transforms using related andlogged convelutions: ACM Trans. on Math. Software, 8, 344368.

1984, Computation of Greens tensor integrals for three-dimensional electromagnetic problems using fast Hankel trans-forms: Geophysics, 49, 17541759.

Brebbia, C. A., 1978, The boundary-element method for engineers:Pentech Press.

Brebbia, C. A., Ed., 1988, Boundary elements X: Springer-VerlagBerlin.

Brebbia, C. A., Umetani, S., and Trevelyan, J., 1985, Critical compari-sonof boundary element andfinite element methods forstressanal-ysis, in Brebbia, C. A., and Noye, B. J., Eds., BETECH 85: Compu-

tational Mechanics Publications, 225256.Coggon, J. H.,1971, Electromagneticand electric modeling by the finiteelement method: Geophysics, 36, 132155.

Daniels, J. J., 1978, Interpretation of buried electrode resistivity datausing a layered earth model: Geophysics, 43, 9881001.

Dey, A., and Morrison,H. F., 1979, Resistivity modeling for arbitrarilyshaped three-dimensional structure: Geophysics, 44, 753780.

Dieter, K., Paterson, N. R., and Grant, F. S., 1969, IP and resistivitytype curves for three-dimensional bodies: Geophysics, 34, 615632.

Fox, R. C., Hohmann, G. W., Killpack, T. J., and Rijo, L., 1980,Topographic effect in resistivity and induced-polarization surveys:Geophysics, 45, 7593.

Lee, T., 1975, An integral equation and its solution for some two andthree-dimensional problems in resistivity and induced polarization:Geophys. J. Roy. Astr. Soc., 42, 8195.

Okabe, M., 1981, Boundary element method for the arbitrary inho-mogeneities problem in electrical prospecting: Geophys. Prosp., 29,3959.

Qian, J. D., and Ma, Q. Z., 1992, The boundary element method forsolving the 2-D geoelectric field of a point source in three-layeredmedium with fluctuated interfaces: Northwestern Seismol. J., 14,

111121 (in Chinese).Schulz, R.,1985, The methodof integral equation in the direct currentresistivity method and its accuracy: J. Geophys., 56, 192200.

Xu, S. Z., Zhao, S. K., and Lou, Y. J., 1984, The boundary elementmethod for solving the three dimensional electric field under theconditions of horizontal ground: Comput. Tech. for Geophys. andGeochem. Expl., 6, 5360 (in Chinese).

Xu, S. Z., and Zhao, S. K., 1985, The boundary element method calcu-lating electric field of a point source on three-dimensional topogra-phy: J. Guilin College of Geol., 5, 163168 (in Chinese).

Xu, S. Z., Gao, Z. C., and Zhao, S. K., 1988, An integral formulationfor three-dimensional terrain modeling for resistivity surveys: Geo-physics, 53, 546552.

Zhao, S. K., and Yedlin, M. J., 1996, Some refinements on the finite-difference method for 3-D dc resistivity modeling: Geophysics, 61,13011307.

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APPENDIX A

EXPLICIT EXPRESSION OF THE FUNDAMENTAL SOLUTIONS

Potentials about a point source of current in a layered earthhave been expressed by Daniels (1978). However, for the sakeof convenience to an applicable technique, the fundamental

solution in this paper is simplified in forms amenable to com-putation.In equation (4a), thefunctionG j canbe expressed as

G j (z,zp, ) = Dj e|zzp| + Aj ()e

(zzp) +Bj ()e(zzp),

(A-1a)

Dj =

m, j = m

0, j = m, (A-1b)

in which Aj (), Bj () can be calculated by

Am = TdTu m

Tu + Td, (A-2a)

Bm = TuTd m

Tu + Td. (A-2b)

Bk = (Dk+1 +Bk+1)Tk + k

Tk + k+1, k= m 1, m 2, . . . , 1,

(A-2c)

Aj = (Dj1 +Aj1)Tj + j

Tj + j1, j = m + 1, m + 2, . . . , n.

(A-2d)

The thickness of the jth layer is denoted by

hj = Hj Hj1, j = 1, . . . , n.

Expressions Tj , Tu , and Td are calculated by the followingrecursion formulas:

T1 =1

tan h(h1), (A-3a)

Tk = kTk1 + k tan h(hk)k + Tk1 tan h(hk)

, k = 2, 3, . . . , m 1, m,

(A-3b)

Tu = Tk, hk = zp Hm1, k = m, (A-3c)

Tn = n, (A-4a)

Tj = jTj+1 + j tan h(hj )j + Tj+1 tan h(hj )

,

j = n 1, n 2, . . . , m + 1, m, (A-4b)

Td = Tj ; hj = Hm zp, j = m. (A-4c)

When z zp, G j , Gn are calculatedby the following formulas:

G j (z,zp,)

= (Dj + Aj )

e(zzp) +

Tj+1 j

Tj+1 + je(2Hjzzp)

,

j = n 1, n 2, . . . , m + 1, (A-6a)

Gn(z,zp, ) = (Dn + An)e(zzp). (A-6b)

When m = 1, formulas (A-3) are invalid for calculating Tu . Sup-pose there is the image of the horizontally stratified mediumover the ground. Then Tu can be replaced by Td. From equa-tion (A-2), we can obtain

A1 = B1 =(T

d

1)

2 , (A-7)

were Td = T1 according to the recursion formulas (A-4).

APPENDIX B

DERIVATION OF THE INTEGRAL EQUATIONS

Greens second identity is used in regions 1, 2, . . . ,m, . . . , n, b, respectively:

j

Uj

2j j 2Uj

d

= j j+1+j

Uj jn

jUj

n d,

b

Ub

2m m2Ub

d

=

b

Ub

m

n m

Ub

n

d,

j = 1, 2, . . . , n, (B-1)

where j represents two segments of which belong to theboundary of region j . When j = m,let j+1 = j+1 b;whenj = n, let j+1 = 0.

In example 1, let p = ps 1, utilizing the relation of bound-ary values, equations (1), and (3). Notice

1 U1[213(ps)] d = 1U1(ps),j

Uj

j

n j

Uj

n

d = 0,

1

1[21I3(A)] d = 1I1(ps, A).

From equation (B-1), we can obtain

U1(ps) = I1(ps, A)

b

mbUb m(ps, q)

nd, (B-2)

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where q is an arbitrary point on boundary b and mb =(1/m) (1/b). Hence, as b m , mb 0.

In example 2, let p = pI b, utilizing the relation of boun-dary values, equations (1) and (3). The boundary surface ofb at each node is not smooth; therefore,

m

Um[m3(pI)] d = mmp

4 Um(pI),b

Ub[m3(pI)] d = mbp

4Ub(pI),

1

1[21I3(A)] d = 1I1(pI, A).

From equation (B-1), we finally obtain

CpUb(pI) = I1(pI, A)

b

mbUbm(pI, q)

nd,

(B-3)where Cp = (4 bpmmb)/(4); q b;

mp is the solid angle subtended by m at pI; and bp is thesolid angle subtended by b at pI.

If the boundary surface at each node is smooth, thenbp = 2 . Let n = n1 = , . . . , = m . The integral equa-tions (B-2) and (B-3), turn out to be the integral equationsof surface potential in Schulz (1985).

In equation (B-3),the surface integral,however, is impropersince {[m(pI, q)]/n} produces a singularity at q = pI. To

avoid the singularity, noticing Okabes method of singular-ity analysis (Okabe, 1981) and utilizing equation (3a), we canchange the surface integral in equation (B-3) into anotherform:

b

mbUb

m(pI, q)

n m3(pI)

2

d, (B-4)

which is now proper everywhere on b. Once the singularity isexcluded theoretically in the integral equation formulation, itis unnecessary to exclude numerically an infinitesimal bound-ary surface around the singular point. Rather, the integra-tion should be carried out as if no singularity in the integrandexisted.