computations of secondary potential for 3d dc resistivity modelling using an incomplete choleski...

Geophysical Prospecting, 2003, 51, 567–577

Computations of secondary potential for 3D DC resistivity modellingusing an incomplete Choleski conjugate-gradient method

Xiaoping Wu,1,2∗ Yifei Xiao,1 Cheng Qi1 and Tongtong Wang3

1Department of Earth and Space Sciences, University of Science and Technology of China, Hefei 230026, China, 2Institute of Geophysics,Freiberg University of Mining and Technology, Gustav-Zeuner Strasse 12, 09596 Freiberg, Germany, and 3Anhui Institute of EnvironmentalProtection Research, Hefei 230061, China

Received October 2001, revision accepted July 2003

ABSTRACTAn accurate and efficient 3D finite-difference (FD) forward algorithm for DC resistiv-ity modelling is developed. In general, the most time-consuming part of FD calculationis to solve large sets of linear equations: Ax = b, where A is a large sparse band sym-metric matrix. The direct method using complete Choleski decomposition is quiteslow and requires much more computer storage. We have introduced a row-indexedsparse storage mode to store the coefficient matrix A and an incomplete Choleskiconjugate-gradient (ICCG) method to solve the large linear systems. By taking advan-tage of the matrix symmetry and sparsity, the ICCG method converges much morequickly and requires much less computer storage. It takes approximately 15 s on a533 MHz Pentium computer for a grid with 46 020 nodes, which is approximately700 times faster than the direct method and 2.5 times faster than the symmetricsuccessive over-relaxation (SSOR) conjugate-gradient method. Compared with 3Dfinite-element resistivity modelling with the improved ICCG solver, our algorithm ismore efficient in terms of number of iterations and computer time. In addition, wesolve for the secondary potential in 3D DC resistivity modelling by a simple manip-ulation of the FD equations. Two numerical examples of a two-layered model and avertical contact show that the method can achieve much higher accuracy than solvingfor the total potential directly with the same grid nodes. In addition, a 3D cubic bodyis simulated, for which the dipole–dipole apparent resistivities agree well with the re-sults obtained with the finite-element and integral-equation methods. In conclusion,the combination of several techniques provides a rapid and accurate 3D FD forwardmodelling method which is fundamental to 3D resistivity inversion.

I N T R O D U C T I O N

In direct current (DC) resistivity modelling, numerical meth-ods are necessary for general 3D model simulation. Thereare three principal numerical methods: the integral-equationmethod (Dieter, Paterson and Grant 1969; Pratt 1972;Hohmann 1975; Lee 1975; Daniels 1977; Okabe 1981;Oppliger 1984; Xu, Guo and Zhao 1988), the finite-elementmethod (Coggon 1971; Fox et al. 1980; Pridmore et al. 1981;

∗E-mail: [email protected]

Holcombe and Jiracek 1984) and the finite-difference method(Dey and Morrison 1979; Scribe 1981; Spitzer 1995; Zhaoand Yedlin 1996). Each method has its own particular ad-vantage and is suitable for specific model geometries. Theintegral-equation method, for instance, only considers thecharge on the surface of a body. The 3D forward calculationusing the integral-equation method is very fast and requiresless computer memory, but is restricted to certain model ge-ometries. The finite-difference and finite-element methods aresuitable for arbitrary 3D structures, and are much more flexi-ble than the integral-equation method. However, they are very

C© 2003 European Association of Geoscientists & Engineers 567

568 X. Wu et al.

time-consuming and a very large amount of computer storageis required to solve the large linear equations,

Ax = b, (1)

resulting from these two numerical methods. A direct methodis used for the complete Choleski decomposition. Efficientsolvers for (1) are preconditioned conjugate-gradient (PCG)methods in combination with compact storage schemes, whichhave been used to deal with 3D resistivity FD problems (Spitzer1995; Zhang, Mackie and Madden 1995; Smith 1996). Thusthe 3D resistivity FD modelling becomes very effective andis widely used in 3D inversions (Zhang et al. 1995; Loke andBarker 1996; Newman and Alumbaugh 2000). Recently Zhouand Greenhalgh (2001) introduced PCG methods to solve the3D resistivity finite-element problem efficiently.

The conjugate-gradient (CG) method was first describedby Hestense and Stiefel (1952). When applied to solve lin-ear equations, the CG method converges very rapidly if thecoefficient matrix A is close to an identity matrix. However,the 3D FD numerical problem results in a very large ma-trix with a high condition number, and thus the CG methodwill converge very slowly. Therefore, preconditioned tech-niques have to be used to improve the linear system of equa-tions (1). Spitzer (1995) presented an efficient 3D FD al-gorithm for DC resistivity modelling using the symmetricsuccessive over-relaxation (SSOR)-preconditioned conjugate-gradient method. Smith (1996) successfully applied the incom-plete Choleski conjugate-gradient (ICCG) method to solve themagnetotelluric 3D FD forward problem.

The incomplete Choleski preconditioner is stable for ma-trices of a special kind, such as the M-matrix (Meijerinkand van der Vorst 1977) or a diagonally dominant matrix(Kershaw 1978), while problems arising from the finite-element method in general cannot be classified in this category.Therefore, the ICCG method is especially effective in solvingthe large sparse linear systems resulting from FD computa-tion because its coefficient matrices are diagonally dominant(Dey and Morrison 1979). Improved incomplete Choleski pre-conditioners are available to solve finite-element problems(Kershaw 1978; Manteuffel 1980; Ajiz and Jennings 1984;Papadrakakis and Dracopoulos 1991; Zhou and Greenhalgh2001), but we will show that ICCG works better in FD compu-tation than the improved ICCG method does in finite-elementcomputation. Compared with the SSOR method, the ICCGmethod is better in terms of number of iterations and com-puter time (Papadrakakis and Dracopoulos 1991; Zhou andGreenhalgh 2001). We introduce the ICCG method and 1Drow-indexed sparse storage mode (Bentley 1986) simultane-

ously to the finite-difference computation of the 3D geoelectricfield, accelerating the convergence rate and greatly reducingthe requirements of storage memory. On a 3D grid with 59 ×39 × 20 = 46 020 nodes, the FD solutions are achieved in ap-proximately 90 iterations or 15 s run time on a 533 MHzPentium computer using the ICCG method, while about200 iterations or 38 s run time are required by the SSORmethod. Using the direct method with complete Choleski de-composition, a time of over 10 000 s is needed.

At the source point, the potential is singular. If we solvefor the total potential in 3D resistivity forward modelling thesingularity will cause large errors, especially in the neighbour-hood of the source point where the singularity effects are thegreatest. Solving for the secondary potential is an effective ap-proach to reducing the error (Zhao and Yedlin 1996). Ourresults show that the average relative errors in FD solutionsare approximately 0.3% compared with the analytical solu-tions for the two-layered model and the vertical contact.

T H E 3 D F I N I T E D I F F E R E N C E

Let I be the current intensity of a point source and let (x0, y0,z0) be the coordinates of the point source of injected charge.Then the equation that governs the 3D geoelectric field yieldedby the point source is

∇ · [σ (x, y, z)∇ϕ(x, y, z)] = −Iδ(x− x0)δ(y− y0)δ(z− z0),∈ �,

(2)

where σ is the 3D conductivity distribution, (x, y, z) are theCartesian coordinates of a point in the computational domain� and δ is the Dirac delta function. φ is the electric poten-tial, subject to the following boundary conditions (Dey andMorrison 1979):

∂φ/∂n = 0, ∈ s,

∂φ

∂n+ cos θ

rφ = 0, ∈ ∞,

where s denotes the flat earth–air interface, ∞ denotes thesubsurface external boundaries, and θ is the angle betweenthe radial distance r from the source point and the outwardnormal spatial coordinate n on the boundary shown in Fig. 1.

The finite-difference method for the computation of the3D geoelectric field, described in detail by Dey and Morrison(1979), is discussed briefly as follows:

The semi-infinite lower half-space with arbitrary con-ductivity distribution is discretized by the 3D grids Nx ×Ny × Nz, which are chosen to be a rectangular prism witharbitrary, irregular spacing of the nodes in the horizontal

C© 2003 European Association of Geoscientists & Engineers, Geophysical Prospecting, 51, 567–577

A rapid and accurate 3D FD forward algorithm for DC resistivity modelling 569

Figure 1 Illustration of the computational domain and boundary forthe boundary-value problem of DC resistivity modelling.

(x- and y-) directions and the vertical z-direction. The nodesin the x-direction are indexed by i = 1, 2, . . . , Nx, those inthe y-direction by j = 1, 2, . . . , Ny, and the nodes in thez-direction by k = 1, 2, . . . , Nz. We integrate (2) over Vi,j,k

which is the elementary volume about a node (i, j, k), andobtain

−∫∫∫ Vi, j,k

∇ · [σ∇φ] dV

=∫∫∫ Vi, j,k

I Vi, j,k

δ(xi − x0)δ(yi − y0)δ(zi − z0) dV.

Using Gauss’s theorem, the volume integral becomes

∫∫∫ Vi, j,k

∇ · [σ∇φ] dV =∫ ∫Si, j,k

σ∂φ

∂nds,

where n is the outward normal and Si,j,k is the surface enclosingthe elemental volume Vi,j,k, and then we obtain

−∫ ∫Si, j,k

σ∂φ

∂nds = I

Vi, j,k

∫∫∫ Vi, j,k

δ (xi − x0) δ (yi − y0)

× δ (zi − z0) dx dy dz

={

I, (x0, y0, z0) ∈ Vi, j,k,

0, (x0, y0, z0) /∈ Vi, j,k.

For an interior node in the discretization grid, the central dif-ference is used to approximate ∂φ/∂n; for the nodes located onthe ground surface and the infinitely distant edge, the bound-ary conditions can be directly implemented. Thus, the self-adjoint difference equation is obtained for each node (i, j, k)

as follows:

Cfφi, j−1,k + Clφi−1, j,k + Ctφi, j,k−1 + Cpφi, j,k + Cboφi, j,k+1

+ Crφi+1, j,k + Cbaφi, j+1,k

={

I, (x0, y0, z0) ∈ Vi, j,k,

0, (x0, y0, z0) /∈ Vi, j,k,

where Ct, Cbo, Cl, Cr, Cf, Cba are the coupling coefficientsbetween the node (i, j, k) and the top, bottom, left, right,front and back nodes, respectively, and Cp is the self-couplingcoefficient at node (i, j, k). The sets of difference equationsfor each node are then assembled into a global matrix form,written symbolically as

A� = s, (3)

where the capacitance matrix A is a large sparse septadiagonal,positive-definite symmetric and diagonally dominant matrix,given by

A =

. . .. . .

. . .. . .

. . .. . .

. . .. . .

. . .. . .

. . .. . .

. . . cba

. . .. . .

. . .. . .

. . .. . .

. . .. . .

. . .. . . cr

. . .. . .

. . .. . .

. . .. . .

. . . ct cp cbo. . .

. . .. . .

. . .. . .

. . .. . . cl

. . .. . .

. . .. . .

. . .. . .

. . .. . .

. . .

cf. . .

. . .. . .

. . .. . .

. . .. . .

. . .. . .

. . .. . .

. . .. . .

.

This shows that there are no more than seven non-zero ele-ments in a row in matrix A. We use the 1D row-indexed sparsestorage mode to store the lower triangular of A. Because ofsymmetry, the storage requirements are no more than 4 × Nx ×Ny × Nz real stores. In addition to the extra storage for fourauxiliary arrays required by the CG method, the total require-ments are no more than 10 × Nx × Ny × Nz. However, the 2Dbanded compact storage requirements for the direct methodare at least (Nx × Ny × Nz) × (Nx × Nz) real stores. Φ isthe voltage vector and s is the current source vector in which


570 X. Wu et al.

only the element at the node p of the point source of injectedcharge is non-zero. That is, s = (0, . . . , 0, sp, 0, . . . , 0)T andsp = I, solving the difference equations (3) to get the potentialφ(x, y, z).

T H E I C C G M E T H O D A N D A C O M PA R I S O NW I T H T H E D I R E C T M E T H O D

Here, we discuss the incomplete Choleski conjugate-gradientmethod briefly. The iterative procedure of the CG method isshown as follows, as it is applied to solving (1).

Let r0 = b − Ax0 and p0 = r0, then

αi = (ri , ri )/(pi , Api ),

xi+1 = xi + αi pi ,

ri+1 = ri − αi Api , (4)

βi = (ri+1, ri+1)/(ri , ri ),

pi+1 = ri+1 + β ipi, where i = 0, 1, 2, . . .

From the above recurrence sequence, we can see that onlythe results of the matrix A multiplying vector are required,and the non-zero elements in matrix A do not contribute tothe product. For the row-indexed sparse storage mode, two1D arrays are set up to represent a large sparse matrix A,the first of these being a real array which stores the non-zeromatrix element values, the second storing the integer valueswhich index the positions of the elements in matrix A. Thus,from the two 1D arrays we can easily obtain the results of thematrix A multiplying vector. In theory, if A is positive definiteas well as symmetric, this leads to a stable convergence forthe CG technique. However, while A is ill-conditioned, theCG algorithm is not suitable for the numerical calculationbecause of its slow convergence rate. Meijerink and van derVorst (1977) described a very simply computed incompleteCholeski decomposition which effectively improves the linearsystem of (1), and greatly accelerates the convergence rate ofthe CG method. Smith (1996) applied it successfully to 3Dmagnetotelluric forward modelling.

The incomplete Choleski decomposition is given by

A ≈ CCT, (5)

where C is a lower triangular matrix, which can be computedfrom diagonal matrix D defined by

djj = ajj −∑k< j

a2jk

/dkk, (6)

where ajk are the elements in matrix A. Obviously, the termswith ajk = 0 are not included in the sum. The matrix D can becomputed readily. Then C is given by

C = UD−1/2,

where U is a lower triangular matrix with ujj = djj for all j,ujk = ajk for k < j.

The incomplete Choleski factorization C based on rejec-tion by position is based on a sparsity pattern of A, which isas sparse as the lower triangular of A. Thus C can be com-puted rapidly without more storage memory because its non-diagonal elements are the same as those of A.

From the incomplete Choleski decomposition in (5), (1) canbe rewritten as

[C−1A(CT)−1]CTx = C−1b. (7)

The incomplete Choleski factorization was first describedby Varga (1960) as a method of constructing a regular split-ting of certain finite-difference operators. A quick calculationshows that CCT and A match on the diagonal and the non-zero set G, in fact, at each non-zero element of A. The graphof CCT. looks like the graph of A with a few more edges(Manteuffel 1980). Thus (CCT)−1 is a good approximationof A−1, and C−1A(CT)−1 will be an approximate identity ma-trix, so the CG method applied to the matrix C−1A(CT)−1

will converge very rapidly. We now apply the CG method tothe modified equation (7) and rearrange equations (4), thusobtaining the incomplete Choleski conjugate-gradient (ICCG)method.

The ICCG method is carried out by the following recurrencesequence:

Let r0 = b − Ax0 and p0 = (CCT)−1r0, then

αi = (ri , (CCT)−1ri

) /(pi , Api ),

xi+1 = xi + αi pi ,

ri+1 = ri − αi Api , (8)

βi = (ri+1, (CCT)−1ri+1

) / (ri , (CCT)−1ri

),

pi+1 = (CCT)−1ri+1 + β ipi, where i = 0, 1, 2, . . . .

In the recurrence sequence above, the key process is to calcu-late the vector g = (CCT)−1r. We can solve equation system(CCT) g = r equivalently to obtain vector g by very simple andrapid forward- and back-substitution of the Gauss method,



Table 1 Comparison of storage requirements and run time for theICCG and direct methods

Calculation time Storage requirements

Direct ICCG Direct ICCGGrid sizes method method method method

15 × 15 × 20 19 s 0.6 s 5 Mb 0.18 Mb19 × 19 × 20 58 s 1 s 11 Mb 0.3 Mb25 × 25 × 20 180 s 2 s 20 Mb 0.5 Mb39 × 39 × 20 1430 s 6 s 97 Mb 1.2 Mb

because matrix C is a lower triangular matrix. Table 1 showsa comparison of storage requirements and run times on aPentium 533 MHz computer for the ICCG method and thedirect method. The convergence criterion of calculations forthe ICCG method is |b − Axk|/|r0| < ε = 10−8, where |r0|is the L2-norm of the residual in the first iteration. It can beseen that with the increase in the number of grid nodes, theICCG method has an increasing advantage in storage require-ments and run times over the direct method for solving the3D FD problem. The run time for the direct method increasesapproximately exponentially, while it increases linearly usingthe ICCG method.

C O M PA R I S O N W I T H I M P R O V E D I C C GU S E D I N T H E F I N I T E - E L E M E N T M E T H O D

As we proceed through algorithm (6), it is crucial that all thedjj are greater than zero. If djj = 0, then the algorithm breaksdown, and if djj < 0, then CCT is not positive definite and theconjugate-gradient method can no longer be used to obtainthe exact solution as in (7) and (8). Because the capacitancematrix A arising from 3D FD modelling is diagonally domi-nant (Dey and Morrison 1979), incomplete Choleski splittingis stable for djj > 0. However, since finite-element matrices donot possess the properties of the M-matrix (Meijerink and vander Vorst 1977), and nor are they diagonally dominant, theICCG method is unstable because djj < 0. Manteuffel (1980)described a shifted incomplete Choleski factorization tech-nique for a positive-definite linear system that arises from theapplication of the finite-element method, which lies betweenJacobi splitting and incomplete Choleski splitting. Kershaw(1978) simply set djj to some positive value and then proceededwith algorithm (6) if djj < 0. Papadrakakis and Dracopoulos(1991) proposed an improved incomplete Choleski factoriza-tion based on rejection by position, while diagonal modifica-

tion for the case djj < 0 was suggested by Ajiz and Jennings(1984). Certainly, altering the diagonal elements of C in theseways worsens the approximate CCT factorization of A, but afairly good approximate factorization can still be obtained aslong as most of the pivots are stable. Thus, it is not difficult tounderstand that ICCG works better in FD computation thanimproved ICCG does in finite-element computation. Using thesame grid with 57 × 57 × 49 nodes and the same convergencecriterion ε = 10−10, Zhou and Greenhalgh (2001) took 140 sto perform 172 iterations in 3D resistivity finite-element mod-elling with the improved ICCG method, while in this studywe take 75 s to perform 129 iterations for the homogeneoushalf-space model of ρ = 1 �m.

C O M P U TAT I O N O F S E C O N D A RYP O T E N T I A L S

At the source point, the potential is singular. If we solve directlyusing (2), the singularity will cause a large error, especially inthe neighbourhood of the source point where the singularityeffects are the greatest. Zhao and Yedlin (1996) split the totalpotential φ into the primary potential u0 due to the currentsource in a uniform half-space and the secondary potential u

due to the conductive inhomogeneities. Thus

φ = u0 + u. (9)

The primary potential u0 is given by

u0 = I

2πσ0

√x2 + y2 + z2

,

where σ 0 is the conductivity of the medium at the source point.Then the boundary-value problem for the secondary potentialu is solved in order to remove the effect of the primary, singularpotential due to the source.

Based on the FD equation (3) of the total potential φ, wesolve for the secondary potential u by a simple manipulationof the FD equations as follows:

From (3) and (9), we can write

A(u0 + u) = s. (10)

We express the FD equation for the potential u0 occurring ina uniform half-space as

A0u0 = s. (11)

Here A0 is its coefficient matrix. Subtracting (11) from (10),we obtain

Au = −(A − A0) u0. (12)


572 X. Wu et al.

Compared with (3), only the right-hand term or source termin (12) differs and this can be calculated analytically. We canuse almost the same programming codes to solve (12) for thesecondary potential u as we do for the total potential φ, andthen a more accurate value of the potential φ can be obtainedusing (9). We call this the second potential finite-difference(SP-FD) method. Zhao and Yedlin (1996) gave an alternativeanalytical computation of the source term derived from the el-ementary volume integrals. However, it seems rather complexfor programming.

R E S U LT S A N D D I S C U S S I O N

To estimate the accuracy of the second potential finite-difference (SP-FD) method, a two-layered model, a verticalcontact and a buried conductive cubic body are simulated. An-alytical solutions are available for the first two models. We willcompute the potentials obtained by the total potential finite-

Figure 2 Two-layered model. The top layer has resistivity ρ1 = 1 �mand thickness h1 = 20 m. The second layer is a uniform half-spacewith resistivity ρ2 = 19 �m.

Table 2 Comparison of analytical solutions and numerical solutions obtained with the total potential finite-difference (TP-FD) and secondarypotential finite-difference (SP-FD) methods, for a two-layered model

Theoretical Numerical potentials Numerical potentials Relative error Relative errorX/m potentials (V) by TP-FD method by SP-FD method by TP-FD (%) by SP-FD (%)

1.0 0.177437 0.187035 0.177318 5.41 0.072.0 0.097871 0.100690 0.097732 2.88 0.143.0 0.071339 0.070021 0.071193 1.85 0.204.0 0.058061 0.055507 0.057912 4.40 0.265.0 0.050082 0.047129 0.049931 5.90 0.306.0 0.044750 0.041659 0.044597 6.91 0.347.0 0.040929 0.037773 0.040774 7.71 0.388.0 0.038050 0.034855 0.037894 8.40 0.419.0 0.035798 0.032581 0.035641 8.99 0.4410.0 0.033984 0.030760 0.033826 9.49 0.4611.0 0.032488 0.029249 0.032329 9.97 0.4912.0 0.031228 0.027955 0.031068 10.48 0.51Average 6.86 0.33

difference method (TP-FD) using (3), and by the secondarypotential finite-difference (SP-FD) method using (12) in com-bination with (9), and we will then compare these results withthe analytical results. For the third model, we will comparedipole–dipole apparent-resistivity pseudosections with the re-sults presented by Pridmore et al. (1981).

The first model consists of two layers as shown in Fig. 2. Thetop layer has resistivity ρ1 = 1 �m and a thickness h1 = 20 m.The second layer is a uniform half-space with resistivity ρ2 =19 �m. The analytical potential can be obtained from O’Neillet al. (1984) by digital linear filter. We computed the potentialdue to a unit current point source for the model, using theTP-FD and SP-FD methods. The forward modelling grid has39 × 39 × 20 nodes, and the source point is located on theground at the centre of the geometrical surface. A comparisonof these results is shown in Table 2. In this case the convergencecriterion is ε = 10−8. The relative errors between the analyt-ical solutions and the numerical solutions are also shown inTable 2, where the x-coordinate represents the distance fromthe receiver point to the source point. It is clear that the SF-FDmethod gives more accurate results than the TP-FD method.The average relative error is 6.86% for the TP-FD method and0.33% for the SP-FD method.

The second model is a 2D vertical contact which has resis-tivities ρ1 = 1 �m and ρ2 = 10 �m on the two sides. The3D grid and the convergence criterion are the same as for thefirst model. The unit point source is located at 5 m offset fromthe contact plane. In this case, an analytical solution exists(Fu 1983). Figure 3 shows the model and the comparison of



Figure 3 Comparison of the analytical and numerical solutions over a vertical contact in (a) the x-domain and (b) the z-domain. The pointsource is located at 5 m offset from the contact plane.

analytical and numerical solutions with the 3D FD method.Figure 3(a) shows the solution in the x-domain. It can be seenthat the numerical solution obtained by the SP-FD methodagrees almost exactly with the analytical values, and a jump

in potential at the contact plane is clearly indicated. The aver-age relative error is 3.77% for the TP-FD method and 0.28%for the SP-FD method. Similarly in Fig. 3(b), which shows thesolution in the z-domain (x = 0), the average relative error


574 X. Wu et al.

Figure 4 Comparison of dipole–dipole apparent resistivities calculated for a 3D body by (a) finite-difference, (b) finite-element and (c) integral-equation methods. The model, as well as finite-element and integral-equation results, was presented by Pridmore et al. (1981).

is 6.23% for the TP-FD method and 0.13% for the SP-FDmethod.

The third model is a buried cubic body of dimension2 units and a depth of burial of 0.5 units. The surroundinghost rock has a resistivity of 100 �m and the inhomogene-ity is assigned a resistivity of 20 �m. Pridmore et al. (1981)showed a comparison of its apparent resistivities calculatedby the finite-element and integral-equation methods: the tworesults agree to within 6%. On a 3D grid consisting of 59 ×39 × 20 nodes, the solution obtained by the SSOR methodtakes approximately 38 s in about 200 iterations for eachsource. The relaxation parameter of the SSOR method is 1.4and the tolerance value in the stopping criterion is ε = 10−10,which is the same as that chosen by Spitzer (1995). The ICCGmethod takes approximately 15 s in about 90 iterations, whichis 2.5 times faster than the SSOR method. Moreover, the di-rect method takes over 10 000 s, which is about 700 timesslower than the ICCG method for this model. The dipole–dipole apparent-resistivity pseudosections for a traverse over

the centre of the body are illustrated in Fig. 4. The results cal-culated from our SP-FD method are shown in Fig. 4(a), whileFigs 4(b) and 4(c) show the results from the integral-equationand finite-element methods presented by Pridmore et al.(1981). Figure 5 is a plot of dipole–dipole apparent re-sistivities in line 4 of the three pseudosections, showing acomparison of the results. The apparent resistivities calcu-lated from the finite-difference method almost all lie be-tween the results obtained with the finite-element and integral-equation methods, although it appears that they are closer tothose from the integral-equation method. Reciprocity testingshowed that the integral-equation solutions achieved higheraccuracy than those from the finite-element method, be-cause they satisfy reciprocity almost exactly, while the resultsfrom the finite-element method satisfy reciprocity to withinabout 6% (Pridmore et al. 1981). The reciprocity testing ofour finite-difference solutions was also performed. They sat-isfy reciprocity almost exactly, as did the integral-equationsolutions.



Figure 5 Plot of dipole–dipole apparent resistivities in line 4 of the three pseudosections in Fig. 4.

Both efficiency and accuracy of forward modelling are fun-damental to inversion. The error in the forward modelling hasa significant influence on model resolution in the inversion. Inorder to discuss the problem, we make a small change in thefirst two-layered model. It now becomes a three-layered modelwith a thin resistivity layer as shown in Fig. 6. We simulatethe updated model with 3D TP-FD and SP-FD methods in or-der to investigate whether the computational errors cover theresponse derived from the small change (the thin resistivitylayer). The results are also shown in Fig. 6. It can be seen thatthe potential curve with 0.57% average relative error obtainedwith the SP-FD method almost exactly matches the theoreticalcurve of the three-layered model containing the thin resistivitylayer. However, the potential curve with 7.85% average rela-tive error obtained with the TP-FD method remains almost thesame as that of the two-layered model, which means that wecannot resolve the thin resistivity layer in the updated modelwith the modelling data using the TP-FD method.

C O N C L U S I O N

Several techniques, such as row-indexed sparse storage mode,the incomplete Choleski conjugate-gradient (ICCG) method

and solving for the secondary potential, are combined to de-velop a rapid and accurate 3D FD forward algorithm for DCresistivity modelling. With the increase in the number of gridnodes, the algorithm has an increasingly noticeable advan-tage over the direct method, as regards storage requirementsand run times. Compared with the SSOR method, it is about2.5 times faster.

A two-layered model and a vertical contact are consideredin order to investigate the accuracy of the algorithm; the re-sults show that solving for the secondary potential for 3D DCresistivity modelling achieves an accuracy 10–20 times higherthan that obtained when solving for the total potential directlyin the neighbourhood of the source point where the singularityeffects are the greatest. For a 3D model, our results agree wellwith those of other numerical methods.

Our algorithm appears more efficient compared with3D finite-element resistivity modelling. However, there isno doubt that Zhou and Greenhalgh (2001) made agreat improvement in 3D finite-element modelling withthe tetrahedron element scheme and the improved ICCGsolver, if we consider that the finite-element matricesare always more dense and complex than their FDcounterparts.


576 X. Wu et al.

Figure 6 Comparison of analytical and nu-merical solutions for the updated three-layered model with a thin resistivity layer.The potential curve calculated by the TP-FD method remains almost the same as thatof the two-layered model without the thinresistivity layer.

The accuracy of forward modelling is also fundamental to3D inversion. The error in the forward computation can re-duce the model resolution significantly. Further investigationswill concentrate on the 3D resistivity inversion using the for-ward algorithm described in this study.

A C K N O W L E D G E M E N T S

This work was supported by funds from the Natural SciencesFoundation (No. 40004005) and the Ministry of TerritorialResources of China. The authors are grateful to ProfessorXu Guoming, University of Science and Technology of China,for his helpful corrections. They also thank Professor Richard

Lagabrielle and another reviewer for their excellent commentson the manuscript.

R E F E R E N C E S

Ajiz M.A. and Jennings A. 1984. A robust incomplete Choleski conju-gate gradient algorithm. International Journal of Numerical Meth-ods in Engineering 20, 949–960.

Bentley J. 1986. Programming Pearls. Addison-Wesley Publishing Co.,Reading, MA.

Coggon J.H. 1971. Electromagnetic and electrical modelling by thefinite element method. Geophysics 36, 132–155.

Daniels J.J. 1977. Three-dimensional resistivity and induced polariza-tion modelling using buried electrodes. Geophysics 42, 1006–1019.

Dey A. and Morrison H.F. 1979. Resistivity modelling for arbitrarilyshaped three-dimensional structures. Geophysics 44, 753–780.



Dieter K., Paterson N.R. and Grant F.S. 1969. IP and resistivity typecurves for three-dimensional bodies. Geophysics 34, 615–632.

Fox R.C., Hohmann G.W., Killpack T.J. and Rijo L. 1980. Topo-graphic effects in resistivity and induced-polarization surveys. Geo-physics 45, 75–93.

Fu Liangkui 1983. Lectures on Geoelectrical Prospecting, pp. 50–56.Geology Press, Beijing.

Hestense M. and Stiefel E. 1952. Methods of conjugate gradients forsolving linear systems. Journal of Research of the National Bureauof Standards 49, 409–436.

Hohmann G.W. 1975. Three-dimensional induced polarization andelectromagnetic modelling. Geophysics 40, 309–324.

Holcombe H.T. and Jiracek G.R. 1984. Three-dimensional terraincorrection in resistivity surveys. Geophysics 49, 439–452.

Kershaw D.S. 1978. The incomplete Choleski conjugate gradientmethod for the iterative solution of systems of linear equations.Journal of Computational Physics 26, 43–65.

Lee T. 1975. An integral equation and its solution for some two- andthree-dimensional problems in resistivity and induced polarization.Geophysical Journal of the Royal Astronomical Society 42, 81–95.

Loke M.H. and Barker R.D. 1996. Practical techniques for 3D re-sistivity surveys and data inversion. Geophysical Prospecting 44,499–523.

Manteuffel T.A. 1980. An incomplete factorization technique for pos-itive definite linear systems. Mathematics of Computation 34, 473–497.

Meijerink J.A. and van der Vorst H.A. 1977. An iterative solutionmethod for linear systems of which the coefficient matrix is a sym-metric matrix. Mathematics of Computation 31, 148–162.

Newman G.A. and Alumbaugh D.L. 2000. Three-dimensional magne-totelluric inversion using non-linear conjugate gradients. Geophys-ical Journal International 140, 410–424.

Okabe M. 1981. Boundary element method for the arbitrary inhomo-geneities problem in electrical prospecting. Geophysical Prospecting29, 39–59.

O’Neill D.J. et al. 1984. A digital linear filter for resistivity soundingwith a generalized electrode array. Geophysical Prospecting 32,105–123.

Oppliger G.L. 1984. Three-dimensional terrain corrections for mise-a-la-masse and magnetometric resistivity surveys. Geophysics 49,1718–1729.

Papadrakakis M. and Dracopoulos M.C. 1991. Improving the effi-ciency of incomplete Choleski preconditionings. Communicationsin Applied Numerical Methods 7, 603–612.

Pratt D.A. 1972. The surface integral approach to the solution ofthe 3D resistivity problem. Bulletin of the Australian Society forExploration Geophysics 3, 33–50.

Pridmore D.F., Hohmann G.W., Ward S.H. and Sill W.R. 1981.An investigation of finite element modelling for electrical andelectromagnetic data in three dimensions. Geophysics 46, 1009–1024.

Scribe H. 1981. Computations of the electrical potential in thethree-dimensional structure. Geophysical Prospecting 29, 790–802.

Smith J.T. 1996. Conservative modelling of 3D electromagnetic fields,Part II: biconjugate gradient solution as an accelerator. Geophysics61, 1319–1324.

Spitzer K. 1995. A 3D finite difference algorithm for DC resistivitymodelling using conjugate gradient methods. Geophysical JournalInternational 123, 903–914.

Varga R.S. 1960. Factorization and normalized iterative methods. In:Boundary Problems in Differential Equations (ed. R.E. Langer),pp. 121–142. University of Wisconsin Press, Madison, WI.

Xu S.Z., Guo Z.C. and Zhao S.K. 1988. An integral formulationfor three-dimensional terrain modelling for resistivity surveys. Geo-physics 53, 546–552.

Zhang J., Mackie R.L. and Madden T.R. 1995. Three-dimensional re-sistivity forward modelling and inversion using conjugate gradients.Geophysics 60, 1313–1325.

Zhao S.K. and Yedlin M.J. 1996. Some refinements on the finite-difference method for 3D DC resistivity modelling. Geophysics 61,1301–1307.

Zhou B. and Greenhalgh S.A. 2001. Finite element three-dimensionaldirect current resistivity modelling: accuracy and efficiencyconsiderations. Geophysical Journal International 145, 679–688.


computations of secondary potential for 3d dc resistivity modelling using an incomplete choleski...

Documents