practical techniques for 3d resistivity surveys and data

Geophysical Prospecting, 1996, 44, 499-523

Practical techniques for 3D resistivitysurveys and data inversionl

M.H. Loke2 and R.D. Barker3

Abstract

Techniques to reduce the time needed to carry out 3D resistivity surveys with a

moderate number (25 to 100) of electrodes and the computing time required to

interpret the data have been developed. The electrodes in a 3D survey are normally

arranged in a square grid and the pole-pole array is used to make the potential

measurements. The number of measurements required can be reduced to about one-

third of the maximum possible number without seriously degrading the resolution of

the resulting inversion model by making measurements along the horizontal, vertical

and 45' diagonal rows of electrodes passing through the current electrode'

The smoothness-constrained least-squares inversion method is used for the data

interpretation. The computing time required by this technique can be greatly

reduced by using a homogeneous half-space as the starting model so that the Jacobian

matrix of partial derivatives can be calculated analytically. A quasi-Newton updating

method is then used to estimate the partial derivatives for subsequent iterations'

This inversion technique has been tested on synthetic and field data where a

satisfactory model is obtained using a modest amount of computer time' On an

80486DX2166 microcomputer, it takes about 20 minutes to invert the data from a 7

by 7 electrode survey grid. using the techniques described below, 3D resistivity

surveys and data inversion can be carried out using commercially available field

equipment and an inexpensive microcomputer'

ln t roduct ion

Although all geological structures

resistivity surveys are carried out

conventional one-dimensional (1D)

are three-dimensional (3D) in nature, 3D

rarely at the present time compared to

resistivity sounding (Koefoed 1979) or even

l Paper presented at the 57th EAEG Meeting, May-June

revision accepted August 1995.2School of Physics, Universiti Sains Malaysia, 11800

Formerly at 3.

I School of Earth Sciences, The University of Birmingham, Birmingham

[email protected])

O 1996 European Association of Geoscientists & Engineers

1995, Glasgow, UK. Received November 1994,

Penang, Malaysia. (Email: [email protected]).

B15 2TT, U.K. (Email :

499

500 M.H. Loke and R.D. Barker

two-dimensional (2D) resistivity tomography (Griffiths and Barker 1993) surveys.This is probably due to the large number of field measurements needed for 3Dsurveys and the lack of suitable automatic inversion software. Significant progresshas been made within the last decade in the development of field equipmentand techniques for carrying out 2D resistivity tomography surveys (Griffiths,Turnbull and Olayinka 1990) and fast inversion algorithms to interpret the data(Loke and Barker 1996). A 2D model for the subsurface gives reasonably accurateresults in areas with elongated geological structures. However, more complexstructures (Park and van 1991) can only be adequately represented by a 3Dsubsurface model.

Most current 3D resistivity inversion techniques are either approximate methodsor they require powerful computing facilities. A one step inversion technique usingthe Born approximation was used by Li and Oldenburg (lgg2) ro give a useful initialmodel for the subsurface. However, as indicated by the authors, it should beregarded as an initial model which can be further improved by using an iterativemethod. A fast iterative inversion technique using alpha centres has also been used(Petrick, Sill and ward 1981). Since this technique uses an approximate methodto calculate the model apparent resistivity values, the results are likely to be lessaccurate than a technique which uses the finite-element or finite-difference methodsfor the forward modelling calculations. The finite-element method was used byShima (1992) to improve an initial model obtained by the alpha centres method. Parkand Van (1991) used the finite-difference method and the total least-squaresinversion method to inverr the data obtained with a 5 by 5 rectangular grid ofelectrodes. Recently, Sasaki (1994) described the use of full and approximate 3Dinversion methods using the finite-element method. The inversion techniques byshima (1992), Park and Van (1991) and Sasaki (1994) were implemented onmainframe computers due to the large number of computations needed. However,such powerful computing resources are not conveniently available to smallgeophysical companies, particularly in third world countries. For the same reason,it would not be practical to carry out the data interpretation in the field during thecourse of a survey.

The two main problems in carrying out 3D surveys on a regular basis are thelack of suitable field equipment and the requirement of powerful computers toprocess the data. To overcome the first problem, we examine ways to adapt existingfield equipment used for 2D surveys to carry out 3D surveys. To overcome thesecond problem, a fast 3D inversion program using a quasi-Newton optimizationmethod (Broyden 1972) was developed which can be used on commonly availablemicrocomputers.

In the next section, the field techniques are described. A brief descriptionof the quasi-Newton inversion method is then given. This is followed by adiscussion on ways to reduce the computing time required by the finite-differencemethod. Finally, two examples of the results obtained by the inversion programare given.

O 1996 European Association of Geoscientists & Engineers, Geophysical Prospecting, 44, 499-523

3D resistiztity suraeYs 501

R E s i s t i v i t yM e l e r

L a p t o pC o m p u l e l

,- 0,rt",,"" I

I E l e

Figure 1. Arrangement of eiectrodes along a multicore cable for a 3D resistivity survey.

Method

3D field surveys

Two-dimensional resistivity surveys are usually carried out with a computer-

controlled multiplexer system with a number of electrodes connected to a multicore

cable. One such system is the Campus Imager System where each electrode along the

cable can be used as a current or potential electrode (Griffiths et al. l99O; Griffiths

and Barker 1993). Our present interest is in 3D surveys using a moderate number (25

to 100) of electrodes which can be carried out with this type of field equipment'

Figure 1 shows one possible arrangement of the electrodes for a 3D survey using a

25-electrode system. For convenience the electrodes are usually arranged in a square

grid with the same unit electrode spacing in the x- andy-directions. In order to map

slightly elongated bodies, a rectangular grid with different numbers of electrodes

and spacings in the x- and 1-directions could be used. The pole-pole electrode

configuration is commonly used for 3D surveys, such as the E-SCAN method (Li

and Oldenbvg 1992). The maximum number of independent measurements '?ma{

that can be made with z. electrodes (Xu and Noel 1993) is given by

i l^ax: n"(n, - 1) l2 '

1996 European Association of Geoscientists & Engineers, Geophysical Prospecting,44' 499-523


a ) .

Ia

16x+

5a

t t . -i(

2+',\

- . 14

. X

7a

' r 1 5f

c u r r e n te l e c t r o d e

p o t e n t i a le l e c t r o d e

. 2 0X

\25

,r(

X

b l _

22x

6

1 1

r c u r r e n t 1 6 . 'T e l e c t r o d r X

7

1 2 t '

X

L7

" 1 4X

2524

I+ - - -

/ i \/ i

/ t \

13

l8x

;23r l z,,\

9 1 0-x------'x

15

1 9 _ r , 2 0

r {

p o l e n t i a l, r ( e l e c t r o d e

zL 22a a

Figure 2' The location of potential electrodes corresponding to a single current electrodein the arrangement used by (a) a survey to measure the complete data set and (b) a cross-diagonal survey.

In this case, each electrode in turn is used as a current electrode and the potentialsat all the other electrodes are measured. Note that because of reciprocity, it is onlynecessary to measure the potentials at the electrodes with a higher index numberthan the current electrode in Fig. 2a. For a 5 by 5 electrode grid, a complete dataset (xu and Noel 1993) will have 300 datum points. For 7 by 7 and l0 by 10 electrodegrids, the numbers of measurements are 1176 and 4500 respectively. It can be verytime-consuming to make such a large number of measurements with typical single-channel resistivity meters commonly used for 2D surveys. For example, it could take

O 1996 European Association of Geoscientists & Engineers, Geophysical Prospecting,44,499-523

3D resistiaity suraeys 503

o E l e c t r o d e

y - d i r e c t i o n

Layer I

Layer 2

Layer 3

Figure 3. The arrangement of the blocks in the 3D model used. In calculating the apparent

resistivity values and their partial derivatives, the blocks at the sides are extended horizontally

sideways and the base ofthe bottom layer ofblocks is extended downwards to the edges ofthe

finite-difference mesh used.

several hours to make the 1176 measurements for a7 by 7 survey grid with a standard

low-frequency earth resistance meter, although speed is partly determined by the

grid spacing and the magnitude of the measured resistances (Griffiths and Barker

1993). To reduce the number of measurements required without seriously degrading

the quality of the model obtained, an alternative measurement sequence has been

tested (Fig. 2b). In this proposed 'cross-diagonal survey' technique, the potential

measurements are only made at the electrodes along the horizontal, vertical and the

45' diagonal lines passing through the current electrode. The number of datum

points with this arrangement for a 7 by 7 grid is reduced to 476.

S mo othne s s - constr ained I e as t - s quar e s inv er sion met ho d

The 3D model used to interpret the data is shown in Fig. 3. The subsurface is divided

into several layers and each layer is further subdivided into a number of rectangular

blocks. The interior blocks within each layer have the same size. The blocks at the

sides are extended horizontally sideways and the base of the bottom layer of blocks

is extended downwards to the edges of the finite-difference mesh used to calculate

O 1996 European Association of Geoscientists & Engineers, Geophysical Prospecting, 44' 499-523

x - d i r e c t i o n


the apparent resistivity values. Note that the upper surfaces of the interior blocks inthe topmost layer have an electrode at each corner. The thickness of the top layer isset at 0.70 times the unit electrode spacing used, which is slightly smaller than themedian depth of investigation (0.87 times) of the pole-pole array for a homogeneoushalf-space (Edwards 1977).From a number of tests that were carried out, we havefound that the inversion program could take a much longer time to converge if thethickness of the top layer is significantly less than 0.70 times the unit electrodespacing. If a thicker top layer is used, the near-surface resistivity variations are notwell resolved. Since the resolution of the resistivity measurements decreases withdepth, the thickness of each subsequent lower layer is increased by l5%. Thenumber of layers in the model used is adjusted such that the depth to the top surfaceof the bottom layer is less than the median depth of investigation of the largestelectrode spacing along a line of electrodes. As an example, for a 7 by 7 survey grid,the largest electrode spacing along the x- ory-direction is 6 times the unit electrodespacing used.

The smoothness-constrained least-squares method has been widely used in theinversion of geoelectrical data using 2D and 3D subsurface models (Sasaki 1989,1994; deGroot-Hedlin and Constable 1990). The system of normal equations used inthis method is given by

(JIJ; + ,\ocrc)po : J-[ et, , )

where r is the iteration number, J; is the Jacobian matrix of partial derivatives, g; isthe discrepancy vector which contains the differences between the logarithms ofthe measured and calculated apparent resistivity values, ); is the damping factor andp; is the perturbation vector to the model parameters for the ith iteration. Theflatness filter C is used to constrain the smoothness of the perturbations to the modelparameters to some constant value. The amplitudes of the elements of the flatnessfilter matrix C are increased by about 15o% for each deeper row to stabilize theinversion process.

The smoothness-constrained least-squares method has proved to be a robusttechnique which converges rapidly. The main problem in using the conventionalleast-squares method on microcomputers for 2D and 3D problems is the time andmemory space required to calculate the elements of the Jacobian matrix J; using thefinite-element or finite-difference method (Sasaki 1989; Park and van 1991). Lokeand Barker (1996) have overcome this problem in 2D resistivity inversion by usinga homogeneous half-space as the starting model so that the partial derivatives can becalculated analytically. The partial derivative of the potential / due to a change inthe resistivity of a volume element Z in a homogeneous half-space with resistivity p isgiven (Loke and Barker 1995) by

x ( x - a ) + y ( y - b ) + 2 2

+ i r + ; T r ( f f i d x d Y d z , (2)I I

4nz .l

oQ

op v ( * 2



z - d i r e c t i o n

( x r . Y r


- d i r e c l i o n

( x2 . y2 .221

Figure 4. The parameters used in the calculation of the partial derivative for a singlerectangular block within the 3D model used.

where the current source 1is located at the origin (0,0,0) and the potential electrode

is located at (a,b,0). For a rectangular block in the model (Fig. 4), the above volume

integral can be determined by using Gaussian quadrature (Churchhouse 1981) which

uses the following approximation:

/ 1 \aoop

r l f n " n t i l t

= "=t \ - \ - r r . r , -- + r 2 k L / . _ r t J t J R l u 2

R = I l = t I : r

u(u - a) -t z;(zt - P) + *t

+ a2 + *t) t t l (u - a)2 + @ - i l ' + wzlr 's ' �

where

y : ( 2 x - x 1 - x 2 ) f @ 1 - x ) ,

a : ( 2y - y r - y ) l ( ! t - ! z ) ,

q : ( 2 a - x 1 - x ) l @ 1 - x ) ,

P : ( 2 b - i r - v ) l ( v r - v ) ,

, 1 : ( 2 2 * 2 1 - z ) f ( 2 1 - z ) ,

V :0.125(xt - x) (y t - y) (z t - zz) .

The top, left, back corner of the block is located at(x1,y1,21) and the opposite corner

isat(x2,y2,2) .Theparameters n, ,noandnrarethenumberof funct ionevaluat ions

in the x-, y- and e-directions respectively and fi, fi and f1, are the weights

(corresponding to the values ef n,, no and n, used) which are multiplied by the

function values to obtain the value of the integral. Tables with the weights (/) and

abscissae (u, a, w) for several values of function evaluations (n) may be found in

Churchhouse (1981). When the rectangular block is next to the current or potential

electrode, n,, n, and n, are all set to 6 which gives a total of 216 function evaluations.

The number of function evaluations in the x-, y- and e-directions are progressively

reduced as the minimum distance of the block from either electrode increases. Since


p


the term within the triple summation in (3) only involves simple rational functions,each function evaluation can be calculated rapidly.

The method described above gives the partial derivative values for the startinghomogeneous earth model. For subsequent iterations, the partials derivatives areestimated by using a quasi-Newton updating method (Bourji and \Talker 1990). TheJacobian matrix J; for the ith iteration is replaced by an approximation B;. Thefollowing updating equation by Broyden (1965) was used:

B;+r : B; + u;pf ,where

e t r ; : (Ay ; *B ;p ) l pTp*

A Y ; : Y t + t - Y t ,

and y; is the model response for the ith iteration. The approximation of the Jacobianmatrix B;*1 for the (f +,/)th iteration is calculated using the Jacobian matrixapproximation B;, the parameter perturbation vector p; and the change in the modelresponse Ay, for the ith iteration. Since the updating matrix u;pf ir a rank-onematrix, the time taken to solve the least-squares equation (1) can be substantiallyreduced by making use of matrix updating techniques (Golub and van Loan 1989;Loke and Barker 1996) associated with the numerical technique used to solve theleast-squares equation.

O p timi zing the finit e- diff er enc e metho d

The potential distribution d(x) due to a point current source /(xs) is given by

V. lo(x)Vt'(*) l : -16(' - x,), (5)

where o(x) is the subsurface conductivity distribution. In the finite-differencemethod, the subsurface is discretized by using a rectangular mesh (Fig. 5) withirregular spacing of the nodes in the x-, y- and a-directions (Dey and Morrison1979). We use a 3D mesh where the spacing between the nodes in the x- and y-directions in the central region of the mesh is one-half the smallest electrode spacing.Towards the sides and bottom of the mesh, the spacings between the nodes areprogressively increased to simulate the 'infinitely distant' edges of the subsurfacemodel. The rectangular elements within the mesh can have different conductivityvalues so that practically any subsurface distribution can be modelled by thismethod. The integrated finite-difference approach by Dey and Morrison (1979),which calculates the volume integral of (5) around each node, leads to the followingmatrix equation:

A d : s , ( 6 )

where the capacitance matrix A is a septadiagonal, positive definite symmetricmatrix. The elements of this matrix depend only on the geometry and conductivity

O 1996 European Association of Geoscientists & Engineers, Geophysical Prospecting, 44,499-523

jD resistiaity sun)eYs 507

a )

b )

N o d e s o n f i r s tO s e t o f d i a g o n a l

p l a n e s

N o d e s o n s e c o n dr ; r s e t o l d i a g o n a l

p l a n e s

Figure 5. Finite difference grid with (a) natural column-by-colurnn ordering and (b)

alternate-plane ordering of the nodes.

o 1996 European Association of Geoscientists & Engineers, Geophysical Prospecting' 44' 499-523

508 M.H. Loke and R.D. Barher

Table 1 ' Comparison of memory required by band-Cholesky with natural column-by-columnnodes ordering and the alternate-plane ordering using a general sparse matrix storage system.The memory ratio gives the ratio of the memory required by the alternate-plane orderingcompared to the band method.

Electrodeconfiguration

Total numberof nodes

MeshSIZC

Memory required in kilobytes MemoryBand method Alternate-plane Ratio

5 b y 57 b y 7

10 by 10

2 3 x 2 3 x 92 7 x 2 7 x 1 l3 3 x 3 3 x 1 3

47618019

r4 t57

3868.39335.8

23779.3

1510 .8 0 .393483.5 0.378584.2 0.36

distribution within the grid, and it does not depend on the current source vector s.The vector @ contains the potentials at the nodes. The most convenient method tocalculate all the possible apparent resistivity measurements for the electrodearrangement shown in Fig. 2 is to calculate the potential distribution due to asingle current electrode at each of the electrode locations. This means that for a 5 by 5survey grid, it is necessary to solve the capacitance matrix equation (6) to determinethe potential vectors for 24 diffetent current source vectors. For such a large numberof different current source vectors, it might be faster to use a direct method (ratherthan an iterative method) to solve the capacitance matrix equation (Dey andMorrison 1979). A commonly used direct method is the band-Cholesky technique(Martin and Wilkinson 1965) which first carries our the decomposition

A : R r R . Q )

The above decomposition is only carried out once for a single subsurface model. Inorder to determine the potential vector @ for each current source vector s, (7) is firstsolved by using the forward substitution step

R ' q : s ,

followed by the backward substitution step

Rd: q . (e )

The time taken for the forward substitution step can be greatly reduced by takingadvantage of the sparsity of the current source vector s. The time taken by thebackward substitution step can be reduced by obtaining a partial solution ofthe potential vector @ (Loke and Barker 1996). As an example, Fig. 5a shows rhenodes in a4 x 4 x 4 finite-difference mesh. If the current electrode is at node 24,thenthe first 23 elements of the s vector are zero. It is only necessary to carry out theforward substitution steps for nodes 24 to 64 since the first 23 elements of the qvector in (8) are equal to zero. If the potential electrode is at node 56 (Fig. 5a), it isonly necessary to carry out the back substitution step for nodes 64 to 56. Note thatto calculate the apparent resistivity values, it is only necessary to determine thepotentials at the nodes with the electrodes. Flowever, if the partial derivatives are


(8)

3D resistiaitY suraeYs 509

directly calculated with the finite-difference method, it would be necessary to

calculate the potentials at all the nodes.

The main disadvantage of using a direct method for solving the capacitance matrix

equation for 3D problems is the huge amount of memory needed compared to

iterative methods. The amount of memory needed depends on the number of nodes

in the 3D mesh used. The total number of nodes depends on the number of nodes per

unit electrode spacing. In general, a finer mesh will result in more accurate potential

values calculated by the finite-difference method. Dey and Morrison (1979) used a

mesh with four nodes between adjacent electrodes, while Park and Van (1991) used

only one node per unit electrode spacing. \We used a mesh with two nodes per unit

electrode spacing in an attempt to strike a balance between reducing the memory

required and the errors in the calculated potential values. The resulting finite-

difference meshes used to model the data for three different survey electrode

configurations are shown in Table 1 which also shows the computer memory

required by the band-Cholesky method (Martin and Srilkinson 1965)' About 23

megabytes of memory would be needed to model the data from a 10 by 10 electrode

survey grid! Thus we have examined the use of sparse matrix techniques (George and

Liu 1981) to reduce the memory needed to a more manageable level. Sparse matrix

methods basically attempt to reduce the number of non-zero elements in the

Cholesky decomposition matrix R in (7) by permuting the ordering of the node

numbers in the mesh. Figure 5a shows the 'natural' ordering for a 4 x 4 x 4

rectangular mesh used for the band-Cholesky method'

One simple but effective method widely used for 2D finite-difference rectangular

grids is the 'black-red' or alternate-diagonal ordering (Woo er al. 1976). A 3D version

or anir type of ordering is shown in Fig. 5b where the nodes are separated into two

sets of diagonal planes. The neighbouring nodes around any node lie on a different

set of diagonal planes. The total memory required for the finite-difference mesh

using the alternate-plane ordering is also listed in Table 1' To store the non-zero

elements of the R matrix, the general sparse matrix allocation scheme described by

Eisenstat, Schultz and Sherman (1976) and George and Liu (1981) is used' The total

storage required by the alternate-plane ordering is less than 40 o/o ofthat required by

the band method.The time required by the forward and backward substitution steps in (8) and (9)

can be further reduced by carefully organizing the location of the nodes with the

electrodes so that they end up with larger node numbers. For the alternate-plane

ordering system, this can be achieved by placing the nodes with the electrodes on

the second set of diagonal planes. As an example, the node numbers for the current

and potential electrodes in Fig. 5b have been increased to 50 and 63 respectively

(compared to 24 and 56 in Fig. 5a) which reduces the time required by the

substitution steps by more than half. The computer time and memory required can

be further reduced by making use of more sophisticated sparse matrix ordering

techniques such as the nested dissection and one-way dissection techniques (George

and Liu 1981).

O 1996 European Association of Geoscientists & Engineers, Geophysical Prospecting,44' 499-523


Table 2. Parameters of rectangular blocks test model. The resistivity of thesurrounding medium is 10f,1m.

Blockx-dimensions

m,y-dimensions

m."a-dimensions Resistivity

m . 0 m

0.0 to 0.700.70 to 2.43

1001

On the 8O486DX/66 microcomputerJ the finite-difference subroutine took about42 s to calculate all the possible potential measurements for a 5 by 5 survey grid, andabout 159 s for a 7 by 7 survey grid. The decomposition and solution steps took about59o/o and 367o respectively of the total time. The remaining5oh of the total time wasused in setting up the sparse matrix array structures and calculating the elements ofthe capacitance matrix. To check the accuracy of the finite-difference program, weused a two-layer test model where the resistivities of the top and bottom layers are 10.0and 1.0 fl m respectively. The thickness of the top layer is 1.51 m and the unit electrodespacing is 1.0 m. The potential values for the pole-pole array were compared with thosecalculated using the method described by Mooney et al. (1966) which has an accuracyof 0.loh. The maximum error in the potential values calculated by the finite-differencemethod was found to be about 4o/o for this test model.

The ina er sion alg orithm

Since we use an algorithm similar to that described in detail by Loke and Barker(1996)' only a very brief description is given here. The resistivity of the startinghomogeneous earth model is calculated by taking the average of the logarithms of themeasured apparent resistivity values. An initial damping factor )e and a minimumdamping factor )- for the least-squares method is then selected. From a numberof tests carried out with synthetic and field data sets, we have found that values of0.20 and 0.03 for )s and )- usually give satisfactory results. However, in thecomputer program used, the damping factor values could be changed by the user ifnecessary. In each iteration, the change in the model resistivity values and Jacobianmatrix are calculated using (1) and (4). After the first two iterations, the dampingfactor ) is reduced by half if it is still larger than the minimum damping factor )mselected. If the inversion process fails to reduce the apparent resistivity root-mean-square (rms) error, a line search using quadratic interpolation is first used to findthe optimum step size for the model perturbation vector (Daniels 1978). If this alsofails to reduce the rms error, the damping factor is increased. The inversion processis usually stopped when the relative change in the rms error is less than 57o (Loke andBarker 1996), or when the maximum number of iterations (usually about 10) isreached. Although a relatively simple algorithm is used, it has given satisfactoryresults for the computer generated and field data sets that we have used. However,


UpperLower

1 .0 t o 5 .01.0 to 4.0

1.0 to 2.02.0 to 5.0

3D resistiaity suroeYs 511

T w o R e c t a n g u l a r E l o c k s m o d e l

y - d i r e c l i o n ,

o - 0

L o w e r B l o c k

3 . 0 4 . 0 5 . 0 m .

f l p p o r e n t S e s i s t i u i t U P s e u d o s e c t i o n


0 - 0 1 . 0 2 - O 3 - 0 4 - 0 5 . 0 6 . 0 m

f l p p a r e n t R e s i s t i u i t U P s e u d o s e c l i o n ( 5 % n o i s e l

f-l L-:f I--:.::l m f:::'::l ffi ffi E I5 , 0 ? , 0 9 . 0 1 1 ' 0

R*lstivllY in om

Figure 6 (a) Apparent resistivity pseudosection along the line of electrodes at x:3.0m for

the test model with two rectangular blocks in a homogeneous medium' (b) The apparent

resistivity pseudosection obtained after adding 57o random noise to the data.

for some data sets, better results might be obtained by using a more sophisticated

inversion algorithm (Dennis and Schnabel 1983).

Resul ts

$7e present the results obtained from the inversion of a synthetic data set and a field

data set. For the synthetic data set, the models obtained using the complete data set

and the reduced data set using the cross-diagonal survey technique are shown. All the

calculations were carried out on an 80486DX2/66 microcomputer with 20 megabytes

of memory.

Example 1: two rectangular blocks

This model consists of two rectangular blocks of resistivit ies 100Om and 1f,lm

embedded in a homogeneous half-space with a resistivity of 10 f,) m. The parameters

of the rectangular blocks are listed in Table 2. The survey grid consists of a7 by 7

(a)

t{

1

2

3

4

5

5

(b)

II

I

2

3

4

5

6

U n i t e l e c t r o d e s P a c i n g 1 - 0 m

A g96 European Association of Geoscientists & Engineers, Geophysical Prospecting, 44, 499-523

512 M.H. Loke and R.D. Barher

I t e r a t i o n 5 - R M S E r r o r 5 - 5 %

L a y e r I0 . 5 0 2 - 5 0 4 . 5 0

0 . 5 0

2 . 5 0

4 . 5 0

D e p t h : 0 - 0 t o

L a y e r 30 . 5 0 2 - 5 0

0 . 7 0 m -

4 - 5 00 - 5 0

2 - 5 0

4 - 5 0

D e p t h - 1 - 5 0 t o 2 - 4 3L a y e r 5

0 - 5 0 2 . 5 0 4 _ 5 00 . 5 0

2 . 5 0

4 - 5 0

T i m e t a k e n - I 3 5 5 s .

L a y e r 20 - 5 0 2 . 5 0 4 - 5 0

D e p t h : 0 . 7 0 t o 1 . 5 0 m .L a y e r 4

[ . 5 0 2 . 5 0 4 _ 5 0

0 _ 5 0

2 _ 5 0

4 - 5 0

0 . 5 0

2 - 5 0

4 . 5 0

m - D e p t h : 2 . 4 3 t o 3 , 4 9 m .L a y e r 5

0 . 5 0 2 . 5 0 4 _ 5 00 - 5 0

2 - 5 0

4 . 5 0

D e p t h : 3 . 4 9 t o 4 . 7 2 m D e p t h . 4 . 7 2 l o 6 . 1 3 m

U n i t e l e c t r o d e s P a c i n g 1 . 0 m -

f--1 E:-] ET.rl F'�iin r::=t ffi m E I2 . 5 1 0 , 0 4 0 . 0 1 6 0 - 0

ResistivitY in Om

Figure 7. The modei obtained from the inversion of the rectangular blocks model completedata set (with 1176 datum points) with 5% noise after 6 iterations. Note that in the true model,the upper block is confined to the top layer while the lower block is conflned to the second andthird layers. The outlines of the rectangular blocks are also shown for comparison.

grid with a unit electrode spacing of 1.0 m. The apparent resistivity values for all thepossible pole-pole combinations were calculated using the finite-difference program.

Figure 6a shows the pole-pole apparent resistivity pseudosection along the line of

electrodes at x equal to 3.0 m (i.e. in the y-z plane) which cuts across both rectangular

O 1996 European Association of Geoscientists & Engineers, Geophysical Prospecting,44, 499-523

' i : i : i i , ,iiiiiit::

ii:ii

:;i:i+::ri-:;-ii: : :: l : : : : : : : : : : : . l : : :. 1 . 1 . . .. : l : : : : : : : : : : : l : : :. . 1 t . . .: l : : : : : : : : : : : f : :I : : : : : : : : : : r " . .{ t , . , '

' . t . t . . .: . 1 : - : : : : : : : : : : . . 1 : . : :. 1 . . . . . t . . .. 1 . . . , . i 1 . . . 1 . : . : . :

, l li i !

ii

. :

i i i i

, ' : :l : l :

: :iiiji i

' : , '

I t e r a t i o n 6 - R M S E r r o r 5

X-Z Pleno I0 . 5 0 2 . 5 0 4 - 5 0

0 . 3 5 . l

1 , 1 0

3D resistiaity sur?q)s 513

T i m e t a k e n - 1 3 6 6 s .

X-ZPlane20 . 5 0 2 . 5 0 4 . 5 0

Y D i s t . : I - 0 t o 2 - 0

X-Z Plane 40 - s 0 2 . 5 0 4 . 5 0

5 %

Depth

m -

l - s 7

2 - 9 6

4 - r 1

0 _ 3 5

l . l 0

I - S 7

2 . S 6

4 . 1 I

5 . 4 ?

De

Pth

Depth

m .

Depth

m -

5 - 4 2 l l i ! i i i ! ; i ! i i i i i ! : i i ! : : i ! : ! i ! , : i : : i i ! i lY D i s t - : 0 . 0 t o 1 . 0 m -

X-Z Plane 30 . 5 0 2 . 5 0 4 . 5 0

0 . 3 5l . t 0

1 - 9 7

2 . S 6

4 . 1 1

5 -42D i s t . : 2 . 0 t o 3 - 0 m .X-Z Plane 5

0 - 5 0 2 - 5 0 4 - 5 0

0 . 3 5

I _ 1 0

l . s 7

2 . 3 5

4 . 1 1

5 . 4 2

.. r+{ii=i:jiii ::'.:l: . ::

: : l l : : : . . . . : : : : ' l : : : : : : : : :. : l : : : : : - : l : : : :

' i l : . l i , , , i . ' ' , : [ , : , , , , , ' ,

; i j . : : : : . : : . : : : . : : : : . '

l i : i : i . : : . . : : . : : : : , i li : i i : i : i ; : : : : : : : l i i i :i l ! : i i : i i : ; r : : : : . : l r r i i ! i i i li i : i i : i i | ! : ! i ! . : , : . . i : i : i : : : : : : !

X-Z Plane 60 . 5 0 2 . 5 0 4 - 5 0

D i s t . : 5 . 0 t o 6 . 0 m

0 - 3 5

1 . 1 0

I _ S 7

2 _ S 5

4 - 1 1

5 . 4 2

0 . 3 5

1 . 1 0

r . 9 7

2 . 9 6

4 - 1 1

5 . 4 2

Depth

Depth

m .

D i s t . : 4 - 0 t o 5 - 0 m -

U n i t e l e c t r o d e s P a c i n g 1 - 0 m -

f----'l f-:f n::i.l ll-ir-ll l!r--=ljTl ffi m E I2 . 5 1 0 . 0 4 0 . 0 1 5 0 - t l

Resistlvlty In om

Figure 8. Vertical cross-sections of the model obtained from the inversion of the complete

data set with 1176 datum points for the rectangular blocks model.

blocks. The upper high resistivity rectangular block causes a prominent low resistivity

anomaly at the first datum level (corresponding to an electrode spacing of 1.0 m) which

is flanked by two linear slanting high resistivity anomalies. This model shows an

example of the 'anomaly inversion' phenomenon frequently observed in borehole logs

where a body with a higher resistivity than the surrounding medium results in lower

aC) 1996 European Association of Geoscientists & Engineers, Geophysical Prospecting,44,499-523

: i i i : i i ii i i : i i : :: . i : ' . . '

:i:i i: i:i ! i i i ! i ii : i i : i i !

i i i i: :',ii: : . :

. i : . :: : : : ii : : : :

illii

i :''

: :; :

. :

ii

: : , : : 1 i

: ! ; . . '

! i i i i : i: i i ! a ! '

i j i ; : i i i : i i. . . . . . : . .

i ; i ; . . 1 : ;

i i l i ; ; : : : :

i : i : i i i : i t ii : i i ! ! : i i i i! ! ! i i : i : i ! i

! : : : i i : : t ! i! : ! i : i ; a i i i

. : :

. . ' :' : :: : : .;',l i

ii

::

ii

:

ii

: .

i : l :' . . .: :

: . : .. : :. . . :. : :. : . :: l : l

: . : . .. i . i : :

: . : . :; i ; . .' : .. : . . i i i

D i s t . : 3 - 0 t o 4 - 0 m


3D Invarsion crror curvaa

- - -3 - - - -

Rectangular block! (Complate det set)Rcctangular blockr (Cro3s.dlagonal rurvey)

Garden field telt (Cross-dlagonal lurvey)

1 5

R M S

t 0

E r r o r

%

5

1 2 ' , , . l . r , o t n " " X , o . r t I I 1 0

Figure 9, Change of the rms error with iteration number for the inversion of the two data setsfor the rectangular blocks model, and for the inversion of the Birmingham field survey data.

apparent resistivity values when it has dimensions smaller than the electrode spacing.The anomaly of the upper block is superimposed on the anomaly caused by the lowerblock on the lower right side of the pseudosection. Due to the complex and overlappinganomalies produced by the two blocks, a simple inspection of the apparent resistivitypseudosection would not be sufficient to obtain an accurate picture of the truesubsurface resistivity distribution.

In order to test the possible behaviour of the inversion method on field data, 5o/orandom noise (Press et al. 1988) was added to the apparent resistivity data whichcaused some distortion of the pseudosection. The result is shown in Fig. 6b. Theinversion model used consists of 216 rectangular blocks. The subsurface is dividedinto 6 layers and each layer is subdivided into 36 blocks. In the model obtained at the6th iteration from the inversion ofthis data set (Fig. 7), the upper rectangular block isclearly shown. The lower rectangular block shows up as a region of low resistivity(less than 5 f2 m) in the 2nd and 3rd model layers. Figure 8 shows the vertical cross-sectional plots (along the x-z plane) of the inverse model. The sides of the upperrectangular block are well resolved. The boundaries of the lower block are not as wellresolved but the model shows the correct location and approximate size of the block.The change in the apparent resistivity rms error with iteration number is shown inFig. 9. The initial homogeneous earth model has a rms error of about 15.57o which


0 . 5 0 2 . 5 r 4 . 5 0

I t e r a t i o n 5 - R M S E r r o t 5 . 5 Y "

L a Y e r I

0 . 5 0

2 - 5 0

4 - 5 0

D e p t h : 0 . 0 t o

L a Y e r0 . 5 0 2 . 5 0

0 - 7 0 m .

34 - 5 [

0 . 5 0

2 . 5 0

4 . 5 0

D e p t h : 1 . 5 0 t o ? - 4 3 m .

L a Y e r 50 . 5 0 2 . 5 0 4 - 5 0

0 . 5 0

2 - 5 0

4 _ 5 0

3D resistittity suraq)s 515

T i m e t a k e n - l l l S s -L a Y e r 2

0 . 5 0 z - 5 0 4 . 5 00 . 5 0

2 - 5 0

4 , 5 0

0 . 5 0

D e p t h : 0 . 7 0 t o 1 - 5 0 m

L a Y e r 40 . 5 0 2 - 5 0 4 . 5 0

2 . 5 0

4 . 5 0

0 . 5 0

2 . 5 0

4 - 5 0

D e p t h : 2 . 4 3 t o 3 - 4 9 m "L a Y e r 6

0 . 5 0 2 . 5 0 4 - 5 0

D e p t h : 4 . 7 2 t o 6 - 1 3 m

U n i t e l e c t r o d e s P a c i n g l - 0 m '

T-1 [-:-l r.il nTj-i] lE-Fl @ q ry Iz . s t o - 0 4 0 . 0 1 6 0 - 0

Reslstlvlty ln Om

Figure 10. Horizontal cross-sections of the model obtained from the inversion of the

rectangular blocks cross-diagonal survey data set with 57o noise after 6 iterations.

reduced to 5.2Yo after 9 iterations. The largest decrease in the rms error occurs in

the first few iterations. The inversion process would normally be stopped at the 6th

iteration since there is only a small change in the rms error after the 5th iteration.

The horizontal and vertical sections of the model obtained from the inversion of

the cross-diagonal survey data set are shown in Figs 10 and 11. The results are similar

O 1996 European Association of Geoscientists & Engineers, Geophysical Prospecting, 44,499-523

- : . : - i ' i

i : i !: i i j',:i !

i : i l ; i i , :i i : i i : i i i

D e p t h : 3 . 4 9 t o 4 - 7 2 m -


I t e r a t i o n 5 - R M S E r r o r 5 . 5 %X-Z Plane 1

0 . 5 0 2 . 5 0 4 . 5 00 _ 3 5 - lI - 1 0

m - 4 - l I

5 . 4 2

Depth

m .

Depth

Depth

m .

Depth

r . 9 7

2 . S 6

0 - 3 5l . l 0

1 _ S 7

z . s 6

4 . 1 1

5 - 4 2

0 . 3 5

l - 1 0

1 . 3 7

2 - 9 5

4 - 1 1

5 . 4 2

0 - 3 5

1 . 1 0

1 . 9 7

2 _ S 6

4 . 1 I

5 . 4 2

Y D i s t . : 0 . 0 t o 1 . 0 mX-Z Plane 3

0 . 5 0 2 . 5 0 4 . 5 0l4

Y D i s t . : 1 . 0 t o 2 . 0 m .X-Z Plane 4

0 - 5 0 2 . 5 0 4 . 5 0|...............-..',g

Depth

m -

Depth

m -

0 . 3 5

1 . 1 0

1 . S 7

2 . S 6

4 . 1 I

5 . 4 2 .Y D i s t . : 2 . 0 t o 3 . 0 m

X-Z Plane 50 . 5 0 2 . 5 0 4 . 5 t t

Y D i s t . : 3 . 0 t o 4 - 0X-Z Plano 6

0 . 5 0 2 . 5 0 4 . 5 0

m -

0 . 3 5l . l 0

1 . 9 7

2 - S 5

4 . 1 I

5 . 4 2

Figure 11. Ver t ica ldiagonal survey data

D i s t . : 4 - 0 t o 5 . 0 m - Y D i s t - : 5 - 0 t o 6 . 0 m .

U n i t e l e c t r o d e s p a c i n g I - 0 m .f---l f--l f.:,-:1 l'iilTl F.:i-A € m E I

2 . 5 1 0 . 0 4 0 . 0 1 6 0 - 0

Reelstivity in Om

cross-sections of the model obtained from the inversion of the cross-set for the rectansular blocks model.

to those obtained from the inversion of the complete data set (Figs 7 and 8). Theupper block is well resolved. The lower block in the complete data set model (Fig. 8)is slightly sharper than in the cross-diagonal survey model (Fig. 11). The minimumresistivity value near the centre of the lower block (at the 6th iteration) is 1.3 f,) m inFig. 8 compared to 2.0 Q m in the cross-diagonal survey model. This is not surprising


r i ! i i i i : !! : i l i : i : i, i : i r i i i :

, : . ; , : . i ,. . : . . . . . .: ; . ; : ; : ; ,: : : : : : : ; l: . . . : . : ; i: : : i r ' i r i i' : : t i i i t :. ' i : i l i : j i

iiiii !: i:iiii i,ii i i;ij j j j

: : : : i : : : : l : lj . : . j : , : . : .

; : ; : ; : ; : ; : ; :i ; i ; i ; i ; i ; i ;

: r " . :

: ; .' ' . ' :; ; ; ,

; : ; : ; :; : ; : ; :

T i m e t a k e n - l 1 l 8 s .X-Z Plane 2

0 - 5 0 2 5 0 4 . 5 0

. t

i ;

:

:

] . , . , ', l : . : . :l : ' : : :; i . i . i

i : i : i :

. : :

. , ' :

. :

: :

irii! i: . : i : ii : r l : i :i i ! i : r !

i iiiii i l : i

r i l i : :

i i : j i ii i : i i i

i i:.:.i : : . : .

i i ' : : ,

iiiiiii i! : ; :

: : : i l .l i i ! l! i i i i

D i s t . : 4 - 0 t o 5 . 0 m -

v -d i r e c t i o n

15

22

29

35

43

t6

37

14

t7 19

26

33

40

47a

13 14

2 B 2 l

2A2723 24 25a a a

34 35a a

30 31 32a a a

5

t210 11

1

18

2

9

38

45a


-..---'..---t}

3D resistiaity surv4)s 517

42

49

39

/t5

Jt1

48

0 . 5 m e l r e g r i d s P a c i n g

Figurel2. Schematicdiagramof thearrangementof theelectrodesintheBirminghamfleldsurvey test.

since the cross-diagonal survey uses only 476 datum points compared to lI76 points

in the complete data set. In many situations, the slight reduction in the model

resolution is more than offset by the large reduction in the field survey time required.

The error curve for the inversion ofthe cross-diagonal survey data set shows a trend

(Fig. 9) similar to that from the inversion of the complete data set-

It took about 20 minutes to obtain a satisfactory model in the inversion of both data

sets. Loke and Barker (1996) have shown that in the 2D inversion of resistivity data

there were no significant differences between the models obtained by the quasi-

Newton and the conventional least-squares methods. llowever, the computer time

and memory required by the quasi-Newton method to produce a satisfactory model

is about an order of magnitude smaller. \We have not attempted to carry out a direct

comparison between the two methods for 3D resistivity inversion because of the large

amount of computer time and memory that would be required by the conventional

least-squares method.

Example 2: Birmingham field test suruey

This field test was carried out in the Moseley area of Birmingham using a multi-

e 1396 European Association of Geoscientists & Engineers, Geophysical Prospecting,44,499-523

a )

N

1

2

3

4

5

6

1 . 5

b )

N

1

2

3

4

5

6



3 . 0 m -

H p p E r e n t B e s i s t i u i t U P s e u d o s e c t i o n I x = 1 . 5 m . )


3 - 0 m .

R p p a r e n t R e s i s t i u i t U P s e u d o $ e c t i o n ( x = 2 . 5 m . f

f----. l r-- f. i :: ' l l ' :TiTi't f lTttt € m E I Unit electrode spacing 0.5 m.5 0 0 6 0 0 7 0 0 8 0 0

R e s a s t i u i t U i n o h m . m

Figure 13. (a) Birmingham field survey apparent resistivity pseudosections along the line ofelectrodes at x : 1.5 m and at (b) x : 2.5 m.

electrode system with 50 electrodes commonly used for 2D resistivity surveys(Griffiths and Barker 1993). The electrodes are arranged in a 7 by 7 square grid witha unit electrode spacing of 0.5 m. There was a large sycamore tree just outside thesurvey area (Fig. 12). The two far electrodes were placed at more than 25 m fromthe grid to reduce their effects on the measured apparent resistivity values. Thecross-diagonal survey technique was used to reduce the survey time. Figure 13 showsthe pole-pole apparent resistivity pseudosections along the line of electrodes at requal to 1.5m (across the middle of the survey area) and at x equal to 2.5m. Thereare significant differences between the two pseudosections although they are only 1 mapart. The subsurface is known to be highly inhomogeneous, consisting of sandsand gravels.

The error curve for the inversion of this data set is also shown in Fig. 9. Theprogram converged in about 6 iterations after which there is only a small change inthe rms error. Figure 14 shows the horizontal sections of the model obtained at the

6th iteration. The two high resistivity zones in the upper left quadrant and the lowerright corner of Layer 2 are probably gravel beds. The two low-resistivity linearfeatures in Layer 1 near the lower edge ofthe survey area could be due to roots from


( x = 2 . 5 m . 1

I t e r a t i o n 6 - R M S E t r o t 4 . 5 7 "L a y e r 1

n - ? 5 1 . 2 5 2 . 2 50 - 2 5

1 . 2 5

2 . 2 5

/aoorct/D e p t h : 0 - 0 t o 0 . 3 5 m .

L a y e r 30 - 2 5 1 . 2 5 2 . 2 5

o . 2 5

1 . 2 5

2 . 2 5

D e p t h : 0 - 7 5 t oL a y e r 5

0 - 2 5 1 - 2 5

1 . 2 2 m .

2 . 2 50 . 2 5

1 . 2 5

2 . 2 5


T i m e t a k e n - 1 1 0 7 s .Layet 2

0 . 2 5 1 . 2 5 2 . 2 5

D e p t h : 0 - 3 5 t o 0 . 7 5 m

L a y e r 40 . 2 5 l - 2 5 2 . 2 5

0 - 2 5

1 . 2 5

2 . 2 5

0 . 2 5

1 . 2 5

2 . 2 5

0 . 2 5

t . 2 5

2 . 2 5

D e p t h : . 1 - 2 2 l o 1 . 7 5 m -L a y e r 6

0 - 2 5 1 . 2 5 2 _ 2 5

D e p t h : 1 . 7 5 t o 2 . 3 5 m - D e p t h : 2 - 3 6 t o 3 . 0 8 m .

U n i t e l e c t r o d e s p a c i n g 0 . 5 m .f----'l f::1 f.i::l li:-iTfi E::fif @ E E I

2 0 0 4 0 0 8 0 0 1 0 0 0

RerlstlvitY ln Om

Figure 14, Horizontal cross-sections of the model obtained from the inversion of the

Birmingham field survey data set.

the sycamore tree (Fig. 12). The vertical extent of the gravel beds is more clearly

shown in the vert ical cross-sections across the model (Fig. 15). The inverse model

shows that the subsurface resistivity distribution is highly inhomogeneous and can

change rapidly within a short distance. In such a situation a simpler 2D resistivity


: l : l

iilii i ! i

: ; : ; : 1 : i

iiiii. :. '

, , .. : :, : :

' : . . . : .l ; i ; l ; i

ilii: :: . :' : : l :

...,.,:: i i i

D e p t h : 1 . 7 5 t o 2 . 3 5 m -

:

. .1

: . :: : :',.....

i i i i i i i li t i ! ! : : :

! i : i r i i :: ! : : : i i :

: i : i : i : ii i ! i ! a i i: i i i : i i ;i i ! i i i i !i i i i : i : i

;iriiiiiiii: i ! i i i i : i : i! : ! ; i i ! i i : :! ' : : i : t i ! i !

i l i i i ; i i : ! ir ! ! ! i ! i ! : ! |! ; ! : ! : ; ; i ! !

' . . ' . . ' i i


I t e r a t i o n 6 - R M S E t r o r 4 . 5 7 "X-Z PIane I

0 . 2 5 1 . 2 5 2 . 2 5

0 - l 7De 0 . 5 5p 0 - s 8th l - 4 8

m . 2 - 0 5

2 . 7 1

T i m e t a k e n - 1 1 0 7 s -X-Z Plane 2

0 . 2 5 1 _ 2 5 2 . 2 5

Depth

m

0 - l 70 - 5 50 - 9 0

I - 4 8

2 . 0 5

2 . 7 1Y D i s t - : 0 . 0 t o 0 . 5 m -

X-Z Plane 30 . 2 5 1 . 2 5 2 . 2 5

t.............4

Y D i s t . : 0 . 5 t o 1 . 0 m -X-Z Plane 4

0 . 2 5 r - 2 5 2 - 2 5|........-4

0 . 1 7

0 . 5 5

0 . s 8

1 . 4 8

2 _ 0 5

2 . 7 1Y D i s t . : 1 . 0 t o I - 5 m .

X-Z Plane 50 . 2 5 1 . 2 5 2 " 2 5

Y D i s t . : 1 . 5 t o 2 . 0 m

X-Z Plane 60 - 2 5 1 . 2 5 2 . 2 5

0 - l 7

D 0 - 5 5* 0 . 9 8pt l . 4 8h

2 _ 0 5m

2 . V 1: 2 . 0 t o 2 . 5 m - Y D i s t - : 2 . 5 t o 3 - 0 m .

U n i t e l e c t r o d e s P a c i n g 0 - 5 m .

r---l [-l r..--:.1 lliri'1l E::::l ffi m [lE I2 0 0 4 0 0 8 0 0 1 6 0 0

Roslstivity In om

Figure 15. Vertical cross-sections of the model obtained from the inversion of the

Birmingham fleld survey data set.

model (and certainly a lD model from conventional sounding surveys) would

probably not be sufficiently accurate.

O 1996 European Association of Geoscientists & Engineers, Geophysical Prospecting, 44' 499-523

Depth

m

Depth

m .

0- ' t 70 - 5 5

0 - s 8

r . 4 8

2 - 0 5

2 . 7 1

0 - l 7D 0 . 5 5* o - g 8Pt r - l Bh

2 . 0 5m .

2 . 7 1

: j ! i i l. :!: i . l: j ! i ! . !' : i i ! i

, : : , i !: : : : :: ; .i i :: : :

i1: .. :

ii. :

{ , R o o t ?

D i s t - : 2 . 0 t o 2 . 5 m -

R o o r ? {

3D resistioitv survevs 521

Conc lus ions

We have presented a field technique and an inversion method which enable 3D

resistivity surveys and data interpretation to be carried out using commercially

available resistivity field equipment and inexpensive microcomputers. The field

survey time required can be greatly reduced by using the cross-diagonal survey

technique at the cost of a slight reduction in the model resolution. A fast inversion

technique using a quasi-Newton optimization method has been developed for the

automatic inversion of 3D resistivity data. In the tests carried out with synthetic and

field data, the quasi-Newton inversion method has proved to be a robust technique

which converges rapidly. This inversion method is able to resolve complex 3D

geological structures in situations where a 2D resistivity survey and inversion is not

adequate. The inversion ofa single data set from a 7 by 7 survey grid takes about 20

minutes on an 80486DX2166 microcomputer.Research is being carried out into different ways to improve further the inversion

method by using more sophisticated schemes to determine the optimum damping

factor and step size for the model perturbation vector (deGroot-Hedlin and

Constable 1990), other, possibly better, quasi-Newton updating techniques(Dennis and Schnabel 1983) and by normalizing the J'J matrix (Lines and Treitel

1984). Research is also being carried out on the use ofsparse least-squares techniques(Bjorck 1972) to reduce the computer time and memory required to solve the least-

squares equation. Newer microcomputers based on the Intel Pentium micropro-

cessor are about 5 to 10 times faster than the 80486DX/66 computer used in this

research. Thus it should be possible to carry out the inversion of data from larger

survey grids (e.g. 14 by I$ on commonly available microcomputers. The main

obstacle to carrying out surveys with such large grids is probably the time needed to

make the field measurements using conventional single-channel resistivity meters. A

multichannel resistivity meter would probably be required to carry out such surveys

within a reasonable amount of time.

Acknowledgements

M.H.L. thanks The Association of Commonwealth Universities and Universiti

Sains Malaysia for the scholarship provided. We also thank two anonymous

reviewers and Prof. D. Oldenburg for critical reviews and comments.

References

Bjorck A. 1972. Metinods for sparse linear least squares problems. In: Sparse MatrixComputations (eds J.R. Bunch and D.J. Rose), pp. 177-200. Academic Press, Inc.

Bourii S.K. and $Talker H.F. 1990. Least-change secant updates of nonsquare matrices.SIAM Journal of Numerical Analysis 27, 7263-7294.

Broyden C.G. 1965. A class of methods for solving nonlinear simultaneous equations.Mathematics of Computation 19, 577-593.

(-r 1996 European Association of Geoscientists & Engineers, Geophysical Prospecting,44,499-523


Broyden C.G. 1972. Quasi-Newton methods. In: Numerical Methods for Unconstrained

Optimization (ed. \7. Murray), pp. 87-106. Academic Press, Inc.

Churchhouse R.F.(ed.) 1981. Handbook of Applicable Marhematics. Vol. III: Numerical

Methods. John Viley & Sons, Inc.

Daniels R.Str. 1978. An Introduction to Numerical Methods and Optimization Techniques.

Elsevier Science Publishing Co.

deGroot-Hedlin C. and Constable S.C. 1990. Occam's inversion to generate smooth, two-

dimensional models from magnetotelluric data. Geophysics 55' 1613-1624'

Dennis J.E. and Schnabel R. 1983. Numerical Methods for Unconstrained Optimization and

N onlinear Equations. Prentice-Hall, Inc.

Dey A. and Morrison H.F. 1979. Resistivity modeling for arbitrarily shaped three-

dimensional shaped structures. G eophy sics 44, 7 53-780.

Edwards L.S. 1977. A modified pseudosection for resistivity and induced-polarization.

Geophysics 42, 1020-1036.

Eisenstat S.C., Schultz M.H. and Sherman A.H. 1976. Considerations in the design of

software for sparse Gaussian elimination. In'. Sparse Matrix Computations (eds J.R. Bunch

and D.J. Rose), pp. 427-438. Academic Press, Inc.

George A. and Liu J.\7. 1981. Computer Solution of Large Sparse Positirte Definite Systems.

Prentice-Hall, Inc.

Golub G.H. andvanLoan C .F.1989. ManixComputations. TheJohnsHopkins UniversityPress.

Griffiths D.H. and Barker R.D. 1993. Two-dimensional resistivity imaging and modelling in

areas of complex geology. Journal of Applied Geophysics 29' 2ll-226.

Griffiths D.H., Turnbull J. and Olayinka A.I. 1990. Two-dimensional resistivity mapping

with a computer-controlled array . First Break 8, 121-129.

Koefoed O. 1979. Geosounding Principles 1: Resistiaity Sounding Measurements. Elsevier

Science Publishing Co.

Li Y. and Oldenburg D.\7. 1992. Approximate inverse mappings in DC resistivity problems.

G e op hy si c al J ournal I nt ernati onal 109, 3 43 -362.

Lines L.R. and Treitel S. 1984. Tutorial: A review of least-squares inversion and its

application to geophysical problems. Geophysical Prospecting 32' 159-186.

Loke M.H. and Barker R.D. 1995. Least-squares deconvolution of apparent resistivity

pseudosections. Geophysics 60, 1682-1690.

Loke M.H. and Barker R.D. 1996. Rapid least-squares inversion of apparent resistivity

pseudosections using a quasi-Newton method. Geophysical Prospecting 44' l3l-152.

Martin R.S. and lTilkinson J.H. 1965. Symmetric decomposition of positive definite band

matrices. Numerische Mathematik 7, 355-361.

Mooney H.M., Orellana E., Pickett H. and Tornheim, L. 1966. A resistivity computation

method for layered earth models. Geophysics 3l' 192-203.

Park S.K. and Van G.P. 1991. Inversion of pole-pole data for 3-D resistivity structures

beneath arrays of electrodes. Geophysics 56,951-960.

Petrick Sfl.R. jr., Sill W.R. and !?ard S.H. 1981. Three-dimensional resistivity inversion using

alpha centers. Geophysics 46, 1148-1163.

Press W.H., Flannery B.P., Teukolsky S.A. and Vetterling \tr.T. 1988. Numerical Recipes

in C. Cambridge University Press.

Sasaki Y. 1989. Two-dimensional joint inversion of magnetotelluric and dipole-dipole

resistivity data. G eophy sics 54, 254-262.



Sasaki Y. 1994. 3-D resistivity inversion using the finite-element method. Geophysics 59,1839-1848.

Shirna H. 1992.2-D and 3-D resistivity image reconstruction using crosshole data. Geophysics57,1270-1281.

\(/oo P.T., Eisenstat S.C., Schultz M.H. and Sherman A.H. 1976. Application of sparsematrix techniques to reservoir simulation. In'. Sparse Matrix Computations (eds J.R.Bunchand D.J. Rose), pp. 427-438. Academic Press, Inc.

Xu B. and Noel M. 1993. On the completeness of data sets with multielectrode systems forelectrical resistivity survey. G eophysical Pro specting 41, 791.-801.

O 1996 European Association of Geoscientists & Engineers, Geophysical Prospecting,44,4gg-523

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practical techniques for 3d resistivity surveys and data

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