METHOD FOR THE INVERSION OF RESISTIVITY SOUNDING AND PSEUDOSECTION DATA

Pergamon
Compurers & Geoscience.~ Vol. 21, No. 2, pp. 321-332, 1995
009%3004(94)ooo75-1
Copyright 8 1995 Elsevier Science Ltd
Printed in Great Britain. Ail rights reserved
0098-3004/95 $9.50 + 0.00
IMPROVEMENTS TO THE ZOHDY METHOD FOR THE
INVERSION OF RESISTIVITY SOUNDING AND
PSEUDOSECTION DATA
M. H. LOKE* and R. D. BARKER
School of Earth Sciences, The University of Birmingham, Edgbaston, Birmingham B15 2TT, U.K.
e-mail: r.d.barker@,bham.ac.uk
(Received and accepted 7 July 1994)
Abstract-Two practical improvements to increase the rate of convergence and to overcome the stability
problem of Zohdy’s method for the inversion of apparent resistivity sounding data have been proposed.
Zohdy uses the measured data as the starting model and then assumes that the necessary correction vector
for a layer resistivity to improve the current model is equal to the logarithmic difference between the
corresponding model response and observed apparent resistivity values. To improve the speed of
convergence, the logarithmic change in the layer resistivity is multiplied by a scaling factor calculated from
the apparent resistivity differences in the previous two iterations. To improve the stability of the inversion,
a weighted average of the apparent resistivity differences is used to determine the correction in the
resistivity of each layer. Many tests with computer generated and field data show that the modifications
make a significant improvement to the inversion; they reduce the computing time needed by a significant
amount with the final model being far less sensitive to noise in the data. The modifications are extended
to the inversion of pseudosection data from two-dimensional resistivity surveys.
Key Words: Geophysics, Inversion, Regression analysis, Resistivity
INTRODUCTION
For the past two decades, several techniques have
been developed for the automatic inversion of appar-
ent resistivity data. Some of the most successful
techniques are based on nonlinear optimization

methods. In the inversion of resistivity sounding data,
the steepest descent method (Koefoed, 1979) and
several variations of the least-squares method (In-
man, Ryu, and Ward, 1973; Constable, Parker, and
Constable, 1987) have been used with some success.
For the two-dimensional (2-D) inversion of resistivity
data from electrical imaging surveys (Griffiths and
Barker, 1993), techniques based on the least-squares
optimization method also have been used widely
(Smith and Vozoff, 1984; deGroot-Hedlin and
Constable, 1990; Sasaki, 1992). These gradient-based
techniques require the calculation of the Jacobian
matrix which consists of partial derivatives with
respect to the model parameters at each iteration
(Lines and Treitel, 1984). The calculation of the
partial derivatives can be time consuming, particu-
larly in the 2-D inversion of apparent resistivity data
using microcomputers.
An interesting new iterative technique which
avoids the calculation of the partial derivatives was
proposed by Zohdy (1989) for the inversion of resis-
tivity sounding data. This technique later was ex-
*Present address: School of Physics, Universiti Sains
Malaysia, 11800 Penang, Malaysia.
tended to the 2-D inversion of resistivity data by
Barker (1992) and to magnetotelluric data by Hobbs
(1992). Two disadvantages of this technique are its
relatively slow convergence and instability when there
is appreciable noise in the data. Practical methods to
overcome these problems are discussed in this paper.

THE ZOHDY TECHNIQUE
The two distinctive features of this technique are
the construction of the initial guess and the calcu-
lation of the correction vector which minimizes the
differences between the measured and model data.
For the inversion of sounding curves, Zohdy (1989)
starts by using a model in which the number of layers
is equal to the data points of the sounding curve. In
the initial model, the resistivity of each layer is set to
be the same as the corresponding sample value of
apparent resistivity (Fig. 1). Also the mean depth of
each layer is set to be the same as the electrode
spacing of the corresponding datum point multiplied
by a constant shift factor. Zohdy used an initial shift
factor value of 1.0, that is the mean depth of each
layer is set to be the same as the electrode spacing of
the corresponding datum point in the initial guess,
and this is reduced progressively until the difference
between the observed and calculated curves reaches a
minimum value. This usually happens when the cal-
culated and observed apparent resistivity sounding
curves are “in phase” (Fig. 1). Barker (1989) recog-
nized that the shift factor used by Zohdy was related
321
322
M. H. Loke and R. D. Barker
model curve
model curve
ELECTRODE SPACING, a, or DEPTH
Figure I. Main steps in Zohdy inversion method. Aabserved data and initial model; B shifted layering

and resulting sounding curve. Logarithic difference e between calculated and observed apparent-resistivity
values is used to apply correction c to layer resistivity; C-final layering and resulting calculated
apparent-resistivity sounding curve, which is closely similar to observed data.
directly to the median depth of investigation
(Edwards, 1977) of the array used for a homogeneous
earth model. In many situations, using a constant
shift factor which is equal to the median depth of
investigation, gives reasonably fast convergence. For
the Wenner array, this shift factor is about 0.5 times
the electrode spacing.
After determining the optimum shift factor, the
resistivities of the layers then are adjusted by using
the differences between the logarithms of the calcu-
lated and observed apparent resistivity values. The
resistivity of a layer is adjusted using the following
equation:
c,(j) = e,(j ),
(1)
where j and i represent the jth layer (and jth spacing)
and the number of iterations respectively.
The logarithmic correction vector is given by:
c,(j ) = log(pi + , (j )I - log( ~7 (j )h
where i equals the key number of iteration step and
p,(j) represents the resistivity of the jth layer during
the ith iteration.
Because:
e,(j) = log&(j)) - log(p,,(j)),
where p,(j) and p,,(j) represent respectively the jth
observed apparent resistivity value and the jth calcu-
lated apparent resistivity value for the ith iteration,

then the resistivity of the jth layer for the (i + 1)th
iteration can be given by:
pi+ I (i I= ~,(i 1 . exp(c,(j 11,
= p,(j). b,(j)/~,,0’)1. (2)
Note that the Zohdy method basically assumes that
the necessary logarithmic correction in the resistivity
of a layer is equal to the logarithmic difference
between the observed and computed apparent resis-
tivities at the corresponding datum point.
As a simple example of the use of the Zohdy
method, Figure 2 shows the results from the inversion
of the Wenner-array sounding curve for a two-layer
model. The apparent resistivity values for this test
model were calculated using the linear filter method
(Koefoed, 1979). The depths to the center of each
layer in the initial model were determined by
Improvements to the Zohdy method
323
Two layer model - Bi: noise
Standard Zohdy inuersion method
ITERf3TION 1 % RIIS Error 12.14
Depth of Layers
1.0 10.0
iee.0
lEn.m~ c E"
I I
I
Cal. CIPP. Res.
Obs. clpp. Res.
+

.___-__
I
Computed Ilode
L
0
El
ie0.e
lectrode Spacing
Clctunl Model
Standard Zohdy inuersion method
ITERCITION 5 % RtlS Error
0.64
Depth of Layers
1.0 1e.e
me.0
1000~
"I' 8
I
1
23 -
c -
._ _
a _
*_
c _
v)
._
v) -
al
a

lee _
” -
c _
a -
i -
a _
eL
a _
m_
1.
4
Cal. Ama. Res.
Ohs. CIPP. Ras.
+
Conputed Model
CIctual Model
g
____I
10.0 100.0
Electrode Spacing
Figure 2. Inversion of two-layer model (Wenner array) sounding curve with Zohdy method. A-Initial
model; B-model obtained by Zohdy method after 5 iterations.
multiplying the electrode spacing of the correspond-
ing sounding curve datum point by the equivalent
depth factor (0.5 for the Wenner array). For the
initial model, the resistivities of the layers were set to
be the same as the measured apparent resistivity
values (Fig. 2A). Whereas the actual model has a
sharp boundary at a depth of 9.1 m (N.B. the model
depth scale is given on the top edge of the figure), the

resistivity of the initial model decreases more gradu-
ally with depth. After 5 iterations, the resistivity
distribution of the computed model shows a better
agreement with the actual model (Fig. 2B). The
change in the root mean squared (r.m.s.) error calcu-
lated from the logarithmic differences between the
computed and measured apparent resistivity values
with iteration number is shown in Figure 3. For the
noise-free sounding curve, the r.m.s. error decreased
rapidly in the first 5 iterations followed by a slower
decline. This slow convergence of the Zohdy method
can be a significant problem, particularly for the
2-D inversion of data on microcomputers. The
calculations for the examples shown in this paper
were performed on an IBM-PC compatible micro-
computer.
It was noted by Zohdy (1989) that for noisy data
the inversion process can become unstable. After a
number of iterations, the r.m.s. error can increase if
the sounding data are contaminated sufficiently by
noise. Then, the model can exhibit anomalous layers
with unusually high or low resistivity values. Figure 3
also shows the change in the r.m.s. error with iter-
ation number when the Zohdy method is used for the
inversion of noisy data. In this example, Gaussian
random noise (Press and others, 1988) with an ampli-
tude of 10% of the apparent resistivity value is added
to each datum point of the sounding curve for the
324
M. H. Loke and R. D. Barker

Two-layer model
Standard Zohdy inversion method
O. Two-layer model with 0~ noise
a Two-layer model with 10~ noise
20
15
RHS
ERROR
%
10
5
0
-I-
1
P
,:’
;,’
a,’
,/
,’
;
,,a’
m
b
\__
.D __.___ I) . __ rl =
,,a’
0
0. .,
‘O.“.o _*,

o o
.o o
I
2 3 4 5 6 7 8 9 10 11
Iteration
Figure 3. Error curves for inversion of two-layer sounding curve data with no noise and with 10% random
noise using Zohdy method.
two-layer model. The r.m.s. error starts to increase
after the 4th iteration (Fig. 3).
To overcome this problem, Zohdy (1989) proposed
that the inversion process be repeated using the
calculated apparent-resistivity sounding curve pro-
Two - layer
12
10
8
RIIS
ERROR
%
6
duced by the model with the lowest r.m.s. error. Thus,
two runs of the Zohdy inversion method are needed
to interpret one set of data. Although this is not a
problem with 1-D sounding data where the sounding
curves can be computed rapidly with the linear filter
model - 0 % noise
0’. Standard Zohdy method
a Zohdy method with Fast Convergence
4 I
1 2 3 4 5 6 7 8 9 10 11

Iteration
Figure 4. Error curves for inversion of two-layer sounding curve data (with no noise) using Zohdy method
with and without fast convergence modification.
Two layer model - 1Bz noise
Standard Zohdy inuersion method
ITERATION 1 % RnS Error 16.02
Depth of Lauers
Cal. C~PP. Res.
Obs. clpp. Res.
+
Computed node1
fictual Model
____
1 ’
____I
10.0
100.0
Electrode Spacing
Standard Zohdy inuersion method
ITERfiTION 5 % RnS Error 9.43
Depth of Layers
1.0 10.0
me.0
me0 m “I’
,
Cal. ~PP. Res.
6
•t
.
Ohs. clpp. Res.

+
Conmated Model
Clctua1 node1
____
I ’
____I
5
._
‘;
-
h
11
__ __
,-
0 10.1
Electrode Spacxng
Smoothed Zohdy inuersion method
ITERCITION 5 % RnS Error 9.01
Depth of Layers
1B.B
me.0
I
I
I
5 6
Cal. CIPP. Res.
Obs. hap. Res.
+
Conputed node1
w2tual Pl0del

-___
I ’
C-
Ml.0
1ee.e
Electrode Spacing
Figure 5. Inversion of two-layer model (Wenner array) sounding curve with 10% random noise. A-Initial
model; B-model obtained with (Standard) Zohdy method after 5 iterations; C-model obtained using
Zohdy method with smoothing modification after 5 iterations.
325
326
M. H. Loke and R. D. Barker
method, this is a disadvantage for 2-D apparent
resistivity data which use the slower finite-element or
finite-difference method for the forward compu-
tations. A possibly better approach is to determine
the cause of the instability and to avoid it.
IMPROVEMENTS AND EXAMPLES
Improvements to the rate of convergence
To improve the rate of convergence, the following
modification was made to Equation (1) which gives
the logarithmic change in the model layer resistivity:
c,(j) =f;(j). 4i). (3)
The multiplication factorf;(j) was set initially to 1.0
for the first two iterations. Then, it was modified by
comparing the logarithmic differences e, for two
successive iterations. The equation used to modify the
multiplication factor f; is given by:
f;(j)=f;~,(i).(l.O+e,(j)le,-,(j)).
(4)

In practice, the value off;(j) is limited to between
1.0 and 3.0 to minimize its effect on the stability of
the inversion process. It is used only if the difference
between e,_,(j) and ei(j) is larger than 0.1%. Fur-
thermore, if the logarithmic difference e,(j) for the
ith iteration is larger than that for the previous
iteration the multiplication factor f;(j) is set back to
1.0. Figure 4 shows the error curve when the modifi-
cation (labeled “Fast Convergence”) is used for the
inversion of the sounding curve for the two-layer
model. With this modification, the r.m.s. error de-
creases at a faster rate after the first two iterations
201
IWO - layer
(particularly between the 2nd and 3rd iterations). By
using the previous multiplication factor, the number
of iterations needed to reduce the r.m.s. error to a
given value, for example l.O%, is reduced by about
one-third. This multiplication factor is similar to the
successive over-relaxation parameter used to acceler-
ate the convergence rate of iterative techniques for
solving linear equations (Golub and van Loan, 1989).
Other methods of modifying the multiplication
factor, for example by using the ratio of the change
in the model layer resistivity to the change in the
logarithmic difference of the corresponding datum
point, also were investigated. Equation (4) generally
gave the best results.
Improvements to stabilize the Zohdy method
It was mentioned earlier that the inversion process

can become unstable for noisy data. The models
obtained using the Zohdy method in the inversion of
the two-layer sounding curve with 10% random noise
are shown in Figure 5. It was noted earlier that in this
situation the r.m.s. error starts to rise after the 4th
iteration (Fig. 3). Figure 5B shows the model ob-
tained at the 5th iteration where some of the model
layers show significant deviations from the actual
resistivity values. The reason for the instability of the
Zohdy inversion process can be determined by con-
sidering the apparent resistivity values at the 5th and
6th data points as an example. The apparent resis-
tivity value at the 5th datum point has been decreased
by the noise added to it while the value at the 6th
datum point has increased compared to the original
noise-free values (cf. Fig. 2). The Zohdy inversion
model - 1Qz noise
Q. Standard Zohdy method
Q Smoothed Zohdy method
-+ Smoothed Zohdy with Fast Conuergence
15
rins
1 2 3 4 5 6 7 8 9 10 11
Iteration
Figure 6. Error curves for inversion of two-layer sounding curve data with 10% noise using Zohdy method
with and without different modifications.
Improvements to the Zohdy method
2-D model used by inuersion method
327
Electrode position Iml

. .
Pseudosection
Model
I _. I
0 40 80
120 160 200
240 280
320 360 400
Death [m]
LC:YT;I
r
1 -
-
2 -
-
3 -
-
2 x
-
-
6 -
-
5
10
15
S!
30
Figure 7. 2-D model
used by Zohdy-Barker method.
process tries to decrease the difference between

the
the 5th layer regardless of what happens to the
calculated and observed apparent-resistivity values at
differences at the other datum points. When the
the 5th datum point by decreasing the resistivity of
resistivity of the 5th layer is decreased, the apparent
Z-D Horst Rode1 - 0 % noise
.
Apparent resistivity datum point
0 Model rectangular block
0.0 80.8
160
240 320
480
488 m.
N
1
nlp rq$q [iiiril
40 -8!= -
Measured apparent resistiuity in ohn-n. Unit Electrode Spacing = 20.0 n.
Standard Zohdy-Barker inuersion method
Iteration
5 completed with 4.9 % RMS Error
0.0 80.0
160
240 320
400
480 II.
Depth
8 I, I I I I I I I I I I I I I,

ie.0 -
28.0 -
30.0 -
40.0 -
50.0 -
B
60.0 -
Model rasistivitr in ohm-n.
Modified Zohdy-Barker inuersion method
Iteration
5 completed with 4.0 % RPLS Error
0.0 80.0
160
240 320
400
480 PI.
J
Depth
60.0’
100
ia. J
20.0 -
30.0 -
40.0 -
58.0 -
Model resistiuity in ohm-n.
Figure 8. A-Apparent resistivity pseudosection for horst model with 0% noise; B-model obtained after
5 iterations with Zohdy-Barker method; C-model obtained with modified ZohdyyBarker method.
328
M. H. Loke and R. D. Barker

Z-D Horst Ilodel - 0 % noise
a- Standard Zohdy
Q Smoothed Zohdy with fast convergence
01
1
1 2 3 4 5 6 7 8 9 10 11 12 13
Iteration
Figure 9. Error curves for inversion of horst model data with 0% noise using standard and modified
Zohdy-Barker methods.
resistivity values of the neighboring points also are
decreased. This increases the difference at the 6th
datum point. The inversion process tries to compen-
sate for this by increasing the resistivity of the 6th
layer in an attempt to increase the apparent-resistivity
value at the corresponding datum point. This process
continues producing an oscillatory-layer resistivity
variation with depth which eventually increases the
apparent resistivity r.m.s. error. By the 5th iteration
(Fig. 5B) resistivity of the 5th layer has decreased to
a value which is too low, whereas the resistivity of the
6th layer has become too high. For the same reason,
the resistivity of the 11 th layer is too low whereas that
for the 12th layer is too high.
One method to avoid this instability is to take into
account the apparent resistivity differences at the
neighboring points when calculating the change in
each layer resistivity. This method is termed the
“Smoothed Zohdy Method” in Figures 5C and 6.
The equation to correct the model resistivity is
modified to the following form:

ciCi)= C,e,(i - I)+ Ge,(.i) + Ge,(j + 1). (5)
Instead of just using the difference at one datum
point to calculate the resistivity change for the jth
layer, a weighted average of the difference of the jth
datum point and the two neighboring points is used.
Normally, the values of the weighting coefficients C,,
C,, and C, used are 0.25, 0.50, and 0.25 respectively.
In this manner, if the weighted average value of the
difference values at the three points is zero, the layer
resistivity is not changed. This insures that the change
in the model layer resistivity calculated does not
increase the overall difference of the three points.
Figure 6 shows the r.m.s. error curves when 10%
random noise is added to the sounding-curve data for
the two-layer model. For the Standard Zohdy
method, the r.m.s. error starts to rise after the 4th
iteration. In comparison, the r.m.s. error for the
Smoothed Zohdy method continues to decline at a
slower rate after the 4th iteration to an asymptotic
value. Figure 5C shows the model obtained with the
Smoothed Zohdy method at the 5th iteration. In this
situation, the effect of the random noise on the
resistivity of the layers is more subdued.
The modifications to improve the rate of conver-
gence and to stabilize the Zohdy method can be
combined. Figure 6 also shows the error curve for the
inversion of the sounding data with 10% noise when
these two modifications are combined. Notice the
improvement in convergence in this situation.
In order to confirm that the results obtained did

not depend on the particular data sets used, the tests
were repeated for different layered models and differ-
ent noise levels. The results were similar with no
significant differences being observed.
Improvements to the Zohdy-Barker method in 2-D
electrical imaging
In the previous section, we have seen that the
model used by the I-D Zohdy method is a I-D
multilayer earth model. The 2-D model used by the
Zohdy-Barker method (Barker, 1992) consists of a
number of rectangular blocks. The arrangement of
0.0
N
80.0
Improvements to the Zohdy method
Z-D Horst Model - 16~ noise
160 240 328
329
400 480 m.
I
1 -
2 -
3 _
4 -
“:A
6
LIT M2F7J ml
40
-8!? -
lleasured apparent resistiuitr in ohm-n.

Unit Electrode Spacing = 20.0 m.
Standard Zohdy -Barker
inuersion
method
Iteration
5 completed with 10.6 % RtlS Error
0.0
80.0 168 240 328
400 480 m.
Depth
I I , I ,
1e.e
20.0
30.0
40.0
50.9
60.8
Model resistiuitr in mhn-n.
Modified Zohdy-Barker inuersion method
Iteration
5 conpleted with 9.5 % RflS Error
0.0 80.0 160 240 320 488 488 II.
Depth
I
60.0
Model resistivitr in ohm-n.
Figure 10. A-Apparent-resistivity pseudosection for horst model with 10% random noise; B-model
obtained after 5 iterations with standard Zohdy-Barker method; C-model obtained with modified
Zohdy-Barker method.
the 2-D rectangular blocks with respect to the

pseudosection data points is shown in Figure 7. The
number of rectangular blocks is the same as the
number of data points. The horizontal location of the
center of each block is placed at the midpoint of the
array used to measure the corresponding apparent
resistivity datum point. The depth of the center of the
block is set at the equivalent depth of the array (0.5
times the electrode spacing for the Wenner array).
Note that the left and the right edges for the blocks
on the left and right sides are extended horizontally
to infinity. The bottom edge of the row of blocks at
the bottom row is extended vertically downwards to
infinity. The top edge of the topmost row of blocks
is extended to the surface. The thickness and width of
each interior block is 0.5 and I .O times the minimum
electrode spacing respectively. Each block is mapped
onto a corresponding datum point on the resistivity
pseudosection in an arrangement which is similar to
that used for the 1-D Zohdy method.
As with the 1-D Zohdy method, the resistivity of
each block is set initially to be the same as the
apparent resistivity value for the corresponding da-
tum point. In the Zohdy-Barker (Barker, 1992)
method, the resistivity of a block is changed at each
iteration by the following equation:
c,(I, n) = e,(4 n)
(6)
where I is the horizontal number of block or datum
point starting from the left-hand side of the model,
and n is the vertical level of the block or datum point.

e,(I, n) is the logarithmic difference between calcu-
lated and observed apparent resistivity values
whereas c,(f, n) is the logarithmic change in the
resistivity of the block (/, n) from the ith to the
(i + I)th iteration.
330
M. H. Loke and R. D. Barker
Z-D Horst llodcl - 10~ noise
0 Standard Zohdy
B Smoothed Zohdy with fast conuergence
5
1 2 3 4 5 6 7 8 9 l0 ill2 13
Iteration
Figure 1 I, Error curves for inversion of horst model data with 10% noise using standard and modified
ZohdyyBarker methods.
The finite-difference method (Dey and Morrison,
1979) is used to calculate the apparent resistivity
values for the 2-D model in consideration. Some tests
already have been made on the inversion of apparent
resistivity pseudosections by Barker (1992) who also
provides a detailed description of the electrical imag-
ing method. The models treated by Barker include a
two-layer model, a faulted block which extends to the
surface, and a horst model. However, Barker (1992)
did not address the issues of stability and conver-
gence. It is recognized that the problems of slow
convergence and instability again exist when the data
are contaminated by noise if one uses the Zohdy-
Barker method for the 2-D inversion of apparent
resistivity data. The modifications made to the 1-D

Zohdy method described earlier were extended there-
fore to the inversion of 2-D apparent resistivity data.
Firstly, the convergence rate can be improved by
using the following modification:
c,U, n) =f;(l, n). e,(l, n).
(7)
As in the 1-D situation, the multiplication factor
f;(l, n) is set initially to 1 .O for the first two iterations
and then modified by comparing the difference values
ei for two successive iterations. A similar equation is
used to modify the multiplication factor f,. It is given
by:
1;(~,~)=f;-,(~,~)~(l.O+e,(~,~)le,-,(~,~)). (8)
To overcome the problem of instability resulting
from noise in the data, a local weighted average of the
logarithmic apparent resistivity differences is em-
ployed. The equation used is:
+ei(l + 1, n - 1) + e,(l + 2, n - 1)
+e,(l-2,n+l)+e,(l-l,n+l)} (9)
where e, is the logarithmic difference at a datum
point, C, is the weight of central datum point, and C,
represents the weight of surrounding points.
The sum of all the weights is normalized to 1.0.
Near the edges of the pseudosection, the weight for
the missing points is distributed to the remaining
values. The weight for the central datum point C, can
have a value between 0.5 (with one-half the total
weight) and 0.15 (with almost the same value as the
surrounding datum points). A value of between 0.20
and 0.30 is used normally in practice. Note that, if the

2-D filter [Eq. (S)] is applied to a 2-D resistivity
pseudosection data because of a 1-D layered earth
structure, it will give the same results as using a 1-D
filter on a sounding-curve data set. In this situation,
the equivalent I-D filter has weights of 2 * C,,
(C, + 2 * C,), and 2 * C,. Thus, a 2-D filter with a
central weight C, of 0.25 is equivalent to a 1-D filter
with weights of 0.25, 0.50, and 0.25 when used for the
inversion of pseudosection data from a 1-D layered
earth structure.
In practice, both the modifications to improve the
convergence rate and stabilize the Zohdy-Barker
method are used together. In the following discus-
sion, this approach, will be referred to as the modified
Zohdy-Barker method.
0.0
N
40.0
Improvements to the Zohdy method
landfill suruey
80.0 120 160
331
200 248 n.
,
1 -
2 -
3 -
4 -
5
6

:A
Measured apparent resistiuitr in ohn-n.
Unit Electrode Spacing = 10.0 II.
Standard Zohdy-Barker inversion method
Iteration 8 conpleted with 5.7 % Rt4S Error
0.0 40.0 80.0 120 160 200 240 II.
Depth 4
,- landfill outline
5.00-
10.8 -
15.6-
20.0 -
25.0.
30.0- '
leachate ‘contamination
Model resistivity in ohm-n.
Modified Zohdy-Barker inuersion method
Iteration 8 completed uith 4.8 % RMS Error
0.0 40.0 80.0 120 160 200 240 II.
Depth 1
I- Landfill outline
5.00 J
10.0 -
15.0 -
20.0-
25.0 -
30.0- c
leachate contamination
Model resistivitr in ohn-n.
Figure 12. A-Landfill survey apparent resistivity pseudosection. Model obtained after 5 iterations with

B standard Zohdy-Barker method and C modified Zohdy-Barker method. Landfill outline was derived
from other sources.
Tests with the modiJied Zohdy-Barker methodfor 2-D
data
The results of some tests with the modifications
made to the Zohdy-Barker method are given in this
section. A horst model (Barker, 1992) is used as an
example. The Wenner-array apparent-resistivity
pseudosection for this model is shown in Figure 8A.
Figure 9 shows the r.m.s. error curves with the
standard and modified Zohdy-Barker methods. The
modified Zohdy-Barker method converges at a
slightly faster rate. Figures 8B and 8C respectively
show the model resistivity sections obtained with the
standard and modified Zohdy-Barker methods after
5 iterations. The model resistivity distribution shows
a progressive sharpening of the sides and top of the
anomaly with each iteration. The resistivity values in
the general area of the actual horst structure also
become increasingly higher. In fact, at the 10th
iteration, the model resistivity values near the bottom
of the structure are higher than the true value of
lOOR-m. The Zohdy method works well for this
example probably because the apparent-resistivity
pseudosection which forms the initial model has a
reasonably similar shape to the true model.
Figure 10A shows the apparent-resistivity pseudo-
section for the same model with 10% random noise.
The pseudosection with 10% noise shows moderate
distortion in the shape of the contours. The apparent

resistivity r.m.s. error curves from the inversion
332
M. H. Loke and R. D. Barker
of this data set are shown in Figure 11. The r.m.s.
error for the standard Zohdy-Barker method starts
to rise after the 7th iteration, whereas that of the
modified Zohdy-Barker method continues to de-
crease slowly after the 7th iteration to about 9%. The
model obtained with the standard Zohdy-Barker
method at the 5th iteration (Fig. IOB) shows severe
distortions resulting from the noise in the data. The
modified Zohdy-Barker method model (Fig. 1OC)
much less distortion.
As a final example, the inversion of the data set
from an electrical imaging survey across a landfill
site in Nottinghamshire, England (Barker, 1992) is
studied with the modified Zohdy-Barker method. The
Wenner array was used for this survey. The apparent-
resistivity pseudosection and the models obtained by
the standard and modified Zohdy-Barker methods
are shown in Figure 12. Figure 12B shows the model
resistivity section obtained after the 5th iteration with
the standard Zohdy-Barker method; similar results
obtained with the modified Zohdy-Barker method
are shown in Figure 12C with the outline of landfill
drawn for comparison. The models obtained with
both methods are similar and agree well with the
outline of the landfill based on existing information.
From the smoothness of the measured apparent-resis-
tivity contours (Fig. 12A), there probably is only a

moderate amount of random noise present such that
the standard Zohdy-Barker method worked reason-
ably well. The main difference between the two
approaches is a high resistivity zone at a depth of
about 10m near the 110-m mark in the model
obtained by the standard Zohdy method. This high
resistivity zone is the result of slightly higher values
at the 1st and 2nd levels near the 110-m mark in the
measured apparent-resistivity pseudosection and
seems to be subdued in the other model. Both
methods show a low resistivity zone beneath the base
of the landfill which is probably the result of leachate
contamination into the sandstone bedrock which has
a higher resistivity than the landfill material (Barker,
1992). Another possible reason why the Zohdy
method works well for this data set is that the
subsurface resistivity varies in a gradational manner
possibly the result of leachate and groundwater seep-
age into the sides of the landfill. This gives rise to a
relatively smooth apparent-resistivity pseudosection
which resembles the actual subsurface resistivity
distribution.
CONCLUSIONS
Two simple modifications to the Zohdy technique
have been introduced. With these modifications, the
Zohdy inversion technique for 1 -D and 2-D apparent
resistivity data are more stable and converges at a
somewhat faster rate. It results in models that are
smoother and less sensitive to the noise in the appar-
ent resistivity data points. It generally reduces the

number of iterations needed to decrease the r.m.s.
error to an acceptable level by about one-third.
Similar modifications probably can be made in the
use of the Zohdy technique for the inversion of other
types of geoelectrical data.
Acknowledgmenfs One of us (MHL) would like to thank
The Association of Commonwealth Universities and Uni-
versiti Sains Malaysia for the scholarship provided.
REFERENCES
Barker, R. D., 1989, Depth of investigation of collinear
symmetrical four-electrode arrays: Geophysics, v. 54,
no. 8, p. 1031-1037.
Barker, R. D., 1992, A simple algorithm for electrical
imaging of the subsurface: First Break, v. 10, no. 2,
p. 53-62.
Constable, S. C., Parker, R. L., and Constable, C. G., 1987,
Occam’s inversion: a practical algorithm for generating
smooth models from electromagnetic sounding data:
Geophysics, v. 52, no. 3, p. 2899300.
Dey, A., and Morrison, H. F., 1979, Resistivity modelling
for arbitrary shaped two-dimensional structures:
Geophys. Prosp., v. 27, no. 1, p. 106-136.
Edwards, L. S., 1977, A modified pseudosection for resis-
tivity and induced-polarization: Geophysics, v. 42, no. 5,
p. 1020-1036.
Griffiths, D. H., and Barker, R. D., 1993, Two-dimensional
resistivity imaging and modelling in areas of complex
geology: Jour. Appl. Geophys., v. 29, no. 3/4,
p. 21 l-226.
Comb, G. H., and van Loan, C. F., 1989, Matrix compu-

tations (2nd ed.): The John Hopkins Univ. Press,
Maryland, 642 p.
deGroot-Hedlin, C., and Constable, S. C., 1990, Occam’s
inversion to generate smooth, two-dimensional models
from magnetotelluric data: Geophysics, v. 55, no. 12,
p. 1613-1624.
Hobbs, B. A., 1992, Application of Zohdy’s method to EM
inversion: paper presented at Developments in Shallow
Electrical and EM Surveys, Joint Assoc. of Geophysics
(13 November 1992), London.
Inman, J. R., Ryu, J. R., and Ward, S. H., 1973, Resistivity
inversion: Geophysics, v. 38, no. 6, p. 1088-1108.
Koefoed, O., 1979, Geosounding principles 1: resistivity
sounding measurements: Elsevier Sci. Publ. Co.,
Amsterdam, 276 p.
Lines, L. R., and Treitel, S., 1984, Tutorial: review of
least-squares inversion and its application to geophysical
problems: Geophys. Prosp., v. 32, no. 2, p. 1599186.
Press, W. H., Flannery, B. P., Teukolsky, S. A., and
Vetterling, W. T., 1988, Numerical recipes in C:
Cambridge Univ. Press, Cambridge, 735 p.
Sasaki, Y., 1992, Resolution of resistivity tomography in-
ferred from numerical simulation: Geophys. Prosp., v.
40, no. 4, p. 453-464.
Smith, N. C., and Vozoff, K., 1984, Two-dimensional DC
resistivity inversion for dipole-dipole data: IEEE Trans.
Geoscience and Remote Sensing, v. 22, no. 1, p. 21-28.
Zohdy, A. A. R, 1989, A new method for the automatic
interpretation of Schlumberger and Wenner Sounding
curves: Geophysics, v. 54, no. 2, p. 245-253.