vention showing the apparent resistivity or phase lag in the centre of both dipoles. Regarding the
real location of the dipoles, it becomes clear that the responses originate within a single re-
gion near the middle of the line where one of the dipoles is placed. The negative apparent IP
effects measured on Line A are due to body polarization effects, common in high-contrast
circumstances involving 2- and 3-D bodies.
Relatively small IP effects are observed on the background line. Other measurements in the
area suggest that the intrinsic polarization effect of the clay, soil, and basalt is less than 5 mrad,
or somewhat less than the largest values ob- served on line B. Some residual IP effect is due
to lateral effects from the ‘large objects’ only one dipole length to the north of the line.
The resistivity data of Fig. 3 show a conduc- tive zone located near the centre of the line,
with apparent resistivities decreased to about one half to one third of background. This fea-
ture forms a viable target for methods based on conductivity contrasts. Another prominent con-
ductive feature on the eastern end of the back-
Ž .
ground line Fig. 4 is not due to any known waste and has no corresponding IP response.
This non-waste-related
resistivity response
shows an apparent resistivity far less than that of the simulated waste, and appears on the short
separations, indicating a shallow causative body. The feature was detected on some of the other
lines as well as by several of the other survey methods used in the EMID work. The unknown
conductive feature may be due to variations in the source, handling, or compaction of the cap
material, or to subsequent modifications such as spills during operations in the area. In any case,
it is clear that buried waste is not the only cause of conductive anomalies at INEL.
5. Forward modelling and inversion
The structures at a waste dump require gener- ally a 3-D data interpretation. The software for
data processing includes forward modelling and inversion procedures. The algorithm used for
forward modelling considers both conductivity s and potential V as complex quantities. The
forward
modelling algorithm
proposed by
Ž .
Weller et al. 1996a is summarized as follows. For a current I fed at a point A represented
by the vector r , the resulting electrical poten-
A
Ž .
tial V r r satisfies the equation of continuity
A
= P s r = V r r q Id r y r
s 0, 11
Ž . Ž
. Ž .
Ž .
A A
where d is the Dirac delta function. This equa- tion is solved in a discrete space of two or three
Ž .
dimensions by the finite difference FD method. In usual modelling geometries, the upper
boundary of the model is given by the earth’s surface, where Neumann boundary conditions
are satisfied. For all other boundaries of the model which are drawn inside the conducting
medium, mixed boundary conditions as pro-
Ž .
posed by Dey and Morrison 1979 are applied. The grid may be presented as an electrical
network. The resulting system of equations is written
in matrix form, where each equation corre- sponds to Kirchhoff’s Current Law for a single
node. The complex matrix of coupling coeffi- cients is symmetric but not Hermitian. A com-
pact storage scheme fully uses the sparcity and symmetry of the matrix while a generalized
conjugate gradient method solves the system of equations. Preconditioning techniques speed up
the solution.
The forward problem is solved for each cur- rent source separately. The apparent complex
resistivities for all configurations can be com- puted easily by a superposition of several
pole–pole configurations. In the case of multi- electrode-arrays, this method reduces the num-
ber of forward problems to the number of cur- rent electrode positions. The number of mea-
sured configurations is generally much higher.
An induced polarization survey aims to pro- vide images of the subsurface distribution of
electrical conductivity and polarization effects in an area of interest. Since each measured
value is influenced by both the conductivity distribution in a large volume of earth and the
electrode configuration, a pseudosection can only be a distorted reflection of the real subsur-
face structures. The reconstruction of the con- ductivity distribution is performed by inversion
techniques. Incorporating all available informa-
Ž tion into the inversion process Olayinka and
. Weller, 1997 can reduce the non-uniqueness
inherent in 2-D and 3-D interpretation. The objective of inversion consists of finding
a conductivity model which can approximate the measured data within the limits of data
errors and which is in agreement with all a priori information. The inversion can be done
manually by forward modelling in which changes in the model parameters are made by
trial and error until a sufficient agreement be- tween measured and synthetic data is achieved.
For more complicated structures, where the number of parameters increases, automatic in-
version procedures are recommended.
The inversion algorithm used is applicable to variable
electrode configurations
including buried electrodes. It is based on a simultaneous
Ž .
iterative reconstruction technique SIRT , which has been applied to several tomographic prob-
Ž lems to solve systems of linear equations e.g.,
Dines and Lytle, 1979; van der Sluis and van .
der Vorst, 1987 . It can be used for both 2-D and 3-D inversion. Since the number of grid
elements is generally much higher than the number of measured data, a strongly underde-
termined system has to be solved.
The steps involved in the iterative procedure are as follows.
Ž . a The subsurface is subdivided into blocks
of constant resistivity. The number of blocks N is equal to the number of model parameters. All
parameters may be described by a parameter Ž
.
T
vector x s x , . . . , x . The parameter x
is
1 N
j
defined as the logarithm of the resistivity of the jth block.
Ž . b The measured data are compiled in a data
Ž .
T
vector y s y , . . . , y where M corresponds
ˆ ˆ
ˆ
1 M
to the number of measurements. The element y
ˆ
i
of the data vector y is the logarithm of the
ˆ
complex apparent resistivity of the ith measure- ment in the data set. The use of the logarithms
of resistivities instead of resistivities has proven to be more appropriate in inversion because
negative resistivities are avoided and relative changes are emphasized.
Ž . c A starting model is chosen. The parameter
vector is initialized x s x
Ž0.
. Ž .
Ž k .
d The forward modelling for the model x
is performed where k denotes the number of the model. The apparent resistivity is calculated for
all M configurations of electrodes used in the survey. The calculated data are compiled in a
data vector y
Ž k .
. The forward modelling is de- scribed by an operator S which is applied to the
parameter vector x
Ž k .
:
y
Ž k .
s S x
Ž k .
. 12
Ž .
Ž .
Although the forward modelling operator S is nonlinear, an attempt was made to use SIRT for
Ž .
a linearization of Eq. 12 in the vicinity of the model x
Ž k . Ž .
Ž k . k
y s y q S x y x
. 13
Ž .
Ž .
The matrix S is the Jacobian or sensitivity ma- trix
4
is1 , . . . , M ,
S s s 14
Ž .
i , j js1 , . . . , N
with the elements E y
i
s s ,
15
Ž .
i , j
E x
j
called sensitivities describing the influence of a slight change of the resistivity of a single grid
element on the measured apparent resistivity of a given configuration. The forward modelling is
performed by the finite difference method.
Ž .
Ž k .
e The residual r between measured and
computed data is determined:
r
Ž k .
s y y y
Ž k .
. 16
Ž . ˆ
5
Ž k .
5
If a norm of the residual r has reached a
predetermined value ´ which corresponds to
the data accuracy the iteration process is stopped. The last model is accepted as a solution of the
inversion. Ž .
f If the residual fails the stopping criterion, the differences are applied to correct the resis-
tivity model according to the general equation 1
s
i , j Ž kq1.
Ž k .
x s x
q v r ,
Ý
j j
i a
2ya
s s
i
Ý Ý
i , l n , j
l n
17
Ž .
with 0 F a F 2 and 0 - v - 2. In the presented Ž
. inversion algorithm, Eq. 17 is used with an
exponent a s 1 which has been proven to be a good choice concerning the rate of convergence.
A relaxation factor 1.2 - v - 1.7 should be used to speed up the procedure.
The next iteration is started with the forward Ž .
modelling in step d . If no starting model is prescribed, one would
be generated by a back projection according to the formula
s y
ˆ
Ý
i , j i
i Ž0.
x s
, 18
Ž .
j
s
Ý
i , j i
using only positive sensitivities for a homoge- neous subsurface resistivity distribution. The
back projection which may be called an approx- imate inversion yields a model showing the
main features of the subsurface at low contrasts.
Ž .
The iterative application of Eq. 17 describes the procedure of a generalized matrix inversion
without explicitly calculating the inverse matrix. Without any smoothness constraint, the SIRT
generates models with low contrasts in the de- termined parameters. The duplicate weighing by
the sums of the absolute values of sensitivities
Ž .
inherent in Eq. 17 is responsible for a stable convergence and the resulting smooth models.
An efficient application of SIRT requires a fast and accurate algorithm to calculate the sen-
sitivity matrix S. Since the logarithms of both the model resistivity r and the apparent resis-
j
Ž .
tivity r are used from Eq. 15 , the following
ˆ
i
results: E y
E ln r r Er
r
ˆ ˆ
i i
j i
j
s s s
s s
s 19
Ž . ˜
i , j i , j
E x E ln r
r Er r
ˆ ˆ
j j
i j
i
with Er
ˆ
i
s s .
20
Ž . ˜
i , j
Er
j
In the case of a homogeneous resistivity dis- tribution with all model resistivities r s r s
j h
const, the resulting apparent resistivity becomes constant r s r , and the sensitivities s
and
ˆ
j h
i, j
s are equal.
˜
i, j
Following an algorithm described by Weller Ž
. et al. 1996b and using the abbreviation
3
s s
= V r r = V r r
d r , 21
Ž .
Ž .
Ž .
ˆ
H
A, M A
M V
j
the sensitivity of any four point configuration with the corresponding geometric factor K re-
i
sults in K
i
s s s
y s y s
q s ,
22
Ž . ˆ
ˆ ˆ
ˆ
½ 5
i , j A , M
A , N B , N
B , M 2
I r r
ˆ
j i
Ž .
Ž .
with V r r and V r r
being the electrical
A M
potentials for a current source at the locations A and M, respectively. The sensitivity calculation
for a given pole–pole configuration with the current electrode A and the potential electrode
M demands two separate forward runs with the source subsequently located at the transmitter
position A and the receiver position M. The decomposition of all configurations in the basic
pole–pole readings has proven to be advanta- geous for the application of multi-electrode-
arrays, because the number of measured config- urations is generally much higher than the num-
ber of electrodes.
This algorithm of sensitivity calculation is valid for arbitrary conductivity distributions. It
can be used for generating the initial sensitivi- ties for a uniform halfspace conductivity and
also for the conductivity model resulting from SIRT. In the special case of a homogenous
halfspace and all electrodes on the surface Eq. Ž
. 21 results in the expression
I
2
r
2
1 1
j 3
s s
= =
d r , 23
Ž . ˆ
H
A, M 2
V
4p r y r
r y r
j
A M
which can be easily calculated. Since a change in the model results in a
change of the sensitivities, these have to be updated. Tests have shown that an update of the
sensitivity matrix in each iteration results only in a slight improvement of the rate of conver-
gence compared with the use of unchanged halfspace sensitivities. Since the computational
effort of a sensitivity update is considerable, current experience suggests that it should be
performed only after every five or ten iterations.
6. Inversion results
