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
The IP survey at the CTP was not originally planned for interpretation by a 3-D inversion
algorithm. All the measurements were made on east–west trending profiles that were not equally
spaced. The distances between the neighbouring profiles vary between 0.76 and 6.1 m. An opti-
mal 3-D survey should be performed on an equally spaced grid with profiles in perpendicu-
lar directions.
The large amount of IP data acquired in a relatively small area over structures which can-
not be regarded as two-dimensional suggested a test of a 3-D inversion algorithm that was devel-
oped to monitor hydraulic experiments with cross-hole measurements of complex conductiv-
Ž .
ity in isolated tanks Weller et al., 1996b . The inversion of the data set from the CTP was the
first test with IP field data. A first grid was generated for the inversion
of all the data acquired with 5-m dipole spacing Ž
. data set 1 . The resulting grid consists of 27,984
Ž nodes with 53 steps in the x-direction west–
. Ž
. east , 33 in the y-direction south–north , and
Ž .
16 in the z-direction depth . Data set 1 includes 638 measurements from all the eleven lines.
Average values are used for the reciprocal re- peat points. The back projection resulted in a
Ž .
model with a root mean square RMS error of 55.0 in the amplitude and 28.2 mrad in the
phase angle compared with the measured val- ues. After 20 steps of iteration performed on a
Pentium Pro 200 microcomputer in 21,437 s, the RMS error has fallen to 22.8 in the ampli-
tude and 11.3 mrad in the phase. The continu- ous decrease of the errors during the iteration
process is presented in Fig. 5. The sensitivity matrix has been updated after the second and
eighth step of iteration. During the first steps, a rapid decrease in error is observed. The last
steps result only in slight improvement of the model. The covergence rate slows down. Since
the residual did not reach the estimated data accuracy the maximum number of iterations
was performed.
The inversion results in a 3-D data matrix with each component describing the amplitude
and phase value of a resistivity grid element. Horizontal and vertical planes of data are ex-
tracted from this 3-D matrix.
The horizontal plane shown in Fig. 6 repre- sents the final distribution of complex resistivity
at a depth of 3.6 to 4.9 m. Three conductive zones are recognized in the image of resistivity
amplitudes. The two western anomalies are re-
Fig. 5. RMS error for 3-D inversion of two data sets.
Fig. 6. Resulting model: horizontal plane at a depth of 4 Ž .
Ž . m. a Resistivity amplitude. b Phase lag.
lated to the waste material in the cell of wooden boxes and the LOP. The third anomaly in the
southeastern part of the survey area is outside the waste trench; it might be caused by conduc-
tive clay-rich material. The phase image is dom- inated by a single strong anomaly with phase
values up to 180 mrad. The width and the length of this highly polarizable zone exhibits
good accordance with the extension of the cell filled with metal, wooden boxes, asphalt and
concrete. Outside this cell, the phase values stay at low level. Only the disperse metallic material
within the waste causes strong polarization ef- fects. The single large metallic objects exhibit
only moderate polarization.
A second inversion has been performed using the data acquired with both 2- and 5-m dipole
Ž .
spacing at the four northern lines data set 2 . The grid with a smaller increment in the x-di-
rection consists of 81 = 17 = 17 nodes in the x-, y- and z-directions. The finer grid results in
a better spatial resolution in this section of the survey area. A total of 393 measurements were
considered by the inversion algorithm. The back projection resulted in a misfit of 44.5 and 30.4
mrad, respectively in amplitude and phase. The RMS error fell to 21.7 and 13.6 mrad after 20
steps of iteration.
Fig. 7 shows a resulting vertical section at the position of line A. The image of the resistivity
amplitude is dominated by a conductive zone which coincides with the waste seam in horizon-
tal extent, taking the isoline of 24 V m as the boundary. The cap thickness of 1.5 to 2 m is
better resolved in the phase image because the contrast between soil and waste is more accen-
tuated in the phase lag. Both the amplitude and the phase images show a steep gradient at the
bottom of the waste structure. Concluding from the amplitude image, the lower boundary of the
trench reaches 9 m. Regarding the phase image, the boundary should be drawn at a depth of
Ž . Fig. 7. Resulting model: vertical section at Line A. a
Ž . Resistivity amplitude. b Phase lag.
approximately 13 m. The comparison with the real depth of 4.5 m reveals that the vertical
resolution of the IP survey is not satisfactory. A 2-D inversion of the data acquired along
Line A was performed by Yaoguo Li and Doug Oldenbourg
of the
University of
British Ž
. Columbia Oldenburg and Li, 1994 . The result-
ing amplitude and phase images are similar to those in Fig. 7, also showing different depths to
the bottom of the waste by resistivity amplitude
Ž .
and phase lag Frangos, 1997 . The finer grid used in 2-D inversion results in a high variabil-
ity of near surface resistivities. The use of a coarse grid and SIRT in the 3-D algorithm
yields smooth images emphasizing the real dominant structures and ignoring noise in the
data. The resolution of the bottom of the waste could not be improved by a 3-D processing.
There are at least three possible reasons for the modest depth resolution.
Ž . a The dipole–dipole configuration is more
suitable for finding vertical discontinuities, as demonstrated by this survey.
Ž . b The SIRT used generates smooth models
fitting the data with the lowest possible contrast in the resistivity amplitude and phase values.
No additional information or constraint is used. Ž .
c The inversion algorithm cannot overcome the ambiguity of both resistivity and IP data. A
high contrast in the intrinsic resistivity and po- larizability is necessary to explain the data.
According to the principle of equivalence, a further increase of the contrast between waste
and soil material would result in a lesser depth to the bottom of the trench.
7. Modelling experiment
