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algorithm through a synthetic and well-defined 2.5D
electrical model. The benchmark model is a 450 × 500m
section constructed with 45 × 50 cells. The conductivity
distribution resembles that of the first case study of the
vertical interface with two units U1 and U2 in the
model. In the first test, we positioned a single point
source at
x
=430mand
z
= 310m, and we recorded
the electrical potentials at
x
= 510 m via an array com-
prised of 50 electrodes with 10 m vertical separation
between adjacent electrodes so that the array extends
from
z
=50mto
z
= 550 m. The initial inverse source
solution is recovered using Equation (4.63),whichis
plotted in Figure 4.19a. This initial solution is diffusive
and does not represent the spatial nature of the true
compact source. Figure 4.19b shows the electric poten-
tial corresponding to the source from the initial inverse
solution versus the observed data with added noise. It is
clear that the algorithm is able to predict, with a great
accuracy, the measured voltages without overfitting
the noise. Nevertheless, the 2D representation of the
model is overly smeared, thus the need to include
compactness as a regularization tool as mentioned
earlier. Upon performing only five iterations of
Equation (4.72), the algorithm was able to locate the
source precisely as shown in Figure 4.19b.
To test the ability of the algorithm to resolve two
localized sources in a heterogeneous conductivity distri-
bution, we conducted another benchmark test. Two
volumetric current sources are located at position (
x
,
z
)
= (250 m, 460 m) and (250 m, 150 m), respectively. The
geometry is kept identical to the previous test. The result-
ing electrical voltage is aliased by the superstition of the
two voltage distributions. The initial (diffusive) solution
in terms of inverted current density is presented in
Figure 4.20a using the observed data contaminated with
added Gaussian noise (same as aforementioned), and in
Figure 4.20b, we show the result of the inversion after
five iterations in the focusing procedure. The algorithm
is able to distinguish and localize two distinct current
sources away from the receivers
examples, the true sources are known to be highly
compact. In other cases, the degree of compactness
may not be known a priori. In other words, the electrical
potential data can be accurately reproduced with diffuse
or compact sources because of the nonunique nature of
the problem; the user must decide how many compact-
ness iterations are appropriate. The choice in focusing
the tomogram is therefore the choice of the use, which
is definitively a drawback of this approach. Our tests
are showing that 5
10 iterations are usually a good
iteration number to properly focus the tomogram. Future
efforts to address this issue will include the use of
auxiliary information from the recorded seismic response
to help constrain the location of the heterogeneities.
-
4.4.6 Numerical case studies
Our goal is to invert the source current densities at
heterogeneities regardless of their magnitudes. However,
we note that our algorithm is able to retrieve the values of
the divergence of the current source density as well.
Hence, inverting for these sources using recorded
voltages in the offset borehole will provide us with a tool
to distinguish and localize heterogeneities from the
background material. We start by sequentially triggering
seismic sources in the source borehole (borehole #1) in
order to produce electric potential time series recorded
in borehole #2 for times belonging to the SC-time
window. The process is repeated for five seismic shots
separated by 75 m in borehole #1 (see Figure 4.15).
Electric potentials at each time event are recorded versus
depth and processed by the inversion algorithm. As
mentioned earlier, the decision to stop the compactness
algorithm is arbitrary. It seems that focusing the
tomograms up to the ninth iteration provides a suitable
compact solution. All of the inversions for different time
events and for different seismic shots are aggregated to
produce the final image, which shows the position of
the heterogeneity between the two boreholes.
Figure 4.21 shows eight voltage recordings as a function
of depth for eight different time steps for shot #3 (localized
in themiddle of the borehole inwhich the seismic sources
are located). This case corresponds to case study #1 in
Figure 4.15a (vertical interface between two porous
media). The eight times we chose belong to the SC-time
window and sample the seismoelectric conversion signal
such as that illustrated in Figure 4.21. At each time
step, the vertical recordings of the potential field are pro-
cessed using the source inversion algorithm to produce
location without
losing sensitivity. Also, the predicted electrical voltages
recorded in borehole #2 (see Figure 4.20b) are almost
identical to the observed one. In other words, the
algorithm is able to retrieve the true amplitude of the
potentials without prior knowledge on the source.
These results highlight the ability of the compact
source inversion to accurately recover the true spatial
distribution of current sources.
'
In these benchmark
