Geoscience Reference
In-Depth Information
estimated variance of the measurement errors. In all the
tests, we consider a Gaussian noise with a standard devi-
ation equal to 10% of the computed data mean. This is a
realistic noise level for this type of experiment accounting
for the amplitude of the signals that are measured (see,
for instance, Ardjmandpour et al., 2011). In addition,
adding data weights helps to stabilize the inversion by
eliminating artifacts that come from overfitting the data.
The vector
now nonlinear since the compactness portion of the
objective function is nonquadratic. The compactness
term is effectively a measure of the number of source
parameters that are greater than
, regardless of their
magnitude. Minimization of this objective function,
Equation (4.70), results in the solution that uses the few-
est number of source parameters that are still consistent
with the measured data, which enforces sparseness of the
source distribution. As model values fall below the
threshold
β
represents the inverse-sensitivity weighting
function. This function is needed because sensitivities
decay quickly away from the receiver locations (typically
as a power law function of the distance from the receiver
in a homogeneous medium, but is also affected by
heterogeneous resistivity distributions). The weighting
function is therefore needed to recover sources that are
distant from the receivers.
Applying the following transform s w =
Λ
, they no longer contribute to the sum in
Equation (4.70) and will be effectively masked from
the solution.
In order to make this compact source problem linear so
that it can be solved in a least-squares framework and to
incorporate the inverse-sensitivity scaling, the model
weighting operator
β
Λ
in Equation (4.69) is modified as
Λ
s and mini-
mizing Equation (4.67) gives the equation
Λ kk 2
s k j 1 2 + β
Ω
= diag
,
4 71
2
Λ 1 G T W d W d G
Λ 1
Λ 1 G T W d W d
obs
+
λ
I s w =
ψ
4 69
where diag( . ) is an operator extracting the diagonal ele-
ments of the argument. Hence, the problem is trans-
formed to a linear one by making the objective
function quadratic in s k by fixing the denominator of
the model objective function
The result of such an inversion is a smooth volumetric
source current distribution. However, we know from
the physics of the problem that the solution should be
spatially compact. In the next section, we describe a
modification to the model regularization term that
promotes this compactness.
Ω
with respect to the previ-
ous solution at step j
1 using an iteratively reweighted
least-squares approach. The vector s j 1 is the initial
model used to compute the first degree of compactness.
A new vector
is determined for every compactness
degree based on the previous model generated from
the immediate previous compactness degree. Using the
renormalization with s w =
Ω
4.4.4 Getting compact volumetric
current source distributions
Source compactness is a relatively classical technique that
suits the nature of the electrical problem because the
volumetric source current densities associated with the
seismoelectric conversion tend to be spatially localized.
The technique has been used in medical imaging and
in geophysics (see, for instance, Last & Kubik, 1983,
and Silva et al . , 2001). Compactness is based on minimiz-
ing the spatial support of the source. The new global
objective function is modified to include compactness
as a regularization term,
s and minimizing the global
model objective function in Equation (4.70) give us the
iterative solution which utilizes compactness
Ω
Ω 1
j
1 G T W d W d G Ω 1
1 + λ I s w , j = Ω 1
1 G T W d W d ψ
obs
j
j
4 72
The process is halted after several iterations. Focusing
the image is a subjective choice. Nine iterations offer a
good compactness level to localize the sources responsi-
ble for the observed self-potential data.
M
s k
s k +
2
2 +
obs
C = W d Gs
ψ
λ
2 ,
4 70
β
4.4.5 Benchmark tests
Before embarking on the process of inverting the
seismoelectric signals generated by the seismoelectric
forward model, it is essential to benchmark our inversion
k =1
where
is the threshold term introduced to provide
stability as s k
β
0. This form of the objective function is
 
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