Geoscience Reference
In-Depth Information
G=PK 1 denotes Green
source current density, can be considered to be small
with respect to the system geometry. Therefore, each
of the occurrences may be considered as a compact or
point source. Additionally, as a seismic wave propagates
through geological heterogeneities, the heterogeneities
are considered to be contacted at single points and time,
and as such, each point of contact can also be considered
to be a compact point of contact, or a point source. There-
fore, at each time step, the seismic source or the resulting
seismic waves impinging on heterogeneities are respon-
sible for a point source current density and, by the ration-
alization stated earlier, by definition can only have
compact support (i.e., the spatial distribution of sources
at any given time is very sparse). Compact electrical point
current densities are vector quantities and therefore can
be represented as electrical dipoles with quantifiable
dipole moments. Based on these source current density
compactness concepts, we recently developed an algo-
rithm, presented in Araji et al. (2012), for the seismoelec-
tric effect to find the location of these compact volumetric
source current densities. The governing equation of
the forward electrical model (Eq. 5.9) can be rearranged
and written in a matrix form:
where
s matrix ( N × M ) com-
puted as a product of the inverse kernel matrix times
the sparse selector operator matrix that contains a single
1 on each row in the column that corresponds to the loca-
tion of that receiver. The rows of G can be computed
using reciprocity, which involves computing the forward
response to a unit source located at each receiver.
Other variables, such as
'
, denote the inverse-sensitivity
weighting function that accounts for the distance from
the receivers as well as the resistivity structure and is
Λ
1 2
2
N
i
1 G kj
formulated as
Λ
= diag
, matrix
W d corre-
sponds to the selection operator related to the expected
noise level
denotes the regularization
parameter, I represents an identity matrix, and
in the data,
α
m w is a
modified model vector, which is
computed from
m w =
in the first stage of inver-
sion is important since we need to have an initial model
to feed the source compactness calculation.
The compact source distribution method is a relatively
classical technique that suits the electrical part of the
problem, because the source current densities associated
with it tend to be spatially localized, as rationalized ear-
lier. This method is basically computed by minimizing
the spatial support for the source current density. In
order to get a compact source solution, the model weight-
ing parameter
Λ m
. The calculation of
m
d = Km
5 29
In Equation (5.29), K denotes the kernel matrix, which
depends on the distance between the source of current
and the receivers and the distribution of the electrical
resistivity. This matrix corresponds to the discrete form
of the leading field operator and accounts for the bound-
ary conditions applied to the system. In this problem, we
used Neumann boundary condition at the interface
between the model and the boundary padding layer
and Dirichlet boundary condition in the outer edge of
the padding layer. The vector
Λ
is modified by
1 2
kk
m k i 1 +
Λ
Ω
= diag
5 31
2
β
Using this modification, the problem now becomes linear
by making the denominator of
Ω
a function of the previ-
ous solution at step ( i
in an iter-
atively reweighted least-squares algorithm. The vector
m i 1 , where i = 1, denotes the initial model used to com-
pute the first solution of the compact source method.
Then, an updated
-
1) and using the new
Ω
is a vector containing
the M source current density terms (right-hand side of
Equation (5.29)), and
m
is the vector of the observed elec-
tric potentials at each time step, at N receiver locations.
Araji et al. (2012) developed a method to invert the
source current density from Equation (5.29) using com-
pactness as a regularization tool.
If we assume that the resistivity structure and bound-
ary conditions for the electrical field are known (via
downhole measurement and/or resistivity tomography),
a stable solution of
d
is determined at each new iteration
based on the previous solution. Using the transformed
parameters
Ω
J , we minimize the new weighted
objective function P α (
m w =
Ω
m
) defined by
M
m k
m k +
2 +
P α m
=
W
Gm d
α
5 32
d
2
β
k =1
m
is obtained by first solving
to get the solution for m w from
Λ 1
G 1
T
d
G Λ 1 + α Ι m w = Λ 1
G 1
T
Ω 1
i
T
T
d
W d G Ω 1
i
Ω 1
i
T
T
d
G
W
1 +
α Ι m w , i =
G
W
W d d
5 33
W
P
Pd
5 30
1
1
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