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the number of point dipoles). We have 3 M >> N (model
unknowns >> actual measurements); therefore, the
inverse problem is strongly underdetermined. In this
deterministic inversion process, the roles of data misfit
and model objective functions are balanced using Tikho-
nov regularization (Tikhonov & Arsenin, 1977; Jardani
et al., 2008) through a global objective function, P λ ( m ),
defined as
(the fewest dipoles with moments that mathematically
support the data). As shown by Last and Kubik (1983),
the depth-weighted and compactness-based stabilizing
functional is expressed as
Ω
x 00
0
Ω
=
Ω y 0
00
5 45
Ω
z
P λ m = d Km 2 + λ s m
5 41
where each of the Ω i i = x , y , z components represents
one orthogonal direction and is a ( M × M ) diagonal
submatrix. The elements of these three submatrices are
computed by
where λ represents a Lagrange regularization parameter
(0 < λ < ) and s ( m ) denotes the (stabilizing) regularizer.
The vector m is the vector described earlier.
Step 3. The inversion process needs to be scaled with
respect to the distance between the sources and the recei-
vers to remove depth-based bias in the solution. This
gives each dipole in the model an equal chance to partic-
ipate in the numerical solution. This will be reversed later
in the process to reflect the actual depth effects on
the solution. Depth weighting is accomplished by using
a(3 M ×3 M ) diagonal matrix, J , that can be computed
from the kernel as follows:
J x , y , z
ii
m x , y , z
i
x , y , z
ii
Ω
=
5 46
2
+
β
x , y , z
ij
Ω
i
j =0
5 47
where
is
explained further in the following text). This compact-
ness matrix structure is also required to match the kernel
matrix, weighting matrix, and model vector organiza-
tions in terms of separating out the three dimensions of
space in three separate subsections.
Step 4. Normalization of the kernel matrix is required
to prevent undesirable depth-based bias while invoking
compactness constraints in the inversion process.
Therefore, we form a new normalized kernel matrix that
is used during compactness computations as follows:
β
is a support parameter (our choice of
β
J x 00
0 J y 0
00 J z
J =
5 42
where each of the J i i = x , y , z components represents
one orthogonal direction and is a ( M × M ) diagonal
submatrix computed by
K = K x , K y , K z
5 48
N
J x , y , z
ii
K x , y , z
ij
=
5 43
Κ x , y , z =
Κ x , y , z Ω 1
5 49
x , y , z
j =1
J x , y , z
ij
i
j =0
5 44
where K is a ( N ×3 M ) matrix that is formed through
the scaling of the original kernel with the compactness
functional as shown in Equation (5.45).
Step 5. We solve the following system of equations
for m using Tikhonov regularization (Tikhonov &
Arsenin, 1977; Jardani et al., 2008) through the global
objective function, P λ ( m ):
This weighting matrix structure matches the kernel
matrix and model vector organizations in terms of separ-
ating out the three dimensions of space in three separate
subsections.
Given that we are looking for a compact source consist-
ing of as few dipoles as possible that would model the
electrical mechanism that generated the voltage mea-
surements, compactness of the source must be assured.
Therefore, we apply a compact source inversion process,
where we seek to find one model (model vector) with
the minimum volume of the source current density
K
λ
d
a
m =
5 50
I
where I represents the (3 M ×3 M ) identity matrix, d is
the ( N × 1) vector of measured electrical potential data,
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