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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,