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solution is called the
S-transformation
. The obtained
S
m
(
z
)
M
can
be smoothed in
x,y
by a spline with a minimum norm of horizontal derivatives of
the conductance.
At the second step we determine
,
m
=
1
,
2
,...
m
(
z
) from
S
m
(
z
)
,
m
=
1
,
2
,...
M
.Amini-
mization problem for
m
(
z
)is
inf
{
(
m
(
z
)
}=
inf
{
I[
m
(
z
)]
+
[
m
(
z
)]
}
m
=
1
,
2
, ...
M
,
(12
.
12)
where
0
m
(
z
)
dz
2
z
S
m
(
z
)
I[
m
(
z
)]
=
−
L
2
2
[
m
(
z
)]
=
p
(
z
)[
m
(
z
)
−
0
(
z
)]
L
2
.
Here
0
(
z
) is the hypothetical model constructed
on the basis of a priori information,
p
(
z
) is a weighting factor that decreases mono-
tonically with depth.
The Euler equation for (12.11) can be reduced to the Volterra integral equation
of the second kind:
is the regularization parameter,
z
z
S
m
(
p
(
z
)[
m
(
z
)
−
0
(
z
)]
+
(
z
−
)
m
(
)
d
=
)
d
z
∈
[0
,
z
max
]
.
0
0
.
(12
13)
On simple rearrangement, we write
z
p
(
z
)
m
(
z
)
+
(
z
−
)
m
(
)
d
=
f
m
(
z
)
z
∈
[0
,
z
max
]
,
(12
.
14)
0
where
m
(
z
)
=
m
(
z
)
−
0
(
z
)
z
z
z
S
m
(
[
S
m
(
f
m
(
z
)
=
)
d
−
(
z
−
)
0
(
)
d
=
)
−
S
0
(
)]
d
0
0
0
z
S
0
(
z
)
=
0
(
)
d
.
0
m
(
z
)
|
m
(
z
)
|
=
Solving this equation for a given
,wefind
and calculate
0
(
z
)
+
m
(
z
)
|
. So, we find a set of approximate one-dimensional solutions
