Information Technology Reference
In-Depth Information
surface in a smoothed diffusive magnetotelluric anomaly with extent that can con-
siderably exceed the body dimensions. The resolution of the magnetotelluric and
magnetovariational soundings is such that we cannot distinguish the smooth body
from the contrasty one. In view of errors in the initial data, both
−
distributions are
equivalent. To decide between the smooth and contrasty
−
distributions, we need
a priori information or some hypotheses.
In regions with sufficiently slow horizontal variations in the conductivity and
thickness of the geoelectric layers, we can take a quasi-homogeneous layered
interpretation model and regularize the magnetotelluric and magnetovariational
inversions by smoothing the solution obtained. This traditional form of Tikhonov's
regularization is named the
smoothing inversion
or
Occam inversion
(after William
of Occam, English philosopher, 1285-1349, whose maxim says that assumptions
introduced to explain a thing must not be multiplied beyond necessity).
The Occam inversion is implemented on the large grids and its stability is ensured
by the smoothing stabilizer. The simplest form of the smoothing stabilizer is
g
1
2
2
dxdydz
2
g
2
(
)
=
+
+
,
(12
.
1)
x
y
z
V
where
g
2
are weights
controlling the horizontal and vertical smoothing. Requirements of the smoothness
of
=
(
x
,
y
,
x
) is a solution of the inverse problem and
g
1
,
(
x
,
y
,
z
) can be readily supplemented with requirement of the closeness of
(
x
,
y
,
z
) to the hypothetical model
0
(
x
,
y
,
z
). In this case we can construct the
stabilizing functional as
g
1
2
−
0
)
2
dxdydz
2
g
2
2
(
)
=
+
+
+
g
3
(
,
x
y
z
V
(12
.
2)
where a weight
g
3
is chosen so small that requirement of the closeness of
to
0
does not dominate over requirements of the smoothness of
.
The mathematical programs OCCAM, REBOCC and NLCG based on the
smoothing inversion are widely used in exploration magnetotellurics (Constable
et al., 1987; DeGroot-Hedlin and Constable, 1990; Siripunvaraporn and Egbert,
2000; Rodi and Mackie, 2001). They have proved to be quite efficient in studying
the regional structure of the sedimentary basins in areas with gentle tectonics.
Unfortunately, there is a limit to what can be done with smoothing inversion.
Magnetotellurics is of frequent use in areas where we look for sharp geoelectric
contrasts between different geological formations (for instance, in areas with ore-
bearing bodies or fluidized layers and faults). When smoothing, we get blurred
images and lose significant information (smear or even miss the real structures).
In that events we have to take a locally inhomogeneous interpretation model
characterized by a contrasty
distribution and regularize the magnetotelluric
and magnetovariational inversions by keeping the solution close to a hypothetical
−
