Information Technology Reference
In-Depth Information
[
M
]
[
M
{
x
,
y
,
z
=
0
,
,
(
x
,
y
,
z
)
}
]
=
c
(10
.
1)
[
M
]
]
≤
−
[
M
{
x
,
y
,
z
=
0
,
,
(
x
,
y
,
z
)
}
,
c
(10
.
2)
M
where [
M
] is operator of the forward problem that calculates the magnetic tensor
from a given electrical conductivity
(
x
,
y
,
z
), it depends parametrically on
x
,
y
,
;
[
M
] is the magnetic tensor determined on the set of surface points (
x
,
y
) and fre-
quencies
M
.
Errors in the initial data
with errors
M
include the measurement and model errors.
The
measurement errors
are commonly random. They arise due to instrumental
noises, external interferences, and inaccuracies in the calculation of [
Z
],
˜
W
,[
M
].
Improvement in instrumentation and field data processing methods decreases these
errors. Presently, due to progress in MT technologies, measurement errors are, as a
rule, fairly small (the problems may be encountered in zones of intense industrial
disturbances). The main difficulty is connected with
model errors
that arise due to
the inevitable deviation of numerical simulations from real geoelectric structures
and real MT fields. As an example, we can cite the errors arising in 2D inversion
of data obtained above 3D structures or the errors typical of polar zones, where
the magnetic field of ionospheric currents has a vertical component contradicting
the plane-wave approximation. Model errors are systematic. They are usually larger
than measurement errors. To estimate the model errors, we need a tentative mathe-
matical modeling.
Strategy and informativeness of the inverse problems depend on the dimension-
ality of models.
The simplest inverse problem is 1D inversion carried out in the class of one-
dimensional models. It applies the mathematics of zero horizontal derivatives. Such
a mathematics provides the local determination of the electrical conductivity along
vertical profiles passing through observation points. The 1D inversion evidently
ignores distortions produced by lateral geoelectric inhomogeneities. It is justified
if horizontal variations in the conductivity are fairly small. Otherwise, it can miss
real structures and give birth to false structures (artefacts).
The transition to 2D and 3D inversions carried out in the classes of two- and
three-dimensional models enables the more or less adequate regard for the horizon-
tal geoelectric inhomogeneities, but calls for horizontal derivatives. This mathemat-
ics substantially complicates the inverse problem.
Z
,
W
,
10.1 On Multi-Dimensional Inverse Problem
Consider three distinguishing features of the multi-dimensional inverse problem.
10.1.1 Normal Background
In solving the multi-dimensional inverse problem, we face the contradiction
between a finite area of MT and MV observations and a mathematical statement
