Information Technology Reference
In-Depth Information
Chapter 10
Statement of Inverse Problem
The inverse problem in magnetotellurics using the plane-wave approximation of
the source field consists in the determination of the geoelectric structure of the
Earth from a dependence of the magnetotelluric and magnetovariational response
functions on observation coordinates
x
,
,
=
of the electro-
magnetic field. Magnetotelluric and magnevariational inversions usually reduce to
solution of the operator equations for the impedance tensor and tipper:
y
z
0 and frequency
[
Z
]
[
Z
{
x
,
y
,
z
=
0
,
,
(
x
,
y
,
z
)
}
]
=
a
(10
.
1)
˜
W
W
{
x
,
y
,
z
=
0
,
,
(
x
,
y
,
z
)
}=
,
b
where [
Z
] and
W
are operators of the forward problem that calculate the impedance
tensor and tipper from a given electrical conductivity
(
x
,
y
,
z
), both the operators
;[
Z
] and
˜
W
are the impedance tensor and tip-
per determined on the set of surface points (
x
depend parametrically on
x
,
y
,
,
y
) and frequencies
with errors
Z
and
W
.
The electrical conductivity
(
x
,
y
,
z
) is found from the conditions
[
Z
]
]
≤
Z
−
[
Z
{
x
,
y
,
z
=
0
,
,
(
x
,
y
,
z
)
}
a
}
≤
W
.
(10
.
2)
W
−
W
{
x
,
y
,
z
=
0
,
,
(
x
,
y
,
z
)
b
Inverse problem (10.1) includes
MT inversion
(10.1
a
) and
MV inversion
(10.1
b
).
It is solved in the class of piecewise-homogeneous or piecewise-continuous models
excited by a plane wave vertically incident on the Earth's surface,
z
0. Inver-
sions (10.1
a
) and (10.1
b
) should be mutually consistent. They result in approximate
conductivity distributions ˜
=
(
x
,
y
,
z
) such that misfits of the impedance tensor and
tipper do not exceed errors,
Z
and
W
, in the initial data. The distributions ˜
(
x
,
y
,
z
)
generate a set
of equivalent solutions of the inverse problem (10.1).
Magnetovariational inversion (10.1
b
), (10.2
b
) can be extended by inversion of
the horizontal magnetic tensor:
