Information Technology Reference
In-Depth Information
The resulting electromagnetic fields on the Earth's surface
E
(1)
E
(1)
x
E
(1)
y
={
,
,
0
}
,
H
(1)
H
(1)
x
H
(1)
y
H
(1)
z
and
E
(2)
E
(2)
x
E
(2)
y
,
H
(2)
H
(2)
x
H
(2)
y
H
(2)
z
={
,
,
}
={
,
,
0
}
={
,
,
}
provide the system of linear equations for the impedance tensor components:
Z
xx
H
(1)
Z
yx
H
(1)
Z
xy
H
(1)
E
(1)
x
Z
yy
H
(1)
E
(1)
y
+
=
+
=
x
y
x
y
Z
xx
H
(2)
x
Z
xy
H
(2)
y
E
(2)
x
Z
yx
H
(2)
x
Z
yy
H
(2)
y
E
(2)
y
+
=
+
=
,
(10
.
24)
whence
E
(1)
x
H
(2)
y
E
(2)
x
H
(1)
y
E
(2)
x
H
(1)
x
E
(1)
x
H
(2)
x
−
−
Z
xy
=
Z
xx
=
H
(1)
H
(2)
H
(2)
H
(1)
H
(1)
H
(2)
H
(2)
H
(1)
−
−
x
y
x
y
x
y
x
y
E
(2)
y
H
(1)
x
E
(1)
y
H
(2)
x
E
(1)
y
H
(2)
y
E
(2)
y
H
(1)
y
−
−
Z
yy
=
H
(1
y
.
(10
Z
yx
=
H
(1)
H
(2)
H
(2)
H
(1)
H
(2)
H
(2)
H
(1)
−
−
x
y
x
x
y
x
y
.
25)
˜
W
. To deter-
2. MV inversion: conductivity
(
x
,
y
,
z
) is found from the tipper
, we use the magnetic fields
H
(1)
mine the operator
W
{
x
,
y
,
z
=
0
,
,
(
x
,
y
,
z
)
}
=
H
(1)
x
H
(1)
y
,
H
(1)
z
and
H
(2)
H
(2)
x
H
(2)
y
H
(2)
z
{
, obtained on the Earth's surface
for two different polarizations of the normal field, and solve the system of linear
equations
,
}
={
,
,
}
W
zx
H
(1)
W
zy
H
(1)
H
(1)
z
+
=
x
y
(10
.
26)
W
zx
H
(2)
+
W
zy
H
(2)
=
H
(2)
z
,
x
y
which yields the tipper components
H
(1)
z
H
(2)
y
H
(2)
z
H
(1)
y
−
H
(2)
z
H
(1)
x
H
(1)
z
H
(2)
x
−
W
zx
=
H
(1
y
,
W
zy
=
H
(1
y
.
.
(10
27)
H
(1)
H
(2)
H
(2)
H
(1)
H
(2)
H
(2)
−
−
x
y
x
x
y
x
10.3 Three Questions of Hadamard
Solving an inverse problem, one should answer three questions of Hadamard:
1. Does the solution of this problem exist?
2. Is it unique?
3. Is it stable with respect to small errors in initial data?
These questions determine the correctness of the inverse problem. If its solution
exists and if it is unique and stable, the problem is well-posed (posed correctly). But
if one of these conditions is violated, the problem is regarded as ill-posed (posed
incorrectly), and it calls for special consideration. We will show that inverse prob-
lems of magnetotellurics are ill-posed.
