Information Technology Reference
In-Depth Information
From (1.29) and (1.32) we can derive another real-valued invariants:
J
8
=
J
5
−
J
6
=
det [Re
Z
]
−
det [Im
Z
]
=
Re det [
Z
]
a
Re
2
I
3
+
Re
2
I
1
−
J
9
=
2
J
5
Re
2
Z
xx
+
Re
2
Z
xy
+
Re
2
Z
yx
+
Re
2
Z
yy
=
=
Re
Z
b
Im
2
I
3
+
Im
2
I
1
−
J
10
=
2
J
6
(1
.
33)
Im
2
Z
xx
+
Im
2
Z
xy
+
Im
2
Z
yx
+
Im
2
Z
yy
=
=
Im
Z
c
2
2
J
11
=
|
I
1
|
+ |
I
3
|
−
2(
J
5
+
J
6
)
2
2
2
2
=
|
Z
xx
|
+|
Z
xy
|
+|
Z
yx
|
+|
Z
yy
|
=
Z
,
d
,
,
where
Re
Z
Im
Z
Z
are the Euclidean norms of the matrices [Re
Z
]
,
,
[Im
Z
]
[
Z
] respectively.
Two more real-valued rotational invariant are
Z
yy
+
Z
xx
Z
yx
)
J
12
=
Im (
Z
xy
,
(1
.
34)
Z
xy
+
Z
yx
Z
yy
)
J
13
=
Im (
Z
xx
,
(1
.
35)
where the bars denote the complex conjugation.
With all these invariants one can construct a standard set of parameters that help
to reveal and classify the geoelectric structures of the Earth.
1.3 Dimensionality of the Impedance Tensor
The general properties of the impedance tensor depend on the dimensionality D of
the magnetotelluric model, that is, on the number of coordinates required for its
description.
We deal with one-dimensional (1D), two-dimensional (2D) and three
-
dimensional (3D) models as well as with superimposition (2D + 3D, 3D + 3D,
2D + 2D) models. Respectively, we consider the one-dimensional, two-dimensional
and three-dimensional impedance tensors as well as the superimposition impedance
tensor.
The number
n
of real quantities which determine the complex-valued impedance
tensor are considered as a number of degrees of its freedom. A simple relation holds
between
n
and D:
2
D
=
.
.
n
(1
36)
