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1
2
arcsin(sin2
H
m
sin
H
m
)
H
m
=
,
−
/
4
≤
H
m
≤
/
4
and
−
1
≤
H
m
≤
1
.
Thus, by the Swift-Eggers method we derive eight independent eigenstate
parameters:
|
1
|
,
1
,
1
=
,
1
=
arg
H
1
H
1
(4
.
59)
|
2
|
,
arg
2
,
2
=
,
2
=
,
H
2
H
2
which fill all eight degrees of freedom possessed by the matrix [
M
].
Take the 2D model with the strike along the
x
-axis. Here
M
xy
=
M
yx
=
0
M
⊥
.
and
M
xx
=
1
,
M
yy
=
Using (4.55) and (4.57), we get
1
=
1
,
1
=
0 and
M
⊥
,
2
=
/
M
⊥
,
1
=
/
2
=
2or
1
=
2 and
2
=
1
,
2
=
0. With (4.58)
we get
0. The principal values of the magnetic tensor [
M
] coincide with its
longitudinal and the transverse components, while the principal directions are the
longitudinal and transverse ones. The magnetic eigenfields are linearly polarized
along the principal directions. Measuring the horizontal magnetic field at observa-
tion and base sites, we define the dimensionality of structure, but cannot distinguish
between the longitudinal and transverse direction.
Asymmetric 3D-structures manifest themselves in the elliptic polarization of the
magnetic eigenfields (
1
,
2
=
1
,
2
=
0) and in the violation of the perpendicularity of their
ellipses (
2).
It is a simple matter to show that in the general case the scalar invariants det [
M
]
and tr [
M
] of the tensor [
M
] can be expressed in terms of geometric and arithmetic
means,
|
1
−
2
| =
/
G
and
A
, of its principal values,
1
and
2
:
G
det[
M
]
=
M
xx
M
yy
−
M
xy
M
yx
=
1
2
=
,
(4
.
60)
tr[
M
]
=
M
xx
+
M
yy
=
1
+
2
=
2
,
A
where
=
√
det[
M
]
=
√
1
2
,
G
(4
.
61)
A
=
1
+
2
2
1
2
tr[
M
]
=
.
A
of the principal
values of the tensor [
M
] can be taken as the invariant parameters characterizing the
change in the intensity and phase of the horizontal magnetic field on the way from
It seems that the geometric and arithmetic means
G
and
