Information Technology Reference
In-Depth Information
E
τ
m
=
[
Z
]
H
τ
m
,
m
=
1
,
2
E
x
1
E
x
2
+
E
y
1
E
y
2
=
P
E
2
=−
E
τ
1
·
E
τ
2
=
0
P
E
1
1
(2
.
61)
H
x
2
+
H
y
2
=
P
H
2
=−
H
τ
1
·
H
τ
2
=
H
x
1
H
y
1
0
P
H
1
1
.
Let us express these magnetotelluric eigenfields in terms of the polarization
descriptors
,
H
(
is an angle made by the major axis of polarization ellipse with
E
,
ε
the
x
-axis) and
H
(
is a ratio between the minor,
b
, and major,
a
, semi-axes of
E
polarization ellipse).
In conformity with (2.13), (2.18) and (2.23), (2.25),
+
2
E
1
cos
E
1
tan 2
=
tan 2
=
E
1
E
2
E
1
(2
.
62)
H
2
=
H
1
+
2
,
H
1
cos
H
1
tan 2
H
1
=
tan 2
=
P
E
1
,
=
P
H
1
,
E
1
E
1
H
1
H
1
where tan
=
arg
P
E
1
and tan
=
arg
P
H
1
.Here
E
1
E
1
is taken within quadrant I (0
≤
E
1
≤
/
2) if cos
≥
0 or within quadrant
E
1
<
0, while
IV (0
>
E
1
≥−
/
2) if cos
H
1
is taken within quadrant I (0
≤
H
1
H
1
≤
/
2) or quadrant III (
≤
H
1
≤
3
/
2) if cos
≥
0, and within quadrant
H
1
<
0
IV (0
>
H
1
≥−
/
2) or quadrant II (
>
H
1
≥
/
2) if cos
.
So, we
determine an acute angle
E
1
and have freedom in choosing either acute or blunt
angle
H
1
to suit proper relationship between electric and magnetic eigenfields.
In the same vein, according to (2.14), (2.19) and (2.22), (2.25),
=
E
1
=−
E
2
=−
tan
E
1
E
2
E
1
E
1
(2
.
63)
=
tan
H
1
=−
H
2
=−
H
1
,
H
1
H
2
H
1
where
1
2
arcsin(sin2
1
2
arcsin(sin2
E
1
sin
E
1
)
H
1
sin
H
1
)
E
1
=
=
H
1
with
1.
Using these definitions, we can introduce orthonormal basises
e
1
,
−
/
4
≤ ≤
/
4 and -1
≤
≤
h
2
into the spaces of electric and ma
gnetic fi
elds. To this end, we normalize the eigen-
fields
E
τ
1
,
e
2
and
h
1
,
H
τ
2
to
√
a
2
b
2
and multiply them by a phase factor
e
−
i
such that their real and imaginary vectors coincide with major
a
and minor
b
semi-
axes of corresponding polarization ellipse. With account for (2.15) and (2.19), we
write
E
τ
2
and
H
τ
1
,
+
