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In-Depth Information
H
where
H
is taken within quadrant I (0
≤
H
≤
/
2) if cos
≥
0 or within
H
<
0.
For the magnetic field
ellipticity
and normalized semi-axes of polarization ellipse
we have
quadrant IV (0
>
H
≥−
/
2) if cos
1
1
2
2
b
H
a
H
=
+ |
P
H
|
+
2Im
P
H
−
+ |
P
H
|
−
2Im
P
H
H
=
1
1
2Im
P
H
=
tan
H
2
2
+ |
P
H
|
+
2Im
P
H
+
+ |
P
H
|
−
(2
.
19)
a
H
a
H
+
b
H
a
H
+
b
H
=
cos
H
b
H
=
sin
H
,
where
1
2
arcsin(sin 2
H
sin
H
)
=
−
/
4
≤
≤
/
4
H
H
and
−
1
<
H
<
1
.
Polarization ellipses offer a geometrical image of the magnetotelluric field. The
major axis of the polarization ellipse gives preferential direction of the field, while
the field ellipticity defines the measure of this preference and the sense of the field
rotation.
Using polarization descriptors, we can readily define the spatial relationships
between complex field vectors.
The complex electric fields,
E
τ
1
and
E
τ
2,
are said to be orthogonal provided that
their scalar product is equal to 0:
E
x
1
E
x
2
+
E
y
1
E
y
2
=
E
τ
1
·
E
τ
2
=
0
.
(2
.
20)
Here
P
E
2
=−
P
E
1
1
(2
.
21)
and, according to (2.13) and (2.14),
E
1
−
E
2
=±
2
E
1
=−
E
2
E
1
=−
E
2
.
(2
.
22)
So, the orthogonal electric fields have the similar polarization ellipses with perpen-
dicular major axes and the opposite sense of field rotation.
The same holds good for the orthogonal magnetic fields,
H
τ
1
and
H
τ
2
:
H
x
2
+
H
y
2
=
H
τ
1
·
H
τ
2
=
,
.
H
x
1
H
y
1
0
(2
23)
