Information Technology Reference
In-Depth Information
Moduli of principal values,
|
1
|
and
|
2
|
, have a simple interpretation:
m
H
ym
+
E
ym
2
2
2
2
|
E
xm
|
=
+ |
m
H
xm
|
H
ym
2
2
=
|
m
|
|
H
xm
|
+
,
m
=
1
,
2
whence
E
ym
2
2
|
E
xm
|
+
|
m
| =
2
,
=
,
.
.
m
1
2
(2
47)
+
H
ym
2
|
H
xm
|
So, the moduli of principal impedances are the ratios between the Euclidean
norms of the electric and magnetic eigenfields.
Next we should determine the principal directions of the tensor [
Z
]
Hereaspe-
cial agreement is required. The point is that the electric and magnetic eigenfields,
E
τ
1
,
.
H
τ
2
, are complex vectors. They can be characterized by direc-
tions of their real and imaginary parts or by orientation of their polarization ellipses.
Let us define the principal directions of the impedance tensor as directions of the
major axes of polarization ellipses for the electric eigenfields
E
τ
1
and
E
τ
2
.
With (2.40) and (2.44), the polarization ratios for
E
τ
1
and
E
τ
2
are
H
τ
1
and
E
τ
2
,
H
ym
=−
m
−
Z
xy
m
+
Z
yx
=−
m
−
Z
xy
−
Z
yy
E
ym
E
xm
=−
H
xm
Z
yy
P
E
m
=
=
m
+
Z
xx
+
Z
yx
,
m
Z
xx
(2
.
48)
=
1
,
2
.
Applying (2.13), we evaluate angles
E
1
and
E
2
made by major axes of the elec-
tric polarization ellipses with the
x
-axis:
E
m
cos
E
m
tan 2
=
tan 2
,
m
=
1
,
2
,
(2
.
49)
E
m
P
E
m
,
E
m
E
m
where tan
=
=
arg
P
E
m
.
Note that the value of
E
m
is taken
E
m
within quadrant I (0
≤
≤
/
2) if cos
≥
0 or within quadrant IV
E
m
E
m
(0
E
2
indicate the directions
of major axes of the electric polarization ellipses. These directions, determined mod-
ulo
>
≥−
/
2) if cos
<
0. The values of
E
1
and
E
m
, are considered as the principal directions of the magnetotelluric impedance
tensor.
Finally we determine the ellipticity parameters
and
E
2
. In accord with
E
1
(2.14),
=
E
m
,
=
,
.
tan
m
1
2
(2
50)
E
m
