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In-Depth Information
Eliminating
H
x
o
,
H
y
o
from (4.40) and (4.41), we get
H
τ
(
r
)
=
[
M
(
r
|
r
B
)] H
τ
(
r
B
)
,
(4
.
42)
where
H
x
(
r
)
H
x
(
r
B
)
H
τ
(
r
)
=
H
τ
(
r
B
)
=
(4
.
43)
H
y
(
r
)
H
y
(
r
B
)
and
⎡
⎣
⎤
⎦
M
xx
(
r
|
r
B
)
M
xy
(
r
|
r
B
)
[
M
(
r
|
r
B
)]
=
M
yx
(
r
|
r
B
)
M
yy
(
r
|
r
B
)
[1
+
J
H
2
x
[1
+
J
H
2
x
−
1
J
H
1
x
J
H
1
x
(
r
)]
(
r
)
(
r
B
)]
(
r
B
)
=
.
J
H
2
y
+
J
H
1
y
J
H
2
y
+
J
H
1
y
(
r
)
[
(
r
)]
(
r
B
)
[
(
r
B
)]
(4
.
44)
Here
J
H
1
J
H
2
are convolutions of the excess currents with the magnetic Green
tensor defined by (1.12).
Let us cite some formulae that can be helpful in analyzing the horizontal mag-
netic tensor.
Rotating the horizontal magnetic tensor clockwise by an angle a (by the same
angle at the observation and base sites), we get
,
[
R
(a)][
M
][
R
(a) ]
−
1
[
M
(a)]
=
,
(4
.
45)
where
cos a sin a
−
cos a
−
sin a
[R(a) ]
−
1
[
R
(a)]
=
,
=
.
sin a cos a
sin a
cos a
The
rotational invariants
are
tr[
M
]
=
M
xx
+
M
yy
det [
M
]
=
M
xx
M
yy
−
M
xy
M
yx
(4
.
46)
tr[
M
]
=
tr[
M
][
R
(
−
/
2)]
=
M
xy
−
M
yx
M
xy
M
yx
M
yy
2
2
2
2
M
=
|
M
xx
|
+
+
+
,
where [
M
]
2)].
The
horizontal magnetic tensor
[
M
] reflects variations in the geoelectric medium
between the base and observation sites. We obtain the most clear image of these
=
[
M
][
R
(
−
/
