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4.4.2 The Schmucker Tensor
Another tensor representation has been proposed by Schmucker in his classical
monograph (Schmucker, 1970). It establishes the relation between the anomalous
magnetic field
H
A
at an observation site and the normal magnetic field
H
N
τ
at a base
site B located in a horizontally homogeneous zone. From (4.1) we derive
H
A
(
r
)
S
(
r
B
)]
H
N
τ
=
[
S
(
r
)
|
(
r
B
)
,
(4
.
63)
where
⎡
⎣
⎤
⎦
,
H
x
(
r
)
H
y
(
r
)
H
z
H
x
(
r
B
)
H
y
(
r
B
)
H
A
(
r
)
H
N
τ
=
(
r
B
)
=
(4
.
64)
(
r
)
and
⎡
⎤
⎡
⎤
J
H
2
x
J
H
1
x
S
xx
(
r
|
r
B
)
S
xy
(
r
|
r
B
)
(
r
)
(
r
)
⎣
⎦
=
⎣
⎦
.
J
H
2
y
J
H
1
y
[
S
(
r
|
r
B
)]
=
S
yx
(
r
|
r
B
)
S
yy
(
r
|
r
B
)
(
r
)
(
r
)
(4
.
65)
J
H
2
z
J
H
1
z
S
zx
(
r
|
r
B
)
S
zy
(
r
|
r
B
)
(
r
)
(
r
)
Here
J
H
1
J
H
2
are convolutions of the excess currents with the magnetic Green
tensor defined by (1.12).
The tensor [
S
] is referred to as the
Schmucker tensor
or the
perturbation tensor.
It falls into the
horizontal Schmucker tensor.
,
S
xx
(
r
J
H
2
x
(
r
)
J
H
1
x
|
r
B
)
S
xy
(
r
|
r
B
)
(
r
)
[
S
τ
(
r
|
r
B
)]
=
=
,
J
H
2
y
(
r
)
J
H
1
y
S
yx
(
r
|
r
B
)
S
yy
(
r
|
r
B
)
(
r
)
(4
.
66)
H
A
τ
(
r
)
=
[
S
τ
(
r
|
r
B
)]
H
N
(
r
B
)
,
τ
which is an analog of the magnetic tensor [
M
], and the
Schmucker matrix
=
S
zx
(
r
|
r
B
)
S
zy
(
r
|
r
B
)
=
J
H
2
(
r
)
,
(
r
)
J
H
1
y
|
[
S
z
(
r
r
B
)]
y
(4
.
67)
H
z
(
r
)
r
B
)]
H
N
=
[
S
z
(
r
|
(
r
B
)
,
τ
which is an analog of the Wiese-Parkinson matrix [
W
].
Consider relationships between tensors [
S
τ
] and [
M
] as well as between matrices
[
S
z
] and [
W
]. Inasmuch as the base site B is located in a horizontally homogeneous
zone, it is evident that
[
S
τ
]
=
[
M
]
−
[
I
]
(4
.
68)
[
S
z
]
=
[
W
][
M
]
,
