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and
Z
N
J
H
S
1
x
J
H
S
2
x
J
E
R
2
y
J
H
R
1
x
J
E
R
1
y
J
H
R
2
x
−
−
+
−
h
xx
=
Z
N
J
E
R
1
J
E
R
y
+
Z
N
+
−
J
E
R
2
x
J
E
R
1
y
−
J
E
R
1
x
J
E
R
2
y
x
Z
N
J
H
R
2
x
J
H
S
1
x
J
E
R
1
x
J
H
S
2
x
J
E
R
2
x
J
H
R
1
x
+
−
+
−
h
xy
=
Z
N
J
E
R
1
x
J
E
R
y
+
J
E
R
2
x
J
E
R
1
y
J
E
R
1
x
J
E
R
2
y
Z
N
+
−
−
Z
N
−
J
H
S
1
y
J
H
S
2
y
(1
.
72)
J
E
R
2
y
J
H
R
1
y
J
E
R
1
y
J
H
R
2
y
−
+
−
h
yx
=
Z
N
J
E
R
1
x
J
E
R
y
+
Z
N
+
−
J
E
R
2
x
J
E
R
1
y
−
J
E
R
1
x
J
E
R
2
y
Z
N
+
J
H
R
2
y
J
H
S
1
y
J
E
R
1
x
J
H
S
2
y
J
E
R
2
x
J
H
R
1
y
−
+
−
h
yy
=
Z
N
J
E
R
1
x
J
E
R
y
+
.
Z
N
+
−
J
E
R
2
x
J
E
R
1
y
−
J
E
R
1
x
J
E
R
2
y
Now we can derive relation between the superimposition impedance, [
Z
S
], and
the regional impedance, [
Z
R
]. Following Zhang et al. (1987), we write:
E
S
[
e
]
E
R
[
e
][
Z
R
]
H
R
[
e
][
Z
R
][
h
]
−
1
H
S
[
Z
S
]
H
S
=
=
=
=
,
(1
.
73)
where
[
Z
S
]
[
e
][
Z
R
][
h
]
−
1
=
.
(1
.
74)
Thus we expand the superimposition impedance, [
Z
S
], in components of the
regional impedance, [
Z
R
]. This decomposition reduces to the left and right-
multiplication of the regional impedance [
Z
R
] by the matrices [
e
] and [
h
]
−
1
reflect-
ing electrical and magnetic anomalies caused by local inhomogeneities. It will be
referred to as the
local-regional decomposition
, LR-decomposition.
Chave and Smith (1994) considered the local-regional decomposition in terms of
the localized Born approximation (Habashy et al., 1993). They believe that the local-
regional decomposition is valid if the regional field is uniform across the distorting
local inhomogeneity. However, in our consideration the regional field does not need
any restriction. Thus, the local-regional decomposition can be applied to rather wide
class of superimposition models.
An important point is that at low frequencies the local-regional decomposition
can be significantly simplified (Bahr, 1985). If the skin-depth is much larger than
dimensions of near-surface inhomogeneities, we can neglect the local induction and
take into account only quasi-static effects caused by excess charges. With this sim-
plification, the electric and magnetic distortion matrices, [
e
] and [
h
], are treated as
real-valued and frequency-independent. Furthermore, we can assume that [
Z
R
]
→
0
and [
h
]
0. Hence, the magnetic anomalies caused by small-scale
near-surface inhomogeneities decay and the low-frequency local-regional decom-
position may be written in the truncated form
→
[
I
]as
→
