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where
Z
N
is the normal impedance of the horizontally layered host medium and
J
F
M
λ
are convolutions
G
F
(
r
r
v
)
j
λ
(
r
v
)
dV
J
F
M
λ
(
r
)
=
|
V
M
with F (field) = E, H , M (model) = S (superimposition), R (regional),
(polariza-
tion) = 1, 2.
Eliminating
H
x
o
,
H
y
o
from (1.67
a,b
) and substituting these values into (1.67
c,d
and
e, f
), we obtain
[
h
]
E
R
[
h
][
Z
R
]
E
S
[
e
]
E
R
H
S
H
R
H
R
[
h
]
H
R
=
=
+
={
[
I
]
+
}
=
,
(1
.
68)
where [
I
] is the identity matrix
10
01
[
I
]
=
and [
e
], [
h
]
[
h
] are matrices of electric and magnetic distortions caused by local
near-surface inhomogeneities:
,
e
xx
h
xx
h
xy
e
xy
[
h
]
[
e
]
=
,
=
(1
.
69)
h
yx
h
yy
e
yx
e
yy
and
1
h
xx
Z
xx
+
h
xy
Z
yx
h
xx
Z
xy
+
h
xy
Z
yy
+
[
h
][
Z
R
]
[
h
]
=
[
I
]
+
=
.
(1
.
70)
h
yx
Z
xx
+
h
yy
Z
yx
h
yx
Z
xy
+
h
yy
Z
yy
1
+
The components of distortion matrices [
e
] and [
h
]are
Z
N
J
E
S
1
x
Z
N
+
J
E
R
2
y
J
E
S
2
x
J
E
R
1
y
J
E
S
1
x
J
E
R
2
y
−
+
−
Z
N
J
E
R
1
x
J
E
R
y
+
e
xx
=
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
E
R
2
x
J
E
S
2
x
J
E
R
2
x
J
E
S
1
x
J
E
S
2
x
J
E
R
1
x
−
+
−
e
xy
=
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
E
S
1
y
(1
.
71)
J
E
R
1
y
J
E
R
1
y
J
E
S
2
y
J
E
S
1
y
J
E
R
2
y
−
+
−
e
yx
=
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
E
R
1
x
Z
N
+
J
E
S
2
y
J
E
R
2
x
J
E
S
1
y
J
E
R
1
x
J
E
S
2
y
−
+
−
Z
N
J
E
R
1
x
J
E
R
y
+
e
yy
=
Z
N
+
−
J
E
R
2
x
J
E
R
1
y
−
J
E
R
1
x
J
E
R
2
y
