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Fig. 7.5
Illustrating the
generalized thin-sheet
approximation
,
layer is approximated by a plane with variable conductance
S
(
x
y
). This plane is
called the
S
−
plane
. The Price-Sheinmann boundary conditions for the
S
−
plane
are
(Berdichevsky and Zhdanov, 1984)
E
x
−
E
x
H
x
−
H
x
SE
y
=
0
=
(7
.
15)
E
l
y
−
E
y
H
y
−
H
y
SE
x
,
=
0
=−
where superscripts
l
and
u
denote the lower and upper sides of the layer.
The Dmitriev generalization introduces into consideration the thickness of the
layer. Figure 7.5 displays a horizontal layer with constant thickness
h
and variable
resistivity
Expanding the horizontal components of the electromagnetic
field in a Taylor series and keeping the first two terms, we write
(
x
,
y
)
.
E
x
,
y
H
x
,
y
h
h
E
l
x
,
y
=
E
x
,
y
+
H
x
,
y
=
H
x
,
y
+
.
z
z
Using the Maxwell equations and substituting the horizontal derivatives for the ver-
tical derivatives, we get
R
H
y
−
H
x
E
l
x
−
E
x
o
hH
y
=
i
+
x
x
y
R
H
y
H
x
−
E
l
y
−
E
y
o
hH
x
=−
i
+
y
x
y
(7
.
16)
2
E
y
2
E
x
h
−
H
x
−
H
x
SE
y
+
=
i
o
x
2
x
y
2
E
y
2
E
x
h
y
−
H
y
−
H
y
SE
x
+
=−
,
i
o
x
y
2
=
/
=
where
S
h
and
R
h
are the
longitudinal conductance
and the
transverse
resistance
of the layer.
