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Here (5.56) and (5.57) are two-dimensional counterparts of (5.47), (5.48) and (5.50),
(5.51). These formulae can be supplemented by the formula
y
E
o
x
(
y
)
H
o
z
(
y
o
)
dy
o
,
=−
i
o
(5
.
58)
−∞
o
H
o
z
.
We have derived the integral relations that enable us to synthesize the syn-
chronous magnetotelluric field from the magnetotelluric and magnetovariational
response functions measured on the Earth's surface.
E
o
x
/
which follows directly from Maxwell's equation
y
=−
i
5.2.2 Synthesis of the Magnetic Field from the Impedance Tensors
Let the impedance tensor [
Z
(
x
,
y
)] be measured on the Earth's surface. Synthesis of
the magnetic field
H
o
(
x
y
)] reduces to the solution of the system
of the integral equations of the second kind.
For the sake of convenience we express the linear relations between horizontal
components of the magnetic and electric fields,
H
o
(
x
,
y
)fromthe[
Z
(
x
,
,
y
) and
E
o
(
x
,
y
)
,
in terms of
[
Z
]
−
1
. Proceeding from (1.19), we write:
the admittance tensor [
Y
]
=
H
o
x
−
Y
xx
E
o
x
−
Y
xy
E
o
y
=
0
,
(5
.
59)
H
o
y
−
Y
yx
E
o
x
−
Y
yy
E
o
y
=
0
,
where
Z
yy
Z
xx
Z
yy
−
Z
xy
Z
xx
Z
yy
−
Y
xx
=
Z
xy
Z
yx
,
Y
xy
=−
Z
xy
Z
yx
,
Z
yx
Z
xx
Z
yy
−
Z
xx
Z
xx
Z
yy
−
Y
yx
=−
Z
xy
Z
yx
,
Y
yy
=
Z
xy
Z
yx
.
Separating the normal and anomalous fields, we get
H
o
x
(
x
y
)
E
o
x
(
x
y
)
E
o
y
(
x
,
y
)
−
Y
xx
(
x
,
,
y
)
−
Y
xy
(
x
,
,
y
)
H
o
x
+
y
)
E
o
x
+
y
)
E
o
y
=−
Y
xx
(
x
,
Y
xy
(
x
,
(5
.
60)
H
o
y
(
x
y
)
E
o
x
(
x
y
)
E
o
y
(
x
,
y
)
−
Y
yx
(
x
,
,
y
)
−
Y
yy
(
x
,
,
y
)
=−
H
o
y
+
Y
yx
(
x
,
y
)
E
o
x
+
Y
yy
(
x
,
y
)
E
o
y
.
