73
Equating E and H fields at z = 0 yields
and
.
The solution of these equations yields the amplitude of the transmitted wave and the reflected
wave:
and
.
These equations lead to expressions for the electric and magnetic field in the air in terms of
impedance of the air and earth give by
and
.
Similarly, in the earth we have
and
H
E
e
y
ik z
1
0
1
0
2
1
.
At the surface of the earth, these two sets of expressions are equal and the ratio of Ex/Hy is
called the surface impedance Z. For the case of a homogeneous earth, Z =
. This is the basis
for defining the apparent resistivity
a: it is the resistivity of a homogeneous earth, which would
yield the same surface impedance as measured over an inhomogeneous earth at a particular
location and frequency. Because
1
4
0
E
H
e
x
y
i
,
we can write
a
x
y
i
E
H
0
2
.
In general, apparent resistivity is complex (e.g. the IP phenomena), but we restrict our attention
to the real component which is defined as
.
Impedance estimation
E
E
E
1
0
0
E
E
E
1
1
0
0
0
0
E
E
1
1
0
1
0
2
E
E
0
1
0
0
1
0
E
E e
e
x
ik z
ik z
0
0
1
0
1
0
0
0
H
E
e
e
y
ik z
ik z
0
0
0
1
0
1
0
0
0
E
E
e
y
ik z
1
0
1
1
0
2
1
a
x
y
i
E
H
0
2
2
