70
,
(5)
where we have also assumed that there is no free charge present (
Ñ·
E
= 0). Equation (5) is the
Helmholtz equation for E and
k
is the propagation constant. The source of MT signals is in the
atmosphere where the conductivity is near 0, and here the propagation constant is
.
In the earth
k
i
because, for earth materials,
at the frequencies we are interested in.
Propagation of electromagnetic waves
A fundamental assumption when interpreting MT measurements is that the source fields
impinge on the earth as uniform plane waves: the
E
and
H
fields are constant in planes
perpendicular to the direction of propagation. For a plane wave propagating into a uniform
earth, where the z direction is positive downward, we need only consider the field components
and
and can set the other components of the
E
and
H
fields to zero. With
equation (5) becomes
(6)
because the plane wave does not vary in the x and y directions. This differential equation has a
general solution
where the coefficients
E
0
and
E
0
are constants which can be found by applying boundary
conditions. Because of the requirement that a wave not gain amplitude in the direction of
propagation, the coefficients
E
0
and
E
0
represent the amplitudes of the electric field waves
traveling in the +z direction (downward) and -z direction (upward) respectively. For a plane
wave propagating in a uniform earth, Ex is 0 at an infinite depth because we are infinitely far
2
2
0
E
E
k
k
0
0
x
E
y
H
2
2
2
2
E
E x
E y
E z
x
y
z
d E
dz
k E
x
x
2
2
2
0
E
E e
E e
x
ikz
ikz
0
0
