74
Modern magnetotelluric systems are designed to record variation of both the electric and
magnetic fields in two orthogonal directions and use these records to calculate the surface
impedance at a measurement site. The surface impedance
Z
is complex, frequency dependent,
and, due to the presence of noise and earth structures, is also a tensor:
or
.
It can be helpful to think of the surface impedance tensor as a two input, two output linear
system where the inputs are the magnetic field components and the outputs are the electrical
field components. This formulation of surface impedance is preferred over the scalar
formulation because, when the source fields are nearly plane waves, the impedance elements
Zij are time invariant. The scalar surface impedance
Z
E
H
ij
i
j
is easier to calculate but can vary as the direction of the source fields vary.
The tensor impedance can be calculated from a number of records (N) using the least-squares
method where the difference between a measured electric field component is minimized
relative to the predicted electric field component. For example just considering Ex, Hx and Hy
and minimization with respect to Zxx and Zxy requires
and
which yields
and
.
where, for example
E H
N
E H
x
x
xi
xi
i
N
*
*
1
1
E
Z
H
( )
( )
( )
E
Z H
Z H
E
Z H
Z H
x
xx
x
xy
y
y
yx
x
yy
y
(
)(
)
*
*
*
*
*
E
Z H
Z H
E
Z H
Z H
xi
xx
xi
xy
yi
i
N
xi
xi
xi
xy
yi
1
(
)
(
)
reZ
imZ
xx
xx
(
)
(
)
reZ
imZ
xy
xy
E H
H H Z
H H Z
x
x
x
x
xx
y
x
xy
*
*
*
E H
H H Z
H H Z
x
y
x
y
xx
y
y
xy
*
*
*
