P. Schmoldt, PhD - MTNet - DIAS

6. Using magnetotellurics to gain information about the Earth
mean cross-spectral density (S xy), which in turn is the product of the respective FT’s X( f )
and Y( f ), i.e.
C 2 xy( f ) =
〈S xy( f )〉 2
〈S xx( f )〉〈S yy( f )〉 =
|〈X ∗ ( f )Y( f )〉| 2
〈X ∗ ( f )X( f )〉〈Y ∗ ( f )Y( f )〉
(6.3)
where the asterisk denotes the complex conjugate of the function. Coherence weighting
is implicitly applied during the impedance estimation using the remote reference method,
in which modern processing schemes also consider partial and multiple coherence as indicators
for the quality of an impedance estimate.
To illustrate the application of this procedure to an MT dataset consider the explicit
form of Equation 3.34 for the electric field component in x-direction (Ex), i.e.
Ex = ZxxHx + ZxyHy
(6.4)
where both, the electric and magnetic channels Ex and Hy are assumed here to contain
noise. Multiplication of this Equation with the complex conjugate of either the electric or
magnetic field component in x-direction (R ∗ x) or in y-direction (R ∗ y) yields
and
〈ExR ∗ x〉 = Zxx〈HxR ∗ x〉 + Zxy〈HyR ∗ x〉 (6.5)
〈ExR ∗ y〉 = Zxx〈HxR ∗ y〉 + Zxy〈HyR ∗ y〉, (6.6)
respectively. Combining these two equations results in an estimate of the Zxy component
Zxy = 〈ExR∗ y〉〈HxR∗ x〉 − 〈ExR∗ x〉〈HxR∗ y〉
〈HxR∗ x〉〈HyR∗ y〉 − 〈HxR∗ y〉〈HyR∗ . (6.7)
x〉
Expressions for all components of the impedance tensor can be derived in a similar manner,
therefore the general form can be stated as
Zi j = 〈EiR∗ ∗
∗
j 〉〈HkRk 〉 − 〈EiRk 〉〈HkR∗ j 〉
, (6.8)
DET
with DET = 〈HxR ∗ x〉〈HyR ∗ y〉 − 〈HxR ∗ y〉〈HyR ∗ x〉, and i, j, k ∈ [x, y], with k j. Whether processing
with the electric or the magnetic component as reference provides superior results
depends on the type of noise and its effect on each channel. The magnetic component is
commonly assumed to be less contaminated by noise and usually chosen as remote, however
the choice is often a rather subjective one. To use a combination of remote references
from different components or stations can often be useful.
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For the case of a perfectly 2D subsurface and according rotation of the magnetic field