P. Schmoldt, PhD - MTNet - DIAS

4. Distortion of magnetotelluric data
and
Y D = DΦY. (4.60)
Without additional information, Z cannot be recovered from Z D for an unknown DΦ,
but, since DΦ contains only real values, the phase relationship between the horizontal
components of the regional electric magnetic and electric fields must be unaffected by
distortion [Caldwell et al., 2004].
For the purpose of exemplification, assume the phase tensor
Φ =
Φ11 Φ12 . (4.61)
Φ21 Φ22
By deducing the phase from the ratio of real and imaginary parts of the impedance tensor
Φ = X −1 Y, (4.62)
with X −1 representing the inverse of the X, the phase tensor for the disturbed case can be
written as
Φ D = (X D ) −1 Y D = (DΦX) −1 (DΦY) (4.63)
= X −1 D −1
Φ DΦY = X −1 Y (4.64)
= Φ. (4.65)
The phase tensor is therefore independent of distortion and can be expressed in terms of
the undistorted X and Y components, i.e.
1 X22Y11 − X12Y21 X22Y12 − X12Y22
Φ =
, (4.66)
det(X) X11Y21 − X21Y11 X11Y22 − X21Y12
or equivalently in terms for the distorted equivalents X D and Y D .
In order to describe the local distortion configuration, Caldwell et al. [2004] define
three rotational invariant parameter
Φmin = (Φ 2
1
Φmax = (Φ 2
1
+ Φ22
)1/2 − (Φ 2
1 + Φ22
− det(Φ))1/2
+ Φ22
)1/2 + (Φ 2
1
βp = 1
2 arctan
Φ2
Φ1
+ Φ2
2
− det(Φ))1/2
(4.67)
(4.68)
(4.69)
(using the corrected form of Moorkamp [2007a]) plus one coordinate dependent parameter
αp = 1
2 arctan
Φ3
, (4.70)
76
Φ4