P. Schmoldt, PhD - MTNet - DIAS

4.4. Removal of distortion effects
Fig. 4.18.: Graphical representation of the magnetotelluric phase tensor defined by Caldwell et al. [2004], with Φmax and Φmin describing
the length of the principle axis of the ellipse and αp − βp the angle between North and the major axis, from Martí [2007] after
Caldwell et al. [2004]
with Φ1 - Φ4 defined 1 as
Φ1 = (Φ11 + Φ22)/2 (4.71)
Φ2 = (Φ12 − Φ21)/2 (4.72)
Φ3 = (Φ12 + Φ21)/2 (4.73)
Φ4 = (Φ11 − Φ22)/2. (4.74)
The four parameters Φmin, Φmax, αp, and βp defined by Caldwell et al. [2004] are minimum
and maximum of the ellipse describing Φ (i.e. the principle, or singular values of
Φ), skew angle, and a coordinate system orientation dependent angle, respectively. A
graphical representation of the four parameters is given in Figure 4.18, illustrating their
use in identifying the present distortion. With these four parameters the phase tensor can
be represented through a Singular Value Decomposition (SVD) as the product of three
matrices
ΦD = R T (αp − βp)
Φmax 0
0 R(αp + βp) (4.75)
Φmin
where R is the rotation matrix with the superscript T indicating the transpose of the matrix.
In a review of the previous work, Bibby et al. [2005] introduced an additional pa-
rameter
λp = (Φ2 3 + Φ2
4 )1/2
(Φ 2
1
, (4.76)
+ Φ2
2
)1/2
describing the degree of ellipticity and therefore indicating whether the structure is 1D
or of higher dimensionality. λp supplements αp and βp in defining the present subsurface
1 Parameter Φ1 used here as defined by Caldwell et al. [2004], whereas Φ2 is replaced by the original
Φ3 owing to the corrections by Moorkamp [2007a]; the additional parameter Φ3, Φ4 are also included
herein for the sake of clearness
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