P. Schmoldt, PhD - MTNet - DIAS

6. Using magnetotellurics to gain information about the Earth
(Sec. 3.3):
⎧
⎪⎨ 2ρa(T) cos
ρS W(δs) = ⎪⎩
2 φ(T), for φ > 45o ρa(T)
2 sin2 φ(T) , for φ < 45o .
(6.19)
The ρ ∗ − z ∗ transform is based on earlier approaches by [Schmucker, 1970; Weidelt, 1972;
Schmucker, 1973; Weidelt et al., 1980] using similar formulations.
ρS W(δNB) and ρNB(δNB) will not exhibit the same resistivity depth profile, except for
the case of a homogeneous halfspace. In this case m(T) = 0 and φ(T) = 45°, thus
ρS W(δs) = ρNB(δNB) = ρ ∀z, where ρ is the resistivity of the halfspace and z is the depth.
6.3.2. Forward modelling
In contrast to analytical approaches (Sec. 6.3.1), which attempt to find a direct mathematical
relation between the measured data and subsurface characteristics, forward modelling
(and the related inversion processes) aim to find a set of model parameters m that reproduce
the measured data d through a functional f :
d = f (m). (6.20)
In MT, the model parameters m are usually logarithmic resistivity log(ρ) or logarithmic
conductivity log(σ) of the subsurface, wherein the logarithmic expression is chosen to
fit with the range of electric conductivity within the Earth as sensed by the MT method
(cf. Sec. 5). The data vector d is usually given in terms of the electric impedance Z or
the apparent resistivity ρa and phase φ for each station (cf. Sec. 3.2). In the following it
is assumed that the vectors m and d are of length M and N, respectively. When vertical
magnetic field data are available, the elements of the magnetic transfer function T can also
be derived, adding further information about the subsurface. Alternatively, the (rotational
invariant) determinant of the impedance matrix det(Z) can be used instead of ρ and φ to
evaluate the model, having the advantage of being unaffected by certain types of distortion
(cf. Sec. 4). However, due to the reduction of the number of degrees of freedom (eight 1
for the complex impedance vector, two for its determinant), certain information about
the subsurface is omitted when using the determinant to evaluate the model; thus the
determinant evaluation is inferior and should not be used on its own.
In MT, the functional f , which relates the model and data vectors, is usually an approximation
of Maxwell’s Equations (Sec. 3.1.1) and is commonly carried out by forward
modelling methods: finite difference (FD), finite element (FE), or integral equation (IE)
[e.g. Avdeev, 2005; Press et al., 2007]. All methods approximate Maxwell’s Equations, or
deduced equations, by reducing the differential terms therein to a system of linear equations
A · x = b (6.21)
1 four in the case of a 2D subsurface
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