P. Schmoldt, PhD - MTNet - DIAS

6. Using magnetotellurics to gain information about the Earth
Fig. 6.5.: Types of minima types, illustrated using of a ball’s behaviour on a curved surface under the effect of gravity; redrawn from
Bahr and Simpson [2002]. A ball released at a point ‘A’ will be trapped in the local minimum instead of the global minimum unless a
sufficient force is applied to push it over the ridge between the minima.
Iterative and stochastic inversion is similar to a ‘trial and error’ process wherein one or
more starting models m0 are created and their responses d ′ are calculated using a forward
modelling process (Sec. 6.3.2). The misfit of each model, i.e. the difference between
model response and measured data, is given in terms of the error function ɛ, viz.
ɛ = ||d ′ − d||. (6.29)
A common choice of the error function is the L2 norm:
ɛ = ||d ′
− d||2 = d ′ − d 2 , (6.30)
i.e. the root mean square (RMS) misfit. Model parameters are subsequently adapted
during the inversion process in order to minimise ɛ. The crucial part of an inversion
approach is to optimise the adaption process in a way that it exhibits a high convergence
rate but also adequately samples the model parameter space. The latter is to ensure that a
global minimum of the error function is obtained, as opposed to a local one (Fig. 6.5).
Non-uniqueness of MT inversion models
Except for the idealistic case of 1D subsurface and noise-free data for the complete frequency
range, MT inversion is non-unique, i.e. a range of models fit the measured data
equally well. The non-uniqueness for the general case of inversion with differential equations
was proven by Langer [1933] and for the MT problem by Tikhonov [1965], Bailey
[1970], and Weidelt [1972] (Fig. 6.6); see also Parker [1983]; Constable et al. [1987];
Vozoff [1987] for illustration of the non-uniqueness problem in MT inversion. Rough
models are superior in terms of data misfit, but often contain resistivity distributions that
are not in agreement with physical laws, e.g. conductivities of a few hundreds of Siemens
per meter. Therefore, more elaborate criteria are required to evaluate a model, incorporating
additional (a priori known) constraints about the characteristics of the model, such as
a limited parameter range or the so-called smoothness of the model. The smoothness is
usually described in terms of first or second order spatial derivatives of the model param-
120
i