P. Schmoldt, PhD - MTNet - DIAS
P. Schmoldt, PhD - MTNet - DIAS 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
6.3. Deriving subsurface structure using magnetotelluric data Fig. 6.6.: The non-uniqueness problem of magnetotelluric (MT) inversion illustrated using of two very different inversion models (top). Both models reproduce the input data reasonably well (bottom), but describe very different subsurface behaviour. Figure and (dashed) inversion model taken from Weidelt [1972]; MT sounding data from Wiese [1965]; (doted) inversion model used for comparison by Fournier [1968]. eters m, which, in the case of MT application, is calculated using ∇ log(σ) and ∇ 2 log(σ) (or ∇σ and ∇ 2 σ) [e.g. Constable et al., 1987; Smith and Booker, 1988; de Groot-Hedlin and Constable, 1990; Smith and Booker, 1991]. Functionals ψ(m), which consider data misfit as well as model smoothness, have the form ψ(m) = ɛ(m) + τ · ξ(m), (6.31) where ɛ(m) is some measure of the data misfit, ξ(m) is some measure of the model smoothness, and τ is a weighting parameter (often referred to as smoothing parameter). The related inversion processes attempt to find a regularised solution of the inverse problem, i.e. models which yield a minimum of ψ(m). A higher degree of smoothing is generally desirable as it usually yields models with less structure, meaning that the persisting structures are more reliable. However, due to the additional constraints on the model parameters a trade-off between smoothness of the model and data misfit is commonly observed (e.g. 121
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6.3. Deriving subsurface structure using magnetotelluric data<br />
Fig. 6.6.: The non-uniqueness problem of magnetotelluric (MT) inversion illustrated using of two very different inversion models (top).<br />
Both models reproduce the input data reasonably well (bottom), but describe very different subsurface behaviour. Figure and (dashed)<br />
inversion model taken from Weidelt [1972]; MT sounding data from Wiese [1965]; (doted) inversion model used for comparison by<br />
Fournier [1968].<br />
eters m, which, in the case of MT application, is calculated using ∇ log(σ) and ∇ 2 log(σ)<br />
(or ∇σ and ∇ 2 σ) [e.g. Constable et al., 1987; Smith and Booker, 1988; de Groot-Hedlin<br />
and Constable, 1990; Smith and Booker, 1991]. Functionals ψ(m), which consider data<br />
misfit as well as model smoothness, have the form<br />
ψ(m) = ɛ(m) + τ · ξ(m), (6.31)<br />
where ɛ(m) is some measure of the data misfit, ξ(m) is some measure of the model smoothness,<br />
and τ is a weighting parameter (often referred to as smoothing parameter). The related<br />
inversion processes attempt to find a regularised solution of the inverse problem, i.e.<br />
models which yield a minimum of ψ(m). A higher degree of smoothing is generally desirable<br />
as it usually yields models with less structure, meaning that the persisting structures<br />
are more reliable. However, due to the additional constraints on the model parameters a<br />
trade-off between smoothness of the model and data misfit is commonly observed (e.g.<br />
121
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