P. Schmoldt, PhD - MTNet - DIAS

6.3. Deriving subsurface structure using magnetotelluric data
Regularisation operator Regularisation order Proposed value for β
Standard grid
∇2 2
m
∇ m
3.0
2 1.0
Uniform grid
∇2 2
m
∇ m
1.0
2 0.3
Tab. 6.1.: Proposed values for the factor β by Mackie [2002].
degree of horizontal smoothness are found), β ∈ [0.3, 3] (dependent on regularisation, see
Tab. 6.1), and τ ∈ [3, 300]. However, the optimal set of values is certainly dependent
on the local subsurface and the mesh used during the inversion; it is therefore strongly
advised to test a range of values in order to identify their effect on the final model.
Stochastic methods
Stochastic methods have the advantage that they randomly search the model space and
are therefore unlikely to be trapped in a local minimum [e.g. Aster et al., 2005; Tarantola,
2005] (Fig. 6.5). Common stochastic methods are Monte Carlo [Metropolis and Ulam,
1949; Press, 1968] (and subsequent techniques, e.g. Markov chain Monte Carlo [Hastings,
1970] algorithms) and Genetic Algorithm [Goldberg, 1989]. Stochastic inversion
methods have been applied to MT problems by different authors, e.g. Jones and Hutton
[1979]; Agarwal and Weaver [1993]; Pérez-Flores and Schultz [2002]; Moorkamp et al.
[2007]; Roux et al. [2009], and Moorkamp et al. [2010]. However, due to the significantly
higher computational cost of stochastic methods, owing to the larger number of models
involved, the application of stochastic methods is usually limited to grids with a small
number of nodes. Accordingly, stochastic methods are usually only used for 1D inversion
(or 2D inversions with a sparse grid).
Iterative MT inversion algorithms
Commonly used iterative inversion algorithms for MT applications, are presented in Table
6.2. In this Section, differences of the approaches are examined and their advantages and
disadvantages are compared. In general, iterative inversion seeks a minimum of the error
function (Eq. 6.31), which can be found as an extremal of its derivative with respect to
the model parameters
min(ψ(m)) → ∂ψ(m)
∂m
= 0 =⇒ ∂ɛ(m)
∂m
+ τ · ∂ξ(m)
∂m
= 0. (6.34)
Iterative approaches can be divided into two main groups according to the implemented
error functional: the unconstrained functional (Eq. 6.33) is mostly used in Occam’s inver-
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