P. Schmoldt, PhD - MTNet - DIAS

6. Using magnetotellurics to gain information about the Earth
sion, whereas the objective functional (Eq. 6.32) is used by most of the other approaches.
The Gauss-Newton (GN) method is based on Newton’s method, which uses
mk+1 = mk − H −1
k ∇T mk, (6.35)
with the Hessian H = ∂ 2 W/∂m 2 and the gradient ∇ T m to derive the model for the (k+1)-th
iteration step. In contrast to Newton’s method the GN method keeps only the first derivative
of H and uses the first order Taylor’s series expansion to linearise the forward response
in the forward model approach f (mk) (cf. Eq. 6.20). The forward response of the
(k+1)-th iteration step can then be expressed in terms of f (m) at the previous step, the
N × M Jacobian Jk = ∂ f /∂mk, and the difference between the models of the two steps:
f (mk+1) = f (mk) + Jk · (mk+1 − mk). (6.36)
Using Equation 6.34 with the objective functional (Eq. 6.33) yields the iteration scheme
mk+1 − mk = τR −1
mm + J T
k R−1
dd Jk + λkI −1 J T
k R−1
dd (d − f (mk)) + τR −1
mm(mk − m0) , (6.37)
where I is the identity matrix and λk is a damping factor introduced for numerical stability
[Marquardt, 1963]. Accordingly, GN methods need to invert the M × M matrix
−1
τRmm + J T
k R−1 ddJk + λkI to calculate the N × M Jacobian Jk, making it computational
very expensive.
Occam’s inversion [e.g. Siripunvaraporn, 2010] is a variation of the classical GN method
and can be divided into two phases: first the RMS misfit is reduced to the specified level
of misfit Υ 2 through varying the smoothing parameter τ, thereafter τ is minimised as
much as possible without increasing the RMS misfit. In the first phase, the first term on
the right hand side in Equation 6.33 is reduced to a minimum specified by Υ, followed
by minimising the second term on the right hand side, which then defines the value of
the error function (together with the covariance matrix Rdd). Derivation of the optimal τ
can be carried out through simple search schemes, e.g. bisection search [Siripunvaraporn
and Egbert, 2000; Press et al., 2007]. A common choice of Υ 2 is 1 RMS, but the value
may vary for certain problems [Siripunvaraporn, 2010]. Occam’s inversion can be further
subdivided into data and model space inversion. The model space algorithm, like the GN
method, uses the first order Taylor’s series expansion to linearise the forward response
(cf. Eq. 6.36). In that case the solution to Equation 6.34 with the error function as defined
in Equation 6.33 yields the iterative sequence
mk+1 − m0 = τR −1
mm + J T
k R−1 ddJk −1 JkR −1
dd d ′ k, (6.38)
where d ′ k = d − f (mk) + Jk · (mk+1 − m0). Equation 6.38 can be solved for a number of
different τ at each iteration step, and the model with the smallest misfit and norm at the
target level (determined in phase I and phase II, respectively) is kept for the next iteration
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