P. Schmoldt, PhD - MTNet - DIAS

6.3. Deriving subsurface structure using magnetotelluric data
[Siripunvaraporn and Egbert, 2000; Siripunvaraporn, 2010]. The advantages of the code
are that it converges in a small number of iterations, and that it searches for a minimum
structure model [Constable et al., 1987; Siripunvaraporn and Egbert, 2000]. The latter
means that all structures in a model are more reliable as they are required by the data and
not unconstrained artefacts of the inversion [Siripunvaraporn, 2010]. The disadvantage
of the code is its large computational cost, similar to the GN methods, i.e. inverting the
M × M matrix τR −1
mm + J T
k R−1 ddJk
and computing the N × M Jacobian.
The issue of large computational cost can be mitigated by transforming the computational
space from the model space into the data space, rearranging Equation 6.38 as
m − m0 = RmmJ Tβ, (6.39)
where β is a unknown coefficient vector of length N [Parker, 1994; Siripunvaraporn and
Egbert, 2000; Siripunvaraporn et al., 2005a]. Inserting Equation 6.39 into Equation 6.32
and differentiating the resulting function with respect to β yields the iterative sequence of
approximate solutions
mk+1 − m0 = RmmJ T
k R−1/2
dd
−1/2 T
τI + Rdd JkRmmJk R−1/2
−1 dd R −1/2
dd d ′ k, (6.40)
where I is the identity matrix. Therefore, only a N × N matrix τI + R −1/2 T
dd
JkRmmJk R−1/2
dd
(instead of the M × M matrix in the model space) needs to be inverted, which significantly
reduces CPU time and memory usage [Siripunvaraporn and Egbert, 2000; Siripunvaraporn
et al., 2005a]. However, the algorithm still requires the costly computation of the
Jacobian.
In the data space Occam’s inversion and in the GN approach, inversion of the matrices
(Eqs. 6.38 and 6.37, respectively) is usually computational the most expensive part of the
process. In the quasi-Newton (QN) method the inverse matrix is therefore approximated
through a recursive process, in which H −1 is replaced by a successively adapted matrix
[Fletcher and Powell, 1963; Shanno, 1970]. As a result the main computation for each
QN iteration is reduced to computing ∇T m and performing the line search, meaning that
the memory requirement of the QN method is insignificant in comparison with the GN
method [Siripunvaraporn, 2010]. Because of the QN method’s slow convergence rate
[Haber, 2005] modifications are made to permit large datasets; e.g. approximating only
the part of H that is related to the data misfit, rather than the full H [Haber, 2005]; or
introducing additional regularisations [Avdeev and Avdeeva, 2009].
Like the QN method, the Gauss-Newton with Conjugate Gradient (GN-CG) method
is set up to reduce the computational load of inverting the large matrices in Equations
6.37, 6.38, and 6.40. The method is referred to as model space GN-CG or data space
GN-CG, respectively, dependent on whether it is set up to solve Equation 6.38 or 6.40.
The advantage of the method is that instead of J only the product of J and J T with an
arbitrary vector V is required, i.e. J Va and J T Vb. Vb and Vb can be computed by solving
one forward problem [Mackie and Madden, 1993; Newman and Alumbaugh, 2000; Rodi
125