Geoscience Reference
In-Depth Information
local structure would constitute a very signifi-
cant progress. This is technically possible thanks
to advanced numerical wave propagation tools for
3D heterogeneous media (e.g. Moczo
et al.
, 2007;
Dumbser
et al.
, 2007; Peter
et al.
, 2011). Poten-
tially, we can calculate the misfit kernels for all
parameters simultaneously and update the com-
plete model. Not all parameters will, of course,
be equally well determined, and trade-offs exit.
This approach will require a detailed resolution
analysis. Tools in the framework of full waveform
inversion are just becoming available (Fichtner &
Trampert, 2011a, b).
While a simultaneous inversion for all possible
parameters is tempting, we must realize that not
all Fr echet derivatives are equally strong. Princi-
pal component analysis (Sieminski
et al.
, 2009)
opens the opportunity to determine a priori those
linear combinations of elastic parameters that are
best constrained by the data set, i.e., that generate
the largest sensitivities. For the elastic problem,
not more than 6 linear combinations mostly ac-
count for about 90% of the total sensitivity,
meaning that all remaining linear combinations
can hardly be constrained. Thus, rather than up-
dating all elastic parameters, only the first 6
linear combinations of parameters could be up-
dated instead. Principal component analysis is a
valuable tool to optimize the design of an inverse
problemby finding themaximumnumber of well-
constrained parameters. While the optimal linear
combinations of elastic parameters may not al-
ways have a specific meaning in terms of wave
propagation, they can still be interpreted in a
mineral physics context, thereby providing robust
insight into the thermochemical and deformation
state of the Earth.
A last promising direction is the incorpora-
tion of new observables, either in the form of
completely new measurements or as specifically
designed misfit functionals. Examples of the for-
mer include measurements of rotational ground
motions (e.g. Ferreira & Igel, 2009), Bernauer
et al.
, 2009), Earth tides (Latychev
et al.
, 2009)
and highly accurate long-period mode splitting
(Deuss
et al.
, 2011). The design of targeted misfit
functionals is based on the realization that not all
parametrizations are equivalent. It is well under-
stood, for instance, that travel times are sensitive
to velocities, but hardly to density. If the problem
is reformulated with elastic parameters and den-
sity instead, the sensitivity to density increases,
but now the problem has to be solved for elas-
tic parameters and density simultaneously, with
possible trade-offs. In the proposed approach it is
further important to realize that the sensitivities
are determined by the definition of the misfit
functional or the measurement. The idea is then
to find a misfit functional which maximizes, or
even better, which diagonalizes the sensitivity to
the parameter of interest. This can be set up as
a design problem in combination with principal
component analysis (Sieminski
et al.
, 2009) or
in a fully nonlinear fashion (van den Berg
et al.
,
2003).
Most of these directions are certain to sig-
nificantly increase our imaging capabilities. In
the more distant future, sampling directly the
posterior (i.e. solving Equation 11.5) including
the full nonlinearity of the forward problem,
should become feasible with exa-computing.
Guided Monte Carlo or Neural Networks (e.g.,
Meier
et al.
, 2007) seem promising.
References
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