Optimization and geophysical inverse problems Page: 25 of 37
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are needed just to solve a linear system of equations. Two general approaches of this type have
been commonly used in geophysics, the class of algebraic reconstruction methods and the class
of projection methods (van der Sluis and van der Vorst, 1987). A more recent approach to
the solution of linear large-scale discrete ill-posed problems that only requires matrix-vector
products is described in Rojas and Sorensen (1999).
5 Appraisal of the Results
An important characteristic of geophysical inverse problems is that the solution to the optimiza-
tion problem is most likely not the true model of the earth that generated the data. Thus, it is
generally recognized in geophysics that a complete solution should include a description of the
optimum model and an evaluation of how this optimum model may be related to the true model.
This latter evaluation typically includes two separate aspects, the sensitivity of the optimum
model to incompleteness in the data and the sensitivity to noise in the data.
For most geophysical inverse problems the amount of information in the data is insufficient to
independently determine all parameters of the model (Jackson, 1972; Alumbaugh and Newman,
1997). This can be expressed as
mopt = R(m) (24)
where the resolution operator R maps the true model m into the model mopt produced by the
optimization procedure. Departure of R from an identity operator signifies imperfect resolution.
Typically it describes a smoothing operation because fine details of the true model can not be
resolved by the available data. Stated in another way, an imperfect resolution operator says
that the solution is non-unique, a characteristic of most geophysical inverse problems.
For linear problems the construction of the resolution operator is straightforward (Jackson,
1972; Wiggins, 1972; Menke, 1989). However, for very large problems this task may represent a
prohibitive computational burden. Recent advances have shown how to compute approximations
to the resolution operator iteratively for such large systems (Nolet, 1985; Zhang and McMechan,
1995; Minkoff, 1997; Berryman, 2000). Alternatively, resolution can be approximated by showing
how well the features of a synthetic model can be reproduced by the inversion method. The use
of such approximate measures of resolution can be misleading (see for instance Levaque et al.,
1993) unless the synthetic model contains a complete range of features.
For nonlinear problems the concept of resolution is still important but general methods for
its estimation are not available. The Occam's razor approach (Constable et al., 1987; deGroot-
Hedlin and Constable, 1990) proceeds by over-parameterizing the model, including a measure
of smoothness in the objective function, and then solving for the smoothest possible model
consistent with the data. Kennett and Nolet (1978) suggested an approach that can be used
with stochastic search methods (Sen and Stoffa, 1995) to produce a suite of successful models.
When noise is present in the data the inverse problem acquires a random nature and the task is
to determine how uncertainty in the data is propagated into uncertainty in the optimum model.
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Barhen, J.; Berryman, J.G.; Borcea, L.; Dennis, J.; de Groot-Hedlin, C.; Gilbert, F. et al. Optimization and geophysical inverse problems, report, October 1, 2000; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc901685/m1/25/: accessed October 20, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.