Optimization and geophysical inverse problems Page: 15 of 37
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3.6 Combined inversions
One of the most effective methods of reducing the fundamental non-uniqueness of geophysical
inverse problems is to combine the results from several different data sets. The inverse problems
for these different data sets can be solved separately and then the results compared, or the data
can be combined and a single inversion performed. So long as the separate data sets all have
similar relationships to the model parameters, this process of combined inversion presents no
new problems, except that the size of the inverse problem grows with the size of the combined
data set. The combined inversion should have advantages over the separate inversions in the
areas of improved resolution and decreased uncertainty, particularly if the different data sets
are distinct in the manner in which they sample the model. The combination of data sets that
sample different types of model parameters is also possible. Here it is necessary to assume that
the different model parameters, such as density and velocity, share a common structure and then
a combined inversion is possible (Haber and Oldenburg, 1997).
There are also situations in geophysics where what appears to be a single inverse problem can
be separated into different inverse problems with improved results. For instance, Xia et al. (1998)
show how the long wavelength and short wavelength parts of a velocity model can be separately
estimated. The primary advantage of such a separation is that different optimization procedures
can be used for the different parts of the inverse problem, allowing the optimization procedure
to be tailored to the particular attribute of the model that is being estimated. Another example
is that of Gritto et al. (1999), where it is shown that a strongly nonlinear inverse scattering
problem can be separated into a linear numerical optimization part and a nonlinear part that
has an analytical solution.
4 Obtaining a Solution
The general outline of a geophysical inverse problem presented in the preceding section has the
form of a mathematical optimization problem. Such problems have been thoroughly studied
and a large selection of methods for solution are available (see for instance Nocedal and Wright,
1999; Dennis and Schnabel, 1996; Fletcher, 1987), depending upon the type of parameterization,
objective function, and constraints. This is indeed fortunate, as, given the variety of geophysical
inverse problems that are encountered, it is unrealistic to think that any one approach would
be optimum for all of them. Thus one of the tasks of solving an inverse problem is to chose the
optimization method that is most appropriate. In this section some of the considerations that
help determine this choice will be discussed.
It is important to point out that considerable resources are already available within the
numerical optimization community for this task of choosing the most appropriate optimiza-
tion algorithm. Compilations such as the Optimization Software Guide (Mor6 and Wright,
1993) present outlines of software available for various types of optimization problems and
provide guidance in making a choice. It is actually possible to do some of this optimization
over the network through the Network-Enabled Optimization Server (NEOS) (Cryzyk et al.,
A general finding of the workshop was that those solving geophysical inverse problems could
benefit from more familiarity and better access to the various optimization methods that are
available. Resources such as those described in the previous paragraph have so far received
very little use in the geophysical community. The concept that no single algorithm is likely
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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/m1/15/: accessed October 21, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.