Optimization and geophysical inverse problems Page: 26 of 37
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This uncertainty is typically represented in terms of a covariance matrix and then
cov[m, t] = U(cov[d]) (25)
where U represents the uncertainty operator.
For linear problems explicit expressions for U are possible (Menke, 1989) and the major
difficulty lies in estimating the statistical properties of the noise. The critical element of the
analysis is that there is a linear mapping between the probability distribution of the noise and
that of the model, which means that meaningful statistics can be derived.
For nonlinear problems a measure of uncertainty is much more problematical. Analytical
expressions for the shape of the objective surface in the vicinity of the optimum model are
generally not available. When sufficient data redundancy is present, a direct exploration of this
surface with resampling methods, such as the jackknife and bootstrap, is possible (Efron, 1982).
In almost all geophysical inverse problems there is a relative weighting between the objectives
of fitting the data and satisfying the regularization condition (the parameter 0 in equations
18 and 19). When the emphasis is on fitting the data the solution is likely to be unstable
and the uncertainty large. When the emphasis is on satisfying the regularization condition the
solution is likely to be inaccurate and the resolution poor. Thus there is a trade-off between
two incompatible objectives and some method of choosing the trade-off parameter is required.
This choice is often rather subjective, depending upon estimates of the accuracy of the data
and expected properties of the solution. It would be a useful contribution to geophysical inverse
problems if methods of optimizing the choice of this trade-off parameter could be included in the
solution of the problem. Lenhart et al. (1997) suggest one way of achieving this using optimal
5.4 Posterior probability
In statistical approaches to the inverse problem, such as Bayesian inference, the concepts of
resolution and uncertainty are lumped together into a posterior probability density function.
For realistic geophysical inverse problems the calculation and display of this probability density
function can represent a major numerical task, particularly for nonlinear problems where an-
alytical approximations are not valid. Sen and Stoffa (1996) have considered several different
methods of making this process more efficient.
6 Computational Needs
Throughout the history of geophysical inverse methods the improvement in computational re-
sources has been just as important as the improvement in methods of analysis for advancement
in the quality and quantity of the solutions that can be obtained. This is likely to be true for fu-
ture advancement also. The situation still exists in geophysics where there are significant inverse
problems that are not being solved primarily because of the lack of the necessary computational
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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/m1/26/: accessed October 18, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.