Resolution analysis of general inverse problems through inverse Monte Carlo sampling

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Abstract
The general inverse problem is characterized by at least one of the following two complications: (1) data can only be computed from the model by means of a numerical algorithm, and (2) the a priori model constraints can only be expressed via numerical algorithms. For linear problems and the so-called `weakly nonlinear problems', which can be locally approximated by a linear problem, analytical methods can provide estimates of the best fitting model and measures of resolution (nonuniqueness and uncertainty of solutions). This is, however, not possible for general problems. The only way to proceed is to use sampling methods that collect information on the posterior probability density in the model space. One such method is the inverse Monte Carlo strategy for resolution analysis suggested by Mosegaard and Tarantola. This method allows sampling of the posterior probability density even in cases where prior information is only available as an algorithm that samples the prior probability density. Once a collection of models sampled according to the posterior is available, it is possible to estimate, not only posterior model parameter covariances, but also resolution measures that are more useful in many applications. For example, posterior probabilities of the existence of interesting Earth structures like discontinuities and flow patterns can be estimated. These extended possibilities for resolution analysis may also provide new insight into problems that are usually treated by means of analytical methods.