Marine electromagnetic inverse solution appraisal and uncertainty using model‐derived basis functions and sparse geometric sampling

Abstract

ABSTRACTWe summarize, for marine electromagnetic inverse problems, a newly developed inverse solution appraisal and non‐linear uncertainty estimation method based on parameter reduction techniques and efficient posterior model space sampling. This method uses model compression methods to decorrelate parameters in an inverse solution and represent all feasible posterior models as linear combinations of a small number of model‐derived basis vectors and corresponding coefficients. This allows us to reduce the posterior sampling problem by orders of magnitude. We further contract this reduced‐dimensional posterior space by confining all acceptable models to a set of bounds mapped from our original parameter space. As a final step to increase efficiency, we implement a geometric sampling scheme that we use to approximate our restricted posterior by generating feasible models on adaptive, optimally‐sparse grids. The sampled equi‐feasible models are accepted according to a data misfit threshold and constitute an optimally‐sparse representation of the restricted posterior model space. Although very efficient, our method imposes a bias in the posterior space by truncating the basis expansion during the model reduction step. To investigate this, we compare two types of fast and scalable bases, the discrete cosine transform and singular value decomposition. We demonstrate that while the choice of base does influence the type of models sampled and the model rejection rates, the posterior statistics are generally compatible between the methods providing confidence in the uncertainty estimations. For the marine electromagnetic problem, we show that a representative ensemble of equivalent inverse solutions can be generated for realistically‐sized inverse problems and that solution appraisal and uncertainty inference follow directly from ensemble statistics.