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

Abstract

We 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.

Comparison of sparse‐grid geometric and random sampling methods in nonlinear inverse solution uncertainty estimation

“…The accepted models represent the space of equivalent models. While the uncertainty of the nonlinear inverse problem is estimated from statistical measures (e.g., mean, covariance, percentile, interquartile range) computed from the posterior (See Tompkins et al . 2011b), our objective here is to focus only on the sampling of the posterior itself; therefore, we leave discussion of uncertainty estimation to future work.…”

“…For the S‐G sampling based on polynomial interpolation, we computed coefficient samples over our 2D polytope using four sparse grid levels (corresponding to polynomial degrees 4–7) containing 29, 65, 155 and 359 equi‐feasible coefficients) (Fig. 4) and interpolated the 2D misfit functions of all these grids (see Tompkins et al . 2011a for details).…”

“…2010), combines four established concepts: 1) parameter reduction by orthogonal factorization (e.g., Principal Component Analysis (PCA)), 2) sample bounding by model parameter constraint mapping, 3) posterior sampling by sparse‐grid interpolation (assuming a uniform prior) and 4) forward simulation for model evaluation. Tompkins et al . (2011a,b) applied this method to both 1D and 2D inverse problems and suggested that the method results in a maximally‐sparse representation, given prior information, of the model posterior.…”

Efficient estimation of nonlinear posterior model covariances using maximally sparse cubature rules

“…While random sampling methods avoid burdensome inversions and can account for nonlinearity, they come at the high cost of inefficiency (e.g., Haario et al, 2001;Tompkins et al, 2011bTompkins et al, , 2012 and thus are limited to modest-sized problems (10 s of unknowns). Recently, a new method has emerged that solves the same posterior sampling problem, but uses sparse-grid interpolation with orthogonal polynomials as opposed to random sampling (i.e., Tompkins et al, 2011aTompkins et al, , 2011b. This work significantly improves the efficiency by which nonlinear uncertainties can be computed for medium-sized problems (10,000s of parameters), but it is still not known whether this geometric sampling approach can be applied to large parameterizations (see Tompkins et al, 2012).…”

Uncertainty analysis and probabilistic segmentation of electrical resistivity images: the 2D inverse problem

“…Tompkins et al . (,b) and Tompkins and Fernández‐Martínez () introduced the geometric sampling approach for EM inversion in 1D and 2D dimensions. This method combines different techniques for model reduction and deterministic sampling using Smolyak grids, providing similar posterior estimates as Monte Carlo methods (Tompkins et al .…”