Efficient Nonlinear Inverse Uncertainty Computation Using Parameter Reduction, Constraint Mapping, and Very Sparse Posterior Sampling

Among the most important aspects of geophysical data interpretation is the estimation and computation of inverse solution uncertainties. We present a general uncertainty estimation method that allows for the comprehensive search of model posterior space while maintaining computational efficiencies similar to deterministic inverse solutions. Integral to this method is the combination of an efficient parameter reduction technique, like Principal Component Analysis, a parameter bounds mapping routine, a sparse geometric sampling scheme, and a forward solver. Parameter reduction, based on prior model covariance estimates, is required to produce both a reduced and orthogonal model space. Parameter constraints are then mapped to this reduced space, using a linear programming scheme, and define a bounded posterior polytope. Sparse deterministic grids are employed to sample this feasible model region, while forward evaluations determine which model samples are equiprobable. The resulting ensemble represents the equivalent model space, consistent with Principal Components that can be used to infer inverse solution uncertainty. Importantly, the number of forward evaluations is determined adaptively and minimized by finding the sparsest sampling set required to produce convergent uncertainty measures. We demonstrate, with a simple surface electromagnetic example, that this method has the potential to reduce the nonlinear inverse uncertainty problem to a deterministic sampling problem in only a few dimensions, requiring limited forward solves, and resulting in an optimally sparse representation of the posterior model space. Depending on the choice of parameter constraints, the method can be exploitative, searching around a given solution, or explorative, when a global search is desired.