Regularized sparse‐grid geometric sampling for uncertainty analysis in non‐linear inverse problems

Abstract

A B S T R A C TThis paper introduces an efficiency improvement to the sparse-grid geometric sampling methodology for assessing uncertainty in non-linear geophysical inverse problems. Traditional sparse-grid geometric sampling works by sampling in a reduceddimension parameter space bounded by a feasible polytope, e.g., a generalization of a polygon to dimension above two. The feasible polytope is approximated by a hypercube. When the polytope is very irregular, the hypercube can be a poor approximation leading to computational inefficiency in sampling. We show how the polytope can be regularized using a rotation and scaling based on principal component analysis. This simple regularization helps to increase the efficiency of the sampling and by extension the computational complexity of the uncertainty solution. We demonstrate this on two synthetic 1D examples related to controlled-source electromagnetic and amplitude versus offset inversion. The results show an improvement of about 50% in the performance of the proposed methodology when compared with the traditional one. However, as the amplitude versus offset example shows, the differences in the efficiency of the proposed methodology are very likely to be dependent on the shape and complexity of the original polytope. However, it is necessary to pursue further investigations on the regularization of the original polytope in order to fully understand when a simple regularization step based on rotation and scaling is enough. I N T R O D U C T I O NGeophysical inverse problems aim to infer Earth's subsurface properties given a limited set of indirect measurements, i.e., d obs ∈ R N . Most of geophysical inverse problems can be summarized by the following equation:where F is the physical system under investigation that relates the n subsurface model parameters m ∈ R N , with the observed data d obs . In non-linear geophysical inverse problems, F −1 is not well defined and is often solved as an optimization problem. These types of problems are ill posed and non-unique *

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“…Another viable strategy to mitigate the curse of dimensionality and to reduce the computational complexity of high‐dimensional inverse problems is to compress the model space through appropriate reparameterization techniques (Fernández‐Martínez et al ., 2011; Azevedo et al ., 2016; Aleardi, 2019; Szabó and Dobróka, 2019; Numes et al ., 2019; Aleardi 2020b). However, it should be noted that the parameterization of an inverse problem must always constitute a compromise between model resolution and model uncertainty (Grana et al ., 2019).…”