Uncertainty estimation for large-scale nonlinear inverse problems using geometric sampling and covariance-free model compression

Abstract

A method for uncertainty estimation for nonlinear inverse problems includes obtaining an inverse model of spatial distribution of a physical property of subsurface formations. A set of possible models of spatial distribution is obtained based on the measurements. A set of model parameters is obtained. The number of model parameters is reduced by covariance free compression transform. Upper and lower limits of a value of the physical property are mapped to orthogonal space. A model polytope including a geometric region of feasible models is defined. At least one of random and geometric sampling of the model polytope is performed in a reduced-dimensional space to generate an equi-feasible ensemble of models. The reduced-dimensional space includes an approximated hypercube. Probable model samples are evaluated based on data misfits from among an equi-feasible model ensemble determined by forward numerical simulation. Final uncertainties are determined from the equivalent model ensemble and the final uncertainties are displayed in at least one map.