Regularized extremal bounds analysis (REBA): An approach to quantifying uncertainty in nonlinear geophysical inverse problems

[1] Geophysical measurements are band-limited in nature, contain noise, and bear a nonlinear relationship to the subterranean features being sampled. These result in uncertainty in data interpretation and necessitate a regularization approach. Although model uncertainty and non-uniqueness can be reduced by combining measurements of fundamentally different physical attributes of a subsurface target under investigation or by using available a priori information about the target, quantifying non-uniqueness remains a difficult task in nonlinear geophysical inversion. This paper develops a new theoretical framework for regularized extremal bounds analysis (REBA) of model uncertainty using the most-squares formalism. The nonlinear most-squares method allows for combining data constraints and their associated uncertainties in an objective manner to determine the model bounds. By making the assumption that the relevant bounding models must sample the same underlying geology, it is proposed that structural dissimilarity between the extreme models can be quantified using the cross-products of the gradients of their property fields and should serve as an objective measure of interpretational uncertainty in multidimensional inversion.