A universal inversion framework for arbitrary regularizations based on the alternating direction method of multipliers

Abstract

Geophysical inversion problems have inherent ambiguity that requires regularization terms in the objective function to add prior information and constraints to reduce non-uniqueness. The types of regularization term introduced are different for different applications, but some regularization terms are complicated or even non-differentiable. We develop a universal inversion framework capable of implementing arbitrary regularizations based on the ADMM (alternating direction method of multipliers). With a generic ADMM formulation, this framework unifies many kinds of regularizations, including bound constraints, L1 norm, L2 norm, total variation, and their combinations. Our framework can be applied to any geophysical data inversion, but in this paper we demonstrate its implementation and performance in 3D magnetic inversion. For inversions with petrophysical information, we constrain the upper and lower bounds of physical properties by adding indicator functions. Then a smooth model and a sharp model subject to the bound constraints can be obtained with L2 and L1 norms, respectively. For non-bound constrained inversions, we show the ADMM inversion results for two hybrid regularizations – the L1-L2 norm and L1-TV (total variation). Those examples show the great flexibility of our ADMM inversion framework.