Joint inversion with petrophysical constraints using indicator functions and the extended alternating direction method of multipliers

Abstract

Joint inversions often need to construct an objective function with multiple complex constraining terms, which usually increase the computational cost, take a long time to converge, and often are nondifferentiable. We have developed a joint inversion framework that implements petrophysical constraints by indicator functions and solves the optimization problem using the alternating direction method of multipliers (ADMM). An indicator function can describe arbitrary ranges and relationships of multiple physical property parameters by a feasible set. Objective functions involving multiple nondifferentiable terms are numerically not easy to solve, so we adopt an extended alternating direction method of multipliers (eADMM) that separates the terms as independent subproblems and select the most straightforward and efficient solving technique for each of them. With the eADMM, every objective function term can generate its own model, and the models are forced to converge through equality constraints. Such a mechanism brings further efficiency from parallelization and the flexibility of allowing the final inversion model to not strictly obey the model constraints because of uncertainties in the data and petrophysical information. Our approach is tested on two synthetic examples of joint inversion of gravity and magnetic data. The first example contains two blocks having the same magnetic susceptibility but different densities. Our synthetic inversions verify that the combination of the indicator function and eADMM method effectively recovers exact boundaries and values. The second example uses the data synthesized from a realistic kimberlite exploration project. Our joint inversion can distinguish two kimberlite facies, image their position, and even delineate the boundary between the two facies with much improved accuracy. Based on the same model, the robustness of our method also has been tested with inexact and incorrect petrophysical information. The simplicity, flexibility, effectiveness, and efficiency make our approach a good candidate for a wide range of applications in joint inversion.