Model Fusion and Joint Inversion

Abstract

Inverse problems are inherently non-unique, and regularization is needed to obtain stable and reasonable solutions. The regularization adds information to the problem and determines which solution, out of the infinitely many, is obtained. In this paper, we review and discuss the case when a priori information exists in the form of either known structure or in the form of another inverse problem for a different property. The challenge is to include such information in the inversion process. To use existing known structure, we review the concept of model fusion, where we build a regularization functional that fuses the inverted model to a known one. The fusion is achieved by four different techniques. Joint inversion of two data sets is achieved by using iterative data fusion. The paper discusses four different methods for joint inversion. We discuss the use of correspondence maps or the petrophysics of the rocks, as well as structure. In particular, we suggest to further stabilize the well-known gradient cross product and suggest a new technique, Joint Total Variation, to solve the problem. The Joint Total Variation is a convex functional for joint inversion and, as such, has favorable optimization properties. We experiment with the techniques on the DC resistivity problem and the borehole tomography and show how model fusion and joint inversion can significantly improve over existing techniques.