Solution to discrete-valued inverse problems using the fuzzy c-means clustering technique

Abstract

The definition of geologic boundaries through inversion can be improved through the incorporation of known physical property values of a small number of geological units. The direct and strict imposition of such physical property values in a Tikhonov regularized inversion, however, leads to an integer programming problem to be solved, which poses significant computational challenges for large problems in practice. As an alternative approach, we present a new solution method for such discrete-valued inverse problems using a guided fuzzy c-means clustering (FCM) technique. This method enforces the discrete values in an inversion by guiding the constructed model to cluster around the known values in the inversion process. The corresponding minimization problem can be solved with any derivative based minimization technique and, therefore, is efficient and applicable to a wide range of problems. In the paper, we present the basics of the method and illustrate it with binary gravity inversions in which the density contrast can only take on a background value of zero or an anomalous value. Applications to inversions of borehole gravity data and combined borehole and surface gravity data demonstrate the efficacy of the method.