Adjoint-based formulation for computing derivatives with respect to bed boundary positions in resistivity geophysics

Abstract

In inverse geophysical resistivity problems, it is common to optimize for certain re-sistivity values and bed boundary positions, as needed, for example, in geosteering applications. When using gradient-based inversion methods such as Gauss-Newton, we need to estimate the derivatives of the recorded measurement with respect to the inversion parameters. In this article, we describe an adjoint-based formulation for computing the derivatives of the electric potential and electromagnetic fields with respect to the bed boundary positions. The key idea to obtain this adjoint-based formulation is to separate the tangential and normal components of the field, and treat them differently. We then apply this method to a 1.5D borehole resistivity problem. We illustrate its accuracy and some of its convergence properties via numerical experimentation by comparing those results vs. both the analytical results when available and a finite differences approximation of the derivative.