Stability estimates for the fault inverse problem

Abstract

We show in this paper stability estimates for the fault inverse problem. In this problem, faults are assumed to be planar open surfaces in a half-space elastic medium with known Lamé coefficients. A traction-free condition is imposed on the boundary of the half space. Displacement fields present jumps across faults, called slips, while traction derivatives are continuous. It was proved in Volkov et al (2017 Inverse Problems 33 055018) that if the displacement field is known on an open set on the boundary of the half space, then the fault and the slip are uniquely determined. In this present paper, we study the stability of this uniqueness result with regard to the coefficients of the equation of the plane containing the fault. If the slip field is known, we state and prove a Lipschitz stability result. In the more interesting case where the slip field is unknown, we state and prove another Lipschitz stability result under the additional assumption, which is still physically relevant, that the slip field is one-directional.