Stability estimates for the fault inverse problem

TL;DR

This paper investigates the stability of the fault inverse problem in elastic media, providing Lipschitz stability estimates for fault and slip reconstruction under certain conditions, advancing understanding of fault detection accuracy.

Contribution

It establishes new Lipschitz stability results for the fault inverse problem, including cases with unknown slip fields under a directional assumption.

Findings

Lipschitz stability when slip is known

Lipschitz stability with unknown slip under a directional assumption

Enhanced understanding of fault reconstruction stability

Abstract

We study 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 Lam\'e 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 \cite{volkov2017reconstruction} 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.