Dislocations in a multi-layered elastic solid with applications to fault and interface identifications

TL;DR

This paper develops a mathematical framework for uniquely identifying faults and interfaces in a layered elastic solid using boundary measurements, addressing the challenges posed by discontinuities and limited data.

Contribution

It establishes local and global uniqueness results for reconstructing faults, interfaces, and displacement jumps in layered elastic media with minimal measurements.

Findings

Proves local uniqueness for faults and interfaces under certain conditions.

Establishes global uniqueness in generic scenarios with geometric prior information.

Handles discontinuities in displacement and traction fields across faults.

Abstract

This paper investigates an elastic dislocation problem within a bounded and multi-layered solid governed by the Lam\'e system. We address the simultaneous reconstruction of the faults, the jumps in displacement and traction fields across the faults, and the interfaces of layers using a single passive boundary measurement. This inverse problem is particularly challenging due to the discontinuities in both the displacement and traction fields across the faults and the inherent difficulty of establishing uniqueness results with limited measurement data. Under the assumptions that the Lam\'e parameters are piecewise constants within each layer, satisfying strong convexity conditions, and that the faults exhibit corner singularities, we establish local uniqueness identifiability results for both the interfaces and the faults, as well as the jumps across the faults. Furthermore, we derive global uniqueness results for reconstructing the interfaces, the faults, and the corresponding displacement and traction jumps in generic scenarios under a priori geometric information, where the faults are geometrically general and may be either open or closed.