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

TL;DR

We address the inverse problem of identifying faults and layer interfaces in a bounded, multi-layered elastic solid from a single passive boundary measurement. Using the time-harmonic Lamé system $\Delta^*\mathbf{u}+\omega^2\mathbf{u}=0$ in $\Omega\setminus\overline{\Sigma}$ with jumps $[\mathbf{u}]_\Sigma=\mathbf{f}$ and $[\mathcal{T}_\nu\mathbf{u}]_\Sigma=\mathbf{g}$, we establish local uniqueness for faults and interfaces under corner-singularity assumptions and strong convexity of the Lamé parameters; globally, we prove uniqueness under a priori polygonal geometry for the interfaces, for both open and closed faults. The analysis combines variational well-posedness in layered media, microlocal corner analysis, and complex geometrical optics (CGO) solutions for the elastic system, with a dimension-reduction approach in 3D to handle edge corners. The results extend elastic dislocation theory to layered bounded domains and demonstrate that, with appropriate priors, a single boundary measurement suffices to uniquely recover the fault geometry, interfaces, and corresponding jumps. This has potential impact for geophysical fault identification and material characterization where boundary data are limited.

Abstract

This paper investigates an elastic dislocation problem within a bounded and multi-layered solid governed by the Lamé 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 Lamé 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.

Figures (1)

• Figure 1: Schematic illustration of some scenarios for 2D and 3D.

Theorems & Definitions (29)

• Theorem 2.1
• proof
• Definition 2.1
• Definition 2.2
• Remark 2.1
• Theorem 2.2
• Definition 2.3
• Theorem 2.3
• Theorem 2.4
• Lemma 3.1