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Dislocations in a layered elastic medium with applications to fault detection

Andrea Aspri, Elena Beretta, Anna L. Mazzucato

TL;DR

This work analyzes forward and inverse problems for elastic dislocations in a bounded, layered elastic medium modeling the Earth's crust, with a buried dislocation surface $S$ and slip $\bm{g}$. It proves well-posedness of the direct elastostatic problem under bounded elastic moduli and establishes uniqueness for the inverse problem from surface displacement data on an open patch $\Xi$ under additional isotropic, piecewise Lipschitz Lamé coefficients and a graph-structured $S$ with $\bm{g}\in H^{\frac{1}{2}}_{00}(S)$. The authors frame the direct problem as a mixed-boundary-value-transmission Lamé system, leverage Neumann-to-Dirichlet maps $N^+$ and $N^-$, and prove the invertibility of $N^+-N^-$ to enforce the prescribed slip. They also show that, under these assumptions, the fault geometry and slip are uniquely determined by boundary measurements, paving the way for numerical reconstruction approaches (e.g., DG/FEM) and future stability analyses.

Abstract

We consider a model for elastic dislocations in geophysics. We model a portion of the Earth's crust as a bounded, inhomogeneous elastic body with a buried fault surface, along which slip occurs. We prove well-posedness of the resulting mixed-boundary-value-transmission problem, assuming only bounded elastic moduli. We establish uniqueness in the inverse problem of determining the fault surface and the slip from a unique measurement of the displacement on an open patch at the surface, assuming in addition that the Earth's crust is an isotropic, layered medium with Lamé coefficients piecewise Lipschitz on a known partition and that the fault surface is a graph with respect to an arbitrary coordinate system. These results substantially extend those of the authors in {Arch. Ration. Mech. Anal.} {\bf 263} (2020), n. 1, 71--111.

Dislocations in a layered elastic medium with applications to fault detection

TL;DR

This work analyzes forward and inverse problems for elastic dislocations in a bounded, layered elastic medium modeling the Earth's crust, with a buried dislocation surface and slip . It proves well-posedness of the direct elastostatic problem under bounded elastic moduli and establishes uniqueness for the inverse problem from surface displacement data on an open patch under additional isotropic, piecewise Lipschitz Lamé coefficients and a graph-structured with . The authors frame the direct problem as a mixed-boundary-value-transmission Lamé system, leverage Neumann-to-Dirichlet maps and , and prove the invertibility of to enforce the prescribed slip. They also show that, under these assumptions, the fault geometry and slip are uniquely determined by boundary measurements, paving the way for numerical reconstruction approaches (e.g., DG/FEM) and future stability analyses.

Abstract

We consider a model for elastic dislocations in geophysics. We model a portion of the Earth's crust as a bounded, inhomogeneous elastic body with a buried fault surface, along which slip occurs. We prove well-posedness of the resulting mixed-boundary-value-transmission problem, assuming only bounded elastic moduli. We establish uniqueness in the inverse problem of determining the fault surface and the slip from a unique measurement of the displacement on an open patch at the surface, assuming in addition that the Earth's crust is an isotropic, layered medium with Lamé coefficients piecewise Lipschitz on a known partition and that the fault surface is a graph with respect to an arbitrary coordinate system. These results substantially extend those of the authors in {Arch. Ration. Mech. Anal.} {\bf 263} (2020), n. 1, 71--111.

Paper Structure

This paper contains 4 sections, 7 theorems, 65 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation and Functional Setting
  3. The direct problem
  4. The Inverse Problem: a uniqueness result

Key Result

Lemma 3.3

Let $\overline{\Omega}=\overline{\Omega^+}\cup\overline{\Omega^-}$, where $\Omega^+$ and $\Omega^-$ are defined in hyp: assum_disl. Let where $f^+=f\lfloor_{\Omega^+}$ and $f^-=f\lfloor_{\Omega^-}$. Then

Figures (3)

  • Figure 1: An example of the geometrical setting. A section of a layered medium with $S$, the dislocation surface, and with $\Sigma$, the buried part of $\Omega$.
  • Figure 2: An example of the geometrical setting in Case \ref{['i.unique1']}. Graphics generated using Wolfram Mathematica$^{\text{\copyright}}$.
  • Figure 3: An example of the geometrical setting in Case \ref{['i.unique2']}. The bounded connected domain $D$ is such that $\partial D=\overline{S_1 \cup S_2}$. Graphics generated using Wolfram Mathematica$^{\text{\copyright}}$.

Theorems & Definitions (17)

  • Definition 2.1: $C^{k,\alpha}$ regularity of domains
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof
  • Theorem 3.6
  • ...and 7 more