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Not AvailableWe consider the inverse problem of determining an elastic dislocation that models a seismic fault in the quasi-static regime of aseismic, creeping faults, from displacement measurements made at the surface of Earth. We derive both a distributed and a boundary shape derivative that encodes the change in a misfit functional between the measured and the computed surface displacement under infinitesimal movements of the dislocation and infinitesimal changes in the slip vector, which gives the displacement jump across the dislocation. We employ the shape derivative in an iterative reconstruction algorithm. We present some numerical test of the reconstruction algorithm in a simplified 2D setting. 

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 determin- ing 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 satisfies certain geo- metric conditions. These results substantially extend those of the authors in [Arch. Ration. Mech. Anal. 236, 71–111 (2020)]. 

We study a model of dislocations in two-dimensional elastic media. In this model, the displacement satisfies the system of linear elasticity with mixed displacement-traction homogeneous boundary conditions in the complement of an open curve in a bounded planar domain, and has a specified jump, the slip, across the curve, while the traction is continuous there. The stiffness tensor is allowed to be anisotropic and inhomogeneous. We prove well-posedness of the direct problem in a variational setting, assuming the coefficients are Lipschitz continuous. Using unique continuation arguments, we then establish uniqueness in the inverse problem of determining the dislocation curve and the slip from a single measurement of the displacement on an open patch of the traction-free part of the boundary. Uniqueness holds when the elasticity operators admits a suitable decomposition and the curve satisfies additional geometric assumptions. This work complements the results in Arch. Ration. Mech. Anal., 236(1):71-111, (2020), and in Preprint arXiv:2004.00321, which concern three-dimensional isotropic elastic media.