  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 Lam\'e 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.
