  We study in this paper stability estimates for the fault inverse problem. In
this problem, faults are assumed to be planar open surfaces in a half space
elastic medium with known Lam\'e coefficients. A traction free condition is
imposed on the boundary of the half space. Displacement fields present jumps
across faults, called slips, while traction derivatives are continuous. It was
proved in \cite{volkov2017reconstruction} that if the displacement field is
known on an open set on the boundary of the half space, then the fault and the
slip are uniquely determined. In this present paper, we study the stability of
this uniqueness result with regard to the coefficients of the equation of the
plane containing the fault. If the slip field is known we state and prove a
Lipschitz stability result. In the more interesting case where the slip field
is unknown, we state and prove another Lipschitz stability result under the
additional assumption, which is still physically relevant, that the slip field
is one directional.
