  In this paper, we investigate well-posedness of time-harmonic scattering of
elastic waves by unbounded rigid rough surfaces in three dimensions. The
elastic scattering is caused by an $L^2$ function with a compact support in the
$x_3$-direction, and both deterministic and random surfaces are investigated
via the variational approach. The rough surface in a deterministic setting is
assumed to be Lipschitz and lie within a finite distance of a flat plane, and
the scattering is caused by an inhomogeneous term in the elastic wave equation
whose support lies within some finite distance of the boundary. For the
deterministic case, a stability estimate of elastic scattering by rough surface
is shown at an arbitrary frequency. It is noticed that all constants in {\it a
priori} bounds are bounded by explicit functions of the frequency and geometry
of rough surfaces. Furthermore, based on this explicit dependence on the
frequency together with the measurability and $\mathbb{P}$-essentially
separability of the randomness, we obtain a similar bound for the solution of
the scattering by random surfaces.
