  Consider the elastic scattering of a plane or point incident wave by an
unbounded and rigid rough surface. The angular spectrum representation (ASR)
for the time-harmonic Navier equation is derived in three dimensions. The ASR
is utilized as a radiation condition to the elastic rough surface scattering
problem. The uniqueness is proved through a Rellich-type identity for surfaces
given by uniformly Lipschitz functions. In the case of flat surfaces with a
local perturbation, we deduce an equivalent variational formulation in a
truncated bounded domain and show the existence results for general incoming
waves. The main ingredient of the proof is the radiating behavior of the Green
tensor to the first boundary value problem of the Navier equation in a half
space.
