  This paper investigates the elastic scattering by unbounded deterministic and
random rough surfaces, which both are assumed to be graphs of Lipschitz
continuous functions. For the deterministic case, an a priori bound explicitly
dependent on frequencies is derived by the variational approach. For the
scattering by random rough surfaces with a random source, well-posedness of the
corresponding variation problem is proved. Moreover, a similar bound with
explicit dependence on frequencies for the random case is also established
based upon the deterministic result, Pettis measurability theorem and Bochner's
integrability Theorem.
