  We study the propagation of surface waves across structured surfaces with
random, localized inhomogeneities. A discrete analogue of Gurtin-Murdoch model
is employed and surface elasticity, in contrast to bulk elasticity, is captured
by distinct point masses and elastic constants for nearest-neighbour
interactions parallel to the surface. Expressions for the surface wave
reflectance and transmittance, as well as the radiative loss, are provided for
every localized patch of point mass perturbation on the surface. As the main
result in the article, we provide the statistics of surface wave reflectance
and transmittance and the radiative loss for an ensemble of random mass
perturbations, independent and identically distributed with mean zero, on the
surface. In the weakly scattering regime, the mean radiative loss is found to
be proportional to the size of the perturbed patch, to the variance of the mass
perturbations, and to an effective parameter that depends on the continuous
spectrum of the unperturbed system. In the strongly scattering regime, the mean
radiative loss is found to depend on another effective parameter that depends
on the continuous spectrum, but not on the variance of the mass perturbations.
Numerical simulations are found in quantitative agreement with the theoretical
predictions for several illustrative values of the surface structure
parameters.
