  The practical applications of surface plasmon polaritons (SPPs) require the
deep understanding of the impact of electrical characteristics variability and
geometrical irregularity of the metal-dielectric interface. Traditional methods
in the theory of wave scattering at rough interfaces fail to treat
simultaneously the interference (Anderson) localization of the SPP, which may
arise due to its multiple back-scattering by random distortions of surface
relief, and the leakage into the uniform dielectric half-space. In our previous
works [Low Temp. Phys. \textbf{42}, 685 (2016); Ann.~Phys.~\textbf{455}, 169378
(2023)], by representing the perturbation of surface impedance as an effective
potential in the Schr\"odinger-like equation, we suggested the way to describe
the interplay between Anderson localization and the leakage of the SPP. In the
present study we show that the problem of SPP scattering from finite
geometrically rough region of the interface can be reduced to the problem of
its scattering from the same region but with effective random impedance. We
calculate the radiation pattern and demonstrate its pronounced anisotropy that
arises due to the interplay between different geometrical parameters of the
interface roughness.
