Rigorous functional integral equation of Dyson for average value and that of Bethe-Salpeter for correlation function of a wave scattered from a random rough Gaussian absolutely reflecting surface are derived on the basis of the Green formulas. Mass and intensity operators are not represented in an ordinary way as series or diagrams but as functional operators. The case of the infinitely small correlation radius is considered. In this case surface roughnesses with arbitrary heights have very steep slopes, and being reflected the waves can't but suffer the multiple scattering on roughnesses. Rigorous expression for an average reflected field is found, the mean surface being plane. It is shown that asymptotically the incident wave energy completely transforms into the coherent component of the field. This result is in accordance with the localization effect of the wave field in strong random media.
Alexander N. Teokharov,
"Strong rough-surface wave reflection", Proc. SPIE 2469, Targets and Backgrounds: Characterization and Representation, (2 June 1995); doi: 10.1117/12.210606; https://doi.org/10.1117/12.210606