Radio Science, Volume 19, Number 2, Pages587-595, March-April 1984 Scattering from two-scale surfaces for large distances Oleg I. Yordanov • Virginia Polytechnic Institute and State University (Received August 3, 1983;revised August 22, 1983;accepted August 22, 1983.) The scatteringof high-frequency scalar waves by a reflecting surface containing two-scale,two- dimensional cylindricalrandom irregularities is considered. All possible specular contributions, as well as the effects of large-scale shadowing, are accounted for. Expressions for the averagefield and the average intensity are derivedand evaluated numerically. The results obtaineddiffer qualitatively from those known in the literature and agreewith the latestexperimental data. 1. INTRODUCTION assume that the distance between the source and the observation point exceeds somecritical value lc relat- In the last three decades the problem of wave scat- ed to the statistical characteristics of the surface. The tering from random rough surfaces has received a great deal of attention. The reasons for this interest averaging over the heights of the large irregularities cannot be done in general.However, for any specific lie in the practical importanceof applications as well as inthe theoretical challenges met inattacking this ' choice of the height distribution function, the averag- problem. ing can be done numerically. In this paper we considerthe reflection of high- frequency scalarwavesfrom surfaces with two types of roughness, small- and large-scaleirregularities. Such surfaces now appear to be the best model for the descriptionof scatteringfrom the ocean surface and bottom, the surface of the moon, etc. Our ap- proach is very closeto that developed by Basset al. [1968a, b] and given in detail by Bass and Fuks [1979]. As in those works, we use a perturbation method to account for the small-scale contribution and the Kirchhoff method to compute the reflection from the large-scaleirregularities. Also, as in the work of Bass and Fuks [1979], we account for the shadowing due to the large-scale irregularities, by ex- ploring the theory of random overshoots. Using a direct consequence of the laws of geometricoptics, however,we are able to averagein general with re- spectto the slopes of the large scalein the case of one-dimensional propagation in the forward direc- tion. The evaluations are made for cylindrical sur- faces only. In section3 we computethe spatial inte- gral usingthe method of stationaryphasewhere we In section 4 we deal similarly with the field inten- sity using the two-dimensionalmethod of stationary phase.Some details of the computations are present- ed in the appendix.The resultsdiffer strikingly from those known in the literature. The field power is found to behave qualitatively as 1-3 for the large- scale terms(wherel denotes the range)and as 1-5 and l -? for the small-scale terms. This gives hope that by using a similar development for the case of radio wave backscattering(which is not treated in this paper) one may explain the experimental data reported by Zuikov et al. [1981]. The receivedpower as a function of the range is found experimentally to behave asl- 5 and l- ?with a further observed depen- dence on 1-3 for distances less than somelo. The model we consideredhere does not predict such a jump, which one should try to explain by including something elsein the shadowing. None of the above behaviors is obtained in the other well-known ap- proachesin the literature [e.g., Kodis, 1965;Barrick, 1968; Valenzuela, 1978]. 2. BRIEF REVIEW OF THE TWO-SCALE MODEL x Permanently with Institute of Electronics, Bulgarian Academy of Sciences. Copyright 1984by the AmericanGeophysical Union. Paper number 3S 1417. 0048-6604/84/003 S-1417508.00 In this section we basically follow Bass and Fuks [1979]. Let q/represent a spherical soundwave radi- ated from a point Ro. Consider the scatteringof q/ from a two-dimensionalrandom surface Z composed of small-scale irregularities(fine ripples) superposed 587