THE PROBABILITY DENSITY-FUNCTION OF OCEAN SURFACE SLOPES AND ITS EFFECTS ON RADAR BACKSCATTER

Based on Longuet-Higgins's theory of the probability distribution of w ave amplitude and wave period and on some observations, a new probabil ity density function (PDF) of ocean surface slopes is derived. It is f (zeta(x), zeta(y)) = n/2 pi(n - 1)sigma(u) sigma(c) x [1 + zeta(x)(2)/ (n - 1)sigma(n)(2) + zeta(y)(2)/(n - 1)sigma(c)(2)](-(n + 2)/2) + skew ness, where zeta(x) and zeta(y) are the slope components in upwind and crosswind directions, respectively; sigma(u)(2) and sigma(c)(2) are t he corresponding mean-square slopes. The peakedness of slopes is gener ated by nonlinear wave-wave interactions in the range of gravity waves . The skewness of slopes is generated by nonlinear coupling between th e short waves and the underlying long waves. The peakedness coefficien t n of the detectable surface slopes is determined by both the spectra l width of the gravity waves, and the ratio between the gravity wave m ean-square slope and the detectable short wave mean-square slope. When n equals 10, the proposed PDF fits the Gram Charlier distribution, gi ven by Cox and Munk, very well in the range of small slopes. When n -- > infinity, it is very close to the Gaussian distribution. Radar backs catter cross sections (RBCS), calculated from specular reflection theo ry using the new PDF of the C-band radar filtered surface slopes, are in keeping with empirically based ERS-1 C-band scatterometer models. I n other words, the proposed PDF can be used successfully in the specul ar reflection theory to predict the RBCS in the range of incidence ang les away from normal incidence. This suggests that the proposed PDF ca n be used to describe the distribution of surface slopes over the full range of slopes. This is an improvement over the Gaussian distributio n and the Gram Charlier distribution. The comparison between the calcu lated RBCS and the ERS-1 C-band scatterometer models indicates that th e peakedness coefficient n should be 5, for wind condition of U-10 les s than or equal to 10 m s(-1). It is also found that the spectral widt h plays an important role on radar backscatter in the range of inciden ce angles less than 30 degrees.