Based on Longuet-Higgins's theory of the probability distribution of wave amplitude and wave period and on some observations, a new probability 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) + skewness, where zeta(x) and zeta(y) are the slope components in upwind and crosswind directions, respectively; sigma(u)(2) and sigma(c)(2) are the corresponding mean-square slopes. The peakedness of slopes is generated by nonlinear wave-wave interactions in the range of gravity waves. The skewness of slopes is generated by nonlinear coupling between the short waves and the underlying long waves. The peakedness coefficient n of the detectable surface slopes is determined by both the spectral width of the gravity waves, and the ratio between the gravity wave mean-square slope and the detectable short wave mean-square slope. When n equals 10, the proposed PDF fits the Gram Charlier distribution, given by Cox and Munk, very well in the range of small slopes. When n --> infinity, it is very close to the Gaussian distribution. Radar backscatter cross sections (RBCS), calculated from specular reflection theory using the new PDF of the C-band radar filtered surface slopes, are in keeping with empirically based ERS-1 C-band scatterometer models. In other words, the proposed PDF can be used successfully in the specular reflection theory to predict the RBCS in the range of incidence angles away from normal incidence. This suggests that the proposed PDF can be used to describe the distribution of surface slopes over the full range of slopes. This is an improvement over the Gaussian distribution and the Gram Charlier distribution. The comparison between the calculated RBCS and the ERS-1 C-band scatterometer models indicates that the peakedness coefficient n should be 5, for wind condition of U-10 less than or equal to 10 m s(-1). It is also found that the spectral width plays an important role on radar backscatter in the range of incidence angles less than 30 degrees.
Journal of Physical Oceanography
(1997). The Probability Density Function of Ocean Surface Slopes and Its Effects on Radar Backscatter. Journal of Physical Oceanography, 27(5), 782-797.
Available at: https://aquila.usm.edu/fac_pubs/5309