Remote sensing of the roughness of a fractal sea surface

Abstract

One use of radar altimeters is to measure surface wind speeds through their effect on the roughness of the sea surface. The specular point reflection model is only appropriate for surfaces with roughness on scales which are large in relation to the radar wavelength. This may not be the case for ocean surfaces. Here we model the sea surface as a fractal on the relevant scales. This is based on Hasselmann's model for nonlinear wave action transfer. The radar cross section for nadir backscatter is derived and its dependence on the radar frequency is determined. The results are compared with the cross section obtained by the specular point model for a smoothed surface, yielding the appropriate cutoff wavenumber. Some existing measurements of the sea surface by altimeter and stereophotogrammetry are discussed, suggesting a smaller fractal dimension for short gravity waves. Dual‐frequency altimeters may help determine the synoptic spectral shape.

Fractal laser glints from the ocean surface

“…Note that we use E ϭ 1 for all of our analysis because we are computing the fractal dimension of timeseries plots that have one Euclidian dimension, regardless of whether the glint counts are from a linear scan with one Euclidian dimension or from an image with two Euclidian dimensions. The recent literature refers to an impressively large and varied collection of fractals in nature, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] including coastlines, clouds, snowflakes, galaxies, and ocean waves. Two approaches have been taken for studying the fractal characteristics of ocean-wave processes.…”

“…[4][5][6] The second approach is to calculate a fractal dimension of the process itself. [7][8][9][10][11][12] We follow the second approach and calculate the fractal dimension of glint-count time series. These time series are related to sea-surface roughness and are shown to contain useful information regarding nonlinear wave-wave interactions.…”

Scattering on Multi-Scale Rough Surfaces

“…Therefore, in the following, we consider a wire source with some mask which stops the waves radiated around normal incidence. Denoting by [-Om, Bm] the darkened angular interval, and assuming that the contributions from the various incidence angles are not correlated in the far field, we obtain According to (2), Ib(k) behaves ask-d. Consequently, the correlation dimension can be deduced from a few measurements. …”

“…An important issue in remote sensing is therefore to understand the impact of fractal characteristics on electromagnetic wave scattering. Recently, several simple models such as Weierstrass functions of fractional Brownian motion have been proposed to describe multiscale rough surfaces and the diffracted field has been studied by means of the usual approximations (Kirchhoff approach [1][2][3][4][5][6][7], the Extended Boundary Condition Method [8], [9] or Integral Equation Method [10]). Some qualitative results relating the scattering amplitude to fractal dimensions of the surface have been exhibited but failed to give a precise and general way of computing the latter quantities.…”

“…We will use formulae ( 1) and (2) to retrieve the correlation dimension of the surface from the scattering pattern. Numerical application will be performed on the so-called band-limited Weierstrass function:…”