## 2 Procedure  
A numerical rather than analytical technique is certainly the best way to take into account the geometrically and physically complex problem of the spatial distribution of the backscattering coefficient over the sea surface; this comes, of course, at the price of a very heavy computational effort, which explains why only recently useful results have become available in this field. An early work along these lines was started by [PERSON] _et al._[12, 13]: even though the techniques were crude at the time, some results managed to reproduce a dependence of \(\mathrm{H_{s}}\) on the peak spectrum period. More recent results for both altimeter and SAR data can be found in [14, 15, 16]. In the present work a more complete algorithm is used. By starting from a pseudo-random realization of the sea state water height and by adding appropriate hypotheses on the dynamics and the positions of whitecaps and foam, radio altimeter waveforms are simulated. A description is reported in Figures 2 and 3. The simulation follows in finite steps \(i=1\),..., \(n\) the physical evolution in time and in elevation of the illuminated surface as the electromagnetic wave front proceeds over the altimeter footprint. Following the pseudo-code reported in Figure 3, the wave form is computed as a succession \(WF_{i}\) of the total backscatter of single resolution steps \(d\), at a range \(R_{i}\) along the nadir direction: each single \(WF_{i}\) is the integral over the beam aperture \(\beta\) of the annuli defined by leading and trailing edge of each resolution step \(i\); this requires a separate definition of \(\sigma(\eta,\,X)\), the average backscattering coefficient of the annulus of radius \(X\) at the local height \(\eta\). The computation of \(\sigma(\eta,\,X)\) function implies all the physical phenomena taking place on the surface and it is therefore critical. Assuming uniform average spatial properties, it becomes simply \(\sigma(\eta)\) and can be calculated from the description of sea surface and for each \(\eta\) as in Equation (1):  
\[\sigma(\eta)=[P(\eta)-F(\eta)]\cdot\sigma_{w}+F(\eta)\sigma_{f} \tag{1}\]  
where \(F(\eta)\) is the fractional foam covered area, \(\sigma_{f}\) and \(\sigma_{w}\) are the vertical backscattering coefficients for foam and blue water respectively, and \(P(\eta)\) is the fractional water surface area as defined above. Figure 3: Simulation procedure for altimeter waveform: pseudo-code (comment in italic). Figure 2: Schematic simulation procedure for altimeter waveform: (**a**) cross section; (**b**) plan view. The example shown in this paper is an attempt to investigate into the effects of foam on such function. Fractional water covered surface \(P(\eta)\) as a function of local water height \(\eta\) can be taken to be purely Gaussian, as in Brown-like models; or it can be easily computed, as it will be shown in the following. The evaluation of \(F(\eta)\) is more complex. A number of results do exist--even if they do not always agree--on the total percentage \(W\) of foam covered sea surface, from the classical work by [PERSON] to more recent work by [PERSON] _et al._[17]  
Its distribution \(F(\eta)\) over the wave, however, is not known from any empirical analysis, and are estimated through an appropriate simulation procedures. In order to do so there are three aspects to consider:  
1. the frequency and the location of foam birth (defined as \"whitecaps\" in the following);
2. the time decay of foam once it is generated (defined as \"floating foam\" in the following);
3. the movement of floating foam over the wave surface. Ample research work has been carried out on these three aspects separately, and by appropriately making use of available results a solution can be found to the problem of defining the average \(F(\eta)\) position of foam for a given height over the sea level. No attempt however is made here to consider the effects of suspended water particles (\"foam\" or \"spray\") which develops for very high wind speeds.