## 3 Determination of Sea Surface Roughness from Measurements  
The Rayleigh roughness parameter is widely used to estimate the degree of roughness in ocean remote sensing of electromagnetic wave scattering from rough surfaces. It is a qualitative estimation of the processes in the reflection zone. The electromagnetic roughness of a surface is directly related to the phase variations of the reflected field caused by surface height variations. Thus, it is the phase variations of the reflected field around its mean value that need be considered. The electromagnetic scattered wave field \(E_{r}\) can be split into the mean \(<\)\(E_{r}\)\(>\) and fluctuating term \(\delta E_{r}\) of the field, where the latter represents the incoherent part of the signal. The total wave field intensity scattered by the surface becomes:  
\[\left\langle\left|E_{r}\right|^{2}\right\rangle=\left|\left\langle E_{r} \right\rangle\right|^{2}+\left\langle\left|\delta E_{r}\right|^{2}\right\rangle. \tag{1}\]  
The first term on the right-hand side represents the coherent intensity, corresponding to the reflection from an almost perfectly flat surface. The second term is the incoherent intensity, due to angular spreading and its weak correlation with the incident wave. The coherent term is largest for small surface height variations and the incoherent term vanishes. For larger ocean waves and winds fields, the coherent term is damped, while the incoherent term increases. Thus, for rough ocean surfaces the incoherent term dominates the electromagnetic scattered wave field, and the coherent term can be neglected. The Rayleigh roughness parameter is related to the coherent scattered intensity (first term on the right-hand side of Equation (1)). It can be shown that the average reflected scattered field intensity \(\left|<\)\(E_{r}\)\(>\)\(\left|\right.^{2}\) can be expressed as follows [22]:  
\[\left|\left\langle E_{r}\right\rangle\right|^{2}=\left|E_{0}\right|^{2}\times \left|\left\langle e^{i\delta\phi_{r}}\right\rangle\right|^{2}. \tag{2}\]  
The term \(\left|\left.E_{0}\right|\right.^{2}\) corresponds to the reflection from a perfectly flat surface, while the second term describes the surface electromagnetic roughness averaged over all surface heights in the reflection zone, equaling the attenuation of the coherent intensity due to the surface roughness. For a Gaussian density probability distribution of the ocean heights in question, the second right-hand term becomes \(\left|\left\langle e^{i\delta\phi_{r}}\right\rangle\right|^{2}=e^{-4\left(R_{ A}\right)^{2}}\), with \(R_{A}\) is the Rayleigh roughness parameter associated with the reflected wave. Thus, \(R_{A}\) can be shown to be equal to:  
\[R_{A}=\frac{1}{2}\;\sqrt{\left\langle\left(\delta\phi_{r}\right)^{2}\right\rangle}. \tag{3}\]  
The Rayleigh roughness parameter, obtained from the root mean square of the phase variations, is a useful and good estimator for assessing the surface roughness of the received electromagnetic signal [22; 23; 24]. A well-developed sea, forced by a wind speed \(U\), generates a wave spectrum for the air-sea interaction [23; 25; 26]. A first-order approximation model to the high-wave-number part of the power spectra states that the mean ocean wave-height \(h\) is proportional to the square of the wind speed \(U\) as given in the Equation (4) [20; 23; 26; 27]. However, short waves are also linked to intermediate and long-scale waves. Thus, a more elaborate spectral model needs to be applied for assessing these processes and the full development of high and low wave numbers of ocean waves. Here, it was omitted in order to simply the retrieval of the ocean sea roughness parameter. \[h=C_{1}\cdot U^{2} \tag{4}\]  
\(C_{1}\) is a constant that, in our calculations, was set to 0.0051 [20]. On the basis of the geometry of the position of the GPS satellite and the GPS receiver, and through the use of Equation (4), it is possible to calculate the rough surface impedance and the rms Rayleigh roughness parameter from the wind speeds.