## 4 Scatterometry Simulations of Ocean Reflected GPS Signals, Sea Surface Roughness, and Wave Heights
It is also seen, for given spectral values, that the corresponding frequency increases as a function of wind speed and ocean wave-height [24; 25; 32]. The spectrograms reveal signs of both the reflected and the direct wave, due to a small difference in the frequencies of the direct and reflected waves, as also shown for the observations in Figure 3.  
The power spectra in a semi-logarithmic presentation are very close to a straight line with decreasing power for increasing frequency. However, the ocean rms wave-height, as a function of the spectral frequency in a given spectral domain, shows increasing Rayleigh roughness as a function of frequency. Here the Rayleigh roughness parameters are calculated from the rms wave-heights [20; 27; 29]. Figures 8 and 9 show how wave-heights and ocean roughness relate to the spectral frequency in the chosen frequency band. The retrieval method builds on the knowledge that the power spectrum has an almost linear slope for the chosen frequency range [20]. The frequency \(f_{0}\) (in Equations (8) and (9)) becomes a constant (determined by the chosen frequency range in the power spectrum) and changes with respect to the used wave heights and surface winds (sea states). Thus, the simulated spectra can be fitted by the function \(W\):  
\[W(f,\,f_{0})=\sqrt{\frac{4}{\pi\cdot f_{0}}}\cdot\left(1+\left(\frac{f}{f_{0}} \right)^{2}\right)^{-1}, \tag{8}\]  
where \(f\) represents a frequency in the frequency band in question, and \(f_{0}\) is a constant frequency related to the parameters used in the simulations. The function \(W\) is similar to the spectrum model applied in the paper [29]. Figure 8: Ocean rms wave-height as a function of the spectral frequency in the frequency domain from 90 to 100 Hz. It should be noted that, when the spectrum model above is used to fit the simulated spectra, an extra constant value has to be added to the model. In our simulations, a spectral value of 33 dB was used for producing the results shown in Figures 8 and 9. The outcome shows clearly that the power spectral values increase as function of wind speed and ocean wave-height. A typical statistical value of \(R^{2}\) for the applied retrieval process was 0.988, meaning that approximately 98% of the variations can be explained by the spectrum model. For most frequency bands (from 0.1 to 500 Hz), the relation of the rms wave-height, roughness, and the spectral frequency follows a hyperbolic relation of the following form:  
\[F(f)=\left[\left(\epsilon^{2}-1\right)\left(\left(f-f_{0}\right)^{2}-a^{2} \right)\right]^{1/2}+C_{0}{}^{2}, \tag{9}\]  
where \(f\) and \(f_{0}\) are spectral frequencies in the frequency band as explained above, and \(C_{0}\) is a constant. The eccentricity, \(\epsilon\), of the hyperbola is given by the major axis, \(a\), and the minor axis, \(b\), of the hyperbolic function. \[\epsilon=\left[1+\left(\frac{b}{a}\right)^{2}\right]^{1/2}. \tag{10}\]  
In the simulations, \(\epsilon\) varies for increasing spectral frequency, and it is also a function of the distance between the phase screens. Thus, for the higher spectral frequencies, \(\epsilon\) may be linked to the turbulence cascading process between the longer- and shorter-wavelength waves. Figure 9: The relation between the Rayleigh roughness parameter and the spectral frequency for the same frequency range as in \[\delta=\sin(\theta)\frac{(1+\rho)\delta_{0}+(1-\rho)\sin(\theta)}{(1-\rho) \delta_{0}+(1+\rho)\sin(\theta)} \tag{5}\]0 The Rayleigh roughness parameters were calculated for a grazing angle of \(-1.5^{\circ}\). Comparing the simulation results (presented in Figures 8 and 9) with the actual parameter calculations from the measurements (given in Figure 7) reveals that the estimated wave heights and the rms Rayleigh roughness parameters are aligned within the error estimations of both approaches.