## 3 Determination of Sea Surface Roughness from Measurements
In this period the variance gave values from 7 Hz\({}^{2}\) and up to 20 Hz\({}^{2}\). The obtained variances applied in all data retrievals have a temporal resolution of 0.1 Hz. Shorter sampling intervals for acquiring the variances have been tested too. Applying 1, 10, and 50 Hz sampling intervals for calculating the variances gave higher variance values mainly due the wide-band noise in the observations. Thus, a 0.1 Hz sampling rate were chosen to minimize this effect. For all the applied sampling intervals the envelope of the variances had the same fixed form and was only shifted by a constant value. Thus, the variance as function of the sampling interval gave only a constant increase independent of the elevation angle and the sea state. For higher ocean surface winds (larger above 10 m/s) the variances became significantly larger. Figure 6 shows the variances for conditions when the mean wind field ranges from 10 m/s to above 15 m/s. The large peak for elevation angles below \(-3.1\) degrees are also here due to the fact that the maximum power of the reflected trace is of the order of the noise power in the measurements. All datasets showing higher mean winds reveal also larger variations in the variances. For example, the variances in Figure 6 change from less than 1 Hz\({}^{2}\) to 7 Hz\({}^{2}\) during a five-minute period for elevation angles going from \(-1.5\) degrees to \(-3.0\) degrees. Here, this may be due to directional changes in the horizontal wind vectors. The meteorological observations and data from a nearby buoy partly support this explanation. But other phenomena need to be considered for understanding the full picture of these variations. Among them are, atmosphere turbulence in the lower boundary layer of the troposphere, ocean evaporation processes, and vertical components of the troposphere wind fields. In the next step, we retrieved the roughness parameter from the mean variance ratio between the reflected and direct signals evaluated for elevation angles in the range from \(-3.0^{\circ}\) to \(+2.0^{\circ}\). Figure 7 shows the derived rms Rayleigh ro  
Figure 5: Sliding power spectra as function of the elevation (GPS SVN15). The observations originate from October 10, 2004 (6:22–6:46 a.m. UTC). function of wind speed based on all the GNSS-R measurements. The obtained curves are piecewise cubic polynomial fits to the calculated roughness parameter as a function of the atmospheric surface wind data from the meteorological forecast model. The measurements cover the observational period in October 2004. The blue graph in Figure 7 represents the mean estimates of all variances from all observations used in the study, while the gray curves represent the envelopes of all variance retrievals. Next to the blue curve is the orange curve, which is the simple model stating that the roughness parameter is proportional to the square of the wind speed [20]. Both the latter graph (orange) and the derived graph for the rms roughness parameter (blue) show a similar shape for the surface wind speed and the ocean roughness. Note that the agreement between the blue and the orange curve is well within the uncertainty margins indicated by the gray curves. However, for wind speeds up to 15 m/s, the simple relation seems to underestimate the Rayleigh roughness parameter (\(R_{A}\)) compared to the observations presented here by the polynomial fit function (blue curve). This can be explained by the fact that \(R_{A}\) is linked to the coherent part of the electromagnetic reflected signal, as described above. For the high-wind domain, the incoherent part of the signal is the dominant cause of GPS waves reflected from the ocean surface [22; 24]. Figure 6: Variances for sea surface states with higher wind fields ranging from 10 m/s to 15 m/s.