## 3 Determination of Sea Surface Roughness from Measurements
The latter is directly related to the variances in the measured signal power spectra (Equation (3)). The experimental geometry (Figure 1) limits the number of reflection zones that is visible for the receiver. The GPS receiver (Rx) is fixed on the Hawaiian mountain looking toward south-southwest with the main lobe of the antennas centered on the rim of the horizon. The GPS satellites (Tx) have rising and setting paths on the sky, which for most cases has a sloping angle with respect to the horizon. The time the GPS receiver is able measure GNSS-R signals varies from 20 min to several hours depending on the skewness of the track in the sky [28]. Satellite tracks off from the vertical result in movements of the reflection zone on the ocean. Thus, we selected only GPS satellite tracks that were close to the vertical with respect to the ocean surface. Such situations gave the shortest observation time of ocean reflected signals and made the temporal analysis of the observations more stringent. The data presented in Figure 3 is from such a situation. Figure 3 shows the measured signal power spectra as function of time for the direct and the reflected GPS signal. Here, time on the \(x\)-axis is transformed into elevation angle. The stack-plot presentation of the spectral content clearly identifies the direct signal for elevation angles spanning from \(-3.5^{\circ}\) to \(5.0^{\circ}\). The track of the direct GPS signal (Rx) is  
Figure 3: Stacked power spectra as function of the elevation (GPS SVN15). The spectra reveal the direct GPS signal and the ocean reflected backscattered GPS signal. The latter is here observed for elevations angles from \(-3.5^{\circ}\) to \(3.0^{\circ}\). Observations originate from 7 October 2004 (4:48–5:09 UTC). horizontal in the plot from 0.0\({}^{\circ}\) to 5.0\({}^{\circ}\). The trace above the direct GPS trace is the ocean reflected signal, which shows up as a significant peak in the power spectra. The direct and reflected signals traces are observed for negative frequencies in the power spectra (\(y\)-axis of Figure 3). This delay is primarily due to the bending of the GNSS signal in the ionosphere and troposphere compared to the power at the straight-line tangent altitude [17; 20]. A negative straight-line tangent altitude refers to the situation when the transmitting GNSS satellite is below the horizon with respect to the receiver. The negative offset in the Doppler model is continuously calculated based on real-time navigation and GNSS ephemerides, and it is included in the retrieval of the amplitude and phase of the sampled signal [17]. The atmospheric conditions in the troposphere during the measurement period in Figure 3 showed slowly varying surface wind fields of 5-10 m/s with high humidity in the boundary layer of the troposphere. Thus, it was a quiet period with constant wave height fields (in a statistical sense). Figure 4 presents the associated variances, centered on the maximum mean power of the ocean reflected trace given in Figure 3. For all elevation angles the variances are below 1 Hz\({}^{2}\) and no changes are observed due to the less than half a kilometer movement of the reflection zone during the observations. The power spectra show a distinct peak centered on the reflected signal trace following a Gaussian distribution function. This is the case for all elevations angles given in Figure 4.  
The spike in the figure for elevation angles less than \(-3.1\) degrees occurs when the peak of the reflected trace descends into or arises from the noise floor. The variances becomes large due to this effect independent of the frequency range. The power in the reflected signal is here low and of the order of the noise level. Figure 4: Retrieved averaged variances (0.1 Hz) for the measurements shown in Figure 3 as a function of the elevation angle. Figure 5 shows such conditions, where the reflected signal is submerged into the broad-band noise of the power spectra. This occurs for elevations from \(-3.2\) degrees to \(-4.6\) degrees.