## 3 Link Budget for the GNSS-R Sea Surface Return
\begin{table}
\begin{tabular}{c l l} \hline \hline
**Symbol** & **Parameter** & **Value** \\ \hline \(\lambda\) & GNSS wavelength & 0.19 m \\ \(\theta\) & Viewing angle & 30\({}^{\circ}\) \\ \(\theta_{s}\) & Zenith scattering angle & Ranging in \([0^{\circ},85^{\circ}]\) \\ \(\phi_{s}\) & Azimuth scattering angle & Ranging in \([0^{\circ},180^{\circ}]\) \\ \(h_{t}\) & Transmitter altitude & 20.200 km \\ \(h_{r}\) & Receiver altitude & 540 km (spaceborne) \\  & & 10 km (airborne) \\ \(\varphi_{sea}^{0}\) & Sea surface NRCS & Evaluated via BA-PTSM \\ \(T_{a}\) & Receiving antenna noise temperature & 99.4 K \\ \(T_{e}\) & Receiver noise temperature & 374.35 K (spaceborne) \\  & & 161.23 K (airborne) \\ \(T_{i}\) & Incoherent integration time & 1 s \\ \(T_{c}\) & Coherent integration time & 1 ms (spaceborne) \\  & & 10 ms (airborne) \\ \(\tau_{c}\) & Chip length & 977.52 ns \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(Cont.\)generated according to the spherical Earth model previously detailed and assuming an initial spatial sampling step of 200 m and 10 m in the spaceborne and airborne configurations, respectively. 3. The coordinates of the transmitter and the receiver, and, then, of the specular reflection point, are computed in the scattering reference frame by simple geometric considerations. 4. Delay-Doppler coordinates (\(f_{d}\), \(\tau\)) are evaluated for each surface point of the sensing region from the position and velocity of the transmitter and the receiver. The delay-Doppler coordinates of the reference frame origin (\(\overline{f_{d}}\), \(\overline{\tau}\)) are computed as well. 5. Surface points falling within the delay-Doppler resolution cell centered in (\(\overline{f_{d}}\), \(\overline{\tau}\)), i.e., such that \[|f_{d}-\overline{f_{d}}|\leq\Delta f_{d}/2\quad\text{and}\quad|\tau-\overline{ \tau}|\leq\Delta\tau/2\] (7) are identified as the sea scattering region \(A_{0}\). 6. Steps 2-5 are repeated until more than 10 surface points are identified as the scattering region. In each iteration, the spatial sampling step is halved to generate a more refined surface grid. Finally, once the sea scattering area is determined for the considered \(\theta_{s}\) and \(\phi_{s}\), the integral in (6) is numerically evaluated as a Riemann sum. In (6), local scattering angles \(\theta_{s,l}\) and \(\phi_{s,l}\) are evaluated for each point of the curved surface grid generated in step 2 by simple geometric considerations from the coordinates of the transmitter, receiver and surface point in the scattering reference frame depicted in Figure 2.  
The thermal noise power at the output of the coherent integration is expressed as:  
\[P_{n}=\frac{k_{B}(T_{a}+T_{e})}{T_{c}} \tag{8}\]  
where \(k_{B}\) is the Boltzmann constant and the bandwidth of the matching filter is approximately equal to \(1/T_{c}\) and, then, \(P_{n}\) is inversely proportional to the coherent integration time \(T_{c}\). Finally, the SNR of the sea surface return at the output of the GNSS-R processing chain can be written as:  
\[SNR=\sqrt{\frac{T_{i}}{T_{c}}}\frac{P_{r,sea}}{P_{n}} \tag{9}\]  
where the square root term is the incoherent integration gain. By combining (8) and (9) it can be noted that the output SNR increases as \(\sqrt{T_{c}}\).