## 3 Link Budget for the GNSS-R Sea Surface Return
It depends on the acquisition geometry--i.e., viewing and scattering angles--, wind speed and direction, among other sensor parameters, e.g., frequency, polarization. \begin{table}
\begin{tabular}{l l l} \hline
**Symbol** & **Parameter** & **Value** \\ \hline \(P_{t}\) & Transmitted power & 26.61 W \\ \(G_{t}\) & Transmitting antenna gain & 13 dBi \\ \(G_{r}\) & Receiving antenna gain & 13.3 dBi (spaceborne) \\  & & 15.05 dBi (airborne) \\ \(HPBW_{al}\) & Receiver HPBW along-track & 60\({}^{\circ}\) \\ \(HPBW_{ac}\) & Receiver HPBW across-track & 30\({}^{\circ}\) \\ \hline \end{tabular}
\end{table} 
Table 1: List of symbols and corresponding definition and value. Figure 2: Scattering geometry and reference system. Sea surface signal strength in (5) can be simplified by noting that, for typical GNSS altitude, variations of the local viewing angle \(\theta_{l}\) within the integration domain can be reasonably neglected, i.e., it can be assumed \(\theta_{l}=\theta\) within the whole integration domain. Additionally, in order to make the link budget analysis independent from the specific receive antenna pattern and reduce the computational complexity of (5), here we assume an ideal radiation pattern, i.e., constant within the main lobe of the antenna and null outside. Moreover, it can be safely assumed that the area illuminated by the GNSS transmitter is much larger than the area sensed by the receive antenna, so that the integration domain in (5) is only limited by the receive antenna beamwidth. We further assume that the receive antenna pattern is pointed towards the center of the resolution cell under test. Transmitter-surface point and receiver-surface point distances are evaluated assuming a flat Earth model, i.e., by assuming that the height deviation of the spherical Earth from its tangent plane is much smaller than the transmitter and receiver altitudes. This is a reasonable assumption for both spaceborne and airborne GNSS-R. Indeed, the much lower altitude in the airborne case is compensated by the much smaller coverage area. Accordingly, (5) can be recast as:  
\[P_{r,sea}(\theta_{s},\phi_{s})=P_{t}G_{t}G_{t}\frac{1}{(4\pi)^{3}}\bigg{(} \frac{\lambda\cos\theta}{h_{t}h_{r}}\bigg{)}^{2}\iint_{A_{0}}\cos^{2}\theta_{ s,l}\varphi_{sea}^{0}(\theta,\theta_{s,l},\phi_{s,l})\ dxdy \tag{6}\]  
where, \(h_{t}/\cos\theta\) and \(h_{r}/\cos\theta_{s,l}\) are the approximate transmitter-surface point and receiver-surface point distances. Additionally, according to previous assumptions, \(A_{0}\) is the intersection region between the delay-Doppler resolution cell and the receive antenna pattern. Its actual shape and area depend on the considered delay-Doppler cell, whose center point, in turn, depends on \(\theta_{s}\) and \(\phi_{s}\). It can be noted that both (5) and (6) provide a more accurate evaluation of the received signal strength respect to the simplified formulations adopted in [27; 28], that can be obtained from (6) by further neglecting variations of the zenith and azimuth scattering angles within the scattering area \(A_{0}\). Such an assumption, despite being reasonable in spaceborne configurations, leads to inaccurate results with airborne GNSS-R due to the much lower receiver altitude [21]. The algorithm for the evaluation of the sea scattering area \(A_{0}\) is as follows. 1. The acquisition geometry, i.e., the values of \(\theta\), \(\theta_{s}\), and \(\phi_{s}\) are assigned as input parameters. This implies that the receive antenna beam is conceptually centered around the input (\(\theta_{s}\), \(\phi_{s}\)) direction. 2. The area sensed by the receive antenna beam is determined according to the half-power beamwidths (HPBWs) defined in Table 1. To this end, a surface points grid is