## 3 Link Budget for the GNSS-R Sea Surface Return  
The evaluation of the strength of the GNSS signal reflected from the sea surface is carried out by assuming the standard two-step GNSS-R processing chain:  
1. _Coherent integration_: The signal received through the reflection path is correlated with a properly delayed and frequency-shifted version of either the direct signal or a locally-generated clean replica of the transmitted GNSS signal. The output of this step is a 2-D map of the signal power distribution in the delay-Doppler domain. It is commonly referred to as _single-snapshot delay-Doppler map (DDM)_. 2. _Incoherent integration_: The coherently-integrated DDM suffers from a very low SNR due to the poor GNSS Effective Isotropical Radiated Power (EIRP), especially at spaceborne altitudes. Accordingly, a strong noise reduction is necessary for enabling meaningful geophysical information retrieval. This is accomplished by incoherently averaging subsequent single-snapshot DDMs, i.e., by temporal multilook, which reduces the thermal noise power by the factor \(\sqrt{N}\), where \(N=\frac{T_{i}}{T_{e}}\) is the number of averaged DDMs. Typical coherent \(T_{c}\) and incoherent \(T_{i}\) integration times are reported in Table 1 for both spaceborne and airborne receivers. The reference system is shown in Figure 2: it is defined such that the incident wavevector is in the \(x\)-\(z\) plane and the \(z\)-axis is normal to the mean sea surface plane. Consequently, the transmitter position is fully defined by the viewing angle \(\theta\), whereas the GNSS-R receiver by the zenith \(\theta_{s}\) and azimuth \(\phi_{s}\) scattering angles. It is worth noting that in this reference system, the backscattering direction is defined by \(\theta_{s}=\theta\) and \(\phi_{s}=180^{\circ}\), while the forward-scattering configuration, typical of conventional GNSS-R, by \(\theta_{s}=\theta\) and \(\phi_{s}=0^{\circ}\)Let's consider the power of the sea surface echo coming from the resolution cell whose central point is the origin of the reference system in Figure 2. It turns out that the scattering reference frame depicted in Figure 2 coincides with the plane tangent to the spherical Earth's surface in the center of the resolution cell under analysis. At the output of the incoherent integration, this received power can be written as  
\[P_{r,sea}(\theta_{s},\phi_{s})=\iint P_{t}G_{t}(\theta_{l})G_{r}(\theta_{s,l}, \phi_{s,l})\frac{1}{(4\pi)^{3}}\frac{\lambda^{2}}{d_{t}^{2}(\theta_{l})d_{r}^{ 2}(\theta_{s,l})}\sigma_{sea}^{0}(\theta_{l},\theta_{s,l},\phi_{s,l})\ dxdy \tag{5}\]  
where \(\theta_{l}\) stands for the _local_ GNSS transmitter viewing angle, \(\theta_{s,l}\) and \(\phi_{s,l}\) are the _local_ zenith and azimuth scattering angles, respectively, see Figure 2. For their evaluation, here we assume a spherical Earth model with radius \(r_{E}=6356.752\) km, so that the generic surface point is \(P(x,y,z)\) with \(z=\sqrt{r_{E}^{2}-x^{2}-y^{2}}-r_{E}\). Actually, for the scope of evaluating local viewing and scattering angles, sea surface roughness is neglected. Remaining symbols are defined in Table 1. In (5), \(d_{t}\) and \(d_{r}\) are the transmitter-surface point and receiver-surface point distances, respectively. Moreover, the integration domain is practically limited by the intersection of the transmit and receive antenna patterns and the delay-Doppler resolution cell. The width of the resolution cell in the delay-Doppler domain is \(A_{cell}=\Delta\tau\Delta f_{d}\), where \(\Delta\tau=2\tau_{c}\) and \(\Delta f_{d}=2/T_{c}\) are approximately the delay and Doppler resolutions, i.e., the widths of the GNSS-R Woodward Ambiguity Function, respectively. They should not be confused with the GNSS-R receiver delay and Doppler sampling steps, that do not affect the link budget analysis. Sea surface NRCS \(\sigma_{sea}^{0}\) coincides with the diagonal terms of the polarimetric covariance matrix in (4) and is evaluated via BA-PTSM according to [26].