## 2 Methodology: Computation of the Altimetry Performance
### Platform Error Sources in GNSS-R Altimetry
#### 2.4.2 Impact of ISS Shadowing and Multiple Scattering
Regarding the polarization, the first order rays (direct ones) can be considered co-polar (LHCP for the down-looking antenna, and RHCP for the up-looking antenna). For the reflected signal, it is assumed that the each reflection over a metallic surface reverses the polarization perfectly, while amplitude of the reflection coefficient is \(|\Gamma_{p}|=1\). Correspondingly, the second order rays are weighted by the cross-polar antenna pattern (RHCP for the down-looking antenna, and LHCP for the up-looking antenna), and the third rays are weighted by the co-polar patterns, as the first order rays. \[ \begin{split}\overset{\rightarrow}{E}_{up}=\overset{ \rightarrow}{E}_{dir}\Big{\{}F_{RH}\Big{(}\theta_{1}^{dir},\phi_{1}^{dir} \Big{)}+F_{LH}\Big{(}\theta_{2}^{dir},\phi_{2}^{dir}\Big{)}e^{-j\omega\tau_{2} ^{dir}}+F_{RH}\Big{(}\theta_{3}^{dir},\phi_{3}^{dir}\Big{)}e^{-j\omega\tau_{3} ^{dir}}\Big{\}}+\\ \overset{\rightarrow}{E}_{scat}\Big{\{}F_{RH}\big{(}\theta_{2} ^{scat},\phi_{2}^{scat}\big{)}e^{-j\omega\tau_{2}^{scat}}+F_{LH}\big{(}\theta_ {3}^{scat},\phi_{3}^{scat}\big{)}e^{-j\omega\tau_{3}^{scat}}\Big{\}},\\ \overset{\rightarrow}{E}_{dn}=\overset{\rightarrow}{E}_{scat} \Big{\{}F_{LH}\big{(}\theta_{1}^{scat},\phi_{1}^{scat}\big{)}+F_{RH}\big{(} \theta_{2}^{scat},\phi_{2}^{scat}\big{)}e^{-j\omega\tau_{2}^{scat}}+F_{LH} \big{(}\theta_{3}^{scat},\phi_{3}^{scat}\big{)}e^{-j\omega\tau_{3}^{scat}}\Big{\}} \\ +\overset{\rightarrow}{E}_{dir}\Big{\{}F_{LH}\Big{(}\theta_{2} ^{dir},\phi_{2}^{dir}\Big{)}e^{-j\omega\tau_{2}^{dir}}+F_{RH}\Big{(}\theta_{3 }^{dir},\phi_{3}^{dir}\Big{)}e^{-j\omega\tau_{3}^{dir}}\Big{\}}\,\end{split}  \tag{11a}\] \[\overset{\rightarrow}{E}_{up}=\overset{\rightarrow}{E}_{dn} \Big{\{}F_{LH}\big{(}\theta_{1}^{scat},\phi_{1}^{scat}\big{)}+F_{RH}\big{(} \theta_{2}^{scat},\phi_{2}^{scat}\big{)}e^{-j\omega\tau_{2}^{scat}}+F_{LH} \big{(}\theta_{3}^{scat},\phi_{3}^{scat}\big{)}e^{-j\omega\tau_{3}^{scat}}\Big{\}}\] (11b) \[+\overset{\rightarrow}{E}_{dir}\Big{\{}F_{LH}\Big{(}\theta_{2} ^{dir},\phi_{2}^{dir}\Big{)}e^{-j\omega\tau_{2}^{dir}}+F_{RH}\Big{(}\theta_{3 }^{dir},\phi_{3}^{dir}\Big{)}e^{-j\omega\tau_{3}^{dir}}\Big{\}}\,\]  
where \(\overset{\rightarrow}{E}_{up}\) and \(\overset{\rightarrow}{E}_{dn}\) are the electric fields received by the up- and down-looking antenna, \(F_{RH}(\theta_{n},\phi_{n})\) is the RHCP antenna pattern (amplitude and phase), for the \(n^{\text{th}}\) order ray coming from the \(\big{(}\theta_{n}^{dir},\phi_{n}^{dir}\big{)}\), and \(\tau_{n}\) is the delay of the \(n^{\text{th}}\) order ray relative to the first order one. Figure 11 shows the directions of arrival to the antenna in the antenna reference frame. Taking into account Equation (11a,b), the cross-correlation of the signals from up-looking and down-looking antennas can be computed (\(\overset{\rightarrow}{E}_{up}\cdot\overset{\rightarrow}{E}_{dn}^{*}\)). An efficient way to compute the final GNSS-R observable consists of the computation of the convolution of the multi-path free waveforms (\(W_{1}(\tau)\)), and the impulse response that models the multiple reflections (\(I_{m}(\tau)\)):  
\[W_{f}(\tau)=\,W_{1}(\tau)|_{(\theta_{1}^{dir},\phi_{1}^{dir};\theta_{1}^{scat}, \phi_{1}^{scat})}\ast I_{m}(\tau)|_{(\theta_{1}^{dir},\phi_{1}^{dir};\theta_{1} ^{scat},\phi_{1}^{scat})} \tag{12}\]  
Figure 11: **(a)** Second order arriving direction to GEROS-ISS in the antenna reference frame, and (b) third order rays arriving direction to GEROS-ISS in the ISS far field reference frame. Top: first matrix and bottom: second matrix, refer to the fact that there are second order and third order rays that have the same far field direction, but enter through different directions in the antenna pattern reference frame. Color code corresponds to the delay in ns. As seen in Equation (12), the multiple reflected rays contribute with additional replicas of the original waveform delayed and weighted by the corresponding antenna pattern of the direction of arrival in the antenna reference frame.