## 2 Methodology: Computation of the Altimetry Performance
### Instrumental Error Sources in GNSS-R Altimetry  
The procedure followed to compute the GEROS ISS experiment performance follows that in [6] for the PARIS In-orbit Demonstrator (IoD) Mission, with the following differences:  
* Maximum and minimum ISS orbital heights considered to compute the maximum off-boresight angle for a given swath of 500 km (i.e., 250 km half swath in each side, nominal off-boresight angle = 35\({}^{\circ}\)).
* Antenna: instead of the hexagonal 19 element antenna array (Figure 5a) foreseen for PARIS IoD [6], a rectangular 31 element antenna array (Figure 5b) was selected for GEROS-ISS, since larger apertures fit in the available space in the upper deck of the Columbus module. This translates into an increased directivity and increased signal-to-noise ratio. The array topology and the numerically computed elementary antenna patterns have been used to estimate the directivities for the up and down-looking antennas, at boresight and at 35\({}^{\circ}\) or the corresponding angle to the maximum off-boresight angle for each orbital height, and swath, and for the lower (L5/E5) and upper (L1/E1) bands. Array parameters are summarized in Table 3.
* Receivers noise figure is NF\({}_{\text{nom}}\) = 3.5 dB.
* Receivers bandwidth is B = 40 MHz.
* Dwell line is 100 km, unless otherwise specified. Results for a 1 ms coherent integration, and 1 s integration (incoherent averaging) time are also provided.
* All other instrumental error sources, etc. as in [6].
* Inter-modulation signal power is taken into account. The inter-Modulation (IM) components are extra signal components transmitted by the GPS navigation satellites to keep the power envelope of the composite signals constant, so as to improve the performance of the solid state power amplifiers (SSPAs). The IM signals do not transmit navigation information, so they have no impact in conventional GNSS-R (cGNSS-R) or reconstructed-code GNSS-R (rGNSS-R) [13], but they do in the interferometric GNSS-R (iGNSS-R), because this later technique cross-correlates the whole  
Figure 4: Simulated One-way ionosphere amplitude scintillations for S\({}_{4}\)= 0.8 and \(\sigma_{\phi}\) = 0.1 rad, at L1/E1 (blue), L2/E2 (red), and L5/E5 (red).  
signal. The IM signal accounts for a 25% of the total transmitted power, so the SNR is increased by 4/3 (1.25 dB). Figure 6a,b show the power spectral density (PSD), and the squared auto-correlation function (ACF) of the composite GPS L1 signal with (red) and without (blue) considering the IM component. As it can be appreciated, not only the power is higher, but it is more concentrated towards the higher frequencies, which increases the Gabor bandwidth (Equation (5)), and the achievable altimetry precision.  
Since both, single frequency or dual-frequency (ionosphere corrected) altimetry estimates depend on the bandwidth and the SNR (Equation (4)), for a given receiver's bandwidth, the impact of any error source reduces to the calculation of its impact on the signal or the noise powers, and therefore, the two most important parameters are the receivers' noise figure and the antenna directivity.  
Figure 5: (**a**) Original antenna in PARIS IoD, and (**b**) antenna of GEROS ISS corresponding to the Phase A study led by Airbus Defense and Military.  
Figure 6: (**a**) Power Spectral Density (PSD), and (**b**) squared auto-correlation function (ACF) of the composite GPS L1 signal with and without considering the IM component (Adapted from [14]).  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline  & \multicolumn{2}{c}{**Upper Band (L1/E1)**} & \multicolumn{2}{c}{**Lower Band (L5/E5)**} \\ \hline \multirow{3}{*}{\(\text{D}_{\text{up-looking}}\)} & Boresight & 35\({}^{\circ}\) & Boresight & 35\({}^{\circ}\) \\  & 24.72 dB & 20.02 dB & 22.42 dB & 19.62 dB \\ \cline{1-1}  & 24.72 dB & 22.12 dB & 22.42 dB & 20.82 dB \\ \hline \hline \end{tabular}
\end{table}
Table 3: Average antenna array parameters used in the estimation of GEROS-ISS experiment performance.