## 2 Methodology: Computation of the Altimetry Performance
### Non-Instrumental Error Sources in GNSS-R Altimetry
#### 2.2.2 Impact of the Ionosphere  
When an electromagnetic wave traverses the ionosphere, the time delay in excess of the propagation time in free space (\(\tau\) [ns]) depends on the frequency (_f_ [Hz]), and the total electron content (TEC) of along the slant propagation path (STEC [e\({}^{-}\)/m\({}^{2}\)]) as:  
\[\tau=134.5\cdot\frac{STEC}{f^{2}} \tag{7}\]  
Then, ionospheric effects can be eliminated by combining the pseudo-range estimates at two different frequencies (e.g., L1/E1 and L5/E5) averaged over \(N\) (\(N_{typ}=3\), [6; 9]) consecutive observations:\[\hat{h}_{15}=-\frac{1}{2}\frac{\rho_{1}+\rho_{5}}{2\cdot sin\theta_{e}}+\frac{1}{2} \cdot\left(f_{1}^{-2}-f_{5}^{-2}\right)\frac{\langle I\rangle_{N}}{2\cdot sin \theta_{e}} \tag{8}\]  
Assuming that the height error is different at both frequency bands (e.g., Table 1), the standard deviation of the resulting altimetry product is then given by [6]:  
\[\sigma_{\hat{h}_{15}}=\sqrt{1+\frac{1.26^{2}}{N}+\frac{2.26^{2}}{N}}.\frac{ \sqrt{\sigma_{h_{1}}^{2}+\sigma_{h_{5}}^{2}}}{2} \tag{9}\]  
However, the accuracy of this correction used in navigation receivers, still needs to be validated, because the ISS height varies in time from 330 to 460 km, and it is in the middle of the ionosphere peak (maximum TEC). Figure 3a shows a global map of TEC computed using the Global Ionospheric Scintillation Model (GISM) [12] for 28 February 2015, at Universal Time UT = 6 AM. Note that maximum TEC content occurs around noon (local time), and around 20\({}^{\circ}\) North/South of the magnetic equator. On the other hand, when an electromagnetic wave transverses the ionosphere, it suffers from intensity (\(I\)) and phase (\(\phi\)) scintillations (rapid variations). The strength of the phase scintillations is characterized by \(\sigma_{\phi}\), the standard deviation of the phase fluctuations, while the strength of the intensity scintillations is characterized by the scintillation index (\(S_{4}\)) defined as:  
\[S_{4}=\sqrt{\frac{\langle I^{2}\rangle-\langle I\rangle^{2}}{\langle I\rangle^ {2}}} \tag{10}\]  
Figure 3b shows a global map of \(S_{4}\) computed using GISM for the same date and UT as in Figure 3a. There are two intense zones of scintillation, one at high latitudes and the other centered within \(\pm 20^{\circ}\) of the magnetic equator, where depth of the scintillation fading ranges from 2 to more than 20 dB depending on the solar activity [12], and has time constants from 0.5 to 2 s. At mid latitudes scintillation occurs exceptionally, e.g., during geomagnetic storms. Scintillation is maximum after sunset (from \(\sim\)19 to 24 h), and around the vernal and autumnal equinoxes, and scintillation events can last from 30 min to hours. Finally, Figure 4 presents simulated amplitude scintillations at L1/E1, L2/E2 and L5/E5 bands computed for \(S_{4}=0.8\) and \(\sigma_{\phi}=0.1\) rad. As it can be observed, large intensity fadings occur, and they are different at different frequency bands (i.e., 1575.42 MHz for L1/E1 at 1176.45 MHz for L5/E5a). They are closer among L2/E2 and L5/E5 because these frequencies are much closer. When the fading is large, the signal-to-noise ratio decreases, the range estimates get worse, and eventually, the signal may be totally lost, although often the SNR degradation is just a few dB. This effect cannot be calibrated using the up-looking antenna since the reflected signal passes through the ionosphere twice (completely in the down-welling path, and through most of it in the up-welling path depending on the orbital height), while the direct signal suffers much smaller ionospheric disturbances as it is only affected by the tenuous fraction of the ionosphere above the receiver. Figure 3: (**a**) Slant Electron Content maps in [TECU] computed with GISM for, and associated, (**b**) Scintillation Index S\({}_{4}\) maps for paths for a receiver on ground. UT = 6 AM.