# Characterizing Ionospheric Effects on GNSS Reflectometry at Grazing Angles from Space
## 3 Methodology
### Ionospheric Group Delay Computation  
The GNSS electromagnetic signal propagation speed in the ionosphere depends on electron density \((N_{c})\), which is influenced by daytime ionization and nighttime recombination processes. According to [13], when considering the signal code measurements, the difference between the measured range (using a signal of frequency \(f\) in \(Hz\)) and the Euclidean distance between the satellite and receiver is expressed as follows:  
\[\Delta_{gr}^{iano}=+\frac{40.3}{f^{2}}\int N_{c}dl \tag{1}\]  
\(\Delta_{gr}^{iano}\) is the term used for the group ionospheric refraction, and the integral is known as the slant total electron content (\(sTEC\)), representing the numerical integration of the electron density along the ray path. \(f\) corresponds to the GNSS signal frequency, and in this study, the GPS L1 frequency is \(1575.42\) MHz. The \(sTEC\) is computed for each ray, including the incident (\(sTEC_{in}\)), reflected (\(sTEC_{re}\)), and direct (\(sTEC_{dr}\)) rays, respectively. The \(sTEC\) is expressed in total electron content units (TECUs) where one TECU corresponds to \(10^{16}\) electrons per square meter \((el/m^{2})\). Finally, the group ionospheric delay in meters (for each ray) is obtained from:  
\[I_{in,re,dr}=+\frac{40.3*10^{16}}{f^{2}}sTEC_{in,re,dr} \tag{2}\]  
As presented in [22], the relative delay between the direct and reflected signals is denoted as \(\Delta_{p}=p_{r}-p_{d}\), where \(p_{r}\) is the cumulative path of the incident and reflected rays, while \(p_{d}\) corresponds to the direct path. The relative delay can be influenced by various contributing factors, such as the standard sources of delay within the GNSS signals. Therefore, the extended version of \(\Delta_{p}\) can be written as:  
\[\Delta_{p}=\Delta_{p_{geo}}+\Delta_{p_{trap}}+\Delta_{p_{lim}}+\Delta_{p_{rgb }}+\Delta_{p_{intr}}+n \tag{3}\]  
where \(\Delta_{p_{geo}}\) represents the relative geometrical delay, and \(\Delta_{p_{trap}}\) and \(\Delta_{p_{lim}}\) correspond to the relative tropospheric and ionospheric delays, respectively. \(\Delta_{p_{rgb}}\) is a bias induced by the surface roughness. The instrumental error is denoted by \(\Delta_{p_{instr}}\), and \(n\) represents unmodeled errors.  
GNSS-R Group Delay Altimetry and Ionospheric Delay Uncertainty  
Based on the analysis conducted in [23], the ionospheric delay constitutes a significant component within the error budget associated with GNSS-R ocean surface altimetry retrievals. At elevation angles above 60\({}^{\circ}\), the uncorrected ionospheric delay can reach \(\sim\)15 m during daytime and \(\sim\)7 at nighttime. The ionospheric group delay bias propagates to an altimetric bias based on the relation between the height offset \(\Delta_{h}\) and the signal path \(\Delta_{p}\). Consequently, when considering only the ionospheric altimetric error, where \(E\) is the elevation angle, it can be expressed as:  
\[\Delta I_{tiono}=\frac{\Delta_{p_{limo}}}{2*\sin E} \tag{4}\]  
Assuming a relative uncertainty of 30% for the ionospheric delay bias, as established in [20], we introduce normally distributed random errors with a respective standard deviation \(\left(\delta_{iono}=\mathrm{N}\left(0,0.3^{2}\right)\right)\).