# Ionospheric Error Models for Satellite-Based Navigation--Paving the Road towards LEO-PNT Solutions
## 3 Methodology  
The ionosphere is home to atoms and molecules that are directly exposed to the sun's intense radiation. When these atoms absorb the radiation, many of them lose electrons through a process called ionization [19; 21], which causes them to become positively charged as follows:  
\[A\to A^{+}+e \tag{2}\]  
where \(A\) is any molecule or atom in its initial (neutral) state, \(A^{+}\) is the positively charged atom (or ion), and \(e\) is the free electron. The ionosphere is packed with these ions and free electrons, hence the term _iono_. Fortunately, this absorption process protects humans and other living beings from harmful solar radiation. However, these free electrons inflict considerable delays that affect the propagation of radio waves [21]. That is why ionospheric delay is one of the most significant atmospheric errors that affect RF signals coming from satellites towards the Earth [24; 41]. The amount of free electrons in the ionosphere is known as TEC, and it can be of two types: Slant Total Electron Content (STEC), which is the electron content measured along a slanted path, or VTEC, which is the electron content measured at the zenith. Figure 3 highlights the geometric relationship between the two, where \(\varphi^{\prime}\) and \(\lambda\) are the latitude and longitude of the receiver and \(R_{E}\) is the radius of the Earth. STEC is calculated using the following formula:  
\[STEC=\int n_{e}(l)\,dl \tag{3}\]  
where \(n_{e}(l)\) is the number of electrons along path \(l\). The mathematical relationship between STEC and VTEC is given by:  
\[VTEC=STEC\times M(E) \tag{4}\]  
where \(M(E)\) is the mapping function or the obliquity factor and \(E\) is the satellite elevation angle in degrees. There are various mapping functions, but the most common one is the following:  
\[M(E)=\sqrt{1-\left(\frac{R_{E}}{R_{E}+H_{max}}\cos{(E)}\right)^{2}} \tag{5}\] \begin{table}
\begin{tabular}{c l l l l l} \hline \hline
**Ref.** & **Model** & **Type** & **Broadcast** & **Input Data** & **Direct Output Data** \\ \hline [40] & Klobuchar & Empirical & Yes & Atmospheric coef., approx. & ID \\  & & & & receiver position & \\  & & & & Atmospheric coef., approx. & \\  & & & & user position & & ED, ID \\
[31; 33; 44; 45] & Neural networks & Data-driven & No & Data from other & \\  & & & & models/stations & \\
[52] & IRI & Empirical & No & Ionosondes, ISRs, in situ data & ED, ID, ionosphere detailed \\  & & & & (satellites) & & composition (see Section 2.5) \\
[53] & IGS (GIMs) & Mathematical & No & Dual-frequency measurements & \\  & & & & from GNSS stations & \\ This work & IAMM & Data-driven & No & Approx. receiver position, \(Y\) & \\  & & & & matrix (see Section 3.2) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of ionospheric models reviewed (ED: electron density, ID: ionospheric delay, ISR: incoherent scatter radar). where \(R_{E}\) is the Earth's radius and \(H_{max}\) is the height of the maximum electron density. Both STEC and VTEC are often expressed in TEC units (TECU), where 1 TECU equals to \(10^{16}\) electrons/m\({}^{2}\). The ionospheric delay corresponding to a satellite's signal is based on STEC and can be expressed as:  
\[I=\frac{\kappa}{f_{c}^{2}}STEC \tag{6}\]  
where \(I\) is the ionospheric delay in meters, \(\kappa\) is a constant that is approximately equal to 40.3 m\({}^{3}\)/s\({}^{2}\), and \(f_{c}\) is the carrier frequency in Hz. The signal path within the ionosphere increases as the satellite's elevation angle decreases. This leads to more free electrons intercepting the passing signal, increasing the ionospheric delay. The ionospheric delay affects the quality of the final positioning solution because it introduces ambiguities to the range calculations that are made by the receiver, known as the pseudorange, which is calculated to determine the user's position, as shown in Equation (7).