# The Implementation of Precise Point Positioning (PPP): A Comprehensive Review
## 2 Error Sources in PPP
### Ionospheric Delays  
The ionosphere is the ionized part of the atmosphere between 50 and 1000 km above sea level. Fluctuations in solar activity induce variations in the ionization levels, thereby impacting the refractive indices of the ionosphere's various layers. This, in turn, alters the transit time of GNSS signals. Based on the solar activity's intensity and the satellite's elevation angle, ionospheric delays can induce ranging errors of up to 100 m [23]. It is essential to note that such delays are frequency-dependent. In the context of the standard DF-PPP, the DF measurements' ionospheric-free (IF) combination is employed to counteract the first-order ionospheric error. The IF pseudorange \(P_{IF}\) and carrier-phase \(\Phi_{IF}\) measurements are articulated as follows:  
\[P_{IF} =\alpha_{IF}P_{1}+\beta_{IF}P_{2}\] \[\Phi_{IF} =\alpha_{IF}\Phi_{1}+\beta_{IF}\Phi_{2} \tag{1}\]  
where \(P_{i}\) and \(\Phi_{i}\) represent the pseudorange and carrier-phase measurements of the GNSS frequency \(f_{i}\), respectively. The IF coefficients are denoted as \(\alpha_{IF}=f_{1}^{2}/(f_{1}^{2}-f_{2}^{2})\) and \(\beta_{IF}=-f_{2}^{2}/(f_{1}^{2}-f_{2}^{2})\), where \(f_{1},f_{2}\) correspond to the carrier frequencies of the combined signals.  
The IF combination effectively eliminates up to 99.99% of the slant delay [25]. However, a trade-off exists: the resulting IF measurements exhibit increased noise levels compared with individual observations. For instance, the GPS IF observations are approximately threefold noisier than the uncombined L1 and L2 code and phase GPS observations [1]. Furthermore, higher-order ionospheric effects can introduce error magnitudes of up to 2 cm. These effects may also be modeled for applications requiring millimeter-level accuracy [7].  
In the case of uncombined GNSS observations, the PPP algorithm can employ the Global Ionospheric Maps (GIMs) provided by the International GNSS Service (IGS) [26; 27]. The IGS GIMs contain grid-based ionospheric Vertical Total Electron Content (VTEC) corrections, which can be interpolated and mapped to calculate the slant TEC (STEC) at the GNSS receiver location. The mapping is typically performed under the thin-layer assumption, which presumes that the ionosphere is just a thin layer at a fixed altitude between 350 and 450 km. The intersection between the ray from the satellite to the receiver and the thin layer is called the Ionosphere Pierce Point (IPP). The thin-layer assumption can introduce up to 10 m vertical range errors, reaching two or three folds at low elevations. More details about the calculation of the IPP and mapping process can be found in [28; 29].  
The IGS ionospheric products are available in a rapid product with a latency of less than 24 h and a final product with a latency of approximately 11 days. Both products can be used in postmission applications and are provided in the IONosphere Map EXchange (IONEX) format [30]. The Centre National d'Etudes Spatiales (CNES) was the first IGS analysis center to broadcast real-time ionospheric corrections [29]. The ionospheric delay \(I\), in meters, is related to the STEC with the formula  
\[I=\frac{40.3\times 10^{16}}{f^{2}}\times STEC \tag{2}\]  
where \(f\) is the signal frequency in Hz.