# The Implementation of Precise Point Positioning (PPP): A Comprehensive Review
## 3 PPP Models
### Ambiguity-Fixed PPP  
While dealing with float ambiguities is more straightforward, it has been well-established that resolving integer ambiguities in PPP can significantly enhance convergence time and solution accuracy, particularly along the longitudinal axis [75]. The standard PPP model combines residual biases with the ambiguity term, as indicated in (38). These biases are known as the Uncalibrated Phase Delays (UPDs) or Fractional Cycle Biases (FCBs). For the sake of simplicity, we focus on GPS ambiguity fixing; nevertheless, the general principles apply to other constellations. To retrieve the integer nature of the ambiguity, UPDs must be estimated separately from the ambiguity term. However, this is not feasible within the traditional PPP frame work. To illustrate this, let us simplify the standard PPP model by introducing Single Differencing (SD) between satellites. The SD operator, represented as \(\Delta\), subtracts the observations of a reference satellite from corresponding observations of the other satellites, effectively eliminating all receiver biases. Assuming all necessary corrections are applied except the satellite clock error, SD PPP observations can be expressed as  
\[\Delta P_{IF} =\Delta\rho-c\Delta dt^{s}_{P_{IF}}+\Delta M_{w}Z_{w}+\Delta e_{P_{IF}} \tag{45}\] \[\Delta\Phi_{IF} =\Delta\rho-c\Delta dt^{s}_{P_{IF}}+\Delta M_{w}Z_{w}+\lambda_{IF} \Delta N_{IF}+(\Delta b^{s}_{\Phi_{IF}}-\Delta b^{s}_{P_{IF}})+\Delta e_{\Phi_{ IF}} \tag{46}\]  
Even with SD, the system remains unsolvable, as the number of unknowns still exceeds the number of observations. Another challenge arises from the fact that the IF ambiguity for GPS has a short wavelength of \(\lambda_{IF}=6.3\) mm, making it comparable to the observation noise. To address this, the IF ambiguity is often decomposed into two components as follows [73]:  
\[N_{IF}=17N_{1}+60(N_{1}-N_{2})=17N_{1}+60N_{WL} \tag{47}\]  
where \(N_{1}\) and \(N_{2}\) represent the ambiguity of the phase observations at the L1 and L2 GPS frequencies, respectively. To overcome the rank deficiency issue, the PPP network model must be adjusted to estimate additional corrections that users can receive to resolve the integer ambiguities [76]. Various models, such as the Decoupled Clock (DC) model [77], the Integer Recovery Clock (IRC) model [78], and the UPD/FCB model [68], have been introduced for this purpose. All these models introduce a third observation to the PPP model--the MW function. The MW function with single differencing can be expressed as  
\[\Delta A_{MW}=\lambda_{WL}\Delta N_{WL}-\Delta b^{s}_{A_{MW}}+\Delta e_{A_{MW}} \tag{48}\]  
where \(\Delta b^{s}_{A_{MW}}\) and \(\Delta e_{A_{MW}}\) denote the FCB and noise for \(\Delta A_{MW}\), respectively. The MW FCB, also known as WL FCB, remains stable over time and can be considered a constant. Nevertheless, smoothing \(\Delta A_{MW}\) still requires around 10 min, after which \(\Delta b^{s}_{A_{MW}}\) can be estimated as the fractional component of \(\Delta A_{MW}\)[68]. CNES, an IGS analysis center, was the first to provide corrections to enable PPP ambiguity fixing, employing the IRC model [79]. Later, other IGS centers and academic organizations also started to provide phase bias products [80]. The IRC model is similar to the DC model but has different parametrization [81]. Both models involve different satellite clock terms for code and phase observations as follows:  
\[\Delta P_{IF} =\Delta\rho-c\Delta dt^{s}_{P_{IF}}+\Delta M_{w}Z_{w}+\Delta e_{P_ {IF}} \tag{49}\] \[\Delta\Phi_{IF} =\Delta\rho-c\Delta dt^{s}_{\Phi_{IF}}+\Delta M_{w}Z_{w}+\lambda_ {IF}(17\Delta N_{1}+60\Delta N_{WL})+\Delta e_{\Phi_{IF}}\] (50) \[\Delta A_{MW} =\lambda_{WL}\Delta N_{WL}-\Delta b^{s}_{A_{MW}}+\Delta e_{A_{MW}} \tag{51}\]  
The IRC model in the above equations estimates and sends to the users three correction parameters, indicated in bold, namely the satellite code clock, satellite phase clock, and WL biases. The satellite phase biases in (50) are absorbed in the phase clock term.