# The Implementation of Precise Point Positioning (PPP): A Comprehensive Review
## 3 PPP Models
### GNSS Code and Phase Observations  
The definition of GNSS pseudorange and carrier-phase observations is the fundamental component of all GNSS models. While the existing literature frequently relies on simplified equations to describe these observations, more detailed formulas are required to understand the impact of the various error sources detailed in Section 2 on GNSS observations. Hence, the pseudorange \(P\) and carrier-phase \(\Phi\) observations from a specific GNSS satellite, both expressed in meters, can be represented as follows:  
\[P_{i} =\rho^{\prime}+c(dt^{\prime}-dt^{s})+b^{\prime}_{P_{i}}-b^{s}_{P_{ i}}+T+I_{i}-v+PCV^{s}_{i}+APC^{\prime}_{i}+\epsilon_{P_{i}} \tag{27}\] \[\Phi_{i} =\rho^{\prime}+c(dt^{\prime}-dt^{s})+b^{\prime}_{\Phi_{i}}-b^{s}_{ \Phi_{i}}+T-I_{i}-v+PCV^{s}_{i}+APC^{\prime}_{i}+\lambda_{i}N_{i}+\frac{ \lambda_{i}}{2\pi}\phi_{w}+\epsilon_{\Phi_{i}} \tag{28}\]  
In the above equations, the subscript \(i\) denotes the frequency index, \(c\) represents the speed of light, and \(dt^{\prime}\), \(dt^{s}\) stand for the receiver and satellite clock errors in seconds, respectively, while \(b^{\prime}_{P_{i}},b^{s}_{P_{i}}\) are the receiver and satellite code biases and \(b^{\prime}_{\Phi_{i}},b^{s}_{\Phi_{i}}\) are the receiver and satellite phase biases. \(T\) denotes the tropospheric delay; \(I\) represents the ionospheric delay; \(v\) accounts for the relativistic effects; \(PCV^{s}_{i}\) signifies the satellite PCV error; \(APC^{\prime}_{i}\) represents the receiver antenna phase center errors; and \(\epsilon_{P_{i}},\epsilon_{\Phi_{i}}\) denote the multipath and receiver noise for code and carrier phase, respectively, all expressed in meters. The integer ambiguity in cycles is symbolized as \(N_{i}\), and the phase wind-up effect \(\phi_{w}\) is measured in radians, as indicated in (21). Each term's sign conforms to the correction models and formulas prevalent in the literature and in Section 2.  
Furthermore, the symbol \(\rho^{\prime}\) in (27) and (28) represents the geometric distance in meters between the receiver and satellite. The prime superscript signifies that this distance is contaminated by influences such as the Sagnac effect, satellite orbit errors, and site displacements. It also encompasses the satellite phase center offset if the satellite position or its corrections are referenced to the satellite CoM rather than APC.  
In addition to individual GNSS observations, GNSS algorithms frequently employ various useful code and phase linear combinations. These linear combinations, for the case of two different frequencies, \(f_{1}\) and \(f_{2}\), generally take the following form [73]:  
\[P_{comb} =\alpha P_{1}+\beta P_{2} \tag{29}\] \[\Phi_{comb} =\alpha\Phi_{1}+\beta\Phi_{2} \tag{30}\]  
where the subscript \(comb\) designates linear combination, while \(\alpha,\beta\) signify the combination coefficients. The factor \(\sqrt{\alpha^{2}+\beta^{2}}\) represents the extent to which the combined observation noise increases or decreases, assuming equal variance and uncorrelated noise from the two combined observations. Subsequently, this section briefly outlines some of these combinations.