# Stochastic Modeling of Smartphones GNSS Observations Using LS-VCE and Application to Samsung S20
## 3 PPP Mathematical Model
### Functional Model  
The PPP functional model used in this contribution is the single-frequency uncombined PPP model (SF-PPP) (L1 only). One can further generalize the model to the dual-frequency case (DF-PPP). The undifferenced GNSS code and carrier phase observations for the satellite \(s\) and the receiver \(r\) on frequency \(j\) are as follows [29]:  
\[ \begin{split} E\left(P_{r,j}^{s}\right)&=\rho_{r}^{ s}+T_{r}^{s}+cdt_{r}-cdt^{s}+\gamma_{j}I_{r,1}^{s}+b_{r,j}+b_{j}^{s}\\ E\left(\Phi_{r,j}^{s}\right)&=\rho_{r}^{s}+T_{r}^{ s}+cdt_{r}-cdt^{s}-\gamma_{j}I_{r,1}^{s}+\lambda_{j}N_{r,j}^{s}+B_{r,j}+B_{j}^{s} \end{split}  \tag{9}\]  
where \(P_{j}\) and \(\Phi_{j}\) denote the pseudorange and carrier phase observations on the frequency \(j\) in meters; \(\rho\) is the geometric range between satellite and receiver as a function of the satellite and the receiver coordinates; \(T\) is the tropospheric delay (m) which can be split into dry and wet parts; \(c\) is the vacuum speed of light (m/s); \(dt_{r}\) and \(dt^{s}\) are the receiver and satellite clock errors (s), respectively; \(I_{r,1}^{s}\) is the first-order slant ionospheric delay on frequency L1 (m);\(\gamma_{j}=f_{j}^{s}/f_{j}^{s}\) is the frequency-dependent multiplier factor (in the case of L1 frequency \(\gamma_{j}=1\)); \(f_{j}\) is the corresponding frequency; \(\lambda_{j}\) is the corresponding carrier phase wavelength (m); \(N_{r,j}^{s}\) denotes the integer carrier phase ambiguity term in the cycle; \(b_{r,j}\) and \(B_{r,j}\) denote the frequency-dependent receiver pseudorange and carrier phase hardware delays (biases), respectively; and \(b_{r,j}^{s}\) and \(B_{j}^{s}\) are the frequency-dependent satellite pseudorange and carrier phase hardware delays (biases), respectively. The satellite clock errors, which are available from the IGS through the precise clock products, are provided based on the ionosphere-free linear combination of code observations on L1 and L2 frequencies, i.e., P1 and P2 [10]. Employing the satellite clock errors provided by the IGS for the original code and carrier phase observations in the PPP model introduces an additional bias in the observations. To compensate for this bias, one must consider the satellite differential code biases (DCB), which are available from the IGS as well. Let us consider the precise satellite clock errors \(cdt^{s,IF}\) provided by the IGS as \(cdt^{s,IF}=cdt^{s}-b_{,IF(1,2)}^{s}\) where \(b_{,IF(1,2)}^{s}=\alpha_{IF}^{1,2}b_{,1}^{s}+\beta_{IF}^{1,2}b_{,2}^{s}\) is the satellite ionosphere-free code bias in which the coefficients \(\alpha_{IF}^{(ij)}=f_{j}^{s}/\left(f_{j}^{s}-f_{j}^{s}\right)\) and \(\beta_{IF}^{i,j}=1-\alpha_{IF}^{i,j}=-f_{i}^{s}/\left(f_{i}^{s}-f_{j}^{s}\right)\). The observation minus calculation (OMC) terms of the uncombined code measurement on frequency \(j\) can be written as follows:  
\[E\left(\Delta P_{r,j}^{s}+cdt^{s,IF}-\left(b_{,j}^{s}-b_{,IF(1,2)}^{s}\right) \right)={\rm{g}_{r}^{s}}^{T}\Delta x_{r}+T_{r}^{s}+\widetilde{cdt}_{r}+\gamma _{j}I_{r,1}^{s} \tag{10}\]  
where \(\widetilde{cdt}_{r}=cdt_{r}+b_{r,j}\), \(b_{,j}^{s}-b_{,IF(1,2)}^{s}\) is a function of satellite differential code bias \(DCB_{12}^{s}=b_{,1}^{s}-b_{,2}^{s}\) which is known (from the IGS products); \(g_{r}^{s}\) is the line-of-sight vector between the satellite and receiver; and \(\Delta x_{r}\) is the receiver position increment error. Similarly, for the carrier phase observation equation, we have  
\[E\left(\Delta\Phi_{r,j}^{s}+cdt^{s,IF}\right)={\rm{g}_{r}^{s}}^{T}\Delta x_{ r}+T_{r}^{s}+\widetilde{cdt}_{r}-\gamma_{j}I_{r,1}^{s}+\lambda_{j}\widetilde{N}_{r,j}^{s} \tag{11}\]  
where \(\lambda_{j}\widetilde{N}_{r,j}^{s}=\lambda_{j}N_{r,j}^{s}+B_{r,j}-b_{r,j}+B_{ j}^{s}-b_{,IF(1,2)}^{s}\). In the case of L1-only observations (\(j=1\)), the linearized observation equations given in Equations (10) and (11) can be written as follows:  
\[\left\{