## 2 Methodology of Real-Time PPP for the Smartphone  
Real-time PPP of the smartphone refers to the use of real-time orbit and clock offset products for high-precision positioning based on pseudorang and carrier-phase observations measured by the smartphone and combined with a tight error correction model [21; 22]. The real-time multi-GNSS undifferenced and uncombined PPP model is shown in Equation (1).  
\[\left\{\begin{aligned} & P_{r,j}^{s,Q}=\rho_{r}^{s,Q}+c\left(dt_{r}-dt^{s,Q} \right)+T_{r}^{s,Q}+I_{r}^{s,Q}+c\left(d_{r,j}^{s,Q}-d_{j}^{s,Q}\right)+\varepsilon _{P_{r,j}}^{s,Q}\\ & L_{r,j}^{s,Q}=\rho_{r}^{s,Q}+c\left(dt_{r}-dt^{s,Q}\right)+T_{r }^{s,Q}-I_{r}^{s,Q}+\lambda_{j}^{s,Q}\left(N_{r,j}^{s,Q}+b_{r,j}^{s,Q}-b_{j}^{s,Q}\right)+\varepsilon_{L_{r,j}}^{s,Q}\end{aligned}\right. \tag{1}\]  
where \(s\) and \(Q\) denote the satellite pseudo-random noise (PRN) and system, in which G, C, R and E represent GPS, BDS, GLONASS and Galileo systems, respectively. \(r\), \(j\) are the receiver and observation frequencies, respectively. \(P_{r,j}^{s,Q}\) and \(L_{r,j}^{s,Q}\) express pseudorang and carrier observations, respectively. \(\rho_{r}^{s,Q}\) shows the geometric distance between the station and the satellite. \(c\) is the speed of light propagation in a vacuum; \(dt_{r}\) and \(dt^{s,Q}\) signify the receiver and the satellite clock offset, respectively. \(T_{r}^{s,Q}\) and \(I_{r}^{s,Q}\) denote tropospheric and ionospheric delay, respectively. \(d_{r,j}^{s,Q}\) and \(d_{j}^{s,Q}\) are the pseudorange hardware delay at the receiver and satellite side, respectively. \(\lambda_{j}^{s,Q}\) represents the carrier phase wavelength. \(N_{r,j}^{s,Q}\) denotes the carrier phase ambiguity. \(b_{r,j}^{s,Q}\) and \(b_{j}^{s,Q}\) express the phase hardware delay at the receiver and the satellite, respectively. \(\varepsilon_{P_{r,j}}^{s,Q}\) and \(\varepsilon_{L_{r,j}}^{s,Q}\) are the sum of observation noise and multipath and other unmodelled errors for pseudorange observations and carrier phase observations, respectively.  
In this contribution, for single-frequency PPP, the undifferenced and uncombined model of the smartphone is applied to estimate the station coordinates, and the ionospheric delay is also estimated [23; 24], while for the dual-frequency PPP of the smartphone, the ionosphere-free combination PPP model is employed [25]. The parameters that should be estimated include station coordinates, receiver clock offset, tropospheric wet delay and ambiguities [26; 27; 28; 29]. Smartphone real-time PPP calculation includes data preprocessing module, error correction module and parameter estimation module. The specific solution process is shown in Figure 1. The detailed processing strategies for smartphone real-time PPP are listed in Table 1.  
Figure 1: Flow chart of real-time PPP solution with smartphones.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Items** & **Strategy/Model** \\ \hline \multirow{5}{*}{Observations} & \multirow{5}{*}{Processing models} & Single frequency: \\  & & GPS: L1; BDS: B11; Galileo: E5a \\  & & Dual frequency: GPS: L1/L5 \\  & Sampling interval & 1s \\  & Cut-off angle & \(15^{\circ}\) \\  & Tropospheric delay & Saastamoinen model and GMF function \\  & Satellite hardware delay & CODE products \\  & Satellite PCO and PCV & igs14.atx \\  & Satellite orbit/clock deviation & GFZ real-time product \\ Correction & Phase windup & Model correction \\  & Relativistic effect & Model correction \\  & Earth rotation & Model correction \\  & Station coordinates & Static: constant; Kinematic: white noise \\  & Tropospheric delay & Random walk \\  & Receiver clock offset & White noise \\ Parameter estimation & ambiguity & Constant (float solution) \\  & Inter-system bias & Random walk \\ \hline \hline \end{tabular}
\end{table}
Table 1: Real-time PPP of the smartphone processing strategy.