## 2 Method
### The Observation Model  
In the time community, the equation of the ionosphere-free (IF) pseudo range and carrier-phase observations is linear, and can be written as follows [3]:  
\[\left\{\begin{array}{ll}P_{r,IF}^{S}&=\rho_{r}^{S}+c\cdot dt_{r}-c\cdot dt^{S}+ mf_{w}(e)\cdot Z_{w}+(d_{r,IF}^{S}-d_{IF}^{S})+\varepsilon_{r,IF}^{S}\\ L_{r,IF}^{S}&=\lambda_{IF}^{S}\cdot\Phi_{r,IF}^{S}\\ &=\rho_{r}^{S}+c\cdot dt_{r}-c\cdot dt^{S}+mf_{w}(e)\cdot Z_{w}+\lambda_{IF}^{S} \cdot(N_{r,IF}^{S}+b_{r,IF}^{S}-b_{IF}^{S})+\varepsilon_{r,IF}^{S}\end{array}\right. \tag{1}\]  
where \(r\) indicates the receiver, \(S\) indicates the satellite; \(P_{r,IF}^{S}\) and \(L_{r,IF}^{S}\) denote the pseudo-range and the phase observables in metres, respectively; \(\lambda_{IF}^{S}\) is the carrier wavelength when ionosphere-free; \(\Phi_{r,IF}^{S}\) is the original carrier phase measurement in cycles; \(\rho\) is the geometric distance in metres; \(c\) denotes the speed of light; \(dt^{S}\) and \(dt_{r}\) are the clock offsets of the satellite and receiver in seconds; \(e\) is the elevation angle of the satellite \(S\); \(mf_{w}(e)\) refers to wet mapping functions [19]; \(Z_{w}\) represents the zenith wet delay; \(d_{r,IF}^{S}\) and \(d_{IF}^{S}\) are the uncalibrated code delays (UCDs) of the IF model at the receiver and satellite end, respectively; \(N_{r,IF}^{S}\) is an integer itself but is estimated as a floating point number from unmodeled effects; \(b_{r,IF}^{S}\) and \(b_{IF}^{S}\) are the uncalibrated phase delays (UPDs) of the IF model at the receiver and the satellite ends in cycles; and \(\varepsilon_{r,IF}^{S}\) and \(\xi_{r,IF}^{S}\) denote the sum of the measurement noise and the multipath error for the pseudo range and carrier phase observations, respectively. Since \(d_{IF}^{S}\) is absorbed by the satellite clock [20], the clock offset of the receiver \(d\overline{t}_{r}\) can be written as follows [8]:  
\[d\overline{t}_{r}=dt_{r}+d_{r,IF}^{S} \tag{2}\]  
The relativistic effects, the Sagnac effect, tidal loadings, the slant hydrostatic delay, satellite and receiver antenna phase centre offsets and variations (PCO and PCV) [21; 22], and the phase wind-up must be corrected with an experimental model [5], even though they are not included in (1). Detailed descriptions are listed in Table 1.  
The least square method is a type of mathematical optimization technology. The optimal result is obtained with the constraint of the minimum square sum of the residuals. The corresponding normal equation is used to estimate the unknown parameters as follows:  
\[A^{T}PA.\hat{X}=A^{T}PL \tag{3}\]  
where \(\hat{X}\) denotes the vector of all estimated parameters, including the receiver clock offset, tropospheric zenith delay and phase ambiguity; \(A\) is the corresponding design matrix; \(P\) is the weight matrix, which  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Item** & **Strategies** \\ \hline Estimator & Kalman filtering \\ \hline Relativistic effect & IERS conventions 2010 \\ \hline Sagnac effect & IERS conventions 2010 \\ \hline Phase wind-up & Corrected by model \\ \hline Tide displacements & IERS conventions 2010 \\ \hline \multirow{4}{*}{Tropospheric delay} & ZHD: corrected with global pressure and temperature \\  & (GPT) model using the formulas of [PERSON] \\ \cline{2-3}  & ZWD: estimated as a continuous piecewise linear function \\ \cline{1-1}  & (2 h parameter spacing), GMF mapping function \\ \hline Satellite antenna PCOS and PCVs & IGS “**.atx” file \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of GPS PPP (global positioning system precise point positioning) processing strategies.  
is determined by the elevation-dependent model; and \(L\) is the difference between the observed quantity and the computed quantity. The equation can be rewritten in the following simple format:  
\[N\cdot\hat{X}=b \tag{4}\]  
where \(N\) denotes \(A^{T}PA\) and \(b\) denotes \(A^{T}PL\). The least square method is applied in the adaptive receiver clock prediction model. We introduce the clock model in the next subsection.