## 2 Model and Parameter Estimation  
### Positioning Function Model  
GNSS observations are affected by various errors related to hardware equipment, the propagation process, and observation environment. The observation models are given,regarding the mathematical relationship between observations, observation errors, and parameters [36; 37; 38]. At present, the widely used PPP models include ionosphere-free PPP and ionosphere-weighted PPP [29; 39; 40; 41; 42]. Each model has its own advantages. In this paper, a combination of ionospheric-free PPP models is used. The basic observation equations of the BDS ionosphere-free carrier phase and pseudo-range can be expressed as  
\[ \begin{split} P_{r,IF}^{s,T}=\rho^{s,T}+c\cdot dt_{r}-c\cdot dt^{s,T}+T_{r}^{s,T}+B_{r,P_{IF}}^{T}-B_{P_{IF}}^{s,T}+Re_{r}^{s,T}+Er_{r}^{s,T}+Pc _{r,IF}^{s,T}\\ +Pc_{r,IF}^{s,T}+Tid_{r}^{s,T}+e_{P_{IF}}^{s,T}\end{split}  \tag{1}\]  
\[ \begin{split} L_{r,IF}^{s,T}=\lambda_{IF}\Phi_{IF}^{s,T}=\rho^{s,T }+cdt_{r}-cdt^{s,T}+T_{r}^{s,T}+\lambda_{IF}(N_{IF}^{s,T}+B_{r,LH}^{s,T}-B_{L _{IF}}^{s,T}+W_{IF}^{s,T})+Re_{r}^{s,T}+Er_{r}^{s,T}\\ +Pc_{r,IF}^{s,T}+Pc_{r,IF}^{s,T}+Tid_{r}^{s,T}+e_{L_{IF}}^{s,T} \end{split}  \tag{2}\]  
where the subscript \(r\) and superscript \(s\) refer to the receiver and satellite, respectively; the subscript \(IF\) denotes ionosphere-free; the superscript \(T\) denotes the satellite system; \(P\) and \(L\) are the pseudo-range observations (m) and the phase observation (m), respectively; \(\rho\) is the geometric distance from the satellite to receiver (m); \(c\) is the speed of light in a vacuum (s); \(dt_{r}\) and \(dt^{s}\) are the receiver clock offset and satellite clock offset (m/s), respectively; \(T\) is the tropospheric delay on the signal propagation path (m); \(\lambda\) is the wavelength (m); \(\Phi\) is the phase observations in cycles; \(N\) is the ambiguity of the carrier phase; \(B_{r}\) and \(B^{s}\) are the pseudo-range hardware delays at the receiver and satellite sides (m), respectively; \(W\) is the phase wind-up error in cycles; \(Re\) is the relativistic effect caused by satellite track eccentricity (m); \(Er\) is the error caused by the Earth's rotation; \(Pco\) and \(Pcv\) are the phase center offset (PCO) and phase center variation (PCV) of the receiver and satellite; \(Tid\) is the earth tide error; and \(\varepsilon\) are the unmodeled errors (multipath, noise). \[\rho=\sqrt{\left(x_{r}-x^{s}\right)^{2}+\left(y_{r}-y^{s}\right)^{2}+\left(z_{ r}-z^{s}\right)^{2}} \tag{3}\]  
where \(\left[x_{r},y_{r},z_{r}\right]\) and \(\left[x^{s},y^{s},z^{s}\right]\) are the coordinates of the receiver and satellite respectively. Then, linearize \(\rho\) at its approximate coordinate \(\left[x_{0},y_{0},z_{0}\right]\), which can be written as  
\[\rho=\rho_{0}+\frac{x_{0}-x^{s}}{\rho_{0}}dx+\frac{y_{0}-y^{s}}{\rho_{0}}dy+ \frac{z_{0}-z^{s}}{\rho_{0}}dz \tag{4}\]  
where \(\rho_{0}\) is the geometric distance from the satellite to the receiver's approximate coordinates. Observations are used to establish the PPP model, but the accuracy of the tropospheric model does not meet the requirements of PPP. Hence, the residual part of the zenith tropospheric wet delay is normally estimated together with station coordinates and carrier phase ambiguity. However, other error corrections, such as PCO, PCV, phase wind-up, relativistic effect, etc., are corrected by corresponding methods. The unknown parameters are the receiver coordinates and clock errors, zenith tropospheric wet delay, and phase ambiguity. Therefore, the observation equation of PPP is formed as:  
\[\left[ \begin{array}{ccccc}P_{1}-\rho_{1}-D_{P1}\\ L_{1}-\rho_{1}-D_{L1}\\ \vdots\\ P_{n}-\rho_{n}-D_{p_{n}}\\ L_{n}-\rho_{n}-D_{Ln}\end{array}