## 3 Multi-GNSS PPP Mathematical Model
### Parameter Estimation Model Based on Robust Kalman Filter  
The Kalman filter is the most commonly used method for parameter estimation in multi-GNSS positioning. The extended Kalman filter (EKF) is a nonlinear version of Kalman filtering; the dynamic and observation models can be expressed as:  
\[X_{k+1}=\Phi_{k+1,k}\hat{X}_{k}+w_{k} \tag{15}\]  
\[L_{k+1}=\widetilde{L}_{k}+H_{k+1}(X_{k}-\widetilde{X}_{k})+v_{k} \tag{16}\]  
where \(X_{k+1}\) is state vector, \(L_{k+1}\) is observation vector, \(\Phi_{k+1,k}\) is state transition matrix, and the vectors \(w_{k}\) and \(v_{k}\) are zero mean Gaussian white sequences having zero cross-correlation with each other [30]:  
\[E[w_{k}w_{i}^{T}]=\left\{\begin{array}{cc}Q_{k},i=k\\ 0,i\
eq k\end{array}\right.;\qquad E[v_{k}v_{i}^{T}]=\left\{\begin{array}{ cc}R_{k},i=k\\ 0,i\
eq k\end{array}\right.;\qquad E[w_{k}v_{i}^{T}]=0 \tag{17}\]  
where \(Q_{k}\) is the process noise covariance matrix and \(R_{k}\) is the measurement noise covariance matrix. With the GPS/BDS/GLOASS priori covariance:  
\[R_{k}=\sigma_{S}^{2}P_{S}^{-1}=\left[\begin{array}{ccc}\sigma_{G}^{2}P_{G}^ {-1}&0&0\\ 0&\sigma_{B}^{2}P_{B}^{-1}&0\\ 0&0&\sigma_{R}^{2}P_{R}^{-1}\end{array}\right] \tag{18}\]  
where the subscript \(S\) denote satellite system, the subscript \(G\), \(B\), and \(R\) denote GPS, BDS, and GLONASS, respectively, \(\sigma_{S}^{2}\) is the measurement error covariance, and \(P_{S}\) is the weight matrix. Although it has been very common to add fictitious process noise to the system model, the best cure for nonconvergence caused by unmodeled states is to correct the model [30].  
To improve the stability of PPP using a smartphone, the robust Kalman filter method is used for parameter estimation, which reduces the effects of observation outliers on positioning accuracy.  
According to the prior weight elements of the observation vector and the robust M estimation principle, the equivalent covariance matrix of observations error \(\overline{R}_{k}\) instead of the original covariance \(R_{k}\) for the robust Kalman filter, and the formula of robust Kalman gain matrix \(\overline{K}_{k}\) follows [31]:  
\[\overline{K}_{k}=P_{k}^{-}H_{k}^{T}(H_{k}P_{k}^{-}H_{k}^{T}+\overline{R}_{k})^{ -1} \tag{19}\]  
where \(P_{k}^{-}\) is the state covariance matrix and \(H_{k}^{T}\) is the coefficient matrix. The equivalent covariance matrix \(\overline{R}_{k}\) can be calculated as [32]:  
\[\overline{R}_{k}=R_{k}/\gamma_{i} \tag{20}\]  
where \(\gamma_{i}\) is the variance inflation factor, which can be used to adjust the variance of observations by the institute of geodesy and geophysics (IGG) III weighting function [33].  
\[\gamma_{i}=\left\{\begin{array}{cc}1&\left|\overline{v}_{i}\right|\leq k_{0 }\\ \frac{k_{0}}{|\overline{v}_{i}|}\left(\frac{k_{1}-|\overline{v}_{i}|}{k_{1}-k _{0}}\right)^{2}&k_{0}<|\overline{v}_{i}|\leq k_{1}\\ 0&\left|\overline{v}_{i}\right|>k_{1}\end{array}\right. \tag{21}\]  
where \(k_{0}\) and \(k_{1}\) are two thresholds, usually chosen as 1.5-3.0 and 3.0-8.0, respectively; \(\overline{v}_{i}\) is the standardized residual, which is defined as [34]:  
\[\overline{v}_{i}=\frac{v_{i}}{\sqrt{\delta_{0}^{2}Q_{vi}}} \tag{22}\]  
where \(v_{i}\) is the observations residual and \(Q_{vi}\) is the corresponding variance. \(\delta_{0}^{2}\) is the estimate of unit weight variance, which can be calculated with generalized least squares principle:  
\[\delta_{0}^{2}=\frac{\xi^{T}Q_{\xi}^{-1}\xi}{n} \tag{23}\]  
where \(\xi\) is predicted residual vector (innovations) and \(n\) is the number of observations; \(Q_{\xi}\) is the corresponding covariance matrix, which can be calculated as follows:  
\[Q_{\xi}=R_{k}+H_{k}P_{k}^{-}H_{k}^{T} \tag{24}\]  
The update state correction vector and error covariance are:  
\[\hat{X}_{k}=\hat{X}_{\overline{k}}+\overline{K}_{k}(L_{k}-H_{k}\hat{X}_{ \overline{k}}) \tag{25}\]  
\[P_{k}=[I-\overline{K}_{k}H_{k}]P_{k}^{-}[I-\overline{K}_{k}H_{k}]^{T}+ \overline{K}_{k}\overline{R}_{k}\overline{K}_{k}^{T} \tag{26}\]