## 2 Influences of Residual Topospheric Delays on Single Point Positioning  
Generally, the GNSS pseudo-range and carrier phase are generally expressed as [28]  
\[\left\{ \begin{array}{c}P_{r,f}^{s}=\rho_{r,f}^{s}+t_{r}-t^{s}+\alpha_{r}^{s} T_{z}+\beta_{f}I_{r}^{s}+b_{r,f}^{s}+\varepsilon_{P}\\ \Phi_{r,f}^{s}=\rho_{r,f}^{s}+t_{r}-t^{s}+\alpha_{r}^{s}T_{z}-\beta_{f}I_{r}^{s }+a_{r,f}^{s}-\lambda_{f}N_{r,f}^{s}+\varepsilon_{\Phi}\end{array} \right. \tag{1}\]  
in which, \(P_{r,f}^{s}\), \(\Phi_{r,f}^{s}\) are pseudo-range and carrier phase from receiver \(r\) to satellite \(s\) (\(s=1\),..., \(j\)) on frequency \(f\)(\(f=1\),..., \(k\)) in length units, respectively; \(\rho\) is the geometric distance,; \(t_{r}\) and \(t^{s}\) are the receiver and satellite clock error in length units, respectively; \(T_{z}\) is the ZTD that can be converted to slant with the mapping function \(\alpha\); \(I\) denotes the line-of-sight (LOS) total electron content (TEC) with the frequency dependent factor \(\beta_{f}\)= 40.3/\(f^{2}\); \(b_{r,f}^{s}\) and \(d_{r,f}^{s}\) are frequency-receiver-satellite dependent signal delays related to pseudo-range and carrier phase observables; \(N\) is the integer ambiguity. For single-epoch single-station point positioning, only pseudo-range observables are applied. Without specifying the frequency, the linearized observation model for positioning is [29]  
\[\mathbf{I}=\mathbf{A}\mathbf{x}+\mathbf{e}\text{ and var}(\mathbf{e})=\mathbf{D}_{\mathbf{e}\mathbf{e}} \tag{2}\]  
where \(\mathbf{I}=\mathbf{P}-\mathbf{\rho}_{0}\) is the observation residual vector, \(\mathbf{A}=(-\mathbf{u}\mathbf{I})\) is design matrix with \(\mathbf{u}\) being the LOS unit vector from the user to satellite; \(\mathbf{e}\) is the random error vector with covariance \(\mathbf{D}_{ee}\). The unknown parameters in conventional SPP model usually contains four elements, as \(\mathbf{x}=\left[ \begin{array}{cccc}\delta\mathbf{x}&\delta\mathbf{y}&\delta\mathbf{z}&t_{ r}\end{array} \right]^{\mathsf{T}}\), with \(\delta\mathbf{x}\), \(\delta\mathbf{y}\) and \(\delta\mathbf{z}\) are three coordinate component corrections and \(t_{r}\) is the receiver clock error. In SPP, the tropospheric delays are usually corrected by empirical models, and ionospheric delays are mitigated either by empirical models or by inter-frequency difference. In this paper, we mainly concentrate on the impacts of residual tropospheric delays on the SPP solutions after model correction. Therefore, the observation residual vectors in Equation (2) are specified as  
\[\left\{ \begin{array}{c}I_{0}=\mathbf{P}-\mathbf{\rho}_{0}-\mathbf{\alpha}T_{Z0}\\ I_{1}=\mathbf{P}-\mathbf{\rho}_{0}-\mathbf{\alpha}T_{Z}\end{array} \right. \tag{3}\]  
where \(T_{Z0}\) and \(T_{Z}\) are the model estimated and unknown true value of the station ZTD in meters, respectively; \(\mathbf{\alpha}\) is the elevation-related mapping coefficient vector. According to least-squares estimation principle, the corresponding unknown parameters are estimated as [29]  
\[\left\{ \begin{array}{c}\hat{\mathbf{x}}_{i}=D_{\hat{x}_{i}\hat{x}_{i}}\mathbf{A}^{ \mathsf{T}}D_{ee}^{-1}I_{i}\\ \text{var}(\hat{x}_{i})=D_{\hat{x}_{i}\hat{x}_{i}}=\left(\mathbf{A}^{\mathsf{T}}D_ {ee}^{-1}\mathbf{A}\right)^{-1}\end{array} \right.\qquad i=0,1 \tag{4}\]  
And the residual error vector of the observables are estimated as [29]  
According to Equations (4) and (5), it is concluded that the residual ZTDs only cause biases on the estimated parameters, and do not influence the estimated covariances. Therefore, corresponding biases on the estimated unknown parameters can be given as  
\[\delta\mathbf{\hat{x}}=\mathbf{\hat{x}}_{1}-\mathbf{\hat{x}}_{0}=\mathbf{D}_{\hat{x},\hat{x}_{1} }\mathbf{A}^{\text{T}}\mathbf{D}_{ee}^{-1}\mathbf{\kappa}\cdot dZTD \tag{6}\]  
where \(dZTD=(T_{\text{Z}}-T_{\text{Z0}})\) is the residual ZTD after model correction.