## 2 Methods
### Measurement Model for Pseudorange, Doppler Shift, and TDCP  
The models for GNSS measurements, including pseudorange, Doppler shift, and TDCP, which are related to the state of a vehicle are considered in the SPP mode in this study. The models of pseudorange and pseudorange rate measurement are well-documented in the literature, which can be found in [38]. The matrix form of the fundamental observations corresponding to the designed EKF can be expressed as,  
\[\mathbf{H}_{\text{\emph{code}}}=\left[ \begin{array}{cccc}e^{i}(t_{k})&0_{n \times 3}&0_{n\times 3}&1_{n\times 1}&0_{n\times 1}&0_{n\times 1}\end{array}  \right]_{n\times 12} \tag{4}\]  
\[\mathbf{H}_{\text{\emph{dappler}}}=\left[ \begin{array}{cccc}0_{n\times 3}&e^{i}(t_{k})&0 _{n\times 3}&0_{n\times 1}&1_{n\times 1}&0_{n\times 1}\end{array} \right]_{n\times 12} \tag{5}\]  
\[\mathbf{H}=\left[ \begin{array}{c}\mathbf{H}_{\text{\emph{code}}}\\ \mathbf{H}_{\text{\emph{dappler}}}\end{array} \right]_{2n\times 12} \tag{6}\]  
\[\delta\mathbf{z}=\left[ \begin{array}{c}\overline{\rho}^{i}-\rho^{i}\\ \overline{\rho}^{i}-\dot{\rho}^{i}\end{array} \right]_{2n\times 1}=\mathbf{H} \delta\mathbf{\hat{x}}+\mathbf{\varepsilon}_{2n\times 1} \tag{7}\]  
\[\mathbf{R}_{\text{\emph{code}}/\text{\emph{dappler}}}=\sigma_{0}^{2}\left[ \begin{array}{cccc}\frac{1}{sin^{2}(e^{i1})}&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&\frac{1}{sin^{2}(e^{i})}\end{array} \right]_{n\times n} \tag{8}\]  
with  
\[e^{i}(t_{k})=\frac{1}{\rho^{i}(t_{k})}\Big{[} \begin{array}{cccc}\left(t_{u, x}^{\prime}(t_{k})-t_{s,x}^{\prime,i}(t_{k})\right)&\left(t_{u,y}^{\prime}(t_{k})-t_{s,y}^ {\prime,i}(t_{k})\right)&\left(t_{u,z}^{\prime}(t_{k})-t_{s,z}^{\prime,i}(t_{ k})\right)\end{array} \right]_{1\times 3}\]  
where \(\mathbf{H}_{\text{\emph{code}}}\), \(\mathbf{H}_{\text{\emph{dappler}}}\), \(\overline{\rho}^{i}\), \(\rho^{i}\), \(\overline{\dot{\rho}}^{i}\), \(\dot{\overline{\rho}}^{i}\), \(\dot{\overline{\rho}}^{i}\), \(\mathbf{R}_{\text{\emph{code}}/\text{\emph{dappler}}}\), \(\sigma_{0}^{2}\), \(e^{i}\), and \(\dot{e}^{i}(t_{k})\) denote the design matrices of the pseudorange and pseudorange rate, the observed pseudorange, geometric range between the receiver and the satellite, the observed pseudorange rate, geometric pseudorange rate, weighting matrix of the pseudorange/pseudorange rate measurement, a priori variance factor, the elevation angle of the \(i^{th}\) satellite with respect to the rover, and LOS unit vector from the satellite to the receiver at \(t_{k}\), respectively. The design of the weighting matrix of pseudorange rate is similar to \(\mathbf{R}_{\text{\emph{code}}}\), but with a different \(\sigma_{0}^{2}\). The TDCP measurements, which are the delta-phase measurements from the same satellites between consecutive epochs, are conventionally used to estimate the average velocity [39; 40; 41; 42; 43]. In other words, the difference between the previous and current positions with mm-level noise can be observed through such phase differences. It could be partially observable if fewer than four satellites are continuously available in the absence of cycle slip. The advantage of this approach over conventional techniques is that carrier-phase measurements from the same satellites are merely available since the previous time epoch, rather than several seconds, to average out the code multipath error [5; 37]. The original measurement equation of the carrier phase is expressed as follows:  
\[\lambda\Phi^{i}=\rho^{i}+c\Big{(}\delta b_{u}-\delta b_{s}^{i}+\delta d_{ \text{\emph{eph}}}^{i}-\delta d_{\text{\emph{iono}}}^{i}+\delta d_{\text{ \emph{trap}}}^{i}\Big{)}+\lambda N^{i}+\dot{\eta}^{i} \tag{9}\]  
where \(\lambda\), \(\Phi^{i}\), and \(N^{i}\) represent the wavelength, measured carrier phase in cycles, and integer ambiguity, respectively. The term \(\eta^{i}\) includes a multipath/NLOS-contaminated source and receiver noise for the carrier phase.