## 2 Methods
### Measurement Model for Pseudorange, Doppler Shift, and TDCP
The difference between the carrier-phase measurements at consecutive epochs \(t_{k}\) and \(t_{k-1}\) is described as follows:  
\[\lambda\Delta\Phi^{i}=\lambda\cdot\Big{[}\Phi^{i}(t_{k})-\Phi^{i}(t_{k-1}) \Big{]}=\Delta\rho^{i}+c\Big{(}\Delta\delta b_{u}+\Delta\delta b_{s}^{i}+ \Delta\delta d_{\text{\emph{eph}}}^{i}-\Delta\delta d_{\text{\emph{iono}}}^{i }+\Delta\delta d_{\text{\emph{trap}}}^{i}\Big{)}+\Delta\eta^{i} \tag{10}\]  
where \(\Delta\) indicates the differencing operation. Under the condition of no cycle slip, the integer ambiguity is eliminated by time differencing. Other common error sources, which include satellite clock bias, ephemeris error, tropospheric error, and ionospheric error, vary slowly within a sampling, and are, \(\Delta\delta b_{s}^{i}\), \(\Delta\delta d_{\text{\emph{eph}}}^{i}\), \(\Delta\delta d_{\text{\emph{iono}}}^{i}\), and \(\Delta\delta d_{\text{\emph{trap}}}^{i}\) are negligibly small. Notably, the transition between the adjacent ephemeris sets causes the discontinuity of \(\Delta\delta d_{\text{\emph{eph}}}^{i}\), especially for \(\Delta\delta b_{s}^{i}\), when different sets are respectively selected between consecutive epochs for TDCP measurements [43]. Selecting the same ephemeris can prevent this problem during the transition. In transforming (10) into a relation of receiver position change, \(\Delta\rho^{i}\) has to be recombined as [43],  
\[\Delta\rho^{i}=\Delta D-\Delta g-\left[e^{i}(t_{k})\times\Delta\mathbf{r}_{u}\right] \tag{11}\]  
with  
\[\Delta g=\left[e^{i}(t_{k})\times\mathbf{r}_{u}^{e}(t_{k-1})\right]-\left[e^{ i}(t_{k-1})\times\mathbf{r}_{u}^{e}(t_{k-1})\right],\]  
where \(\mathbf{r}_{u}^{e}\) and \(\mathbf{r}_{s}^{e,i}\) are the receiver position vector and satellite position vector, respectively, including the \(x\), \(y\), and \(z\) components in the e-frame. The term \(\Delta g\) reflects the orientation change of the LOS vector in the relative satellite-receiver geometry, and \(\Delta D\) represents a change in the range and is proportional to the average Doppler shift caused by the relative satellite-receiver motion along the LOS vector [44]. The derivation of (11) is presented in detail in [39; 43]. By substituting (11) into (10), the TDCP measurement equation becomes (12), and the corresponding measurement model is expressed by (13) and (14). The weighting matrix of the TDCP measurements, \(\mathbf{R}_{\text{TDCP}}\), is also similar to the design of (17) with a different \(\sigma_{0}^{2}\). \[\lambda\Delta\Phi^{i}-\Delta D+\Delta g=-\left[e^{i}\Big{(}t_{j}\Big{)}\times \Delta\mathbf{r}_{u}^{e}\right]+c\Delta\delta b_{u}+\Delta\eta^{i} \tag{12}\]  
\[\mathbf{H}_{\text{TDCP}}=\left[ \begin{array}{ccccc}&e^{i}(t_{k})&0_{1\times 3 }&-e^{i}(t_{k-1})&1&0&-1\\ \end{array} \right]_{m\times 12} \tag{13}\]  
\[\delta\mathbf{z}_{\text{TDCP}}=\left[\Delta\Phi^{i}-\Delta\rho^{i}\right]_{m \times 1} \tag{14}\]  
The effectiveness of TDCP is affected by cycle slip. The cycle-slip detection method applied in this study is \"Doppler integration\" for a single-frequency receiver. This method uses an observed Doppler shift to predict the growth in a carrier phase observation as an indicator of a cycle slip, whose fundamental equation is presented in [38]. Nonetheless, due to the vehicular dynamics and the environment, the signal tracking quality of Doppler shift strongly affects the accuracy of detection. When a cycle slip occurs with false detection, new ambiguities are introduced in delta-phase measurements. On the other hand, the path delay of the carrier phase measurements due to NLOS reception could be similar to the pseudorange error, as mentioned previously in [8]. These errors could be introduced into carrier phase ambiguities and further degrade GNSS positioning if the phase-smoothing approach is applied. In this situation, standalone GNSS positioning cannot satisfy the requirement of accuracy and reliability in vehicular navigation.