## 2 Materials and Methods
### A-VTTCES-KF Algorithm
#### 2.1.1 Preliminary Results and Analysis through Stationary Field Tests
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(B_{pll}\) [Hz] & **VT** LR [Hz] & \(\bar{x}_{GM}\) **[s]** & \(\bar{y}_{GM}\) **[s/s]** & \(q_{c}\) **[s/s]** & \(T_{c}\) **[s]** \\ \hline
5 & 20 & 0.130 & \(2.34\times 10^{-10}\) & \(2.04\times 10^{-9}\) & 0.069 \\  & 10 & 0.170 & \(1.92\times 10^{-10}\) & \(1.46\times 10^{-9}\) & 0.090 \\  & 5 & 0.200 & \(1.77\times 10^{-10}\) & \(1.25\times 10^{-9}\) & 0.10 As listed in Tables 1 and 2, it can be found that the parameters in terms of the same VT settings in two situations are very close to each other such that they can be exploited to build up a model to reduce the Markov process or time-correlated error disturbance on the VT tracking loop. The way to build up the system model will be subsequently explained. On the other hand, it can be noticed that, as \(\eta=1000\) ms and \(B_{pll}=1\) Hz, there are some other unexpected noise sources added into the Allan deviation plot of the VT-based clock drift error estimations. Since the dynamic stress on the tracking loop may be not adequately removed by the very slow feedback rate in terms of the LOS Doppler compensation combined with the user's and the SV's velocities. In this case, the Allan deviation curve might be dramatically contaminated by the dynamic stress residuals. However, analysis for this type of noise term is not our contribution in this research, it will not be discussed in this paper. * Clock modelling based on Allan variance The curve of the timing stability for the local clock integrated in the front-end for the GNSS SDR is plot in Figure 7 by which the clock drift characteristics can be studied and depicted [63]. The test spot related to Situation #3 is given in Figure 10 in Section 3. Referring to Figure 7, the coefficient of the clock drift random walk can be estimated by (A8) with \(\tau=1\), and the numerical value is approximated as \(7.675\times 10^{-11}\); the coefficient related to the rate of the clock drift random walk will be obtained from (A10) with \(\tau=3\), and the numerical approximation is estimated as \(1.069\times 10^{-10}\). The flicker noise terms related to the flat segment will be not contained in the subsequent system model for the KF, and this omission is reasonable referring to previous researches [66]. It has been studied through previous mathematical analysis and experimental implementations, that the PSD related to the noise with the highly stable frequency terms can be expressed by a power-law noise model as follow [71; 72]  
\[S_{y}\left(f\right)=\sum_{a=-2}^{+2}h_{a}f^{\alpha} \tag{2}\]  
subject to \(0\leq f\leq f_{h}\), where the PSD varies with the variable of \(f\) which is corresponded to the signal power, and \(f_{h}\) stands for the upper bound of the cutoff frequency; \(h_{a}\) is an amplitude coefficient [73] by which the range of \(f\) can be constrained; every separate numerical value of \(\alpha\) corresponds to an  
Figure 7: Allan variance for the clock drift based on ST towards Situation #3.  
existing noise source in the signal. The noise sources of white frequency and random walk frequency, which corresponds to \(\alpha=0\) and \(\alpha=-2\), respectively, will be subsequently taken into consideration for the receiver clock noise modelling. These two noise sources are more than enough to afford a stochastic model of the receiver clock [66]. Table 3 summaries the Allan variance results related to different types of coefficients. In Table 3, the coefficient for the white frequency term in the noise of the clock drift is estimated as \(Q=7.675\times 10^{-11}\), while the coefficient for the random walk frequency term can be approximated as \(K=1.069\times 10^{-10}\), as illustrated in Figure 1:0 Under this circumstance, the amplitude coefficients in terms of the power-law model can be attained as  
\[\left\{ \begin{array}{l}\hat{h}_{0}=2Q^{2}\approx 1.178\times 10^{-20}\\ \hat{h}_{-2}=\frac{K^{2}}{2\pi^{2}}\approx 5.789\times 10^{-22}\end{array} \right.