## 2 Materials and Methods
### A-VTTCES-KF Algorithm
#### 2.1.1 Preliminary Results and Analysis through Stationary Field Tests
* VT time-correlated error modelling based on Allan variance As mentioned earlier, two groups of the stationary experiments have been done to help identify and quantify the characteristics of the VT loop in terms of the time-correlated error. The estimation series of the clock drift can be estimated by the Doppler measurements through the WNLS method. The tests are carried out with different feedback modes. For example, the VT feedback intervals which range from 50 ms to 1000 ms are combined with different scales of the bandwidths for the 3 rd-order phase lock loop (PLL), i.e., 1 Hz, 3 Hz, and 5 Hz. The feedback interval is the reciprocal of the LR as mentioned earlier. The results of the clock drift estimations related to Allan variance or Allan deviation for Situation #1 and Situation #2 are illustrated in Figures 5 and 6, respectively. Some annotations have been depicted in the legends of these two figures as the format of \(\tau_{l}\)-\(B_{pll}\), where \(\tau_{l}\) denotes the VT feedback interval in millisecond, and \(B_{pll}\) stands for the bandwidth of the PLL in Hertz; st in the legends means that the ST mode is activated in the test. Figure 4: Scenarios for stationary field tests in Situation #2 including testing spot (**left**) and sky plot for of the SVs (**right**). Figure 3: Scenarios for stationary field tests in Situation #1 including testing spot (**left**) and sky plot for of the SVs (**right**). Comparing the curves out of the ST loop with the ones from all the tested VT loops shown in Figures 5 and 6, it should be noted that there is a distinctive segment appearing in the VT curves. Figure 5: Plot of the square root of Allan variance related to the clock drift estimations towards Situation #1. Numbers in the legends correspond to the format as \(\tau_{l}\)-\(B_{pll}\), where \(\tau_{l}\) denotes the VT feedback interval in millisecond, and \(B_{pll}\) stands for the bandwidth of the PLL in Hertz; st means that the ST mode is activated in the test. Figure 6: Plot of the square root of Allan variance related to the clock drift estimations towards Situation #2. Numbers in the legends correspond to the format as \(\tau_{l}\)-\(B_{pll}\), where \(\tau_{l}\) denotes the VT feedback interval in millisecond, and \(B_{pll}\) stands for the bandwidth of the PLL in Hertz; st means that the ST mode is activated in the test. The shape of this sort of segment is very close to the curve in Figure A3. Such similarity implies that the VT loop brings in a certain time-correlated error which is coupled in the clock drift estimations of the GNSS receiver. As we know, the performance of the tracking loop based on the VT technique is highly dependent on the quality of the clock drift estimations, and it can be inferred that the accuracy of the PVT solution for the receiver will certainly be reduced by this sort of the time-correlated disturbance error. Referring to the curves in Figures 5 and 6, and (A13), the parameters for Markov process in different cases of the VT architectures can be approximated as the numerical values in Tables 1 and 2 which are summarized for Situation #1 and Situation #2, respectively; \(\bar{x}_{GM}\) and \(\bar{y}_{GM}\) represent the time cluster and Allan deviation with respect to the peak of the Allan deviation plot, respectively, and they can be approximately read from the given plots; \(q_{c}\) and \(T_{c}\) can be referred to (A11) as well as the descriptions below for detailed information. Furthermore, \(q_{c}\) and \(T_{c}\) can be calculated through \(\bar{x}_{GM}\) and \(\bar{y}_{GM}\) estimations based on (A13).