*4.2. Analysis of ISB for Different Station Attributes*

From the Equation (10), we know that the ISB contains two components, the time difference of two receiver clock offsets with different GNSS observations, and the difference in the receiver hardware delays for two GNSS systems. For the latter, the receiver type, antenna, and frequency standard are the important factors to affect the receiver hardware delays. For further analyses of the relationship between the station attribute and ISB stability, the stations shown in Table 3 were regrouped into five comparative schemes according to three indicators, namely receiver type, antenna, and frequency standard. Table 3 shows the comparative strategies and corresponding stations, where • means the same, and ż means different. Further, to alleviate the effect of systematic bias in the results from different stations of three analysis centers on ISB stability analysis, the ISB difference of stations is more focused, which is also related to the calibration of the time-transfer links.


**Table 3.** Comparative schemes and corresponding stations.

As 10 stations were involved in the 20 d experimental period, we randomly used the results of DOY 101, 106, 111, 115, and 119 of 2021 in the five comparison schemes, as shown in Figures 3–7. To clearly plot and compare the receiver type, antennas, and frequency standards in the figures, we used abbreviations, namely "Rec", "Ant", and "Fre".

**Figure 4.** ISB difference series for scheme 1 on DOY 101, 2021.

**Figure 5.** ISB difference series for scheme 2 on DOY 106, 2021.

Figure 4 shows the ISB difference series for scheme 1, using the same receiver type, antenna, and frequency standard for SEPT POLARX5, AOAD/M\_T, and H-MASER, respectively. One can see that the variation trends of ISB\_com, ISB\_wum, and ISB\_gbm agree very well. The standard deviation (STD) values are all 0.05 m for the three analysis center products. The ranges between the minimum and maximum are 0.2 m.

Figure 5 shows the ISB difference series for scheme 2, which uses the same type of antenna and frequency standard, but different receiver types for one time-transfer link. The left panel shows the time link of station NYAL-ONSA, with the same type of antenna AOAD/M\_B and frequency standard H-MASER. The right panel shows station BOR1-SPT0, with the same type of antenna TRM59800.00 and frequency standard H-MASER. The ISB difference series of ISBcom and ISBwum agree relatively well, whereas the ISBgbm difference series shows obvious bias compared with that of ISBcom and ISBwum for the two timetransfer links. The bias for NYAL-ONSA is approximately 0.07 m and that for BOR1-SPT0 is approximately 0.12 m. The standard deviation of divergence is approximately 0.014 m

and 0.013 m, respectively. The above analyses show that the ISB difference series has a close relationship with the type of receiver of the different analysis center products.

**Figure 6.** ISB difference series for scheme 3 on DOY 111, 2021.

**Figure 7.** ISB difference series for scheme 4 on DOY 115, 2021.

Figure 6 shows the ISB difference series for scheme 3, which uses the same type of receiver and frequency standard, but a different antenna type. This scheme is similar to scheme 2, and the difference series of ISBcom and ISBwum show good agreement, although the antenna type differs. Obvious bias exists between ISBgbm, ISBcom, and ISBwum, with the corresponding values ranging from 0.1 m to 0.25 m. This finding indicates that the ISB difference series has a certain relationship with the antenna type for the three analysis centers' satellite orbits and clock products. Compared with ISBcom, the mean value of divergence is 0.010 m for ISBwum and 0.167 m for ISBgbm, and the standard deviations are 0.013 m and 0.050 m.

Figure 7 shows the ISB difference series for scheme 4, which uses the same type of receiver and antenna, but a different frequency standard. The left panel is the time link of station BOR1-TRO1, with the same type of receiver TRIMBLE NETR9 and antenna TRM59800.00. The right panel is station KIRU-GOP6, with the same type of receiver SEPT POLARX5 and antenna SEPCHOKE\_B3E6. The variations in the ISB difference series for the ISBcom and ISBwum solutions are in extremely good agreement. Although the general trend of ISBgbm is somewhat similar, the divergence between ISBwum and the former two solutions does not show constant bias, but indicates significant variation for the two time-transfer links.

Figure 8 shows the ISB difference series for scheme 5, which uses different receivers, antennas, and frequency standards. Although the general trend is somewhat similar, the divergence among ISB\_com, ISB\_wum, and ISB\_gbm shows not only obvious bias but also the variation term. As indicated by the analyses and discussions of the five schemes, the ISB difference of ISB\_com and ISB\_wum agrees well. The mean value of divergence is 0.023 m for ISB\_wum and 0.030 m for ISB\_gbm, and the standard deviations are 0.030 m and 0.036 m compared with those of ISB\_com.

**Figure 8.** ISB difference series for scheme 5 on DOY 119, 2021.

#### *4.3. Influence of Different ISB Stochastic Models on Time and Frequency Transfer*

To assess the characteristics of different ISB stochastic models in the time and frequency transfer, the previous three modes are applied in the experiment. Then, the ISB were modeled as constants for one hour in a model of random constant process, marked as "constant." For the random walk process, it was defined as "walk" and the power density was 1 mm/s0.5, whereas the white noise process was marked as "noise." Figure 9 shows the results of time and frequency with three ISB stochastic models on the time link PTBB– WTZZ, using GFZ precise products. The variations in the clock difference agree well for the three ISB stochastic models. Figure 10 shows a comparison of Allan deviations of time-transfer results for the three ISB stochastic models at the PTBB–WTZZ time link. It is clear that the "constant" and "walk" schemes show slightly superior performances for frequency stability compared with that of "noise" at different time intervals. The average values within 10,000 s among the solutions of the three models are 1.08 × <sup>10</sup>−<sup>13</sup> for "noise", 1.00 × <sup>10</sup>−<sup>13</sup> for "constant," and 9.95 × <sup>10</sup>−<sup>13</sup> for "walk," with the improvements being 5.08% and 5.67%, respectively.

**Figure 9.** Result of time and frequency transfer with three ISB stochastic models on time link PTBB– WTZZ. For plotting purposes, the overall values for "walk" results were translated up to 5 ns, whereas those for results using the constant were translated down to 5 ns.

**Figure 10.** Comparison of Allan deviations in time-transfer results for the three ISB stochastic models at the PTBB–WTZZ time link.

#### **5. Discussion**

The multi-GNSS time and frequency transfer is essential for UTC comparison and traceability services, particularly for the existing BDS-2 regional and BDS-3 global system. However, the character of intersystem biases in the BDS-2/BDS-3 GNSS time and frequency transfer is still unclear. Therefore, the spatiotemporal characterization, different station attributes, and stochastic models of ISB were focused.

One can note that the daily ISB in BDS-2/BDS-3 is relatively stable, but exhibits obvious discrepancies among the three IGS analysis centers. Considering that the current BDS daily products have an obvious day-boundary jump, the daily stability of ISB helps to precisely estimate parameters. From the results of ISB for different station attributes, it can be see that common receiver type, antenna, and frequency standard can contribute to improving consistency for the three different analysis centers, these mainly being caused by the relationship between the attribute of the station (receiver DCB [35], receiver calibration, type of frequency standard) and the strategies of satellite products (i.e., sample interval, data-processing strategy, used stations, and so on). In addition, the three ISB stochastic models are compared in the time and frequency transfer, which agree with the results in previous research [32].

Of course, this study proposes only the first step of this research, and several topics still require further investigation in our near future work; for example, how to use a functional model to improve the estimation precision of ISB, and how to calibrate the ISB delay in the time link based on multisystem time and frequency transfer.

#### **6. Conclusions**

To maintain the consistency and continuity from BDS-2 to BDS-3 time transfer for one time link, we analyzed the ISBs in BDS-2/BDS-3. We deduced the mathematical model of BDS-2/BDS-3 time and frequency transfer, including observation and stochastic models. The temporal characteristics of ISB for different types of receivers, antennas, and frequency standards, with different IGS analysis center products were discussed. The three stochastic models of ISB were evaluated using one time link.

Our results indicated that the ISB series exhibit obvious discrepancies among the three analysis centers, but relatively stable characteristics. The mean values of the daily results of differ markedly for the three analysis centers. The receiver type, antenna, and frequency standard have a certain influence on the ISB difference in time and frequency transfer. The receiver type, antenna, and frequency standard are different for the two ends of the time link; the obvious system bias exists among the com, gbm, and wum analysis centers. As the only different receiver type scheme, the ISB difference series of ISB\_com and ISB\_wum agree relatively well, whereas the ISB\_gbm series shows obvious bias compared with ISB\_com and ISB\_wum for the two time-transfer links. The bias differs for the two time links. The bias for station NYAL-ONSA is approximately 0.07 m, and that for station BOR1-SPT0 is approximately 0.12 m. The ISB difference series of ISB\_com and ISB\_wum agree relatively well for the only different antenna type scheme, whereas the ISB\_gbm series shows obvious bias compared with ISB\_com and ISB\_wum for the two time-transfer links. It should be noted that the bias is not a constant but varies with time. As the only different-frequency-standard scheme, the general trend of ISB\_gbm is somewhat similar; the divergence between ISB\_wum and the other two solutions is not constant but shows significant variations for the two time-transfer links. The effect of the three different ISB stochastic models was assessed with respect to time and frequency transfer. The "walk" and "constant" schemes were slightly superior to the "noise", with the improvements in frequency stability being approximately 5.08% and 5.67%, respectively, compared with that of "noise".

This study proposes only the first step of this research, and several topics still require further investigation in our near-future work; for example, how to use a functional model to improve the estimation precision of ISB, and how to calibrate the ISB delay in the time link based on multisystem time and frequency transfer.

**Author Contributions:** P.Z. and R.T. conceived and designed the experiments; P.Z. performed the experiments, analyzed the data and wrote the paper. L.T., B.W., Y.G. and X.L. contributed to discussions and revisions. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by National Natural Science Foundation of China (Grant No: 11903040, 41674034, 41974032, 42030105) and Chinese Academy of Sciences (CAS) programs of "Youth Innovation Promotion Association" (Grant No: 2022414), "Western young scholars" (Grant No.: XAB2019B21), China Postdoctoral Science Foundation (Grant No: 2020M683763).

**Data Availability Statement:** The datasets analyzed in this study are managed by the MGEX and National Time Service Center, Chinese Academy of Sciences, which can be available on request from the corresponding author.

**Acknowledgments:** Many thanks go to the IGS MGEX for providing multi-GNSS ground tracking data, precise orbit, and clock products.

**Conflicts of Interest:** The authors declare no conflict of interest.
