The GPS space segment currently has a total of 31 satellites in orbit providing services, including four types of navigation satellites from old to new, namely Block IIR, Block IIR-M, Block IIF, and Block IIIA. Except for Block IIR, which is a traditional system satellite, the rest are modernized GPS satellites. GPS extensively uses rubidium clocks to maintain the time reference on board, with only two GPS Block IIF satellites launched in 2015 using cesium clocks, while the rest use rubidium clocks ^[10–11]; The BDS space segment has 30 satellites in orbit providing services, including 3 GEO satellites, 3 IGSO satellites, and 24 MEO satellites, which are equipped with hydrogen clocks and rubidium clocks, produced by the China Academy of Space Technology (CAST) and the Shanghai Engineering Center for Microsatellites (SECM) ^[12]; Galileo has 23 satellites in orbit providing services, extensively using hydrogen clocks, with only three satellites launched in 2011 using rubidium clocks. The GLONASS global satellite navigation system is designed with 24 satellites, and currently, all GLONASS satellite clocks have been updated to cesium clocks.

In order to conduct a comparative analysis of GNSS onboard atomic clocks by type and system, two units of each type from the four major satellite navigation systems were selected for evaluation. The analysis primarily includes three types of onboard atomic clocks: hydrogen masers, rubidium clocks, and cesium clocks. The latest launched and commissioned satellites were chosen, all of which are in a healthy state, available, and operating stably in orbit. The selected hydrogen masers are from the BDS system (C28 and C29) and the Galileo system (E10 and E34), the rubidium clocks are from the GPS system (G11 and G28) and the BDS system (C36 and C42), and the cesium clocks are from the GLONASS system (R16 and R22) and the GPS system (G08 and G10). Due to the fact that the Galileo system's onboard rubidium clocks have been in operation for over 13 years, they were not evaluated. A total of 12 onboard atomic clocks from 3 types were involved in the performance evaluation, with specific information as shown in Table 1 below.

To objectively analyze the actual time and frequency performance of the satellite clock in orbit, it is necessary to unify the satellite clock error data source and choose the precise clock error products released by the Multi-GNSS Experiment (MGEX) analysis center of the International GNSS Service (IGS), which is highly recognized internationally, has the highest accuracy, and has good continuity ^[4–6]. The data time period is from January 1, 2024, to February 1, a total of 31 days, extracting 30-second post-event clock error data, which can evaluate the main performance of time and frequency in the medium and short term. After completing the extraction of the original data, data preprocessing is required, which is divided into three steps: the first step is to integrate the GNSS satellite clock error data for each day; the second step is to deduct the systematic errors introduced by the observation station clock error; the third step is to analyze, identify, and process the abnormal values for each satellite clock, including step jumps, outliers, and missing points, mainly using the Median Absolute Deviation (MAD) method to identify and eliminate the above abnormal points, using linear interpolation to fill in the missing or already eliminated data, obtaining a healthy and complete GNSS satellite clock error sequence, the results are shown in Fig. 1, the left side shows the phase difference values of the satellite clocks participating in the evaluation, and the right side shows their frequency difference values.

Satellites typically maintain their onboard time using atomic clocks carried on their respective platforms. They measure the time difference between the onboard time and the system time through monitoring stations, and then use the broadcast of navigation messages for correction. When the time difference exceeds the designed threshold, frequency modulation or phase adjustment instructions are sent to adjust the onboard time. Therefore, each satellite only needs to maintain its own continuous and stable onboard time reference, and there is usually no correlation between the phase differences of the various satellite clocks, as shown on the left side of Fig. 1. The frequency difference is calculated by converting the time domain into the frequency domain through a formula, which is the difference between adjacent phase differences divided by the smoothing time. The smoothing time is 30 seconds, and the calculation result is shown on the right side of Fig. 1. The frequency difference can reflect certain basic time-frequency domain performance of the satellite clock. The frequency difference of the hydrogen clock in the figure is at the order of magnitude of 1e-12, which is about one order of magnitude lower than that of the cesium and rubidium clocks. The smaller this value, the smaller the variation of the adjacent clock difference, that is, the better the short-term stability. Compared with the cesium clock, the range of frequency difference variation of the rubidium clock is significantly smaller, that is, the rubidium clock has better short-term stability than the cesium clock. The above derivation results can be verified through the frequency stability analysis in the following text.

Calculate the Root Mean Square Error (RMSE) values for the onboard hydrogen clock, cesium clock, and rubidium clock separately, and observe the fluctuation between the measured values and the fitted values of the satellite clocks. Place their RMSE calculation results on the same graph to obtain the result of Fig. 2.

From Fig. 2, it can be seen that the RMSE fluctuation range of the hydrogen clock is between 0.91 and 1.13, with an average of 1.03, while the cesium clock fluctuates between 2.10 and 6.39, with an average of 3.98. The rubidium clock fluctuates between 0.92 and 2.67, with an average of 1.61. The hydrogen clock is approximately 1/4 of the cesium clock, indicating that the stability of the hydrogen clock is better, followed by the rubidium clock, and the cesium clock is worse. Furthermore, from the RMSE results of the hydrogen clocks of the two systems BDS and Galileo on the left side of Fig. 2, it can be observed that the variation trends of the four hydrogen clocks are quite consistent. According to the principle of the influence of measurement resolution noise on the time-domain stability ^[13], it can be judged that this trend is mainly caused by the system uncertainty introduced by measurement errors. It can be considered that the system measurement errors have covered the original short-term stability, resulting in very similar fluctuation amplitudes. The root mean square error curves and statistical values of the above satellite clocks are all within a reasonable range, all at the same order of magnitude and without any abnormal frequency variations or jumps, indicating that the data processing results have obtained reasonable and effective satellite clock difference data. This data can be used for further evaluation and analysis of the time-frequency characteristics of GNSS satellite clocks. The specific statistical results of the root mean square error values are shown in Table 2.

Atomic clock frequency deviation refers to the degree of complexity between its actual output frequency and the standard frequency. IGS time maintains a high degree of consistency with International Atomic Time. Here, the frequency deviation measured by IGS is used to replace frequency accuracy, and the most commonly used least squares linear fitting method is used to calculate the frequency accuracy. The specific method is to use the preprocessed data of this month, calculate the frequency accuracy once every 5 days, calculate once every day, and a total of 25 days of calculation results. The calculation results are shown in Fig. 3.

In this coordinate system, the horizontal axis represents the calculation time, measured in days, and the vertical axis represents the frequency deviation results. The smaller the value, the closer it is to the standard frequency, indicating better frequency accuracy. Blue represents the hydrogen clock, yellow represents the cesium clock, and red represents the rubidium clock. From the figure, it can be seen that the cesium clock has the best frequency accuracy, followed by the hydrogen clock, and the rubidium clock is relatively poor.

The rate of change in frequency accuracy within a unit of time is known as the frequency drift rate, which is typically caused by factors such as environmental changes or aging of electronic components. Depending on the value of the unit of time, it can be divided into second drift, daily drift, and monthly drift, among others. Navigation systems have a high real-time requirement for time, and here the calculation of the more significant daily drift is considered. The frequency drift rate is calculated using the standard least squares second-order fitting method. Generally, it is required to collect data for more than 15 days to calculate the daily drift. The specific method involves taking the previous 20 days of data to calculate the frequency drift rate once per day. By doing this daily, the results for 10 days can be obtained after processing the data for the current month, as shown in Fig. 4.

In this coordinate system, the horizontal axis represents the calculation time, measured in days, and the vertical axis represents the frequency drift rate calculation result. The smaller the value, the less obvious the frequency drift, indicating better performance. Blue represents the hydrogen clock, yellow represents the cesium clock, and red represents the rubidium clock. From the figure, it can be seen that the daily drift of the hydrogen clock is smaller, followed by the cesium clock, and the rubidium clock is worse.

Frequency stability refers to the magnitude of the random fluctuations in the output frequency signal, characterizing the ability to produce the same time or frequency over a given period. It is related to both internal noise in actual clocks and atomic clocks and measurement noise. There are multiple methods for calculating frequency stability, with the most common being the Allan variance and the Hadamard variance. The latter can significantly reduce the effects of frequency linear drift. If overlapping samples are used, the confidence level is higher, known as the overlapping Allan variance and the overlapping Hadamard variance. The calculation methods for both are based on the second and third differences of time difference data ^[14–16]. Using these two methods to calculate and analyze the medium-term frequency stability of satellite clocks, with smoothing times ranging from 30 seconds to 200,000 seconds, the results are shown in Fig. 5.

In the coordinate system of Fig. 5, the horizontal axis represents the smoothing time interval, and the vertical axis represents the overlapping Allan variance and the calculated values of the overlapping Allan variance, respectively. Blue indicates the hydrogen clock, yellow indicates the cesium clock, and red indicates the rubidium clock. It can be seen that under both calculation methods, the frequency stability of the hydrogen clock is the best, followed by the rubidium clock, and the cesium clock is worse. However, the rubidium clock shows a significant frequency linear drift after 4000s, as indicated by the red line in Fig. 5(a). The overlapping Hadamard variance can eliminate its impact on the frequency stability results, as shown by the red line in Fig. 5(b).

Statistical results of key timing and frequency performance calculations for GNSS satellite clocks, including frequency accuracy, frequency drift rate, and frequency stability, are shown in Table 2.

In Table 2, the average results of frequency accuracy and frequency drift rate for each star clock are taken, and the frequency stability results are taken from the overlapping Hadamard variance. It can be seen that hydrogen clocks and rubidium clocks with smaller root mean square errors also perform better in frequency stability, but there is no direct correlation with the goodness of frequency accuracy and frequency drift rate. For example, rubidium clocks G11 and G28 have root mean square errors of 0.92 and 1.27 respectively, and their frequency stability is at the same order of magnitude as the optimal hydrogen clock E10, but their frequency accuracy and frequency drift rate are 1–2 orders of magnitude lower than those of hydrogen clocks and cesium clocks. In terms of frequency accuracy, cesium clocks perform the best, reaching the magnitude of 1e-12. Although there is no frequency drift, their frequency stability index is poor. When used in real-time satellite navigation systems, the high accuracy of cesium clocks cannot be fully utilized. On the contrary, due to the fact that both frequency accuracy and frequency drift rate can be predicted and eliminated through modeling of the star clocks, the characteristic of poor stability still affects the measurement accuracy to some extent. In terms of frequency drift rate, hydrogen clocks perform the best, all at the magnitude of 1e-16. This is somewhat different from the previous understanding that hydrogen clocks have poor frequency drift rate and are lower than cesium clocks. The analysis may have two factors: (1) The iterative update of the onboard hydrogen clock technology has overcome the problem of large frequency drift rate to some extent; (2) The frequency drift rate is calculated based on daily drift, without calculating the longer weekly and monthly drift. Within this time unit, hydrogen clocks do not show significant frequency drift. In terms of frequency stability, hydrogen clocks are the best, rubidium clocks perform similarly, with a stability of 1e-14 at the thousand-second level and 1e-15 at the hundred-thousand-second level, while cesium clock stability is about 1 order of magnitude lower in all smoothing time periods.