STABILITY OF GPS AND GLONASS ONBOARD CLOCKS ON A MONTHLY BASIS

This paper presents the stability of the GPS and GLONASS system clocks’ stability. It describes the construction of these two systems and calculated four different Allan variances (AVAR), based on the MGEX (the Multi-GNSS Experiment) clock products. Four used variances allowed making a better analysis of each GNSS system clock. The results are shown at different averaging times from 5 s as successive multiples to 655,360 s in a monthly period. The stability of GPS and GLONASS clocks is included in the range of 10-12~10-14 s. The results showed that GLONASS clocks are stable (10-12~10-14 s) and are affected with white frequency noise (WFM). The GPS clock stability models have more fluctuations for τ > 40,960 s and the mean stability is concluded between 10-12~1013 s. Mean frequency accuracy for GPS clocks is related with WFM and Random Walk Frequency (RWF). The differences in clock stability are caused by several factors – block type, type of clock and the time of a satellite in orbit. These factors have an influence on stability results.


INTRODUCTION
Currently, GNSS systems are used in many areas of human life, for example, safety, transport, shipping, or in many professions like geodesy, mapping and many other engineering studies ( Currently, the GPS space segment consists of 31 satellites from which 4 satellite blocks (IIR IIRM, IIF, III) are in use (Tab. 1), from the 8 satellite blocks manufactured so far. A total of 39 satellites were launched in the already inactive Blocks I, II and IIA. One of the Block I had a launch failure. In the current blocks, 38 satellites were launched into orbit. Again, one of the IIR Block did not perform the mission. The first of the 2R block was launched on July 23, 1997. Since September 25, 2006, an upgraded 2R-M block has been introduced. 2F satellites were launched between 2010 and 2016, while the latest Block III started launches on December 23, 2018 (Quasi-Zenith Satellite System -2020; Steigenberger, Thoelert, and Montenbruck -2020 I  11  0  0  10  1  Block II  9  0  0  9  0  Block IIA  19  0  0  19  0  Block IIR  13  7  2  3  1  Block IIRM  8  7  1  0  0  Block IIF  12  12  0  0  0  Block III  5  4  1  0  0  Block IIIF  0  0  0  0  0  Total  77  30  4 IIv  56  0  0  56  0  Uragan Block III  1  0  0  1  0  Uragan  50  22  1  21  6  Uragan-K1  3  1  2  0  0  Total  141  23  3  103 12 Source: https://en.wikipedia.org/wiki/List_of_GLONASS_satellites GPS and GLONASS differ from each other by several factors. As can be seen in the two tables above, GPS has 7 more operational satellites than GLONASS. For GPS, there are 4 operational blocks, and there are 2 for GLONASS. Historically, the Russian system has launched 141 satellites, which is 64 more than the GPS has launched. These GNSS systems also have technical differences. GPS has better position accuracy, which is 3.5-7.8 m, while GLONASS is 5-10 m (Rutkowski, 2019). In the case of frequencies, GLONASS operates at 1.602 GHz, whereas GPS uses the L1 signal, operating at 1.57542 GHz (Yunck et al., 1985; Marin et al., 2017). All GLONASS satellites are equipped with caesium clocks, unlike GPS satellites, where the majority are rubidium clocks. Despite these differences, there is one common factor without which the GNSS systems could not work. It is time, which is the key to work in each GNSS system (Meng et al., 2020). The user position is generated based on the measuring time at which the signals travel from the satellite to the receiver, with the assumption of the known speed of the signal, distance is calculated (Galleani and Tavella, 2015). To make it possible there are precise, atomic clocks onboard each satellite (Daly et al., 1991). In the case of internal clocks, we have quartz, rubidium, caesium and hydrogen masers (Roberts, 2019). After 2000, the most common were rubidium clocks because of their lower costs (Brouwer, 1951). Authors in several articles have provided research on GNSS clock performance and described the differences in stability between some satellite blocks. As research shows, the highest stability was confirmed on rubidium clocks of GPS Block IIA. The rubidium clocks from Block IIA had lower noise than the clocks from newer Blocks IIR and IIRM for up to 100 s (Hauschild et al., 2013). Furthermore, Block IIA granted greater stability than GPS Block IIA and IIF caesium clocks (Thoelert et al., 2014). Whereas GLONASS uses caesium clocks, which reveal similar stability as GPS Block IIA caesium clocks and rubidium clocks from IIR and IIRM at above 20 s (Griggs et al., 2014; Xu and Xu, 2016) In this paper, GPS satellites are mostly equipped with the rubidium type, only numbers 8 and 24 have caesium clocks. The results show that caesium clocks from GPS Block IIF present lower stability (10 -12~1 0 -13 s) than some of the GPS rubidium clocks (10 -12~1 0 -14 s). The GLONASS clocks are all caesium type, and their stability is in (10 -12~1 0 -14 s) level. To estimate the stability of GNSS clocks, the author made a frequency stability analysis. Already in the mid-1960's, the new modern frequency stability analysis has begun. In those days, new statistics that could better suit common clock noises became available and better methods were developed for high-resolution measurements (

METHODS
In this paper, the author analysed short-term satellite clock stability (in terms of 1 month) of two fully operational GNSS -GPS and GLONASS. The special calculations of a few variances were done to obtain better certainty of the analysis.
Variances are used to characterise fluctuations in the frequency domain. The most common time domain measure of frequency stability is the classic Allan Variance (AVAR) and there are also related versions (variances), that provide better confidence or have other specifications (Stefani, 2004). The Allan (and related) variances allow analysing the stability of atomic clocks, including GPS and GLONASS satellites (Zucca and Tavella, 2005; Galleani and Tavella, 2009; Gambis, 2002). Equation (1) presents frequency data as the difference of the phase data and dividing them by the sampling interval : The formula needed to determine the variance are shown in Figure 1. The x(t) function presents a simulated plot of time fluctuations calculated from frequency data y_i. At the bottom left, are equations for the standard deviation, and at the bottom right are equations, through which we can determine stability in the field of time ("Section Four", 1980). The y_i in those calculations is the ith of M fractional frequency values. The τ interval is the time in which the data is averaged and analysed. There are several versions of Allan variance that provide greater statistical accuracy or allow the determining of other noises. In this paper, the author uses the classic Allan deviation (ADEV), overlapping Allan deviation (ODEV) and Modified Allan deviation (MDEV) and Hadamard Allan deviation (HDEV) (Howe and Peppler, 2001). The classic ADEV is the most common variance type used to measure frequency stability. It has the advantage of being convergent for most types of clock noises. Allan's deviation is expressed as equation (2): ODEV variance, 2 ( ), provides more confidence than Allan's classic variance. It uses sample overlap during averaging time , which improves the certainty of estimating the resulting stability. It can be estimated from a set of M frequency measurement for averaging time = 0 , where m is the averaging factor and is the basic measurement 0 interval (Stefani, 2004). The disadvantage of this type of variance is the longer calculation time.
Overlapping Allan variance can be obtained from the formula: The modified Allan variance type is used to distinguish between white noise and flicker PM (Phase Modulation) and can be expressed as equation (4): Hadamard's variance is characterised by insensitivity to linear drift. It has a much narrower transfer function, making it useful for the analysis of rubidium atomic clocks (Stefani, 2004). Hadamard variance can be calculated using the formula: The   Figure 2 shows noise types detected by ODEV and MDEV. An additional advantage of Allan's modified variance is that one can distinguish white phase and flicker phase noise.
In this paper, Allan (and related) variations were calculated based on the MGEX (the Multi-GNSS Experiment) clock products. Time span covers the first 31 days of the year 2021 (1st-31st January). Daily 30 s clock products for GPS and GLONASS observations were combined and based on equations 1-5, for each GNSS system were calculated four different types of the Allan variations for combined clock products for the whole GPS and GLONASS space segment separately. Moreover, on each graph were added a graph with noises characterising each space segment.

RESULTS
A total of 8 figures was presented showing the stability of GPS and GLONASS clocks using 4 different types of Allan's variances. The stability is shown at different averaging times from 5 as successive multiples to 655,360 seconds. The clock stability was calculated for 32 GPS satellites from four operational blocks: IIR, IIRM, IIF and III. The oldest operational satellite is from Block IIR, and it has been orbiting for 24 years (since 1997). In the GLONASS case, calculations were made for 21 satellites from blocks M and K1. The oldest satellite has been flying for over 14 years (since 2007). Figures 3 and 4 present values of ADEV. In Figure 3, we related mean-variance (yellow line) to white frequency modulation (WFM) in the range of 5 < τ < 40,960 s. WFM is marked with a longer dashed black line and random walk frequency (RWF) is presented with a dotted black line. We can see that stability fluctuates for τ > 40,960 s and its connection with (RWF). The GPS clocks stability is increasing from level 10-12 s to 10-14 s in the range of τ = 5 s to τ = 40,960 s, and for τ > 40,960 s, the stability drops to 10-13~10-12 s. It is mostly visible for satellites G14, G23, G32. Figure 4 shows GLONASS clocks stability using ADEV. Mean-variance is related with WFM and stability is about 10-12~10-13 s for 5 < τ < 40,960 s, and for τ > 40,960 s, it is increasing even to 10-14 s. We can see that the three clocks from satellites R11, R13, R2 have worse stability compared to the others and R11 has a fluctuating frequency drift. The rest GLONASS clocks stabilities are close to each other and the lines presenting frequency accuracy have a similar course. Figures 5 and 6 present stability results using HDEV. As the charts show, for both GPS and GLONASS satellites, the mean stability has convergence with WFM. In Figure 5, it is visible that stability fluctuates respectively after τ > 40,960 s. It is shown that GPS satellites, G24 and G8 have worse stability, which is caused by the caesium clock type, for τ = 5 s, it is between 10-11~10-12 s, and it rises to 10-13 s for τ = 655,360 s. The clocks from satellites G27 and G30 reach the best stability, which is 10-15 at τ = 655,360 s. The mean frequency drift for GPS clocks increases from 10-12 to 10-14 s, respectively, with τ increase. Whereas in Figure 6, there are lower fluctuations compared to the GPS drift. We also can see fluctuations after τ = 40,960 s, but they are not as high as GPS ones. The mean stability drift is associated with WFM. GLONASS clocks stability is between 10-11~10-12 s for τ = 5 s, and it increases to the range of 10-13~10-14 s at τ = 655,360 s. Further, it should be noted that the clock from the R21 satellite reaches the best 10-15 s stability at the end of τ range. The differences in stability between the HDEV and ADEV methods are little for GLONASS satellites.  G1  G2  G3  G4  G5  G6  G7  G8  G9  G10  G11  G12  G13  G14  G15  G16  G17  G18  G19  G20  G21  G22  G23  G24  G25  G26  G27  G28  G29  G30  G31  G32  Mean  WFM   The next two charts represent clock stability after modified Allan variance. We can see a similar stability model in this method and ADEV. In Figure 7, mean-variance is fitted with WFM and RWF noises. It is visible that half of the GPS clocks have worse stability, which is close to 10-12 s, and the rest have better stability on the level between 10-12~10-13 s, where 5 < τ < 1.280 s. The biggest changes in line course begin for τ > 40,960 s, where the fluctuation starts, and the stability result drops from 10-13 to 10-12 s for the clock from Satellite G14. Mean frequency accuracy of GPS satellite clocks starts at level 10-12 s for τ = 5 s and reach almost 10-14 s for τ = 40,960 s, and then decrease, respectively, to 10-13 s.  G1  G2  G3  G4  G5  G6  G7  G8  G9  G10  G11  G12  G13  G14  G15  G16  G17  G18  G19  G20  G21  G22  G23  G24  G25  G26  G27  G28  G29  G30  G31 G32  Fig. 7. ADEV of GPS space segment with mean value (yellow line) and assigned noise types (black lines) In Figure 8, we can see GLONASS clock stability and again it has a more uniform course than the GPS. Similarly, in the MDEV method, it is visible that the clocks frequency accuracy slowly getting better from 10-12 to 10-14 s, respectively, with τ increase. The stability is slightly better at τ = 5 s than it was in HDEV and ADEV variations. The stability of the R11 satellite clock stands out from the rest mostly in the range 320 < τ < 20,480 s, and it is around 10-12 s. The mean-variance has convergence with WFM in the studied τ range. The last type of variation used by the author is the ODEV variance. Figure 9 shows a similar stability model of GPS clocks to the rest Allan variances. The difference between the clock G24 with the worst stability, and presenting the higher accuracy G30, is 10-1 s in the range where 5 < τ < 40,960 s. The divergence between these two clocks is caused by the time each satellite is  G1  G2  G3  G4  G5  G6  G7  G8  G9  G10  G11  G12  G13  G14  G15  G16  G17  G18  G19  G20  G21  G22  G23  G24  G25  G26  G27  G28  G29  G30  G31  G32  Mean  WFM  in orbit and also, the clock type is the reason. The higher fluctuations in stability are visible after τ > 40,960 s, where the differences in stability between satellites grow, and the most reach 10-2 s for G14 and G20 satellites. The mean-variance has convergence with WFM and RWF noises. Fig. 9. ODEV of GPS space segment with mean value (yellow line) and assigned noise types (black lines) In Figure 10, we can see the similar GLONASS clock stability fitted to WFM noise as in previous related variances. The last variation method again shows sticked linear drift, the most similar to the MDEV variance model. Fluctuations after τ > 40,960 s seconds are not that much visible as they were in the ADEV and HDEV variations. The most visible difference between the lowest stability R11 satellite clock and the highest accuracy R21 is almost 10-1 s in the range 2,560 < τ < 5,120 s. The mean stability for GLONASS clocks using the ODEV method concluded between 10-11~10-12 s for τ = 5 s, and then increase, respectively, with time to level 10-13~10-14 s at τ = 655,360 s. Table 4 shows individual values of clock stability depending on the variance type and the GNSS system. It is visible that as with τ multiplies, the stability of clocks is increasing. In the range 10< τ < 2,560 s, the accuracy of clocks stands on level 10-12~10-13 s. As the τ increase to 40,960 s and then gets 655,360 s, the stability is improving up to 10-14 s and stands on a similar level. It should be noticed that the mean stability of GPS is higher for each variance used than GLONASS clocks, where 10< τ < 40,960 s. The change appears at τ = 163,840 s, where GLONASS has better accuracy than GPS using MDEV, and at τ = 655,360 s GLONASS presents higher stability than GPS clocks for ADEV, MDEV and ODEV.
The presented figures show us the differences in the stability of GLONASS and GPS satellite clocks. We can conclude that GLONASS figures present a more consistent course of variations than the GPS figures. In the GPS, the fluctuations and differences between some satellite frequency stabilities are more visible and its mean stability is associated with WFM and RWF for 3 variances. The GLONASS clock stability is related only to WFM for 4 used variations. The mean stability for both satellite systems clocks is similar and varies in the range 10-12~10-13 s for GPS and 10-12~10-14 s for GLONASS clocks. From the four methods of variances used, the most distinctive stability model for GPS is HDEV, while the rest three are similar. Whereas for GLONASS, we can see the resemblance in the various models between ADEV and HDEV and between MDEV and ODEV.

DISCUSSION
In this article, GPS and GLONASS clocks stability were analysed. Four variances Allan (and related) were calculated based on the MGEX clock products. Operational satellites from two GNSS systems were chosen for this research and each satellite clock performance was presented at different averaging times, from 5 as successive multiples to 655,360 s. The analysed 32 satellites from GPS concluded mean stability between 10-12~10-13 s and influenced WFM and RWF. The exception was HDEV calculation, where mean variation was associated with WFM, and the stability was approximately 10-12~10-14. The 21 GLONASS satellites also had stability at the level 10-12~10-14 s and were related with WFM noise. The main differences between these two systems are that GLONASS clocks frequency is more consistent, and stability improves with averaging time increasing, whereas GPS clocks are not as compatible as their Russian counterpart, having higher fluctuations for τ = 40,960 s and above. The reason for these differences is that GLONASS satellites use only caesium atomic clocks free from frequency drift. The differences between some GPS satellite clocks are visible in the range 5 < τ < 1,280 s, where half of all satellites have better stability about 10 times than the others. This may be caused by the different times that each satellite is in orbit. This means that the satellites from older blocks have worse stability than the recently launched ones. From the four variances calculated for GPS clocks, three of them gave very similar results. Only HDEV variation differs more from the others, cause its stability consists of averaging time increasing also after τ = 40,960 s, and it is only affected by the WFM noise. For GLONASS satellite clocks, the ADEV and HDEV variations showed higher fluctuations. Whereas the other two methods, ODEV and MDEV, presented similar and more consistent stability. The analyses showed that the mean stability of the presented GNSS system clocks is on a similar level, but GLONASS is more stable and consistent. Four calculated variances were compared to each other and allowed for even deeper analysis.