## Abstract

A characterization of the short-term stability of the atomic frequency standards onboard GNSS satellites is presented. Clock performance is evaluated using two different methods. The first method derives the temporal variation of the satellite’s clock from a polynomial fit through 1-way carrier-phase measurements from a receiver directly connected to a high-precision atomic frequency standard. Alternatively, three-way measurements using inter-station single differences of a second satellite from a neighboring station are used if the receiver’s clock stability at the station tracking the satellite of interest is not sufficient. The second method is a Kalman-filter-based clock estimation based on dual-frequency pseudorange and carrier-phase measurements from a small global or regional tracking network. Both methods are introduced and their respective advantages and disadvantages are discussed. The analysis section presents a characterization of GPS, GLONASS, GIOVE, Galileo IOV, QZSS, and COMPASS clocks based on these two methods. Special focus has been set on the frequency standards of new generation satellites like GPS Block IIF, QZSS, and IOV as well as the Chinese COMPASS/BeiDou-2 system. The analysis shows results for the Allan deviation covering averaging intervals from 1 to 1,000 s, which is of special interest for real-time PPP and other high-rate applications like processing of radio-occultation measurements. The clock interpolation errors for different sampling rates are evaluated for different types of clocks and their effect on PPP is discussed.

### Keywords

Clock analysis Allan deviation GNSS GPS Block IIF QZSS COMPASS Galileo IOV Real-time PPP## Introduction

The onboard atomic frequency standard (AFS) is a key component of a GNSS satellite. Uncorrected satellite clock offsets directly affect the range measurement and thus limit the navigation accuracy. Therefore, a high predictability of the onboard frequency standards is required to overcome this limitation and enable an extended validity of clock correction parameters. Since the early days of GPS, an improved accuracy of the onboard clocks has been striven for with every new generation of satellites. Significant effort has been put into the analysis of the clock stability and the comparison between laboratory tests and flight results (McCaskill et al. 1981; Oaks et al. 2004; Vannicola et al. 2010).

The current GPS constellation can be grouped into five families of clocks: the oldest satellites are the Block IIA satellites, which are either operated on a Cesium or a Rubidium AFS. The Block IIR and Block IIR-M satellites carry rubidium clocks, exclusively. Finally, the newest generation of Block IIF satellites uses improved Cesium and Rubidium clocks. GLONASS, the second long-standing navigation system, has reached a complete constellation again in 2011, after a significant drop in the number of operational satellites one decade ago. Comparably little information, however, is available on the onboard frequency standards of the new generation of satellites and their performance. It has been published though, that the current generation of GLONASS-M satellites depends exclusively on Cesium clocks from Russia (Gouzhva et al. 1993; Bassevich et al. 1996). The test satellites for the future European satellite navigation system Galileo, GIOVE-A and -B, are equipped with two Rubidium clocks. GIOVE-B additionally has a highly precise passive hydrogen maser (PHM) (Waller et al. 2010). The first two In-Orbit Validation (IOV) satellites, which have been recently launched, are equipped with two Rubidium clocks and two PHMs.

Two new navigation systems are currently under development in the asian–pacific region: Japan has launched the first satellite of its Quasi-Zenith Satellite System (QZSS). It is equipped with the same Rubidium clock, which is also used in the Block IIF satellites (Inaba et al. 2009; Steigenberger et al. 2012). Finally, China has made remarkable progress in establishing its COMPASS/BeiDou-2 system. As of April 2012, the constellation consists of five geostationary (GEO) satellites, five inclined geosynchronous (IGSO) satellites, and one satellite in medium altitude earth-orbit (MEO). Even though the MEO satellite is unusable due to a suspected clock problem (Hauschild et al. 2012) and one of the GEO satellite has orbit control problems and ceased the transmission of signals, the constellation can already provide a service for regional users (Chen et al. 2009). The COMPASS GEO and MEO satellites use Rubidium clocks from Chinese and European manufacturers (Mallette et al. 2010; Han et al. 2011).

So far, the onboard frequency standards of the COMPASS, QZSS, and Galileo IOV satellites as well the newer GLONASS satellites have not yet been extensively analyzed. Furthermore, most of the clock stability analyses in the literature are focused on longer timescales of hours or even days. Very few publications extend to averaging intervals of 30 s or less (Gonzalez and Waller 2007; Hesselbarth and Wanninger 2008; Senior et al. 2008). The analysis provided here intends to close this gap and presents results for intervals between 1 s and 15 min for GPS, GLONASS, GIOVE, Galileo IOV, QZSS, and COMPASS. The short-term intervals are particularly interesting for the upcoming real-time precise-point-positioning applications, where high-rate clock corrections are transmitted to users via Internet, satellite, or radio links. Knowledge of the magnitude of clock extrapolation errors for different systems allows for a trade-off between maintaining accuracy and saving bandwidth for the broadcast corrections in a quickly growing GNSS environment.

The high-rate clocks have been analyzed using two different methods, which will be introduced in the next chapter. Clock results obtained with both methods are compared. The main part of the analysis presents the short-term Allan deviation (ADEV) of the different GNSS clocks and compares the different systems. Finally, results for clock extrapolation errors for selected types of clocks are presented.

## Clock analysis concepts

The following sections introduce the clock analysis concepts used for this study, starting with the one-way and 3-way carrier-phase methods followed by the Kalman-filter-based clock estimation.

### 1-way/3-way carrier-phase method

*k*and receiver

*m*can be written as

*c*is the speed of light, \( {\text{d}}t_{m} \) and \( {\text{d}}t^{k} \) are the receiver and the satellite clock offsets, respectively, \( T_{m}^{k} \) and \( I_{m}^{k} \) are the signal delays due to troposphere and ionosphere, respectively, and \( b_{m}^{k} \) comprises the constant carrier-phase ambiguity and all other unconsidered effects like phase center variations or phase wind-up. Finally, \( \varepsilon_{m}^{k} \) denotes receiver noise and multipath errors. Subtracting the modeled range \( \tilde{r}_{m}^{k} \) on both sides of (1) and substituting the difference between ground and satellite clock with ∆

*t*leads to:

The term \( r - \tilde{r} \) corresponds to the projection of the error in the orbit product on the line-of-sight vector and is therefore slowly varying. The signal delays caused by troposphere and the ionosphere do also not exhibit quick temporal variations under quiet atmospheric conditions. Unless cycle slips occur, the ambiguity is constant and the other terms summarized in \( b_{m}^{k} \) can also be considered to vary slowly over time. Therefore, the entire term in square brackets in (2) can be removed by a polynomial fit, leaving only the variation of the satellite clock with respect to the ground station clock and measurement errors in \( \varepsilon_{m}^{k} \). Since the noise contributions of these two clocks cannot be separated, the satellite clock under investigation must have the dominant noise contribution. Furthermore, the impact of receiver noise and multipath can be minimized by limiting the analysis to parts of the satellite pass with high elevation angles.

The choice of the polynomial for the removal of the nuisance parameters in (2) mainly depends on the orbit quality. Experience has shown that a second- or third-order polynomial is sufficient for data arcs of 1 h and orbits with a quality comparable to broadcast ephemerides or better. Tests using a third- and fourth-order polynomial for broadcast orbits have shown that marginal differences appear for averaging times of 1,000 s. If only two-line elements (Kelso 2007) are available, the use of a fourth-order polynomial is recommended. With higher polynomial orders, the differences in the resulting Allan deviation become more pronounced and appear at shorter averaging intervals. Therefore, the order should generally be selected as low as possible since true physical behavior of the clock could otherwise be removed from the data, which leads to overly optimistic results in the clock assessment.

*k*are available at a station

*n*, which is operated using an unstable clock:

*l*at both stations

*m*and

*n*, we can form the single difference of pseudorange measurements:

*l*has been eliminated in (4) and does not affect the results. Note that the effects due to troposphere and ionosphere do not necessarily cancel out in the single difference, as they decorrelate with increasing spatial distance between the stations. Taking the difference of (3) and (4) leads to:

*t*and subtracting the modeled ranges \( \tilde{r}_{n}^{k} \), \( \tilde{r}_{n}^{l} \), and \( \tilde{r}_{m}^{l} \) on both sides leads to:

*o*denotes the orbit error terms. Since the noise of the measurements for satellite

*l*affects (6), it should be tracked with high elevation angles at both stations. Assuming that the carrier-phase measurement noise is statistically independent and equal for all three satellite links, the noise on the clock estimates in (6) is higher by a factor of \( \sqrt 3 = 1.7 \) compared to the 1WCP approach. Finally, the single-frequency carrier-phase observations in (2) and (6) can also be replaced by the ionosphere-free combination in order to eliminate the first-order effects of the ionosphere.

### Kalman-filter-based clock estimation

In addition to the 1WCP and 3WCP methods, clock solutions have also been computed based on measurements from a network of approximately 20 tracking stations. The algorithm for clock estimation is based on a Kalman-filter and estimates a clock offset and a drift parameter for each satellite, a clock offset and a troposphere zenith delay parameter for each station as well as float carrier-phase ambiguities. The filter processes dual-frequency pseudorange and carrier-phase observations from daily RINEX files and can be operated in forward and backward mode with additional smoothing of the filter state (Hauschild and Montenbruck 2008, 2009). The process noise model in the Kalman-filter for the clock and the drift assumes an integrated white noise process. The standard deviation of the clock and drift process noise has been set to 300 ps over 10 s and 2 ps over 900 s, respectively. These settings are sufficiently relaxed to not affect the resulting Allan deviation of the clock. In order to make the clock estimation problem solvable, one ground station clock is fixed to zero in the processing causing all other clocks to be estimated with respect to this master clock.

## Results for GNSS satellite clocks

The results for selected satellite clocks computed with the different clock estimation methods are compared to analyze their individual particularities and advantages. Then, all currently active satellite clocks of GPS, GLONASS, QZSS, COMPASS, GIOVE, and Galileo IOV satellites are compared in terms of Allan deviation.

### Comparison of different clock estimation strategies

High-rate clocks have been produced using the different methods introduced in the previous section. The Kalman-filter-based clock estimation has been computed for a 12-h period on March 19, 2012, based on the measurements from the network depicted in Fig. 2. The station at USN3 has been selected as master clock for the clock estimation. The first and the last 15 min of the clock results have been omitted in the Allan deviation computation to avoid potential effects due to filter convergence.

The clock estimates generated with the carrier-phase-based methods are based on precise GPS orbits from IGS, for all other satellites precise ephemerides from IAPG/TUM are used. The H-maser at the HOB2 GPS reference station in Australia is used for the 1WCP-based GPS clock analysis and (along with a multi-GNSS receiver at UNSX) for the 3WCP analysis of Galileo, GIOVE, COMPASS, and QZSS. In all cases, a third-order polynomial is fitted to a 1-h slice of the satellite pass near highest elevation.

The upper right plot in Fig. 3 depicts the results of the Kalman-filter-based method for the GPS satellites G17 and G25 and the QZSS satellite J01. The same satellites have also been processed with the 3WCP carrier-phase method and dual-frequency observations. The station HOB2 is again used as the source for the reference clock. It is coupled via a single-difference link of a commonly observed satellite to the station UNSX, which tracks the satellite of interest. As expected, the ADEV of the 3WCP method is higher at short timescales compared to the 1WCP method due to the increase in measurement noise from the additional satellite links. However, for timescales of 10 s and higher, this method yields virtually identical results compared to the 1WCP method, which confirms that the ADEV of G17 at these averaging intervals is indeed dominated by clock noise and not measurement noise. For the Block IIF satellite, however, an increase of the Allan deviation is also visible at higher averaging intervals, which corresponds to the expected factor of 1.7. The Kalman-filter-based method yields better results for time intervals up to 8 s compared to the 3WCP method, but is still inferior for longer intervals. It is interesting to note that the results for the ADEV of the QZSS clock and the Block IIF clock show almost identical performance up to averaging intervals of 100 s. Between 100 and 1,000 s, QZSS exhibits an increased ADEV, which is attributed to an effect caused by the satellite’s time keeping system (TKS) (Steigenberger et al. 2012). The Kalman-filter-based ADEV for QZSS at time intervals of 5 s and less is higher compared to G25. This is an expected result, however, since the QZSS satellite is observed from fewer stations, on average, compared to G25.

The bottom left plot of Fig. 3 depicts the ADEV for two selected COMPASS satellites, the GEO satellite C01 and the ISGO satellite C06, based on the 3WCP and the Kalman-filter-based approach. For C01, the results of both methods agree for averaging intervals of 20 s and longer. Below this threshold, the 3WCP method has a notably higher ADEV. A similar observation can be made for C06, however, both methods already agree for averaging intervals of 10 s and longer.

The bottom right plot of Fig. 3 depicts the results for the Rubidium AFS of the Galileo IOV satellite E11 and the Rubidium clock of the GIOVE-A satellite E51, again computed with the 3WCP and the Kalman-filter-based method. Both show comparable results for the Allan deviation of GIOVE-A for averaging intervals of 30 s and longer. For shorter intervals, the ADEV based on the 3WCP method exceeds that of the Kalman-filter-based, which is consistent with the results obtained for COMPASS. For the IOV satellite, the results slightly are more unambiguous. The ADEV based on the 3WCP method exceeds that of Kalman-filter-based method for short timescales, as expected, but for averaging intervals longer than 10 s, its ADEV is even lower. At any rate, it becomes obvious that the Rubidium clock of the IOV satellite has a noticeable better performance than the GIOVE-A clock.

It can be summarized that the 1WCP single-frequency method yields the lowest ADEV results even for the most precise clocks for averaging intervals from 1 s to few hundreds of seconds under favorable atmospheric conditions. With dual-frequency measurements, the range of averaging intervals can be extended to 1,000 s at the expense of slightly increased noise at short timescales. The 3WCP method suffers from increased noise due to the additional satellite links, which affects the Allan deviation from 1 s to higher averaging intervals depending on the accuracy of the satellite clock. For timescales close to 1,000 s and above, the 1WCP and 3WCP methods tend to be too optimistic, since too much of the physical clock behavior is removed with the polynomial fit. The Kalman-filter method provides more realistic results for most of the current satellite clocks, except for the highly stable Block IIF and QZSS AFS. Since this method can provide continuous clock estimates over longer time intervals, the ADEV analysis can also extend to longer averaging intervals compared to the 1WCP and 3WCP methods. All methods are limited by the carrier-phase measurement noise, which can dominate the ADEV results for the high-precision clocks at short timescales. The clock noise of the reference station, on the other hand, is negligible when an active hydrogen maser is selected, which reaches an ADEV on the order of 10^{−13} at 1 s intervals (Allan 1983).

### Comparison of different GNSS clocks

The majority of GPS satellites currently consists of Block IIR satellites. Their ADEV for averaging intervals of 100 s and less is very similar to the Block IIA Cesium clocks; however, for larger averaging intervals τ, the Allan deviation decreases with τ^{−1}, which corresponds to white phase noise. For the Block IIA Cesium clocks, the decrease in ADEV is closer to τ^{−0.5}, which corresponds to white frequency noise. The Block IIR-M satellites have a very similar performance in terms of ADEV as the Block IIR satellites. However, it is interesting to note that the newer Block IIR and Block IIR-M clock have a higher noise compared to the Block IIA satellite clocks for averaging intervals up to approximately 100 s. This effect has been attributed to the influence of the TKS, which couples the atomic frequency standard to an oscillator and introduces high-frequency noise through a phase meter comparator (Wu 1996; Petzinger et al. 2002).

The Block IIF and the QZSS satellite clocks have a significantly better performance over the entire range of averaging intervals compared to all other GPS clocks. Furthermore, the satellite G01 exhibits an increased Allen deviation compared to G25 for averaging intervals of 1,000 s, which is due to the increased line bias variations during the eclipse phase of this satellite (Montenbruck et al. 2011b). It should be noted, however, that the true clock performance of the Block IIF Rubidium clocks and the QZSS clock can be expected to be a factor of 3 better as discussed in the previous section. Since both Block IIF satellites were operated on Rubidium clocks in the time period considered here, results for the Cesium AFS are not available. The last plot in Fig. 4 depicts the Allan deviation for GIOVE-A (E51), GIOVE-B (E52) as well as the first Galileo IOV satellites E11 and E12. All four satellites were operated on the Rubidium frequency standards in the time period of interest. It is interesting to note that the Galileo IOV Rubidium AFS obviously has an improved stability compared to the frequency standard for GIOVE-A and -B. The passive hydrogen maser used for GIOVE-B and Galileo IOV was not active during the test period, but has a comparable performance (Waller et al. 2010).

^{−11}belong to the oldest spacecraft of the constellation and are on orbit between 3.5 and 5.25 years. The next group of satellites in the top right plot, which reach an Allan deviation of 1.5 × 10

^{−11}, consists of three satellites with service duration between 2.3 and 4.3 years. The majority of the satellites belong to the third group, which reaches an ADEV of 1.0 × 10

^{−12}and is depicted in the middle left plot. The newest spacecraft of this group is approximately 0.5 years old and the oldest one has an operational lifetime of 4.5 years. The three satellites with the best performance shown in the middle right plot reach an ADEV of 7.0 × 10

^{−13}. These satellites belong to the most recent satellites launched in October and November of 2011. The overall trend for GLONASS satellites is that the stability of the onboard clocks has increased for the more recently launched satellites. However, some of the older satellites still have a reasonably good ADEV performance, for example R11 and R19, which have a lifetime of approximately 4.5 years, but are still among the second best clocks on orbit.

Allan deviation at selected averaging intervals for Kalman-filter-based clock results. Satellite IDs are GPS SVN number, GLONASS satellite number, GEO or IGSO satellite number for COMPASS and commonly used satellite names for GIOVE, IOV, and QZSS. Specific information about the atomic frequency standard has been added where available. Note that the Rubidium clocks used for Block IIF (G01/SVN63 and G25/SVN62) and for QZSS have an even lower ADEV when analyzed with the 1WCP or 3WCP method (compare also Fig. 3)

PRN | Sat-ID | Clock | Averaging intervals of Allan deviation | ||||
---|---|---|---|---|---|---|---|

1 s | 10 s | 100 s | 1,000 s | 10,000 s | |||

G01 | SVN63 | Rb1 | 4.625e−12 | 1.377e−12 | 2.900e−13 | 6.945e−14 | 5.677e−14 |

G02 | SVN61 | Rb1 | 3.394e−12 | 2.118e−12 | 1.492e−12 | 1.854e−13 | 3.672e−14 |

G03 | SVN33 | Cs4 | 3.789e−12 | 2.300e−12 | 2.015e−12 | 7.660e−13 | 2.155e−13 |

G04 | SVN34 | Rb1 | 6.729e−12 | 2.822e−12 | 8.379e−13 | 2.991e−13 | 8.619e−14 |

G05 | SVN50 | Rb1 | 4.029e−12 | 2.542e−12 | 1.581e−12 | 2.200e−13 | 4.025e−14 |

G06 | SVN36 | Rb1 | 4.098e−12 | 2.195e−12 | 7.244e−13 | 2.103e−13 | 4.335e−14 |

G07 | SVN48 | Rb3 | 4.167e−12 | 2.461e−12 | 1.578e−12 | 2.021e−13 | 4.068e−14 |

G08 | SVN38 | Cs4 | 4.417e−12 | 2.068e−12 | 2.227e−12 | 9.535e−13 | 2.686e−13 |

G09 | SVN39 | Cs4 | 3.567e−12 | 1.664e−12 | 1.780e−12 | 7.371e−13 | 2.654e−13 |

G10 | SVN40 | Cs3 | 3.943e−12 | 1.979e−12 | 1.956e−12 | 7.904e−13 | 1.826e−13 |

G11 | SVN46 | Rb1 | 4.446e−12 | 2.257e−12 | 1.495e−12 | 1.985e−13 | 4.908e−14 |

G12 | SVN58 | Rb3 | 3.662e−12 | 2.606e−12 | 1.757e−12 | 1.979e−13 | 3.658e−14 |

G13 | SVN43 | Rb1 | 7.512e−12 | 5.103e−12 | 1.682e−12 | 2.781e−13 | 4.301e−14 |

G14 | SVN41 | Rb1 | 4.025e−12 | 2.267e−12 | 1.491e−12 | 1.867e−13 | 2.450e−14 |

G15 | SVN55 | Rb3 | 3.655e−12 | 2.259e−12 | 1.513e−12 | 1.954e−13 | 3.870e−14 |

G16 | SVN56 | Rb3 | 3.656e−12 | 2.438e−12 | 1.659e−12 | 3.001e−13 | 3.878e−14 |

G17 | SVN53 | Rb3 | 4.041e−12 | 2.276e−12 | 1.503e−12 | 2.241e−13 | 4.232e−14 |

G18 | SVN54 | Rb1 | 4.017e−12 | 2.395e−12 | 1.686e−12 | 2.326e−13 | 3.428e−14 |

G19 | SVN59 | Rb3 | 4.212e−12 | 2.179e−12 | 1.538e−12 | 1.883e−13 | 2.746e−14 |

G20 | SVN51 | Rb1 | 3.478e−12 | 2.168e−12 | 1.422e−12 | 3.330e−13 | 4.330e−14 |

G21 | SVN45 | Rb3 | 4.666e−12 | 2.976e−12 | 2.114e−12 | 4.286e−13 | 5.063e−14 |

G22 | SVN47 | Rb3 | 3.375e−12 | 2.219e−12 | 1.420e−12 | 1.864e−13 | 3.534e−14 |

G23 | SVN60 | Rb2 | 4.288e−12 | 2.100e−12 | 1.389e−12 | 1.830e−13 | 4.315e−14 |

G25 | SVN62 | Rb1 | 4.137e−12 | 1.497e−12 | 2.876e−13 | 6.393e−14 | 1.439e−14 |

G26 | SVN26 | Rb2 | 4.340e−12 | 1.667e−12 | 4.244e−13 | 1.227e−13 | 5.367e−14 |

G28 | SVN44 | Rb2 | 4.690e−12 | 2.186e−12 | 1.447e−12 | 2.036e−13 | 2.705e−14 |

G29 | SVN57 | Rb3 | 3.757e−12 | 2.270e−12 | 1.366e−12 | 2.251e−13 | 2.301e−14 |

G30 | SVN35 | Rb1 | 4.105e−12 | 1.899e−12 | 4.163e−13 | 1.046e−13 | 4.236e−14 |

G31 | SVN52 | Rb3 | 3.675e−12 | 2.189e−12 | 1.476e−12 | 1.959e−13 | 3.114e−14 |

G32 | SVN23 | Rb2 | 4.449e−12 | 1.683e−12 | 4.237e−13 | 1.511e−13 | 1.097e−13 |

R01 | R730 | Cs | 6.863e−12 | 3.474e−12 | 9.218e−13 | 3.158e−13 | 6.027e−14 |

R02 | R728 | Cs | 6.541e−12 | 3.412e−12 | 9.687e−13 | 2.619e−13 | 8.982e−14 |

R03 | R744 | Cs | 8.013e−12 | 2.600e−12 | 7.090e−13 | 1.840e−13 | 3.043e−14 |

R04 | R742 | Cs | 9.087e−12 | 2.709e−12 | 7.127e−13 | 2.195e−13 | 2.760e−14 |

R05 | R734 | Cs | 7.071e−12 | 4.147e−12 | 1.094e−12 | 3.392e−13 | 1.404e−13 |

R06 | R733 | Cs | 6.959e−12 | 5.907e−12 | 1.603e−12 | 4.834e−13 | 7.415e−14 |

R07 | R745 | Cs | 8.985e−12 | 3.475e−12 | 9.376e−13 | 2.453e−13 | 6.465e−14 |

R08 | R729 | Cs | 7.931e−12 | 6.447e−12 | 1.796e−12 | 5.841e−13 | 1.592e−13 |

R09 | R736 | Cs | 8.042e−12 | 4.006e−12 | 1.095e−12 | 3.720e−13 | 9.781e−14 |

R11 | R723 | Cs | 6.429e−12 | 3.838e−12 | 1.070e−12 | 3.430e−13 | 9.566e−14 |

R12 | R737 | Cs | 1.239e−11 | 3.374e−12 | 9.160e−13 | 2.438e−13 | 3.884e−14 |

R13 | R721 | Cs | 7.486e−12 | 5.066e−12 | 1.514e−12 | 4.685e−13 | 1.020e−13 |

R14 | R715 | Cs | 6.182e−12 | 7.058e−12 | 3.011e−12 | 8.056e−13 | 3.196e−13 |

R15 | R716 | Cs | 7.312e−12 | 1.276e−11 | 3.544e−12 | 9.471e−13 | 2.057e−13 |

R16 | R738 | Cs | 1.264e−11 | 3.917e−12 | 1.088e−12 | 2.979e−13 | 1.014e−13 |

R17 | R746 | Cs | 9.895e−12 | 2.474e−12 | 6.404e−13 | 1.635e−13 | 6.931e−14 |

R18 | R724 | Cs | 1.175e−11 | 9.982e−12 | 2.966e−12 | 8.647e−13 | 2.000e−13 |

R19 | R720 | Cs | 5.693e−12 | 3.188e−12 | 9.263e−13 | 3.132e−13 | 1.267e−13 |

R20 | R719 | Cs | 6.142e−12 | 8.413e−12 | 3.004e−12 | 8.693e−13 | 4.293e−13 |

R22 | R731 | Cs | 5.956e−12 | 3.307e−12 | 9.204e−13 | 2.366e−13 | 1.124e−13 |

R23 | R732 | Cs | 6.703e−12 | 3.841e−12 | 1.114e−12 | 3.204e−13 | 5.786e−14 |

R24 | R735 | Cs | 7.449e−12 | 3.239e−12 | 9.009e−13 | 3.020e−13 | 7.161e−14 |

C01 | GEO-01 | – | 7.829e−12 | 2.131e−12 | 5.018e−13 | 1.455e−13 | 5.623e−14 |

C03 | GEO-03 | – | 7.346e−12 | 1.482e−12 | 3.699e−13 | 1.121e−13 | 7.137e−14 |

C04 | GEO-04 | – | 1.064e−10 | 3.515e−11 | 3.525e−12 | 3.623e−13 | 5.490e−14 |

C06 | IGSO-06 | – | 7.531e−12 | 1.933e−12 | 5.263e−13 | 1.662e−13 | 4.266e−14 |

C07 | IGSO-07 | – | 7.862e−12 | 1.979e−12 | 5.226e−13 | 1.471e−13 | 5.140e−14 |

C08 | IGSO-08 | – | 8.942e−12 | 2.422e−12 | 5.987e−13 | 1.609e−13 | 4.734e−14 |

C09 | IGSO-09 | – | 7.973e−12 | 2.248e−12 | 1.075e−12 | 3.840e−13 | 1.482e−13 |

C10 | IGSO-10 | – | 9.565e−12 | 2.516e−12 | 6.524e−13 | 2.372e−13 | 1.013e−13 |

E11 | IOV-PFM | Rb | 5.364e−12 | 1.302e−12 | 3.644e−13 | 7.491e−14 | 3.441e−14 |

E12 | IOV-FM2 | Rb | 5.636e−12 | 1.328e−12 | 3.883e−13 | 8.477e−14 | 5.623e−14 |

E51 | GIOVE-A | Rb | 8.450e−12 | 1.980e−12 | 5.111e−13 | 1.535e−13 | 6.786e−14 |

E52 | GIOVE-B | Rb | 1.162e−11 | 1.673e−12 | 4.318e−13 | 1.183e−13 | 2.251e−14 |

J01 | QZS-1 | Rb2 | 6.059e−12 | 1.355e−12 | 3.146e−13 | 7.880e−14 | 1.010e−13 |

## Linear clock extrapolation error

The analysis in the previous section has illustrated that a variety of satellite atomic frequency standards with different clock stability is employed in the current satellite navigation systems. The analysis in this section focuses on the effect of AFS stability on the clock prediction error. This is of interest especially for real-time applications and in particular for real-time precise-point-positioning services, which provide accurate satellite clock estimates to their users via Internet, radio link, or satellite broadcast. Bandwidth limitations of these transmission channels may put constraints on the minimal possible sampling rate of the correction data. Therefore, the sampling rate should be selected as long as possible without compromising the positioning accuracy due to clock extrapolation errors. To get insight into the maximum possible sampling interval, the prediction errors for a representative selection of satellite clocks has been computed based on the high-rate clock solutions of the previous sections. For the analysis, two clock values with a certain time interval have been used for a linear extrapolation over the same interval into the future. This means, for example, that the extrapolated clock over 100 s has been computed based on two clock estimates with 100 s spacing. The extrapolated clock correction has then been compared with the estimated value to compute the extrapolation error.

## Summary and conclusions

Different methods to assess the short-term stability of GNSS atomic frequency standards have been introduced and compared. The 1WCP carrier-phase method requires a highly stable clock (typically a hydrogen maser) to be connected to the receiver tracking the satellite of interest and at least coarse knowledge of the satellite orbit to remove the major part of the observation geometry from the carrier-phase measurements. Residual effects are removed with a polynomial fit. In order to avoid that the clock estimates are dominated by measurement noise, a data arc of about 1 h near the highest elevation of the satellite pass is selected. With the 3WCP method, this clock estimation strategy can even be used if the satellite of interest is not tracked by a receiver with a sufficiently stable oscillator. Both methods are straightforward to implement and allow the assessment of satellite clocks for short timescales in a quick and simple manner. Limitations are the maximum averaging interval of about 1,000 s and, for the 3WCP method, the effects of unfiltered measurement noise at very short timescales.

The second method is Kalman-filter-based clock estimation using a small ground station network. This method is more demanding in terms of algorithmic complexity and computational load, but still simpler to set up than a traditional least-squares estimation approach. The method provides realistic clock results over timescales from 1 to 10,000 s for all current types of satellite clock except for the most precise Block IIF and QZSS AFS. This limitation is most likely due to the comparably small network size, the estimation of float instead of fixed ambiguities, and the fact that only one ground clock could be constrained. Future work will investigate whether a constrained ensemble of hydrogen masers on the ground can improve the results for the Block IIF/QZSS clocks.

The comparison of Allan deviation results for GPS, GLONASS, GIOVE, Galileo IOV, COMPASS, and QZSS has shown that the Block IIF clocks, which are also used for QZSS, are the most stable clocks currently on orbit. Other publications have shown that a similar performance is achieved by the Galileo PHM. The satellite clocks of the current GLONASS constellation can be divided into four different groups depending on the ADEV results. As expected, more recently launched satellites show improved clock stability. Comparisons of the clock extrapolation errors have shown that all satellite clocks, except for the worst performing GLONASS clocks, have extrapolation errors of less than 1 cm for a 10-s interpolation interval.

## Notes

### Acknowledgments

The IGS is acknowledged for providing high-rate tracking data and GPS/GLONASSS orbit. JAXA and Curtin University of Technology are acknowledged for providing high-rate data from GNSS receivers.

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