An investigation of GNSS atomic clock behavior at short time intervals

A technique for obtaining clock measurements from individual GNSS satellites at short time intervals is presented. The methodology developed in this study allows for accurate satellite clock stability analysis without an ultra-stable clock at the ground receiver. Variations in the carrier phase caused by the satellite clock are isolated using a combination of common GNSS carrier-phase processing techniques. Furthermore, the white phase variations caused by the thermal noise of the collection and processing equipment are statistically modeled and removed, allowing for analysis of clock performance at subsecond intervals. Allan deviation analyses of signals collected from GPS and GLONASS satellites reveal distinct intervals of clock noise for timescales less than 100 s. The clock data collected from GPS Block IIA, IIR, IIR-M, and GLONASS satellites reveal similar stability performance at time periods greater than 20 s. The GLONASS clock stability in the 0.6–10 s range, however, is significantly worse than GPS. Applications that rely on ultra-stable clock behavior from the GLONASS satellites at these timescales may therefore require high-rate corrections to estimate and remove oscillator-based errors in the carrier phase.     

Influence of clock jump on the velocity and acceleration estimation with a single GPS receiver based on carrier-phase-derived Doppler

Since the Selective Availability was turned off, the velocity and acceleration can be determined accurately with a single GPS receiver using raw Doppler measurements. The carrier-phase-derived Doppler measurements are normally used to determine velocity and acceleration when there is no direct output of the raw Doppler observations in GPS receivers. Due to GPS receiver clock drifts, however, a GPS receiver clock jump occurs when the GPS receiver clock resets itself (typically with 1 ms increment/decrement) to synchronize with the GPS time. The clock jump affects the corresponding relationship between measurements and their time tags, which results in non-equidistant measurement sampling in time or incorrect time tags. This in turn affects velocity and acceleration determined for a GPS receiver by the conventional method which needs equidistant carrier phases to construct the derived Doppler measurements. To overcome this problem, an improved method that takes into account, GPS receiver clock jumps are devised to generate non-equidistant-derived Doppler observations based on non-equidistant carrier phases. Test results for static and kinematic receivers, which are obtained by using the conventional method without reconstructing the equidistant continuous carrier phases, show that receiver velocity and acceleration suffered significantly from clock jumps. An airborne kinematic experiment shows that the greatest impact on velocity and acceleration reaches up to 0.2 m/s, 0.1 m/s² for the horizontal component and 0.5 m/s, 0.25 m/s² for the vertical component. Therefore, it can be demonstrated that velocity and acceleration measurements by using a standalone GPS receiver can be immune to the influence of GPS receiver clock jumps with the proposed method.     

Short-term analysis of GNSS clocks

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.     

神经网络在卫星钟差短期预报中的应用研究

本文针对卫星钟差的特点,提出了基于神经网络的卫星钟差短期预报模型,给出了基于径向基函数(RBF)网络进行卫星钟差预测的基本思想、预测模型和实施步骤,并对比分析了神经网络模型与灰色系统理论模型的区别.为验证本文提出的预报模型的可行性和有效性,利用GPS卫星钟差数据进行钟差预报精度分析,并与灰色系统模型进行对比分析.仿真结...     

Assessing antenna field of view and receiver clocks of COSMIC and GRACE satellites: lessons for COSMIC-2

We provide suggestions for the approved COSMIC-2 satellite mission regarding the field of view (FOV) and the clock stability of its future GNSS receiver based on numerical analyses using COSMIC GPS data. While the GRACE GPS receiver is mounted on the zenith direction, the precise orbit determination (POD) antennas of COSMIC are not. The COSMIC antenna design results in a narrow FOV and a reduction in the number of GPS observations. To strengthen the GPS geometry, GPS data from two POD antennas of COSMIC are used to estimate its orbits. The phase residuals of COSMIC are at the centimeter level, compared to the millimeter level of GRACE. The receiver clock corrections of COSMIC and GRACE are at the microsecond and nanosecond levels, respectively. The clock spectra of COSMIC at the frequencies of 0–0.005 Hz contain significant powers, indicating potential systematic errors in its clock corrections. The clock stability, expressed by the Allan deviation, of COSMIC ranges from 10^?9 to 10^?11 over 1 to 10⁴ s, compared to 10^?12 to 10^?14 for GRACE. Compared to USO-based clock of GRACE, the clock of COSMIC is degraded in its stability and is linked to the reduction of GPS data quality. Lessons for improvement of COSMIC-2 over COSMIC in FOV and receiver clock stability are given.     

利用SLR和伪距资料确定导航卫星钟差

提出了综合利用SLR和GPS伪距资料测定导航卫星钟差的方法,采用2002年10月的SLR和伪距实测数据计算了GPS 35卫星的钟差,并对GPS 35卫星的钟差进行了预报,为了验证计算结果的精度,将本文计算的卫星钟差与IGS精密钟差进行了比较.通过比较分析发现:综合利用SLR和伪距资料测定的导航卫星钟差精度优于3 ns,测定的导航卫星钟差与实际卫星钟差不存在系统差;导航卫星钟差的预报精度与计算卫星钟速的时间跨度有关;可以分离卫星坐标和卫星钟差之间的相互影响,便于对卫星钟差的研究.     

(Near-)real-time orbit determination for GNSS radio occultation processing

The processing of GPS radio occultation measurements for use in numerical weather predictions requires a precise orbit determination (POD) of the host satellite in near-real-time. Making use of data from the GRAS instrument on Metop-A, the performance of different GPS ephemeris products and processing concepts for near-real-time and real-time POD is compared. While previous analyses have focused on the achievable along-track velocity accuracy, this study contributes a systematic comparison of the resulting estimated bending angles. This enables a more rigorous trade-off of different orbit determination methodologies in relation to the end-user needs for atmospheric science products. It is demonstrated that near-real-time GPS orbit and clock products have reached a sufficient quality to determine the Metop-A along-track velocity with an accuracy of better than 0.05 mm/s that was formerly only accessible in post-processing. The resulting bending angles are shown to exhibit standard deviation and bias differences of less than 0.3 % compared with post-processed products up to altitudes of at least 40 km, which is notably better than 1 % accuracy typically assumed for numerical weather predictions in this height regime. Complementary to the analysis of ground-based processing schemes, the potential of autonomous on-board orbit determination is investigated for the first time. Using actual GRAS flight data, it is shown that a 0.5 m 3D rms position accuracy and a 0.2 mm/s along-track velocity accuracy can in fact be obtained in real-time with the currently available GPS broadcast ephemeris quality. Bending angles derived from the simulated real-time processing exhibit a minor performance degradation above tangent point heights of 40 km but negligible differences with respect to ground-based products below this altitude. Onboard orbit determination and, if desired, bending angle computation, can thus enable a further simplification of the ground segment in future radio occultation missions and contribute to reduced product latencies for radio occultation data assimilation in numerical weather predictions.     

Impact of GPS differential code bias in dual- and triple-frequency positioning and satellite clock estimation

The features and differences of various GPS differential code bias (DCB)s are discussed. The application of these biases in dual- and triple-frequency satellite clock estimation is introduced based on this discussion. A method for estimating the satellite clock error from triple-frequency uncombined observations is presented to meet the need of the triple-frequency uncombined precise point positioning (PPP). In order to evaluate the estimated satellite clock error, the performance of these biases in dual- and triple-frequency positioning is studied. Analysis of the inter-frequency clock bias (IFCB), which is a result of constant and time-varying frequency-dependent hardware delays, in ionospheric-free code-based (P1/P5) single point positioning indicates that its influence on the up direction is more pronounced than on the north and east directions. When the IFCB is corrected, the mean improvements are about 29, 35 and 52% for north, east and up directions, respectively. Considering the contribution of code observations to PPP convergence time, the performance of DCB(P1–P2), DCB(P1–P5) and IFCB in GPS triple-frequency PPP convergence is investigated. The results indicate that the DCB correction can accelerate PPP convergence by means of improving the accuracy of the code observation. The performance of these biases in positioning further verifies the correctness of the estimated dual- and triple-frequency satellite clock error.     

Estimation of satellite position,clock and phase bias corrections

Precise point positioning with integer ambiguity resolution requires precise knowledge of satellite position, clock and phase bias corrections. In this paper, a method for the estimation of these parameters with a global network of reference stations is presented. The method processes uncombined and undifferenced measurements of an arbitrary number of frequencies such that the obtained satellite position, clock and bias corrections can be used for any type of differenced and/or combined measurements. We perform a clustering of reference stations. The clustering enables a common satellite visibility within each cluster and an efficient fixing of the double difference ambiguities within each cluster. Additionally, the double difference ambiguities between the reference stations of different clusters are fixed. We use an integer decorrelation for ambiguity fixing in dense global networks. The performance of the proposed method is analysed with both simulated Galileo measurements on E1 and E5a and real GPS measurements of the IGS network. We defined 16 clusters and obtained satellite position, clock and phase bias corrections with a precision of better than 2 cm.     

Performance of the BDS3 experimental satellite passive hydrogen maser

Various types of onboard atomic clocks such as rubidium, cesium and hydrogen have different frequency accuracies and frequency drift rate characteristics. A passive hydrogen maser (PHM) has the advantage of low-frequency drift over a long period, which is suitable for long-term autonomous satellite time keeping. The third generation of Beidou Satellite Navigation System (BDS3) is equipped with PHMs which have been independently developed by China for their IGSO and MEO experimental satellites. Including Galileo, it is the second global satellite navigation system that uses PHM as a frequency standard for navigation signals. We briefly introduce the PHM design at the Shanghai Astronomical Observatory (SHAO) and detailed performance evaluation of in-orbit PHMs. Using the high-precision clock values obtained by satellite-ground and inter-satellite measurement and communication systems, we analyze the frequency stability, clock prediction accuracy and clock rate variation characteristics of the BDS3 experimental satellites. The results show that the in-orbit PHM frequency stability of the BDS3 is approximately 6 × 10^?15 at 1-day intervals, which is better than those of other types of onboard atomic clocks. The BDS3 PHM 2-, 10-h and 7-day clock prediction precision values are 0.26, 0.4 and 2.2 ns, respectively, which are better than those of the BDS3 rubidium clock and most of the GPS Block IIF and Galileo clocks. The BDS3 PHM 15-day clock rate variation is ? 1.83 × 10^?14 s/s, which indicates an extremely small frequency drift. The 15-day long-term stability results show that the BDS3 PHM in-orbit stability is roughly the same as the ground performance test. The PHM is expected to provide a highly stable time and frequency standard in the autonomous navigation case.     

Precise orbit determination of the Sentinel-3A altimetry satellite using ambiguity-fixed GPS carrier phase observations

The Sentinel-3 mission takes routine measurements of sea surface heights and depends crucially on accurate and precise knowledge of the spacecraft. Orbit determination with a targeted uncertainty of less than 2 cm in radial direction is supported through an onboard Global Positioning System (GPS) receiver, a Doppler Orbitography and Radiopositioning Integrated by Satellite instrument, and a complementary laser retroreflector for satellite laser ranging. Within this study, the potential of ambiguity fixing for GPS-only precise orbit determination (POD) of the Sentinel-3 spacecraft is assessed. A refined strategy for carrier phase generation out of low-level measurements is employed to cope with half-cycle ambiguities in the tracking of the Sentinel-3 GPS receiver that have so far inhibited ambiguity-fixed POD solutions. Rather than explicitly fixing double-difference phase ambiguities with respect to a network of terrestrial reference stations, a single-receiver ambiguity resolution concept is employed that builds on dedicated GPS orbit, clock, and wide-lane bias products provided by the CNES/CLS (Centre National d’Études Spatiales/Collecte Localisation Satellites) analysis center of the International GNSS Service. Compared to float ambiguity solutions, a notably improved precision can be inferred from laser ranging residuals. These decrease from roughly 9 mm down to 5 mm standard deviation for high-grade stations on average over low and high elevations. Furthermore, the ambiguity-fixed orbits offer a substantially improved cross-track accuracy and help to identify lateral offsets in the GPS antenna or center-of-mass (CoM) location. With respect to altimetry, the improved orbit precision also benefits the global consistency of sea surface measurements. However, modeling of the absolute height continues to rely on proper dynamical models for the spacecraft motion as well as ground calibrations for the relative position of the altimeter reference point and the CoM.     

灰色模型在快速卫星钟差预报中的应用

GPS实时精密单点定位需要实时的、精确的、可靠的预报卫星钟差预报,因此卫星钟差的预报是一项非常重要的工作,它对实时的高精度导航定位具有重要意义。为导航定位提供时间标准的导航卫星原子钟是非常精密的仪器,对外界环境非常敏感,无法将卫星钟差作为普通的白噪声处理,可以但可将卫星钟差看作是灰色系统来进行研究。本文根据灰色系统相关理论,将灰色系统模型GM(1,1)应用到卫星钟差的预报,并用IGS超快速星历建立了预报卫星钟差的灰色预测模型,研究了卫星钟差的变化规律。结果表明:灰色模型可用于卫星钟差的短期预报,它对超快速星历的预报精度与IGS产品中的IGU超快速星历本身的预报精度相当。     

SBAS orbit and satellite clock corrections for precise point positioning

The quality of real-time GPS positions based on the method of precise point positioning (PPP) heavily depends on the availability and accuracy of GPS satellite orbits and satellite clock corrections. Satellite-based augmentation systems (SBAS) provide such corrections but they are actually intended to be used for wide area differential GPS with positioning results on the 1-m accuracy level. Nevertheless, carrier phase-based PPP is able to achieve much more accurate results with the same correction values. We applied SBAS corrections for dual-frequency PPP and compared the results with PPP obtained using other real-time correction data streams, for example, the GPS broadcast message and precise corrections from the French Centre National d’Etudes Spatiales and the German Deutsches Zentrum für Luft- und Raumfahrt. Among the three existing SBAS, the best results were achieved for the North American wide area augmentation system (WAAS): horizontal and vertical position accuracies were considerably smaller than 10 cm for static 24-h observation data sets and smaller than 30 cm for epoch-by-epoch solutions with 2 h of continuous observations. The European geostationary navigation overlay service and the Japanese multi-functional satellite augmentation system yield positioning results with biases of several tens of centimeters and variations larger by factors of 2–4 as compared to WAAS.     

基于高频观测值的不同GNSS卫星钟稳定性分析

在GNSS高精度数据处理中,卫星钟差往往是决定结果精度的核心因素之一。采用20 Hz的双频观测数据对GNSS星载原子钟0.05~100 s平滑时间下的短期稳定性进行分析,通过星间单差的方法消除接收机钟差,采用无电离层组合及夜间观测避免电离层高阶项短期变化的影响,同时采用经验模型和映射函数来进行对流层延迟改正。通过Lag 1自相关函数分析了影响GNSS卫星钟稳定性的主要噪声类型,并使用阿伦方差计算分析GPS、GLONASS及BDS各自系统内不同卫星组合之间的钟差。结果表明,GPS、GLONASS及BDS系统钟差稳定性0.05秒稳均可达到10-10量级,秒稳可达10-11量级。可以认定,GPS、GLONASS及BDS在短期内的稳定性量级相当,从而验证了基于星间单差的BDS掩星数据处理方案的可行性。     

A modified mix-differenced approach for estimating multi-GNSS real-time satellite clock offsets

A modified mixed-differenced approach for estimating multi-GNSS real-time clock offsets is presented. This approach, as compared to the earlier presented mixed-differenced approach which uses epoch-differenced and undifferenced observations, further adds a satellite-differenced process. The proposed approach, based on real-time orbit products and a mix of epoch-differenced and satellite-differenced observations to estimate only satellite clock offsets and tropospheric zenith wet delays, has fewer estimated parameters than other approaches, and thus its implementing procedure is efficient and can be performed and extended easily. To obtain high accuracy, the approach involves three steps. First, the high-accuracy tropospheric zenith wet delay of each station is estimated using mixed-differenced carrier phase observations. Second, satellite clock offset changes between adjacent epochs are estimated using also mixed-differenced carrier phase observations. Third, the satellite clock offsets at the initial epoch are estimated using satellite-differenced pseudorange observations. Finally, the initial epoch clock results and clock offset changes are concatenated to obtain the clock results of the current epoch. To validate the real-time satellite clock results, multi-GNSS post-processing clock products from IGS ACs were selected for comparison. From the comparison, the standard deviations of the GPS, GLONASS, BeiDou and Galileo systems clock results are approximately 0.1–0.4 ns, except for the BeiDou GEO satellites. The root mean squares are about 0.4–2.3 ns, which are similar to those of other international real-time products. When the clock estimates were assessed based on a pseudo-kinematic PPP procedure, the positioning accuracies in the East, North and Up components reach 5.6, 5.5 and 7.6 cm, respectively, which meet the centimeter level and are comparable to the application of other products.     

Real-time clock offset prediction with an improved model

The GPS orbit precision of the IGS ultra-rapid predicted (IGU-P) products has been remarkably improved since 2007. However, the satellite clock offsets of the IGU-P products have not shown sufficient high-quality prediction to achieve sub-decimeter precision in real-time precise point positioning (RTPPP), being at the level of 1–3 ns (30–90 cm) RMS in recent years. An improved prediction model for satellite clocks is proposed in order to enhance the precision of predicted clock offsets. First, the proposed prediction model adds a few cyclic terms to absorb the periodic effects, and a time adaptive function is used to adjust the weight of the observation in the prediction model. Second, initial deviations of the predictions are reduced by using a recomputed constant term. The simulation results have shown that the proposed prediction model can give a better performance than the IGU-P clock products and can achieve precision better than 0.55 ns (16.5 cm) in real-time predictions. In addition, the RTPPP method was chosen to test the efficiency of the new model for real-time static and kinematic positioning. The numerical examples using the data set of 140 IGS stations show that the static RTPPP precision based on the proposed clock model has been improved about 22.8 and 41.5 % in the east and height components compared to the IGU-P clock products, while the precisions in the north components are the equal. The kinematic example using three IGS stations shows that the kinematic RTPPP precision based on the proposed clock model has improved about 30, 72 and 44 % in the east, north and height components.     

Relativistic Time Transformations in GPS

Since Selective Availability was permanently switched off on 7 May 2000, most of the GPS satellite clocks have been well behaved. During a 24-h period precise satellite clock solutions, corrected for GPS conventional relativistic corrections, follow straight lines within a few nanoseconds. The linear clock fit RMS for the best satellite clocks are well below the 1-ns level, which is consistent with the nominal stability of the GPS frequency standards. Typically, the GPS satellite clocks show an Allan variance at or below one part in 10¹¹/100 s for the Cesium frequency standards and a few parts in 10¹²/100 s for the Rubidium frequency standards. These results correspond to clock RMSs for 15-min sampling at or below 3 and 0.3 ns, respectively. This already confirms experimentally that the conventional periodic relativity correction of the GPS system, also adopted for all the IGS clock solution products, is precise and correct to 0.6 ns or better. To establish the precision limits of the GPS conventional relativity treatment, the relativistic time transformations of GPS satellite frequency and clocks are critically reviewed, taking into account all the contributions larger than the 10⁻¹⁸ (or 0.001 ns). The conventional GPS relativity treatment was found to be accurate, i. e., correctly modeling the actual relativistic frequency (clock rate) effects of GPS satellites at about the 10⁻¹⁴ level. However, it is also affected by small periodic errors of the same magnitude. The integration of these small periodic frequency relativistic errors gives the approximation errors of the conventional periodic relativistic clock correction with amplitudes of about 0.1 ns and a predominant period equal to a half of the orbital period (∼ 6 h). These approximation errors of the conventional GPS relativistic clock correction are at about the same level as the current precision of the IGS clock solutions. ? 2002 Wiley Periodicals, Inc.     

基于指数平滑法的GPS卫星钟差预报

提出了一种基于指数平滑法的GPS卫星钟差预报方法。该方法可采用少量数据建模,且计算过程简单、方便,尤其是在缺少相关历史数据或数据变化趋势不明显、不稳定的情况下,用该方法仍可取得较好的效果。通过与GPS卫星钟差预报中常用的二次多项式模型和灰色预测模型的对比分析,结果表明:指数平滑法适用于GPS卫星钟差的中、短期预报,其预报精度可达ns级;在利用小数据量建模的情况下,其预报效果优于二次多项式模型,与灰色模型的预报效果基本相当;该方法还可用于GPS卫星钟差的长期预报,其预报精度可达μs级,与灰色预测模型的精度相当。     

Multifrequency algorithms for precise point positioning: MAP3

We present the new MAP3 algorithms to perform static precise point positioning (PPP) from multifrequency and multisystem GNSS observations. MAP3 represents a two-step strategy in which the least squares theory is applied twice to estimate smoothed pseudo-distances, initial phase ambiguities, and slant ionospheric delay first, and the absolute receiver position and its clock offset in a second adjustment. Unlike the classic PPP technique, in our new approach, the ionospheric-free linear combination is not used. The combination of signals from different satellite systems is accomplished by taking into account the receiver inter-system bias. MAP3 has been implemented in MATLAB and integrated within a complete PPP software developed on site and named PCube. We test the MAP3 performance numerically and contrast it with other external PPP programs. In general, MAP3 positioning accuracy with low-noise GPS dual-frequency observations is about 2.5 cm in 2-h observation periods, 1 cm in 10 h, and 7 mm after 1 day. This means an improvement in the accuracy in short observation periods of at least 7 mm with respect to the other PPP programs. The MAP3 convergence time is also analyzed and some results obtained from real triple-frequency GPS and GIOVE observations are presented.     

Improved method for estimating the inter-frequency satellite clock bias of triple-frequency GPS

Considering the contribution of the hardware biases to the estimated clock errors, an improved method for estimating the satellite inter-frequency clock bias (IFCB) is presented, i.e., the difference in the satellite clock error as computed from ionospheric-free pseudorange and carrier phase observations using L1/L2 and P1/P2 versus L1/L5 and P1/P5. The IFCB is composed of a constant and a variable part. The constant part is the inter-frequency hardware bias (IFHB). It contains the satellite and receiver hardware delays and can be expressed as a function of the DCBs [DCB (P1 ? P2) and DCB (P1 ? P5)]. When a reference satellite is selected, the satellite IFHB can be computed but is biased by a reference satellite IFHB. This bias will not affect the utilization of IFCB in positioning since it can be absorbed by the receiver clock error. Triple-frequency observations of 30 IGS stations between June 1, 2013, and May 31, 2014, were processed to show the variations of the IFHB. The IFHB values show a long-term variation with time. When a linear and a fourth-order harmonic function are used to model the estimated IFCB, which contains contributions of the hardware delays and clock errors, the results show that 89 % of the IFCB can be corrected given the current five triple-frequency GPS satellites with the averaged fitting RMS of 1.35 cm. Five days of data are processed to test the estimated satellite clock errors using the strategy presented. The residuals of P1/P5 and L1/L5 have a STD of <0.27 m and 0.97 cm, respectively. In addition, most predicted satellite IFCBs reach an accuracy of centimeter level and its mean accuracy of 5 days is better than 7 cm.