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.     

Analysis of BDS satellite clocks in orbit

The products of Wuhan University with 5-min sampling are used to analyze the characteristics of BeiDou satellite clocks. Two nanoseconds root-mean-square (RMS) variations are obtained for 1-day quadratic fits in the sub-daily region. The relativistic effects of BDS clocks are also studied. General relativity predicts that linear variation of the semimajor axes of geostationary and inclined geosynchronous satellites causes a quadratic clock drift with a magnitude at the 10^?16/day level. The observed drift is higher than what general relativity theory would produce. Several periodic terms are found in the satellite clock variations through spectrum analysis. In order to identify the origin of the BDS clock harmonics, a correlation analysis between the period or amplitude of the harmonics and properties of the satellite orbits is performed. It is found that the period of the harmonics is not exactly equal to the orbit period, but rather the ratio of the orbit period to clock period is almost the same as that of a sidereal day to solar day. The BDS clocks obey white frequency noise statistics for intervals from 300 s to several thousands seconds. For intervals greater than 10,000 s, all the BDS satellites display more complex, non-power-law behavior due to the effects of periodic clock variations.     

Precise orbit determination for the FORMOSAT-3/COSMIC satellite mission using GPS

The joint Taiwan–US mission FORMOSAT-3/ COSMIC (COSMIC) was launched on April 17, 2006. Each of the six satellites is equipped with two POD antennas. The orbits of the six satellites are determined from GPS data using zero-difference carrier-phase measurements by the reduced dynamic and kinematic methods. The effects of satellite center of mass (COM) variation, satellite attitude, GPS antenna phase center variation (PCV), and cable delay difference on the COSMIC orbit determination are studied. Nominal attitudes estimated from satellite state vectors deliver a better orbit accuracy when compared to observed attitude. Numerical tests show that the COSMIC COM must be precisely calibrated in order not to corrupt orbit determination. Based on the analyses of the 5 and 6-h orbit overlaps of two 30-h arcs, orbit accuracies from the reduced dynamic and kinematic solutions are nearly identical and are at the 2–3 cm level. The mean RMS difference between the orbits from this paper and those from UCAR (near real-time) and WHU (post-processed) is about 10 cm, which is largely due to different uses of GPS ephemerides, high-rate GPS clocks and force models. The kinematic orbits of COSMIC are expected to be used for recovery of temporal variations in the gravity field.     

广域实时精密定位与时间服务系统

广域实时精密定位与时间服务已成为GNSS应用领域研究热点,目前国内外学者围绕其模型算法已展开大量的研究。本文重点论述广域实时精密定位与时间服务数据的处理方法和服务系统,给出了基于不同基准约束的卫星钟差解算数学模型,提出通过引入外接原子钟测站、标准时间源(UTC/BDT)等不同时间基准,构建卫星拟稳基准、外接原子钟跟踪站拟稳基准及标准时间源等约束下的钟差解算模型,分析了时间基准对精密单点定位和精密单点授时的影响。本文采用实时卫星轨道、钟差、相位偏差、电离层延迟等服务产品及跟踪站实时数据,验证了系统产品可靠性及终端定位与时间服务性能。实测结果表明:GPS轨道径向精度1.8 cm,钟差STD精度约0.05 ns;BDS-3轨道径向精度6.7 cm,钟差STD精度优于0.1 ns;GPS和BDS-2电离层改正精度分别为0.74 TECU与1.03 TECU。基于该产品实现了用户端PPP、PPP-RTK及PPT、PPT-RTK服务,满足了用户实时厘米级定位和优于0.5 ns的单站时间传递服务,当采用GPS+BDS-2 PPP-RTK解算时,平面收敛至5 cm约需要12 min。     

卫星钟差单差的小波神网络预报

针对现有卫星钟差预报模型对非平稳过程预报的局限性,提出基于卫星钟差一次差值的小波神经网络预报模型。对在轨卫星钟差求取一次差值的基础上,运用小波神经网络模型预报GPS卫星钟差,同时与GM(1,1)模型预报的结果进行比较。得出BlockΠA Cs短期预报的精度能达到0.690ns,14d预报的精度最差时依然优于1ns;其余稳定性良好的卫星钟,一天预报的结果均要优于0.207ns,预报14d卫星钟差的平均精度优于0.183ns,部分卫星钟差预报精度可以达到0.050ns,预报得到的结果可以达到GPS对实时精密单点定位的要求。     

顾及钟差变化率的GPS卫星钟差预报法

提出了一种基于钟差变化率拟合建模的卫星钟差预报方法。以附加周期项的线性或二次多项式作为基础模型对钟差变化率序列进行拟合,最优估计卫星钟差的趋势项系数,然后直接使用精密定轨得到的相应时刻的卫星钟差计算预报初始时刻的基准项系数,来建立卫星钟差的预报模型。以IGS发布的快速星历(IGR)的卫星钟差为试验数据,对GPS星座中各种型号的所有卫星钟差进行预报。结果表明:本文方法3、6、12与24h的预报精度分别可达0.43、0.58、0.90与1.47ns,相比于传统的基于钟差拟合的预报方法,精度分别提高69.3%、61.8%、50.5%与37.2%;与IGS发布的超快速星历(IGU)的预报钟差相比,钟差精度分别提高15.7%、23.7%、27.4%与34.4%。     

基于TIN全球选站方法的GPS卫星钟差实时解算

解算所有GPS卫星钟差时要求选用地面跟踪站能够观测到每颗卫星,而组成该网的跟踪站数量对卫星钟差的解算效率有较大影响。跟踪站数量越多,卫星钟差的解算效率就越低,不利于实时应用。本文利用不规则三角网对全球跟踪站进行建模,提出一种新的全球均匀选站方法,并应用于卫星钟差实时解算。试验结果表明:当跟踪站个数达到25个时,卫星钟差解算精度优于0.3 ns,且随着跟踪站的增加,精度无明显提升。此跟踪站分布可作为卫星钟差实时解算的一种选站分布参考。     

顾及码频间偏差的GPS/GLONASS实时卫星钟差估计

在进行GPS/GLONASS联合卫星钟差估计时,GLONASS码频间偏差（inter-frequency bias,IFB）因卫星频率间的差异而无法被测站接收机钟差参数吸收,其一部分将进入GLONASS卫星钟差估值中。通过引入多个"时频偏差"参数（inter-system and inter-frequency bias,ISFB）及附加基准约束对测站GLONASS码IFB进行函数模型补偿,实现其与待估卫星钟差参数的有效分离,并对所估计实时卫星钟差和实时精度单点定位（real-time precise point positioning,RT-PPP）进行精度评估。结果表明,在卫星钟差估计观测方程中忽略码IFB,会明显降低GLONASS卫星钟差估值精度;新方法能有效避免码IFB对卫星钟差估值的影响,所获得GPS、GLONASS卫星钟差与ESA（European Space Agency）事后精密钟差产品偏差平均均方根值分别小于0.2 ns、0.3 ns。利用实时估计卫星钟差进行静态RT-PPP,当观测时段长为2 h时,GPS单系统、GPS/GLONASS组合系统的3D定位精度优于10 cm,GLONASS单系统3D定位精度约为15 cm;三种模式24 h单天解的3D定位精度均优于5 cm。     

GNSS广播星历轨道和钟差精度分析

为了对多个全球导航卫星系统（global navigation satellite system, GNSS）当前的广播星历精度进行一个全面的分析,对比了2014—2018年共5 a的GNSS广播星历与精密星历,并对全球定位系统（global positioning system, GPS）、格洛纳斯卫星导航系统（global navigation satellite system, GLONASS）、伽利略卫星导航系统（Galileo satellite navigation system, Galileo）、北斗卫星导航系统（BeiDou navigation satellite system, BDS）、准天顶卫星系统（quasi-zenith satellite system, QZSS）等5个系统的广播星历长期精度变化进行了分析。结果表明:5 a中GPS的广播星历轨道及钟差精度最稳定;GLONASS的广播星历轨道精度稳定性较好,但其钟差精度存在较大的离散度;Galileo得益于具备全面运行能力（full operational capability, FOC）卫星的大量发射及运行,其广播星历轨道、钟差精度大幅度变好,切向轨道、法向轨道与钟差精度已赶超GPS;BDS的广播星历轨道精度离散度较大,钟差精度出现不稳定现象;QZSS的广播星历轨道与钟差精度的稳定性与离散度相对最差。以2018年1 a的广播星历与精密星历为例分析了各个系统当前的广播星历精度,结果表明,当前GPS、GLONASS、Galileo、BDS、QZSS的考虑轨道误差与钟差误差贡献的空间信号测距误差（signal-in-space ranging error,SISRE）分别为0.806 m、2.704 m、0.320 m、1.457 m、1.645 m,表明Galileo广播星历整体精度最高,GPS次之,其次分别是BDS、QZSS和GLONASS。只考虑轨道误差贡献的SISRE分别为0.167 m、0.541 m、0.229 m、0.804 m、0.675 m,表明GPS广播星历轨道精度最高,其次分别是Galileo、GLONASS、QZSS和BDS。GPS卫星广播星历中新型号卫星的钟差精度总体要优于旧型号卫星。     

A simplified yaw-attitude model for eclipsing GPS satellites

A simplified yaw-attitude modeling, consistent with Bar-Sever (1996), has been implemented and tested in the NRCan PPP software. For Block IIR GPS satellite it is possible to model yaw-attitude control during eclipsing periods by using the constant hardware yaw rate of 0.20°/s. The Block IIR satellites maintain the nominal yaw attitude even during a shadow crossing (Y. E. Bar-Sever, private communication, 2007), except for the noon and shadow midnight turn maneuvers, both of which can be modeled and last up to 15 min. Thus, for Block IIR satellites it is possible to maintain continuous satellite clock estimation even during eclipsing periods. For the Block II/IIA satellites, it is possible to model satisfactorily the noon turns and also shadow crossing, thanks to the permanent positive yaw bias of 0.5°, implemented in November 1995. However, in order to model the Block II/IIA shadow crossings, satellite specific yaw rates should be used, either solved for or averaged yaw-rate solutions. These yaw rates as estimated by the Jet Propulsion Laboratory (JPL) can differ significantly from the nominal hardware values. The Block II/IIA post-shadow recovery periods, which last about 30 min, should be considered uncertain and cannot be properly modeled. Data from post-shadow recovery periods should, therefore, not be used in precise global GPS analyses (Bar-Sever 1996). For high-precision applications, it is essential that users implement a yaw-attitude model, which is consistent with the generation of the satellite clocks. Initial testing and analyses, based on the IGS and AC Final orbits and clocks have revealed that during eclipsing periods, significant inconsistencies in yaw-attitude modeling still exist amongst the IGS Analyses Centers, which contribute to the errors of the IGS Final clock combinations.     

不同姿态控制模式下的北斗卫星定轨策略研究

北斗卫星的姿态控制分为动态偏置、零偏置和连续动偏3种,不同类型卫星、不同姿态控制模式、不同时段下定轨精度不一致,影响了北斗系统的连续性。详细研究了北斗不同类型卫星在不同姿态控制模式下的最优定轨策略,并基于实测数据进行试验,结果表明,BeiDou-2 IGSO（inclined geosynchronous orbit）/MEO（medium earth orbit）卫星采用基于星地钟差约束下多星定轨方法和ECOM（extended CODE model）5参数模型相结合的方法定轨精度最优,零偏期间,用户等效距离误差值为2.08 m,全球激光评估轨道视向精度约为1 m;BeiDou-3 IGSO/MEO卫星采用常规多星定轨和ECOM 5参数模型相结合的方法定轨精度最优;连续动偏期间,用户等效距离误差值为1.22 m,全球激光评估轨道视向精度为0.23 m,与动偏期间精度一致;GEO（geostationary earth orbit）卫星在春秋分附近时段采用基于星地钟差约束下多星定轨方法和ECOM 9参数模型相结合的方法定轨精度最优,用户等效距离误差值为0.72 m。     

GPS/BDS卫星姿态异常对PPP相位缠绕的影响及其改正模型

在精密单点定位中,相位缠绕是一项不可忽略的误差。相位缠绕的计算严格依赖于卫星姿态的确立,不同的卫星类型产生不同的异常。本文给出了卫星在正常情况下的姿态模型和在异常情况下的姿态改正模型。使用真实数据测试以验证本文所提出模型的正确性。观察滤波收敛后出现异常情况的卫星观测值的残差,结果表明:在异常时期残差最大可能超过20 cm,然而使用本文的改正模型,残差可降低到5 cm以下。使用不同分析中心的精密轨道和钟差产品,效果存在微小差异。II/IIA卫星通过地影区域的时间最长可达1 h,此期间卫星姿态完全受航向角偏差(II/IIA为+0.5°)控制,出了地影区域后30 min,姿态难以模型化,因此这30 min的观测数据不建议采用。     

Improved GRACE kinematic orbit determination using GPS receiver clock modeling

Kinematic positions of Low Earth Orbiters based on GPS tracking are frequently used as pseudo-observations for single satellite gravity field determination. Unfortunately, the accuracy of the satellite trajectory is partly limited because the receiver synchronization error has to be estimated along with the kinematic coordinates at every observation epoch. We review the requirements for GPS receiver clock modeling in Precise Point Positioning (PPP) and analyze its impact on kinematic orbit determination for the two satellites of the Gravity Recovery and Climate Experiment (GRACE) mission using both simulated and real data. We demonstrate that a piecewise linear parameterization can be used to model the ultra-stable oscillators that drive the GPS receivers on board of the GRACE satellites. Using such a continuous clock model allows position estimation even if the number of usable GPS satellites drops to three and improves the robustness of the solution with respect to outliers. Furthermore, simulations indicate a potential accuracy improvement of the satellite trajectory of at least 40 % in the radial direction and up to 7 % in the along-track and cross-track directions when a 60-s piecewise linear clock model is estimated instead of epoch-wise independent receiver clock offsets. For PPP with real GRACE data, the accuracy evaluation is hampered by the lack of a reference orbit of significantly higher accuracy. However, comparisons with a smooth reduced-dynamic orbit indicate a significant reduction of the high-frequency noise in the radial component of the kinematic orbit.     

精密卫星钟差加密方法及其对星载GPS低轨卫星定轨精度影响

系统分析、比较了几种精密卫星钟差加密方法，研究了利用全球分布的IGS永久跟踪站的GPS观测数据估计高采样率卫星钟差参数的原理与方法，并将各种卫星钟差加密方法得到的结果与IGS数据分析中心估计的卫星钟差结果相比较。最后将不同加密方法得出的精密卫星钟差结果用于基于星载GPS双频非差观测值的CHAMP低轨卫星的定轨，并将不同方法得到的定轨精度进行比较。结果表明，利用地面跟踪站的GPS观测数据，可高精度、高密度地估计GPS卫星钟差，估计精度可达0．1～0．5ns。经地面GPS跟踪站数据估计的GPS卫星钟差，应用于基于PPP方法的低轨卫星定轨，其定轨精度在10cm以内。     

Modeling GPS satellite attitude variation for precise orbit determination

High precision geodetic applications of the Global Positioning System (GPS) require highly precise ephemerides of the GPS satellites. An accurate model for the non-gravitational forces on the GPS satellites is a key to high quality GPS orbit determination, especially in long arcs. In this paper the effect of the satellite solar panel orientation error is investigated. These effects are approximated by empirical functions to model the satellite attitude variation in long arc orbit fit. Experiments show that major part of the long arc GPS orbit errors can be accommodated by introducing a periodic variation of the satellite solar panel orientation with respect to the satellite-Sun direction, the desired direction for solar panel normal vector, with an amplitude of about 1 degree and with a frequency of once per orbit revolution.     

Single epoch estimation of the Galileo integrity chain sensor station clock offsets

The Galileo integrity chain depends on a number of key factors, one of which is contamination of the signal-in-space errors with residual errors other than imperfect modelling of satellite orbits and clocks. A potential consequence of this is that the user protection limit is driven not by the errors associated with the imperfect orbit and clock modelling, but by the distortions induced by noise and bias in the integrity chain. These distortions increase the minimum bias the integrity chain can guarantee to detect, which is reflected in the user protection limit. A contributor to this distortion is the inaccuracy associated with the estimation of the offset between the Galileo sensor station (GSS) receiver clocks and the Galileo system time (GST). This offset is termed the receiver clock synchronization error (CSE). This paper describes the research carried out to determine both the CSE and its associated error using GPS data as captured with the Galileo System Test Bed Version 1 (GSTB-V1). In the study we simulate open access to a time datum using IGS data. Two methods are compared for determining CSE and the corresponding uncertainty (noise) across a global network of tracking stations. The single-epoch single-station method is an ‘averaging’ technique that uses a single epoch of data, and is carried out at individual sensor stations, without recourse to the data from other stations. The global network solution method is also single epoch based, but uses the inversion of a linearised model of the global system to solve for the CSE simultaneously at all GSS along with a number of other parameters that would otherwise be absorbed into the CSE estimate in the averaging technique. To test the effectiveness of various configurations in the two methods the estimated synchronisation errors across the GSS network (comprising 25 stations) are compared to the same values as estimated by the International GPS Service (IGS) using a global tracking network of around 150 stations, as well as precise orbit and satellite clock models determined by a combination of global analysis centres. The results show that the averaging technique is vulnerable to unmodelled errors in the satellite clock offsets from system time, leading to receiver CSE errors in the region of 12 ns (3.7 m), this value being largely driven by the satellite CSE errors. The global network approach is capable of delivering CSE errors at the level of 1.5 ns (46 cm) depending on the number of parameters in the linearised model. The International GNSS Service (IGS) receiver clock estimates were used as a truth model for comparative assessment.     

Precise orbit determination of BeiDou constellation: method comparison

Chinese BeiDou navigation satellite system is in official service as a regional constellation with five geostationary earth orbit (GEO) satellites, five inclined geosynchronous satellite orbit (IGSO) satellites and four medium earth orbit (MEO) satellites. There are mainly two methods for precise orbit determination of the BeiDou constellation found in the current literatures. One is the independent single-system method, where only BeiDou observations are used without help from other GNSS systems. The other is the two-step GPS-assisted method where in the first step, GPS data are used to resolve some common parameters, such as station coordinates, receiver clocks and zenith tropospheric delay parameters, which are then introduced as known quantities in BeiDou processing in the second step. We conduct a thorough performance comparison between the two methods. Observations from the BeiDou experimental tracking stations and the IGS Multi-GNSS Experiment network from January 1 to March 31, 2013, are processed with the Positioning and Navigation Data Analyst (PANDA) software. The results show that for BeiDou IGSO and MEO satellites, the two-step GPS-assisted method outperforms the independent single-system method in both internal orbit overlap precision and external satellite laser ranging validation. For BeiDou GEO satellites, the two methods show close performances. Zenith tropospheric delays estimated from the first method are very close to those estimated from GPS precise point positioning in the second method, with differences of several millimeters. Satellite clock estimates from the two methods show similar performances when assessing the stability of the BeiDou on board clocks.     

Geodetic methods to determine the relativistic redshift at the level of 10$$^{}$$ in the context of international timescales: a review and practical results

The frequency stability and uncertainty of the latest generation of optical atomic clocks is now approaching the one part in \(10^{18}\) level. Comparisons between earthbound clocks at rest must account for the relativistic redshift of the clock frequencies, which is proportional to the corresponding gravity (gravitational plus centrifugal) potential difference. For contributions to international timescales, the relativistic redshift correction must be computed with respect to a conventional zero potential value in order to be consistent with the definition of Terrestrial Time. To benefit fully from the uncertainty of the optical clocks, the gravity potential must be determined with an accuracy of about \(0.1\,\hbox {m}^{2}\,\hbox {s}^{-2}\), equivalent to about 0.01 m in height. This contribution focuses on the static part of the gravity field, assuming that temporal variations are accounted for separately by appropriate reductions. Two geodetic approaches are investigated for the derivation of gravity potential values: geometric levelling and the Global Navigation Satellite Systems (GNSS)/geoid approach. Geometric levelling gives potential differences with millimetre uncertainty over shorter distances (several kilometres), but is susceptible to systematic errors at the decimetre level over large distances. The GNSS/geoid approach gives absolute gravity potential values, but with an uncertainty corresponding to about 2 cm in height. For large distances, the GNSS/geoid approach should therefore be better than geometric levelling. This is demonstrated by the results from practical investigations related to three clock sites in Germany and one in France. The estimated uncertainty for the relativistic redshift correction at each site is about \(2 \times 10^{-18}\).     

New IGS Station and Satellite Clock Combination

Following the principles set forth in the Position Paper #3 at the 1998 Darmstadt Analysis Center (AC) Workshop on the new International GPS Service (IGS) International Terrestrial Reference Frame (ITRF) realization and discussions at the 1999 La Jolla AC workshop, a new clock combination program was developed. The program allows for the input of both SP3 and the new clock (RINEX) format (ftp://igsch.jpl.nasa.gov//igscb/data/format/rinex_clock.txt). The main motivation for this new development is the realization of the goals of the IGS/BIPM timing project. Besides this there is a genuine interest in station clocks and a need for a higher sampling rate of the IGS clocks (currently limited to 15 min due to the SP3 format). The inclusion of station clocks should also allow for a better alignment of the individual AC solutions and should enable the realization of a stable GPS time-scale. For each input AC clock solution the new clock combination solves and corrects for reference clock errors/instabilities as well as satellite/station biases, geocenter and station/satellite orbit errors. External station clock calibrations and/or constraints, such as those resulting from the IGS/BIPM timing pilot project, can be introduced via a subset of the fiducial timing station set, to facilitate a precise and consistent IGS UTC realization for both station and satellite combined clock solutions. Furthermore, the new clock combination process enforces strict strict conformity and consistency with the current and future IGS standards. The new clock combination maintains orbit/clock consistency at millimeter level, which is comparable to the best AC orbit/clock solutions. This is demonstrated by static GIPSY precise point positioning tests using GPS week 0995 data for stations in both Northern and Southern Hemispheres and similar tests with the Bernese software using more recent data from GPS week 1081. ? 2001 John Wiley & Sons, Inc.     

Characterization of periodic variations in the GPS satellite clocks

The clock products of the International Global Navigation Satellite Systems (GNSS) Service (IGS) are used to characterize the timing performance of the GPS satellites. Using 5-min and 30-s observational samples and focusing only on the sub-daily regime, approximate power-law stochastic processes are found. The Block IIA Rb and Cs clocks obey predominantly random walk phase (or white frequency) noise processes. The Rb clocks are up to nearly an order of magnitude more stable and show a flicker phase noise component over intervals shorter than about 100 s. Due to the onboard Time Keeping System in the newer Block IIR and IIR-M satellites, their Rb clocks behave in a more complex way: as an apparent random walk phase process up to about 100 s and then changing to flicker phase up to a few thousand seconds. Superposed on this random background, periodic signals have been detected in all clock types at four harmonic frequencies, n × (2.0029 ± 0.0005) cycles per day (24 h coordinated universal time or UTC), for n = 1, 2, 3, and 4. The equivalent fundamental period is 11.9826 ± 0.0030 h, which surprisingly differs from the reported mean GPS orbital period of 11.9659 ± 0.0007 h by 60 ± 11 s. We cannot account for this apparent discrepancy but note that a clear relationship between the periodic signals and the orbital dynamics is evidenced for some satellites by modulations of the spectral amplitudes with eclipse season. All four harmonics are much smaller for the IIR and IIR-M satellites than for the older blocks. Awareness of the periodic variations can be used to improve the clock modeling, including for interpolation of tabulated IGS products for higher-rate GPS positioning and for predictions in real-time applications. This is especially true for high-accuracy uses, but could also benefit the standard GPS operational products. The observed stochastic properties of each satellite clock type are used to estimate the growth of interpolation and prediction errors with time interval.