GPS satellite clock determination in case of inter-frequency clock biases for triple-frequency precise point positioning

Significant time-varying inter-frequency clock biases (IFCBs) within GPS observations prevent the application of the legacy L1/L2 ionosphere-free clock products on L5 signals. Conventional approaches overcoming this problem are to estimate L1/L5 ionosphere-free clocks in addition to their L1/L2 counterparts or to compute IFCBs between the L1/L2 and L1/L5 clocks which are later modeled through a harmonic analysis. In contrast, we start from the undifferenced uncombined GNSS model and propose an alternative approach where a second satellite clock parameter dedicated to the L5 signals is estimated along with the legacy L1/L2 clock. In this manner, we do not need to rely on the correlated L1/L2 and L1/L5 ionosphere-free observables which complicates triple-frequency GPS stochastic models, or account for the unfavorable time-varying hardware biases in undifferenced GPS functional models since they can be absorbed by the L5 clocks. An extra advantage over the ionosphere-free model is that external ionosphere constraints can potentially be introduced to improve PPP. With 27 days of triple-frequency GPS data from globally distributed stations, we find that the RMS of the positioning differences between our GPS model and all conventional models is below 1 mm for all east, north and up components, demonstrating the effectiveness of our model in addressing triple-frequency observations and time-varying IFCBs. Moreover, we can combine the L1/L2 and L5 clocks derived from our model to calculate precisely the L1/L5 clocks which in practice only depart from their legacy counterparts by less than 0.006 ns in RMS. Our triple-frequency GPS model proves convenient and efficient in combating time-varying IFCBs and can be generalized to more than three frequency signals for satellite clock determination.     

GPS inter-frequency clock bias modeling and prediction for real-time precise point positioning

To ensure the consistent use of the current GPS precise satellite clock products, the inter-frequency clock bias (IFCB) should be carefully considered for triple-frequency precise point positioning (PPP). It is beneficial to investigate the modeling of the IFCB for multi-frequency PPP, especially for real-time users suffering from difficulties in real-time IFCB estimations. Our analysis is based on datasets from 129 stations spanning a whole year. A harmonic analysis is performed for all single-day IFCB time series, and periodic IFCB variations with periods of 12, 8, 6, 4.8, 4 and 3 h are identified. An empirical model composed of a sixth-order harmonic function and a linear function is presented to describe daily variations in the IFCB, and the modeling accuracy is 4 mm. A least squares fit is adopted to estimate the single-day harmonic coefficients phase and amplitude. The prediction accuracy of the IFCB models degrades from 7.2 to 12.3 mm when the time span of prediction is increased from a day to a week. When using IFCB models of the previous day to obtain the IFCB correction values, the positioning accuracy of triple-frequency PPP is improved by 21, 11 and 16% over the triple-frequency PPP neglecting the IFCB in the post-processing mode in the east, north and up directions, respectively. As to the real-time triple-frequency PPP, the corresponding accuracy improvement is 24, 9 and 10% in the three directions, respectively.     

Fast estimation and analysis of the inter-frequency clock bias for Block IIF satellites

The inter-frequency bias of PRN25 was noticed by the scientific community and considered to be caused by thermal variations. The inter-frequency bias leads to an apparent inter-frequency clock bias (IFCB), which could be obtained using the difference of two ionosphere-free phase combinations (L1/L2 and L1/L5). We present an efficient approach derived from the epoch-differenced strategy for fast estimation of IFCBs for Block IIF satellites. For the analysis, data from 32 stations from the IGS network spanning 10 months (DOY 213, 2011–153, 2012) are processed. The processing times show that the epoch-differenced method is more efficient than the undifferenced one. In order to study the features of IFCB, a harmonic analysis is performed by using a FFT (fast Fourier transformation), and significant periodic variations with the periods of 12, 6 and 8 h are noticed. The fourth-order period is determined by comparing the performances of the model with different periods. After determination, a harmonics-based function of order 4 is used to model the IFCB, and the single-day amplitudes and phases are estimated for the 10 months from a least squares fit. Based on the estimated results, the characterization of IFCB is discussed. The algorithm is incorporated into the MGPSS software developed at SHAO (Shanghai Astronomical Observatory, Chinese Academy of Sciences) and used to monitor the IFCB variations of GPS and COMPASS systems in near real time.     

GNSS多频融合精密单点定位频率间卫星钟偏差研究

全球导航卫星系统(GNSS)提供多频信号,多频融合已经成为一种趋势。在精密钟差估计(PCE)的过程中,卫星钟差参数会吸收卫星端稳定的伪距偏差和时变的相位偏差,这些偏差均与频率相关。因而使用不同的观测值进行PCE时,得到的卫星钟差估值是不同的,它们之间的差值被定义为频率间卫星钟偏差(IFCB)。按组成成分,IFCB可以分成伪距相关的IFCB(CIFCB)和相位相关的IFCB(PIFCB)两部分。国际GNSS服务(IGS)提供的精密卫星钟差产品是基于双频消电离层(IF)组合观测值生成的。由于IFCB的存在,导致IGS卫星钟差产品不能直接应用于多频精密单点定位(PPP)。IFCB的精确考虑已经成为多频PPP的一个关键问题。本研究旨在对IFCB特性和估计方法开展系统深入的研究,并评估其对多频PPP解的影响。     

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.     

A comparison of three widely used GPS triple-frequency precise point positioning models

GPS Solutions - The EUREF Permanent Network (EPN) is a network of continuously operating GNSS stations installed throughout the European continent. The EPN Central Bureau (CB) performs the...     

GPS/BDS/Galileo三频精密单点定位模型及性能分析

在精细考虑伪距和载波相位硬件偏差时变特性的基础上,导出了更为严谨的非差非组合观测方程,并给出了非组合模式下两类GNSS偏差的数学表达形式.基于此,本文详细研究了3种常用的三频精密单点定位(PPP),即无电离层两两组合IF1213、单个无电离层组合IF123与非组合UC123函数模型的独立参数化方法,系统分析了3种PPP...     

精密单点定位中卫星钟差加密方法的探讨与分析

本文系统地分析了多种精密卫星钟差加密方法,以IGS提供的GPSweek 1421周第2天的2号、11号、21号和28号卫星的30s间隔的精密卫星钟差为基准,然后以从中提取的15min和5min间隔的钟差为例,将其用内插或拟合等不同方法分别加密到30s,将加密结果与30s间隔的精密卫星钟差基准进行比较分析,得出样条函数内插法精度较高、较可靠,拟合法精度较差,在个别历元处,内插或拟合卫星钟差的误差仍然很大。     

Generating GPS satellite fractional cycle bias for ambiguity-fixed precise point positioning

With the development of precise point positioning (PPP), the School of Geodesy and Geomatics (SGG) at Wuhan University is now routinely producing GPS satellite fractional cycle bias (FCB) products with open access for worldwide PPP users to conduct ambiguity-fixed PPP solution. We provide a brief theoretical background of PPP and present the strategies and models to compute the FCB products. The practical realization of the two-step (wide-lane and narrow-lane) FCB estimation scheme is described in detail. With GPS measurements taken in various situations, i.e., static, dynamic, and on low earth orbit (LEO) satellites, the quality of FCB estimation and the effectiveness of PPP ambiguity resolution (AR) are evaluated. The comparison with CNES FCBs indicated that our FCBs had a good consistency with the CNES ones. For wide-lane FCB, almost all the differences of the two products were within ±0.05 cycles. For narrow-lane FCB, 87.8 % of the differences were located between ±0.05 cycles, and 97.4 % of them were located between ±0.075 cycles. The experimental results showed that, compared with conventional ambiguity-float PPP, the averaged position RMS of static PPP can be improved from (3.6, 1.4, 3.6) to (2.0, 1.0, 2.7) centimeters for ambiguity-fixed PPP. The average accuracy improvement in the east, north, and up components reached 44.4, 28.6, and 25.0 %, respectively. A kinematic, ambiguity-fixed PPP test with observation of 80 min achieved a position accuracy of better than 5 cm at the one-sigma level in all three coordinate components. Compared with the results of ambiguity-float, kinematic PPP, the positioning biases of ambiguity-fixed PPP were improved by about 78.2, 20.8, and 65.1 % in east, north, and up. The RMS of LEO PPP test was improved by about 23.0, 37.0, and 43.0 % for GRACE-A and GRACE-B in radial, tangential, and normal directions when AR was applied to the same data set. These results demonstrated that the SGG FCB products can be produced with high quality for users anywhere around the world to carry out ambiguity-fixed PPP solutions.     

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.     

BDS-3三频精密单点定位精度分析

我国北斗卫星导航系统(BDS)处于最后阶段BDS-3的建设之中,其定位精度一直是国内外研究的热点．本文基于IGS连续跟踪站实测数据,阐述了BDS三频精密单点定位模型,初步对比分析了BDS-2与BDS-3的观测数据质量以及精密单点定位精度．经研究发现,BDS-3数据质量良好,相比BDS-2有所提高,单独利用BDS-3卫星进行精密单点定位精度相比于BDS-2略差,收敛时间也略慢,二者结合下的精密单点定位精度与收敛时间相比单独定位有很大的提升．      

不同星历和钟差产品对车载PPP定位的影响分析

星历误差和星钟误差是GPS动态精密单点定位的主要误差源,不同的星历和钟差产品会对结果产生影响。利用IGS网站给出的不同星历和钟差数据及广播星历数据进行计算,采用Waypoint软件进行不同的组合计算,对不同星历和钟差组合对机载GPS数据定位结果进行分析。     

北斗卫星伪距偏差标定及对用户定位精度影响

伪距偏差是指卫星导航信号非理想特征导致的不同技术状态接收机产生的伪距测量常数偏差。本文将伪距偏差作为一种用户段误差,提出基于并置接收机的伪距偏差计算方法和基于DCB参数的伪距偏差计算方法,以实现伪距偏差与其他误差的分离。然后利用实测数据测量了北斗卫星伪距偏差,结果表明伪距偏差标定序列波动STD约为0.1 m,不随时间明显变化,不同地点接收机测量的伪距偏差具有较好的一致性。在1.5 G频段,北斗卫星B1I频点伪距偏差最大。北斗卫星新体制信号B1C伪距偏差最小,较北斗卫星B1I频点伪距偏差明显改善,也明显好于GPS卫星L1C/A频点伪距偏差。在其他频段,GPS卫星L2C伪距偏差略大于北斗卫星B3I伪距偏差,L5C频点伪距偏差次之,B2a频点伪距偏差最小。最后,利用实测数据分析了伪距偏差对定位精度的影响。结果表明伪距偏差与卫星群延迟参数高度相关。若用户接收机与群延迟参数计算采用的接收机技术状态差异较大,用户接收机定位精度将明显恶化。     

Method for real-time self-calibrating GLONASS code inter-frequency bias and improvements on single point positioning

Utilization of frequency-division multiple access (FDMA) leads to GLONASS pseudorange and carrier phase observations suffering from variable levels inter-frequency bias (IFB). The bias related with carrier phase can be absorbed by ambiguities. However, the unequal code inter-frequency bias (cIFB) will degrade the accuracy of pseudorange observations, which will affect positioning accuracy and convergence of precise point positioning (PPP) when including GLONASS satellites. Based on observations made on un-differenced (UD) ionospheric-free combinations, GLONASS cIFB parameters are estimated as a constant to achieve GLONASS cIFB real-time self-calibration on a single station. A total of 23 stations, with different manufacturing backgrounds, are used to analyze the characteristics of GLONASS cIFB and its relationship with variable receiver hardware. The results show that there is an obvious common trend in cIFBs estimated using broadcast ephemeris for all of the different manufacturers, and there are unequal GLONASS inter-satellite cIFB that match brand manufacture. In addition, a particularly good consistency is found between self-calibrated receiver-dependent GLONASS cIFB and the IFB products of the German Research Centre for Geosciences (GFZ). Via a comparative experiment, it is also found that the algorithm of cIFB real-time self-calibration not only corrects receiver-dependent cIFB, but can moreover eliminate satellite-dependent cIFB, providing more stable results and further improving global navigation satellite system (GNSS) point positioning accuracy. The root mean square (RMS) improvements of single GLONASS standard point positioning (SPP) reach up to 54.18 and 53.80% in horizontal and vertical direction, respectively. The study’s GLONASS cIFB self-estimation can realize good self-consistency between cIFB and stations, working to further promote convergence efficiency relative to GPS?+?GLONASS PPP. An average improvement percentage of 19.03% is observed, realizing a near-consistent accuracy with GPS?+?GLONASS fusion PPP.     

GPS/BDS三频信号频间时钟偏差估计与定位验证

在三频GNSS应用中,受精密产品频率基准不一致的影响,会引入系统性偏差,即频间时钟偏差（IFCB）。本文首先通过对IFGF组合观测值进行历元差分,利用全球分布的80个MGEX观测站及中国区域内100个连续运行参考站,在2021年年积日（DOY）153—160 d的实测数据,进行了IFCB的估计并分析了其时变特性;然后将IFCB的估计结果运用到非差非组合PPP中。结果表明：GPS BLOCK Ⅱ-F的IFCB较大,幅值可达14 cm,GPS BLOCK Ⅲ与BDS的IFCB则较小,一般不超过5 cm。在定位验证中,经过IFCB改正后,GPS/BDS-2/BDS-3-IGSO在第3频点L5、B2I、B2a的相位残差分别减小了59.54%、26.31%、10.98%。其中,动态定位的GPS、BDS-2/BDS-3-IGSO、GPS/BDS-2/BDS-3-IGSO 3种方案的点位精度分别提升了56.55%、29.16%、20.72%,改善效果显著。     

一种基于BDS三频观测量的精密单点定位算法

针对传统双频BDS精密单点定位收敛速度及定位精度如何进一步提高的问题,该文提出了一种基于两组消电离层组合的BDS新三频精密单点定位(PPP)定位算法,并且对由于引入第3个频段观测量所导致的函数和随机模型与传统双频PPP模型存在的差异进行了公式推导。最后利用实测数据以动态和静态模式对新三频PPP模型进行了测试,以传统双频PPP解算结果为参照,对新三频PPP模型的收敛速度及定位精度进行了评估分析。基于实测数据的测试结果表明,新三频算法有利于提升定位解算的精度并有效缩短初始收敛过程,而且这一改善效果在动态模式下较静态模式更为显著。     

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.