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.     

卫星钟差解算及其星间单差模糊度固定

整数相位模糊度解算可以显著提高GNSS精密单点定位（PPP）的精度。本文提出一种解算卫星钟差的方法,通过固定星间单差模糊度恢复出能够支持单台接收机进行整数模糊度解算的卫星钟差,即所谓的“整数”钟差。为了实现星间单差模糊度固定,分别通过卫星端宽巷FCB解算和模糊度基准的选择与固定恢复出宽巷和窄巷模糊度的整数性质。为了证明本文方法的可行性,采用IGS测站的GPS数据进行卫星钟差解算试验。结果表明,在解算钟差时,星间单差模糊度固定的平均成功率为73%。得到的卫星钟差与IGS最终钟差产品相比,平均的RMS和STD分别为0.170和0.012 ns。448个IGS测站的星间单差宽巷和窄巷模糊度小数部分的分布表明本文得到的卫星钟差和FCB产品具备支持PPP用户进行模糊度固定的能力。基于以上产品开展了模拟动态PPP定位试验,结果表明模糊度固定之后,N、E、U和3D的定位精度（RMS）分别达到0.009、0.010、0.023和0.027 m,与不固定模糊度或采用IGS钟差的结果相比,分别提高了30.8%、61.5%、23.3%和37.2%。     

Rapid PPP ambiguity resolution using GPS+GLONASS observations

Integer ambiguity resolution (IAR) in precise point positioning (PPP) using GPS observations has been well studied. The main challenge remaining is that the first ambiguity fixing takes about 30 min. This paper presents improvements made using GPS+GLONASS observations, especially improvements in the initial fixing time and correct fixing rate compared with GPS-only solutions. As a result of the frequency division multiple access strategy of GLONASS, there are two obstacles to GLONASS PPP-IAR: first and most importantly, there is distinct code inter-frequency bias (IFB) between satellites, and second, simultaneously observed satellites have different wavelengths. To overcome the problem resulting from GLONASS code IFB, we used a network of homogeneous receivers for GLONASS wide-lane fractional cycle bias (FCB) estimation and wide-lane ambiguity resolution. The integer satellite clock of the GPS and GLONASS was then estimated with the wide-lane FCB products. The effect of the different wavelengths on FCB estimation and PPP-IAR is discussed in detail. We used a 21-day data set of 67 stations, where data from 26 stations were processed to generate satellite wide-lane FCBs and integer clocks and the other 41 stations were selected as users to perform PPP-IAR. We found that GLONASS FCB estimates are qualitatively similar to GPS FCB estimates. Generally, 98.8% of a posteriori residuals of wide-lane ambiguities are within \(\pm 0.25\) cycles for GPS, and 96.6% for GLONASS. Meanwhile, 94.5 and 94.4% of narrow-lane residuals are within 0.1 cycles for GPS and GLONASS, respectively. For a critical value of 2.0, the correct fixing rate for kinematic PPP is only 75.2% for GPS alone and as large as 98.8% for GPS+GLONASS. The fixing percentage for GPS alone is only 11.70 and 46.80% within 5 and 10 min, respectively, and improves to 73.71 and 95.83% when adding GLONASS. Adding GLONASS thus improves the fixing percentage significantly for a short time span. We also used global ionosphere maps (GIMs) to assist the wide-lane carrier-phase combination to directly fix the wide-lane ambiguity. Employing this method, the effect of the code IFB is eliminated and numerical results show that GLONASS FCB estimation can be performed across heterogeneous receivers. However, because of the relatively low accuracy of GIMs, the fixing percentage of GIM-aided GPS+GLONASS PPP ambiguity resolution is very low. We expect better GIM accuracy to enable rapid GPS+GLONASS PPP-IAR with heterogeneous receivers.     

长距离GPS/BDS双系统网络RTK方法

基于区域参考站网的网络实时动态定位(real-time kinematic,RTK)方法是实现全球定位系统(global positioning system,GPS)、北斗卫星导航系统(BeiDou satellite navigation system,BDS)高精度定位的主要手段.研究了一种长距离GPS/BDS双...     

单星模糊度固定的整数相位钟法及在低轨卫星定轨中的应用

单星GPS相位模糊度固定可以显著提升低轨卫星的定轨精度。目前,CNES/CLS、武汉大学和CODE 3家机构都已公开发布用于单星模糊度固定的GPS整数相位钟产品。本文首先利用整数相位钟方法实现单星模糊度固定,并应用于低轨卫星精密定轨中;然后,对比分析了不同机构提供的整数相位钟产品在低轨卫星单星模糊度固定和精密定轨中的应用性能;最后,通过对GRACE-FO编队卫星数据进行处理,发现基于不同机构产品的窄巷模糊度固定成功率都可以达到94%左右。不同机构产品获得的模糊度固定解轨道的SLR(satellite laser ranging)检核残差RMS约为0.9 cm,与模糊度浮点解的定轨结果相比,单星绝对轨道精度提高了约30%。在分别利用CNES/CLS、武汉大学和CODE产品实现单星模糊度固定后,双星相对轨道的KBR(K-band ranging)检核残差RMS分别从5.7、5.4和5.3 mm减小到2.1、2.0和1.5 mm。结果表明,不同整数相位钟产品在GRACE-FO卫星单星模糊度固定和精密定轨中的效果相当。     

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.     

长距离GPS/BDS参考站网多频载波相位整周模糊度解算方法

长距离网络RTK是实现GPS/BDS高精度实时定位的主要手段之一,其核心是长距离参考站网GPS/BDS整周模糊度的快速准确确定。本文提出了一种长距离GPS/BDS参考站网载波相位整周模糊度解算方法,首先利用GPS双频观测数据计算和确定宽巷整周模糊度,同时利用BDS的B2、B3频率观测值确定超宽巷整周模糊度。然后建立GPS载波相位整周模糊度和大气延迟误差的参数估计模型,附加双差宽巷整周模糊度的约束,解算双差载波相位整周模糊度,并建立参考站网大气延迟误差的空间相关模型。根据B2、B3频率的超宽巷整周模糊度建立包含大气误差参数的载波相位整周模糊度解算模型,利用大气延迟误差空间相关模型约束BDS双差载波相位整周模糊度的解算。克服了传统的使用无电离层组合值解算整周模糊度的不利影响。采用实测长距离CORS网GPS、BDS多频观测数据进行算法验证,试验结果证明该方法可实现长距离参考站网GPS/BDS载波相位整周模糊度的准确固定。     

A computationally efficient approach for estimating high-rate satellite clock corrections in realtime

Realtime satellite clock corrections are usually estimated using undifferenced phase and range observations from a global network. Because a large number of ambiguity parameters must be estimated, the computation is time-consuming. Consequently, only a sparse global network of limited number of stations is processed by most IGS Realtime Analysis Centers with an update rate of 5 s. In addition, it is very desirable to build the capability to simultaneously estimate clock corrections for multi-GNSS constellations. Although the estimation can be sped up by epoch-differenced observations that eliminate ambiguities, the derived clocks can contain a satellite-specific bias that diminishes the contribution of range observations. We introduce a computationally efficient approach for realtime clock estimation. Both the epoch-differenced phase and undifferenced range observations are used together to estimate the epoch-differenced satellite clocks and the initial clock bias for each satellite and receiver. The biased clock corrections accumulated from the estimated epoch-differenced clocks are then aligned with the estimated clock biases and provided as the final clock corrections to users. The algorithm is incorporated into the EPOS-RT software developed at GFZ (GeoForschungsZentrum) and experimentally validated with the IGS global network. The comparison with the GFZ rapid products shows that the accuracy of the clock estimation with the new approach is comparable with that of the undifferenced approach, whereas the computation time is reduced to one-tenth. As a result, estimation of high-rate satellite clocks from a large reference network and tracking satellites of multi-GNSS constellations becomes achievable.     

北斗卫星导航系统双差网络RTK方法

针对北斗卫星导航系统常规实时动态差分(RTK)定位中,整周模糊度的快速解算和流动站位置信息的解算精度问题,该文研究了一种北斗卫星导航系统双频双差网络RTK方法,首先解算参考站网B1、B2载波相位整周模糊度,利用固定的参考站网载波相位整周模糊度计算参考站的观测误差,使用区域误差内插法计算流动站的综合误差影响,改正流动站的观测误差并进行流动站的整周模糊度解算,最后使用实测数据进行算法实验。实验结果表明,该文的方法可以利用北斗卫星导航系统双频观测数据实现网络RTK流动站的厘米级定位。     

Quality assessment of GPS rapid static positioning with weighted ionospheric parameters in generalized least squares

Precise GPS positioning requires the processing of carrier-phase observations and fixing integer ambiguities. With increasing distance between receivers, ambiguity fixing becomes more difficult because ionospheric and tropospheric effects do not cancel sufficiently in double differencing. A popular procedure in static positioning is to increase the length of the observing session and/or to apply atmospheric (ionospheric) models and corrections. We investigate the methodology for GPS rapid static positioning that requires just a few minutes of dual-frequency GPS observations for medium-length baselines. Ionospheric corrections are not required, but the ionospheric delays are treated as pseudo-observations having a priori values and respective weights. The tropospheric delays are reduced by using well-established troposphere models, and satellite orbital and clock errors are eliminated by using IGS rapid products. Several numerical tests based on actual GPS data are presented. It is shown that the proposed methodology is suitable for rapid static positioning within 50–70 km from the closest reference network station and that centimeter-level precision in positioning is feasible when using just 1 min of dual-frequency GPS data.     

BDS参考站三频整周模糊度单历元确定方法

参考站载波相位整周模糊度的准确确定是实现BDS网络RTK定位的关键。本文研究了BDS参考站三频载波相位整周模糊度单历元确定方法。首先推导了参考站三频载波相位整周模糊度之间的多个整数线性关系,根据双频载波相位整周模糊度的整数线性关系,以及B1载波相位整周模糊度备选值,确定B1/B2和B1/B3载波相位整周模糊度的备选组合。然后利用不受误差影响的三频载波相位整周模糊度间整数线性关系,对整周模糊度备选值进行约束和确定。根据大气误差的空间相关性,采用以卫星高度角和方位角为依据的基准卫星选择方法,降低了对流层延迟误差残差对多频载波相位整周模糊度之间线性关系约束能力的影响。试验结果表明,本文方法能够实现参考站三频载波相位整周模糊度的单历元准确确定,且计算效率高,算法简单。     

GOCE gravitational gradiometry

GOCE is the first gravitational gradiometry satellite mission. Gravitational gradiometry is the measurement of the second derivatives of the gravitational potential. The nine derivatives form a 3 × 3 matrix, which in geodesy is referred to as Marussi tensor. From the basic properties of the gravitational field, it follows that the matrix is symmetric and trace free. The latter property corresponds to Laplace equation, which gives the theoretical foundation of its representation in terms of spherical harmonic or Fourier series. At the same time, it provides the most powerful quality check of the actual measured gradients. GOCE gradiometry is based on the principle of differential accelerometry. As the satellite carries out a rotational motion in space, the accelerometer differences contain angular effects that must be removed. The GOCE gradiometer provides the components V _xx , V _yy , V _zz and V _xz with high precision, while the components V _xy and V _yz are of low precision, all expressed in the gradiometer reference frame. The best performance is achieved inside the measurement band from 5 × 10^–3 to 0.1 Hz. At lower frequencies, the noise increases with 1/f and is superimposed by cyclic distortions, which are modulated from the orbit and attitude motion into the gradient measurements. Global maps with the individual components show typical patterns related to topographic and tectonic features. The maps are separated into those for ascending and those for descending tracks as the components are expressed in the instrument frame. All results are derived from the measurements of the period from November to December 2009. While the components V _xx and V _yy reach a noise level of about \({10\;\rm{\frac{mE}{\sqrt{Hz}}}}\), that of V _zz and V _xz is about \({20\; \rm{\frac{mE}{\sqrt{Hz}}}}\). The cause of the latter’s higher noise is not yet understood. This is also the reason why the deviation from the Laplace condition is at the \({20 \;\rm{\frac{mE}{\sqrt{Hz}}}}\) level instead of the originally planned \({11\;\rm{\frac{mE}{\sqrt{Hz}}}}\). Each additional measurement cycle will improve the accuracy and to a smaller extent also the resolution of the spherical harmonic coefficients derived from the measured gradients.     

PERFORMANCE ANALYSIS OF INTEGRATED GPS/GLONASS CARRIER PHASE-BASED POSITIONING

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Performance analysis of integrated GPS/GLONASS carrier phase-based positioning

Due to the different signal frequencies for the GLONASS satellites, the commonly-used double-differencing procedure for carrier phase data processing can not be implemented in its straightforward form, as in the case of GPS. In this paper a novel data processing strategy, involving a three-step procedure, for integrated GPS/GLONASS positioning is proposed. The first is pseudo-range-based positioning, that uses double-differenced (DD) GPS pseudo-range and single-differenced (SD) GLONASS pseudo-range measurements to derive the initial position and receiver clock bias. The second is forming DD measurements (expressed in cycles) in order to estimate the ambiguities, by using the receiver clock bias estimated in the above step. The third is to form DD measurements (expressed in metric units) with the unknown SD integer ambiguity for the GLONASS reference satellite as the only parameter (which is constant before a cycle slip occurs for this satellite). A real-time stochastic model estimated by residual series over previous epochs is proposed for integrated GPS/GLONASS carrier phase and pseudo-range data processing. Other associated issues, such as cycle slip detection, validation criteria and adaptive procedure(s) for ambiguity resolution, is also discussed. The performance of this data processing strategy will be demonstrated through case study examples of rapid static positioning and kinematic positioning. From four experiments carried out to date, the results indicate that rapid static positioning requires 1 minute of single frequency GPS/GLONASS data for 100% positioning success rate. The single epoch positioning solution for kinematic positioning can achieve 94.6% success rate over short baselines (<6 km).     

Precise GPS Positioning by Applying Ionospheric Corrections from an Active Control Network

In this article, initial results are presented of a method to improve fast carrier phase ambiguity resolution over longer baselines (with lengths up to about 200 km). The ionospheric delays in the global positioning system (GPS) data of these long baselines mainly hamper successful integer ambiguity resolution, a prerequisite to obtain precise positions within very short observation time spans. A way to correct the data for significant ionospheric effects is to have a GPS user operate within an active or permanently operating network use ionospheric estimates from this network. A simple way to do so is to interpolate these ionospheric estimates based on the expected spatial behaviour of the ionospheric delays. In this article such a technique is demonstrated for the Dutch Active Control Network (AGRS.NL). One hour of data is used from 4 of the 5 reference stations to obtain very precise ionospheric corrections after fixing of the integer ambiguities within this network. This is no problem because of the relatively long observation time span and known positions of the stations of the AGRS.NL. Next these interpolated corrections are used to correct the GPS data from the fifth station for its ionospheric effects. Initial conclusions about the performance of this technique are drawn in terms of improvement of integer ambiguity resolution for this baseline. ? 1999 John Wiley & Sons, Inc.     

星间单差模糊度固定的低轨卫星精密定轨精度分析

高精度、高可靠性的卫星轨道是实现低轨卫星精密应用的重要前提,而模糊度固定技术是提高卫星定轨精度的关键途径。研究了基于整数钟的星间单差模糊度固定原理和方法,并利用2019年4月—5月的两颗GRACE-FO（gravity recovery and climate experiment follow on）卫星数据（GRACE-C/D）系统评估了固定解对低轨卫星简化动力学和运动学定轨的精度提升效果。结果表明,两颗卫星简化动力学和运动学定轨的宽巷模糊度固定率均达到99%,而窄巷模糊度固定率在95%左右。对于简化动力学定轨,GRACE-C/D固定解轨道的重叠轨道的3D均方根误差（root mean square error, RMSE）分别从7.1 mm和7.4 mm减小到了4.2 mm和3.6 mm;卫星激光测距（satellite laser ranging, SLR）残差标准差（standard deviation, STD）分别从15.9 mm和14.4 mm降低到了10.8 mm和11.0 mm,精度提升了32%和24%;K波段测距残差RMSE从8.0 mm减小到2.9 mm,进一步表明固定解还能有效提升低轨卫星间相对位置精度。对于运动学定轨,与精密科学轨道产品互差3D RMSE,浮点解分别为37.5 mm和36.4 mm,固定解分别为27.7 mm和25.5 mm,精度提升约28%,SLR残差STD也减小了约20%。     

Ambiguity resolved precise point positioning with GPS and BeiDou

This paper focuses on the contribution of the global positioning system (GPS) and BeiDou navigation satellite system (BDS) observations to precise point positioning (PPP) ambiguity resolution (AR). A GPS + BDS fractional cycle bias (FCB) estimation method and a PPP AR model were developed using integrated GPS and BDS observations. For FCB estimation, the GPS + BDS combined PPP float solutions of the globally distributed IGS MGEX were first performed. When integrating GPS observations, the BDS ambiguities can be precisely estimated with less than four tracked BDS satellites. The FCBs of both GPS and BDS satellites can then be estimated from these precise ambiguities. For the GPS + BDS combined AR, one GPS and one BDS IGSO or MEO satellite were first chosen as the reference satellite for GPS and BDS, respectively, to form inner-system single-differenced ambiguities. The single-differenced GPS and BDS ambiguities were then fused by partial ambiguity resolution to increase the possibility of fixing a subset of decorrelated ambiguities with high confidence. To verify the correctness of the FCB estimation and the effectiveness of the GPS + BDS PPP AR, data recorded from about 75 IGS MGEX stations during the period of DOY 123-151 (May 3 to May 31) in 2015 were used for validation. Data were processed with three strategies: BDS-only AR, GPS-only AR and GPS + BDS AR. Numerous experimental results show that the time to first fix (TTFF) is longer than 6 h for the BDS AR in general and that the fixing rate is usually less than 35 % for both static and kinematic PPP. An average TTFF of 21.7 min and 33.6 min together with a fixing rate of 98.6 and 97.0 % in static and kinematic PPP, respectively, can be achieved for GPS-only ambiguity fixing. For the combined GPS + BDS AR, the average TTFF can be shortened to 16.9 min and 24.6 min and the fixing rate can be increased to 99.5 and 99.0 % in static and kinematic PPP, respectively. Results also show that GPS + BDS PPP AR outperforms single-system PPP AR in terms of convergence time and position accuracy.     

Network-based geometry-free three carrier ambiguity resolution and phase bias calibration

Continuously operating reference stations (CORS) are increasingly used to deliver real-time and near-real-time precise positioning services on a regional basis. A CORS network-based data processing system uses either or both of the two types of measurements: (1) ambiguity-resolved double-differenced (DD) phase measurements, and (2) phase bias calibrated zero-differenced (ZD) phase measurements. This paper describes generalized, network-based geometry-free models for three carrier ambiguity resolution (TCAR) and phase bias estimation with DD and ZD code and phase measurements. First, the geometry-free TCAR models are constructed with two Extra-Widelane (EWL)/Widelane (WL) virtual observables to allow for rapid ambiguity resolution (AR) for DD phase measurements without distance constraints. With an ambiguity-resolved WL phase measurement and the ionospheric estimate derived from the two EWL observables, an additional geometry-free equation is formed for the third virtual observable linearly independent of the previous two. AR with the third geometry-free model requires a longer period of observations for averaging than the first two, but is also distance-independent. A more general formulation of the geometry-free model for a baseline or network is also introduced, where all the DD ambiguities can be more rigorously resolved using the LAMBDA method. Second, the geometry-free models for calibration of three carrier phase biases of ZD phase measurements are similarly defined for selected virtual observables. A network adjustment procedure is then used to improve the ZD phase biases with known DD integer constraints. Numerical results from experiments with 24-h dual-frequency GPS data from three US CORS stations baseline lengths of 21, 56 and 74 km confirm the theoretical predictions concerning AR reliability of the network-based geometry-free algorithms.

Regional reference network augmented precise point positioning for instantaneous ambiguity resolution

Integer ambiguity fixing can significantly shorten the initialization time and improve the accuracy of precise point positioning (PPP), but it still takes approximate 15 min of time to achieve reliable integer ambiguity solutions. In this contribution, we present a new strategy to augment PPP estimation with a regional reference network, so that instantaneous ambiguity fixing is achievable for users within the network coverage. In the proposed method, precise zero-differenced atmospheric delays are derived from the PPP fixed solution of the reference stations, which are disseminated to, and interpolated at user stations to correct for L1, L2 phase observations or their combinations. With the corrected observations, instantaneous ambiguity resolution can be carried out within the user PPP software, thus achieving the position solutions equivalent to the network real-time kinematic positioning (NRTK). The strategy is validated experimentally. The derived atmospheric delays and the interpolated corrections are investigated. The ambiguity fixing performance and the resulted position accuracy are assessed. The validation confirms that the new strategy can provide comparable service with NRTK. Therefore, with this new processing strategy, it is possible to integrate PPP and NRTK into a seamless positioning service, which can provide an accuracy of about 10 cm anywhere, and upgrade to a few centimeters within a regional network.     

长距离网络RTK参考站间双差模糊度快速解算算法

提出了一种长距离网络RTK参考站间双差整周模糊度快速解算方法,该方法利用双频载波相位模糊度间的线性关系确定宽巷模糊度,然后选取双频载波相位的备选模糊度组合,通过计算参考站间对流层误差和轨道误差等非色散误差,对双频载波相位整周模糊度进行搜索。实验结果表明,此方法能够快速、可靠地解算长距离参考站间的双差整周模糊度。