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.     

An approach to GLONASS ambiguity resolution

 When processing global navigation satellite system (GLONASS) carrier phases, the standard double-differencing (DD) procedure cannot cancel receiver clock terms in the DD phase measurement equations due to the multiple frequencies of the carrier phases. Consequently, a receiver clock parameter has to be set up in the measurement equations in addition to baseline components and DD ambiguities. The resulting normal matrix unfortunately becomes singular. Methods to deal with this problem have been proposed in the literature. However, these methods rely on the use of pseudo-ranges. As pseudo-ranges are contaminated by multi-path and hardware delays, biases in these pseudo-ranges are significant, which may result in unreliable ambiguity resolution. A new approach is addressed that is not sensitive to the biases in the pseudo-ranges. The proposed approach includes such steps as converting the carrier phases to their distances to cancel the receiver clock errors, and searching for the most likely single-differenced (SD) ambiguity. Based on the results from the theoretical investigation, a practical procedure for GLONASS ambiguity resolution is presented. The initial experimental results demonstrate that the proposed approach is useable in cases of GLONASS and combined global positioning system (GPS) and GLONASS positioning. Received: 19 August 1998 / Accepted: 12 November 1999     

A modified geometry- and ionospheric-free combination for static three-carrier ambiguity resolution

The reliability of the classical geometry- and ionospheric-free (GIF) three-carrier ambiguity resolution (TCAR) degrades when applied to long baselines of hundreds of kilometers. To overcome this deficiency, we propose two new models, which are used sequentially to resolve wide-lane (WL) and narrow-lane (NL) ambiguities and form a stepwise ambiguity resolution (AR) strategy. In the first model, after a successful extra-wide-lane AR, the pseudorange and phase observations are combined to estimate WL ambiguities, in which the residual ionospheric delays and geometry effects are eliminated. In the second model, using the resolved ambiguities from the first step, the two WL ambiguities are combined to remove ionospheric and geometry effects. The unknown coefficients in the two models are determined in such that they minimize the formal errors in the ambiguity estimates to optimize the ambiguity estimation. Using experimental BeiDou triple-frequency observations, we evaluate our method and identify three advantages. First, the two models use double-differenced phase observations that are not differences across frequency. Second, the two models are entirely free from ionospheric delay and geometry effects. Third, the unknown estimates in the two models satisfy the minimum noise condition, which makes the formal errors in the float NL ambiguity estimates much lower than those obtained with common GIF TCAR methods, thereby directly and significantly increasing the success rate of AR compared to the cascaded integer resolution method and two other GIF combinations.     

GLONASS CDMA L3 ambiguity resolution and positioning

A first assessment of GLONASS CDMA L3 ambiguity resolution and positioning performance is provided. Our analyses are based on GLONASS L3 data from the satellite pair SVNs 755-801, received by two JAVAD receivers at Curtin University, Perth, Australia. In our analyses, four different versions of the two-satellite model are applied: the geometry-free model, the geometry-based model , the height-constrained geometry-based model, and the geometry-fixed model. We study the noise characteristics (carrier-to-noise density, measurement precision), the integer ambiguity resolution performance (success rates and distribution of the ambiguity residuals), and the positioning performance (ambiguity float and ambiguity fixed). The results show that our empirical outcomes are consistent with their formal counterparts and that the GLONASS data have a lower noise level than that of GPS, particularly in case of the code data. This difference is not only seen in the noise levels but also in their onward propagation to the ambiguity time series and ambiguity residuals distribution.     

GLONASS phase bias estimation and its PPP ambiguity resolution using homogeneous receivers

Integer ambiguity resolution (IAR) appreciably improves the position accuracy and shortens the convergence time of precise point positioning (PPP). However, while many studies are limited to GPS, there is a need to investigate the performance of GLONASS PPP ambiguity resolution. Unfortunately, because of the frequency-division multiple-access strategy of GLONASS, GLONASS PPP IAR faces two obstacles. First, simultaneously observed satellites operate at different wavelengths. Second and most importantly, distinct inter-frequency bias (IFB) exists between different satellites. For the former, we adopt an undifferenced method for uncalibrated phase delay (UPD) estimation and proposed an undifferenced PPP IAR strategy. We select a set of homogeneous receivers with identical receiver IFB to perform UPD estimation and PPP IAR. The code and carrier phase IFBs can be absorbed by satellite wide-lane and narrow-lane UPDs, respectively, which is in turn consistent with PPP IAR using the same type of receivers. In order to verify the method, we used 50 stations to generate satellite UPDs and another 12 stations selected as users to perform PPP IAR. We found that the GLONASS satellite UPDs are stable in time and space and can be estimated with high accuracy and reliability. After applying UPD correction, 91 % of wide-lane ambiguities and 99 % of narrow-lane ambiguities are within (?0.15, +0.15) cycles of the nearest integer. After ambiguity resolution, the 2-hour static PPP accuracy improves from (0.66, 1.42, 1.55) cm to (0.38, 0.39, 1.39) cm for the north, east, and up components, respectively.     

Network-based triple-frequency carrier phase ambiguity resolution between reference stations using BDS data for long baselines

Network-based ambiguity resolution (AR) between reference stations is the prerequisite to realize a precise real-time kinematic positioning service. With the help of BDS triple-frequency signals, we can efficiently deal with the ionospheric delay and tropospheric delay, and achieve rapid and reliable AR. To overcome the inaccurate ionospheric delay estimated by the geometry-free three carrier ambiguity resolution (GF TCAR) technique, which leads to failure in the original ambiguity resolution, we propose an ionospheric-free (IF) TCAR method to resolve the ambiguity between the reference stations over long baselines. Taking full advantage of the known positions of the reference stations, the easily resolved extra-wide-lane (EWL) ambiguity, and the IF phase combinations, we can reliably fix the wide-lane (WL) ambiguity. A Kalman filter is applied to estimate precise IF ambiguities and the original ambiguity is resolved with the fixed WL ambiguity. A numerical analysis with triple-frequency BDS data from three long baselines of a CORS network is provided to compare the AR performance of GF TCAR with that of IF TCAR. The results show that both methods can reliably resolve the WL ambiguity with a remarkable correctly-fixed rate of higher than 99%, and the reliably-fixed rates of the IF TCAR slightly increase from 92.19, 94.67 and 94.61–98.26, 99.54 and 97.51% for the three baselines. Herein “correctly-fixed” and “reliably-fixed” mean the difference between the float ambiguity and the true one are less than ± 0.5 and ± 0.25 cycles, respectively. On the other hand, the AR performance of the original signals with the IF TCAR method is much better than that with the GF TCAR method attaining a 100% correctly-fixed rate, while the GF TCAR method can hardly fix the original ambiguity with the largest bias being as much as 4 cycles because of the amplified systematic bias.     

GLONASS ambiguity resolution of mixed receiver types without external calibration

GLONASS processing from mixed receiver types is typically subject to unmodeled inter-frequency phase biases which prevent carrier phase ambiguity parameters from converging to integers. Receiver-dependent values have been proposed to mitigate the contribution of these biases, but are still subject to a number of issues, such as firmware updates. Recent studies have demonstrated that the origin of inter-frequency biases is a misalignment between phase and code observations, and could be calibrated to first order by manufacturers. In this contribution, a calibration-free method for GLONASS ambiguity resolution is presented in which ambiguities naturally converge to integers. A mandatory condition is that two GLONASS satellites with adjacent frequency numbers are observed simultaneously, although this condition can be relaxed once a fixed solution has been obtained. This approach then permits the integration of different receiver types and firmware versions into seamless processing.     

基于单差模糊度的 GPS／GLONASS 组合相对定位模型研究

采用频分多址技术(FDMA),GLONASS系统双差模糊度固定存在两个问题:不同卫星波长不一致,双差后不能保持模糊度整数特性;共视卫星频率不同,不同卫星之间存在大小不同的频间偏差(IFB)。传统的双差不能很好处理GLONASS相对定位模糊度固定问题。文中考虑将双差所涉及的两颗卫星的站间单差模糊度分别求解,不受共视卫星波长不一致的影响。同时采用参数估计法消除不同厂商接收机的频间偏差影响。试验结果表明采用文中方法可以正确固定GLONASS模糊度,并且达到与GPS相当的解算精度,GPS/GLONASS组合定位精度和可靠性也比GPS单系统有所提高。     

Assessment of PPP integer ambiguity resolution using GPS,GLONASS and BeiDou (IGSO,MEO) constellations

Although integer ambiguity resolution (IAR) can improve positioning accuracy considerably and shorten the convergence time of precise point positioning (PPP), it requires an initialization time of over 30 min. With the full operation of GLONASS globally and BDS in the Asia–Pacific region, it is necessary to assess the PPP–IAR performance by simultaneous fixing of GPS, GLONASS, and BDS ambiguities. This study proposed a GPS + GLONASS + BDS combined PPP–IAR strategy and processed PPP–IAR kinematically and statically using one week of data collected at 20 static stations. The undifferenced wide- and narrow-lane fractional cycle biases for GPS, GLONASS, and BDS were estimated using a regional network, and undifferenced PPP ambiguity resolution was performed to assess the contribution of multi-GNSSs. Generally, over 99% of a posteriori residuals of wide-lane ambiguities were within ±0.25 cycles for both GPS and BDS, while the value was 91.5% for GLONASS. Over 96% of narrow-lane residuals were within ±0.15 cycles for GPS, GLONASS, and BDS. For kinematic PPP with a 10-min observation time, only 16.2% of all cases could be fixed with GPS alone. However, adding GLONASS improved the percentage considerably to 75.9%, and it reached 90.0% when using GPS + GLONASS + BDS. Not all epochs could be fixed with a correct set of ambiguities; therefore, we defined the ratio of the number of epochs with correctly fixed ambiguities to the number of all fixed epochs as the correct fixing rate (CFR). Because partial ambiguity fixing was used, when more than five ambiguities were fixed correctly, we considered the epoch correctly fixed. For the small ratio criteria of 2.0, the CFR improved considerably from 51.7% for GPS alone, to 98.3% when using GPS + GLONASS + BDS combined solutions.     

Single receiver phase ambiguity resolution with GPS data

Global positioning system (GPS) data processing algorithms typically improve positioning solution accuracy by fixing double-differenced phase bias ambiguities to integer values. These “double-difference ambiguity resolution” methods usually invoke linear combinations of GPS carrier phase bias estimates from pairs of transmitters and pairs of receivers, and traditionally require simultaneous measurements from at least two receivers. However, many GPS users point position a single local receiver, based on publicly available solutions for GPS orbits and clocks. These users cannot form double differences. We present an ambiguity resolution algorithm that improves solution accuracy for single receiver point-positioning users. The algorithm processes dual- frequency GPS data from a single receiver together with wide-lane and phase bias estimates from the global network of GPS receivers that were used to generate the orbit and clock solutions for the GPS satellites. We constrain (rather than fix) linear combinations of local phase biases to improve compatibility with global phase bias estimates. For this precise point positioning, no other receiver data are required. When tested, our algorithm significantly improved repeatability of daily estimates of ground receiver positions, most notably in the east component by approximately 30% with respect to the nominal case wherein the carrier biases are estimated as real values. In this “static” test for terrestrial receiver positions, we achieved daily repeatability of 1.9, 2.1 and 6.0 mm in the east, north and vertical (ENV) components, respectively. For kinematic solutions, ENV repeatability is 7.7, 8.4, and 11.7 mm, respectively, representing improvements of 22, 8, and 14% with respect to the nominal. Results from precise orbit determination of the twin GRACE satellites demonstrated that the inter-satellite baseline accuracy improved by a factor of three, from 6 to 2 mm up to a long-term bias. Jason-2/Ocean Surface Topography Mission precise orbit determination tests results implied radial orbit accuracy significantly below the 10 mm level. Stability of time transfer, in low-Earth orbit, improved from 40 to 7 ps. We produced these results by applying this algorithm within the Jet Propulsion Laboratory’s (JPL’s) GIPSY/OASIS software package and using JPL’s orbit and clock products for the GPS constellation. These products now include a record of the wide-lane and phase bias estimates from the underlying global network of GPS stations. This implies that all GIPSY–OASIS positioning users can now benefit from this capability to perform single-receiver ambiguity resolution.     

Multi-GNSS phase delay estimation and PPP ambiguity resolution: GPS,BDS, GLONASS,Galileo

This paper focuses on the precise point positioning (PPP) ambiguity resolution (AR) using the observations acquired from four systems: GPS, BDS, GLONASS, and Galileo (GCRE). A GCRE four-system uncalibrated phase delay (UPD) estimation model and multi-GNSS undifferenced PPP AR method were developed in order to utilize the observations from all systems. For UPD estimation, the GCRE-combined PPP solutions of the globally distributed MGEX and IGS stations are performed to obtain four-system float ambiguities and then UPDs of GCRE satellites can be precisely estimated from these ambiguities. The quality of UPD products in terms of temporal stability and residual distributions is investigated for GPS, BDS, GLONASS, and Galileo satellites, respectively. The BDS satellite-induced code biases were corrected for GEO, IGSO, and MEO satellites before the UPD estimation. The UPD results of global and regional networks were also evaluated for Galileo and BDS, respectively. As a result of the frequency-division multiple-access strategy of GLONASS, the UPD estimation was performed using a network of homogeneous receivers including three commonly used GNSS receivers (TRIMBLE NETR9, JAVAD TRE_G3TH DELTA, and LEICA). Data recorded from 140 MGEX and IGS stations for a 30-day period in January in 2017 were used to validate the proposed GCRE UPD estimation and multi-GNSS dual-frequency PPP AR. Our results show that GCRE four-system PPP AR enables the fastest time to first fix (TTFF) solutions and the highest accuracy for all three coordinate components compared to the single and dual system. An average TTFF of 9.21 min with \(7{^{\circ }}\) cutoff elevation angle can be achieved for GCRE PPP AR, which is much shorter than that of GPS (18.07 min), GR (12.10 min), GE (15.36 min) and GC (13.21 min). With observations length of 10 min, the positioning accuracy of the GCRE fixed solution is 1.84, 1.11, and 1.53 cm, while the GPS-only result is 2.25, 1.29, and 9.73 cm for the east, north, and vertical components, respectively. When the cutoff elevation angle is increased to \(30{^{\circ }}\), the GPS-only PPP AR results are very unreliable, while 13.44 min of TTFF is still achievable for GCRE four-system solutions.     

BDS-3四频参考站间宽巷模糊度解算方法

针对中长基线参考站间宽巷模糊度固定较慢的问题,提出了一种基于BDS-3四频信号(B1I/B2a/B3I/B1C)的分步无电离层(SIF)单历元解算方法.利用BDS-3四频超宽巷/宽巷模糊度易于固定的优点,通过无几何无电离层(GIF)法固定两个超宽巷模糊度,将模糊度固定的双差载波观测值与待求的宽巷观测值组成无电离层模型消...     

利用迭代加权模糊度函数法的单历元模糊度固定

为提高全球导航卫星系统GNSS在复杂环境下的定向成功率,将迭代加权的思想与模糊度函数法（AFM）相结合,提出迭代加权模糊度函数法（IWAFW）。基于AFM算法,通过设计新的适应度函数,利用残差计算权重,实现对不同卫星信号权重的自动调节,并结合惯性传感器给出的姿态信息,压缩搜索空间,降低计算量。通过实测试验验证,在复杂环境中,相比于AFM和带基线长度约束的最小二乘模糊度去相关算法（CLAMBDA）,该算法能够有效提高模糊度固定成功率。     

GNSS three carrier ambiguity resolution using ionosphere-reduced virtual signals

This paper presents a general modeling strategy for ambiguity resolution (AR) and position estimation (PE) using three or more phase-based ranging signals from a global navigation satellite system (GNSS). The proposed strategy will identify three best “virtual” signals to allow for more reliable AR under certain observational conditions characterized by ionospheric and tropospheric delay variability, level of phase noise and orbit accuracy. The selected virtual signals suffer from minimal or relatively low ionospheric effects, and thus are known as ionosphere-reduced virtual signals. As a result, the ionospheric parameters in the geometry-based observational models can be eliminated for long baselines, typically those of length tens to hundreds of kilometres. The proposed modeling comprises three major steps. Step 1 is the geometry-free determination of the extra-widelane (EWL) formed between the two closest L-band carrier measurements, directly from the two corresponding code measurements. Step 2 forms the second EWL signal and resolves the integer ambiguity with a geometry-based estimator alone or together with the first EWL. This is followed by a procedure to correct for the first-order ionospheric delay using the two ambiguity-fixed widelane (WL) signals derived from the integer-fixed EWL signals. Step 3 finds an independent narrow-lane (NL) signal, which is used together with a refined WL to resolve NL ambiguity with geometry-based integer estimation and search algorithms. As a result, the above two AR processes performed with WL/NL and EWL/WL signals respectively, either in sequence or in parallel, can support real time kinematic (RTK) positioning over baselines of tens to hundreds of kilometres, thus enabling centimetre-to-decimentre positioning at the local, regional and even global scales in the future.