Triple-frequency Precise Point Positioning Research Articles

Characteristics and modelling of BDS satellite inter-frequency clock bias for triple-frequency PPP

Aiming at the problem of the inter-frequency clock bias (IFCB) in Beidou Navigation Satellite System (BDS) triple-frequency observations, the variation characteristics with time are analysed in detail. The IFCB models for all the three kinds of satellites, i.e. GEO, IGSO and MEO, are proposed. Then the attenuation and long-term forecasting performance of the models are evaluated. Finally, the validity and benefit of the model are verified by triple-frequency Precise Point Positioning (PPP) experiments. The IFCB results from consecutive two-month BDS triple-frequency observation data of 44 globally distributed Multi-GNSS Experiment (MGEX) stations show that the IFCBs of GEO satellites have prominent periodic variation in general. The correlation coefficient and the determination coefficient of predicted IFCB by the model for GEO satellites are up to 0.957 and 0.915 respectively, which shows the model has a high stability and is suitable for long-term prediction. The IFCBs of IGSO satellites have the same periodicity as those of GEO satellites. Although the model is not as good as that of GEO satellites, it still performs well overall and can be applied to long-term prediction in most instances. The model of MEO satellites performs worse than GEO and IGSO due to the numerical instability and less obvious periodicity of IFCBs. Moreover, the effect of the IFCB models is better than the substitution method of using the IFCB at the corresponding moment in the previous period as the IFCB forecast value of the current period. In the triple-frequency Precise Point Positioning (PPP) experiments, the modelled IFCB correction can improve the positioning accuracy by 21.1%, 9.0% and 9.9% in the north, east and up directions and also shorten the convergence time in these three directions by 26.8%, 10.4% and 24.4% respectively compared with the observation model without IFCB correction.

Multi-GNSS triple-frequency differential code bias (DCB) determination with precise point positioning (PPP)

Differential code biases (DCBs) account for the most significant systematical biases when sensing the earth’s ionosphere with GNSS observations and are also important correction parameters in GNSS applications of positioning, navigation and timing. With the continuous modernization of the American GPS and Russian GLONASS systems, and also the rapid developments of the European Galileo and Chinese BeiDou systems, there is a strong demand of precise satellite DCB products for multiple constellations and frequencies. This study proposes a new method for the precise determination of multi-GNSS triple-frequency DCBs, which can be divided into three steps. The first step is to precisely retrieve slant ionospheric delays and “additional code biases” based on a newly established full-rank triple-frequency precise point positioning (PPP) model with raw observations. Both the slant ionospheric delays and “additional code biases” containing the DCBs need to be estimated. Then, an enhanced IGGDCB (IGG stands for Institute of Geodesy and Geophysics) method is used to estimate the DCBs between the first and second frequency bands with the PPP-derived slant ionospheric delays. At last, the previously estimated DCBs between the first and second frequency bands are substituted into the “additional code biases” and DCBs between the first and third frequency bands are estimated. Multi-GNSS slant ionospheric delays from the triple-frequency PPP method are compared with those from the traditional dual-frequency carrier-to-code level (CCL) method, in terms of formal precision and zero-baseline experiment. Quad-system average formal precisions are 0.08 and 0.41 TECU, for PPP and CCL methods, respectively, indicating the obvious improvements of PPP over CCL. One month of data from 60 globally distributed multi-GNSS experiment stations are selected, and totally eight types of DCBs are estimated for GPS, GLONASS, Galileo and BeiDou. Multi-GNSS satellite DCBs generated with the proposed method are compared with the products from different agencies, including Center for Orbit Determination in Europe (CODE), Deutsches zentrum fur Luft-und Raumfahrt (DLR) and Chinese Academy of Sciences (CAS). For GPS C1WC2 W DCBs, RMS values with respect to CODE products are 0.24, 0.07 and 0.09 ns for DLR, CAS and IGG (this study), respectively. RMS values are 0.31/0.25 and 0.19/0.15 ns, for GPS C1WC5X and C1WC5Q DCBs and with respect to DLR/CAS, respectively. For GLONASS C1PC2P DCBs, RMS values with respect to CODE are 0.68, 0.49 and 0.33 ns for DLR, CAS and IGG, respectively. For Galileo, RMS values are 0.16/0.20 and 0.13/0.14 ns, for C1XC5X and C1XC7X DCBs and with respect to DLR/CAS, respectively. For BeiDou, RMS values are 0.32/0.25 and 0.34/0.41, for C2IC7I and C2IC6I DCBs and with respect to DLR/CAS. These results show that the proposed method can provide multi-GNSS and multi-frequency satellite DCB estimation with high precision, processing efficiency and flexibility.

GPS inter-frequency clock bias estimation for both uncombined and ionospheric-free combined triple-frequency precise point positioning

The time-varying biases within carrier phase observations will be integrated with satellite clock offset parameters in the precise clock estimation. The inconsistency among signal-dependent phase biases within a satellite results in the inadequacy of the current L1/L2 ionospheric-free (IF) satellite clock products for the GPS precise point positioning (PPP) involving L5 signal. The inter-frequency clock bias (IFCB) estimation approaches for triple-frequency PPP based on either uncombined (UC) observations or IF combined observations within a single arbitrary combination are proposed in this study. The key feature of the IFCB estimation approaches is that we only need to obtain a set of phase-specific IFCB (PIFCB) estimates between the L1/L5 and L1/L2 IF satellite clocks, and then, we can directly convert the obtained L1/L5 IF PIFCBs into L5 UC PIFCBs and L1/L2/L5 IF PIFCBs by multiplying individual constants. The mathematical conversion formula is rigorously derived. The UC and IF triple-frequency PPP models are developed. Datasets from 171 stations with a globally even distribution on seven consecutive days were adopted for analysis. After 24-h observation, the UC and IF triple-frequency PPP without PIFCB corrections can achieve an accuracy of 8, 6 and 13 mm, and 8, 5 and 13 mm in east, north and up coordinate components, respectively, while the corresponding positioning accuracy of the cases with PIFCB consideration can be improved by 38, 33 and 31%, and 50, 40 and 23% to 5, 4 and 9 mm, and 4, 3 and 10 mm in the three components, respectively. The corresponding improvement in convergence time is 17, 1 and 22% in the three components in UC model, respectively. Moreover, the phase observation residuals on L5 frequency in UC triple-frequency PPP and of L1/L2/L5 IF combination in IF triple-frequency PPP are reduced by about 4 mm after applying PIFCB corrections. The performance improvement in UC triple-frequency PPP over UC dual-frequency PPP is 7, 4 and 2% in terms of convergence time in the three components, respectively. The daily solutions of UC triple-frequency PPP have a comparable positioning accuracy to the UC dual-frequency PPP.

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.

Uncombined precise point positioning with triple-frequency GNSS signals

With the modernisation of GPS and the development of GALILEO/BDS, more satellites can transmit multi-frequency signals, which brings new opportunities and challenges for data integration in multiple systems. This work focuses on the performance of triple-frequency precise point positioning (PPP) and the phase anomaly phenomenon due to the introduction of the third frequency. First, three PPP models, particularly the triple-frequency PPP model using uncombined observations, are derived. The new biases, inter-frequency biases (IFBs), are estimated to compensate for the pseudorange hardware delays in the triple-frequency PPP model. Then the reason for the phase anomaly on the third frequency is analysed theoretically. Finally, three PPP models with real GPS/GALILEO/BDS triple-frequency data are performed to evaluate the performance of triple-frequency PPP and the influence of time-variant phase hardware delays. Due to the smaller magnitude of the time-dependent phase hardware delays for GALILEO and BDS, even regardless of them, the triple-frequency PPP model can achieve a similar or better positioning accuracy. However, this is not the case for GPS. The results show that the traditional clock products based on dual-frequency ionosphere-free (IF) combinations cannot be directly used in the GPS triple-frequency PPP because of the time-dependent part of the phase hardware delays. Ignoring the time-dependent part will cause a much poorer positioning performance compared with dual-frequency PPP. After adding a small amount of process noise to the GPS ambiguities on the third frequency, the 3D positioning accuracy of the GPS/GALILEO/BDS triple PPP model can achieve marginal improvement for both static and kinematic mode.

Three-frequency BDS precise point positioning ambiguity resolution based on raw observables

All BeiDou navigation satellite system (BDS) satellites are transmitting signals on three frequencies, which brings new opportunity and challenges for high-accuracy precise point positioning (PPP) with ambiguity resolution (AR). This paper proposes an effective uncalibrated phase delay (UPD) estimation and AR strategy which is based on a raw PPP model. First, triple-frequency raw PPP models are developed. The observation model and stochastic model are designed and extended to accommodate the third frequency. Then, the UPD is parameterized in raw frequency form while estimating with the high-precision and low-noise integer linear combination of float ambiguity which are derived by ambiguity decorrelation. Third, with UPD corrected, the LAMBDA method is used for resolving full or partial ambiguities which can be fixed. This method can be easily and flexibly extended for dual-, triple- or even more frequency. To verify the effectiveness and performance of triple-frequency PPP AR, tests with real BDS data from 90 stations lasting for 21 days were performed in static mode. Data were processed with three strategies: BDS triple-frequency ambiguity-float PPP, BDS triple-frequency PPP with dual-frequency (B1/B2) and three-frequency AR, respectively. Numerous experiment results showed that compared with the ambiguity-float solution, the performance in terms of convergence time and positioning biases can be significantly improved by AR. Among three groups of solutions, the triple-frequency PPP AR achieved the best performance. Compared with dual-frequency AR, additional the third frequency could apparently improve the position estimations during the initialization phase and under constraint environments when the dual-frequency PPP AR is limited by few satellite numbers.

Considering Inter-Frequency Clock Bias for BDS Triple-Frequency Precise Point Positioning

The joint use of multi-frequency signals brings new prospects for precise positioning and has become a trend in Global Navigation Satellite System (GNSS) development. However, a new type of inter-frequency clock bias (IFCB), namely the difference between satellite clocks computed with different ionospheric-free carrier phase combinations, was noticed. Consequently, the B1/B3 precise point positioning (PPP) cannot directly use the current B1/B2 clock products. Datasets from 35 globally distributed stations are employed to investigate the IFCB. For new generation BeiDou Navigation Satellite System (BDS) satellites, namely BDS-3 satellites, the IFCB between B1/B2a and B1/B3 satellite clocks, between B1/B2b and B1/B3 satellite clocks, between B1C/B2a and B1C/B3 satellite clocks, and between B1C/B2b and B1C/B3 satellite clocks is analyzed, and no significant IFCB variations can be observed. The IFCB between B1/B2 and B1/B3 satellite clocks for BDS-2 satellites varies with time, and the IFCB variations are generally confined to peak amplitudes of about 5 cm. The IFCB of BDS-2 satellites exhibits periodic signal, and the accuracy of prediction for IFCB, namely the root mean square (RMS) statistic of the difference between predicted and estimated IFCB values, is 1.2 cm. A triple-frequency PPP model with consideration of IFCB is developed. Compared with B1/B2-based PPP, the positioning accuracy of triple-frequency PPP with BDS-2 satellites can be improved by 12%, 25% and 10% in east, north and vertical directions, respectively.

Handling the satellite inter-frequency biases in triple-frequency observations

The new generation of GNSS satellites, including BDS, Galileo, modernized GPS, and GLONASS, transmit navigation sdata at more frequencies. Multi-frequency signals open new prospects for precise positioning, but satellite code and phase inter-frequency biases (IFB) induced by the third frequency need to be handled. Satellite code IFB can be corrected using products estimated by different strategies, the theoretical and numerical compatibility of these methods need to be proved. Furthermore, a new type of phase IFB, which changes with the relative sun–spacecraft–earth geometry, has been observed. It is necessary to investigate the cause and possible impacts of phase Time-variant IFB (TIFB). Therefore, we present systematic analysis to illustrate the relevancy between satellite clocks and phase TIFB, and compare the handling strategies of the code and phase IFB in triple-frequency positioning. First, the un-differenced L1/L2 satellite clock corrections considering the hardware delays are derived. And IFB induced by the dual-frequency satellite clocks to triple-frequency PPP model is detailed. The analysis shows that estimated satellite clocks actually contain the time-variant phase hardware delays, which can be compensated in L1/L2 ionosphere-free combinations. However, the time-variant hardware delays will lead to TIFB if the third frequency is used. Then, the methods used to correct the code and phase IFB are discussed. Standard point positioning (SPP) and precise point positioning (PPP) using BDS observations are carried out to validate the improvement of different IFB correction strategies. Experiments show that code IFB derived from DCB or geometry-free and ionosphere-free combination show an agreement of 0.3ns for all satellites. Positioning results and error distribution with two different code IFB correcting strategies achieve similar tendency, which shows their substitutability. The original and wavelet filtered phase TIFB long-term series show significant periodical characteristic for most GEO and IGSO satellites, with the magnitude varies between – 5cm and 5cm. Finally, BDS L1/L3 kinematic PPP is conducted with code IFB corrected with DCB combinations, and TIFB corrected with filtered series. Results show that the IFB corrected L1/L3 PPP can achieve comparable convergence and positioning accuracy as L1/L2 combinations in static and kinematic mode.

Triple-Frequency GPS Precise Point Positioning Ambiguity Resolution Using Dual-Frequency Based IGS Precise Clock Products

With the availability of the third civil signal in the Global Positioning System, triple-frequency Precise Point Positioning ambiguity resolution methods have drawn increasing attention due to significantly reduced convergence time. However, the corresponding triple-frequency based precise clock products are not widely available and adopted by applications. Currently, most precise products are generated based on ionosphere-free combination of dual-frequency L1/L2 signals, which however are not consistent with the triple-frequency ionosphere-free carrier-phase measurements, resulting in inaccurate positioning and unstable float ambiguities. In this study, a GPS triple-frequency PPP ambiguity resolution method is developed using the widely used dual-frequency based clock products. In this method, the interfrequency clock biases between the triple-frequency and dual-frequency ionosphere-free carrier-phase measurements are first estimated and then applied to triple-frequency ionosphere-free carrier-phase measurements to obtain stable float ambiguities. After this, the wide-lane L2/L5 and wide-lane L1/L2 integer property of ambiguities are recovered by estimating the satellite fractional cycle biases. A test using a sparse network is conducted to verify the effectiveness of the method. The results show that the ambiguity resolution can be achieved in minutes even tens of seconds and the positioning accuracy is in decimeter level.

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.

Characteristics of inter-frequency clock bias for Block IIF satellites and its effect on triple-frequency GPS precise point positioning

The latest generation of GPS satellites, termed Block IIF, provides a new L5 signal. Multi-frequency signals open new prospects for precise positioning and fast ambiguity resolution and have become the trend in Global Navigation Satellite System (GNSS) development. However, a new type of inter-frequency clock bias (IFCB), i.e., the difference between the current clock products computed with L1/L2 and the satellite clocks computed with L1/L5, was noticed. Consequently, the L1/L2 clock products cannot be used for L1/L5 precise point positioning (PPP). In order to solve this issue, the IFCB should be estimated with a high accuracy. Datasets collected at 129 globally distributed Multi-GNSS Experiment (MGEX) stations from 2015 are employed to investigate the IFCB. The results indicate that the IFCB is satellite dependent and varies with the relative sun---spacecraft---earth geometry. Other factors, however, may also contribute to the IFCB variations according to the harmonic analysis of the single-day IFCB time series. In addition, the results show that the IFCB exhibits periodic signal with a notable period of 43,080 s and the peak-to-peak amplitude is 0.023---0.269 m. After considering a time lag of 240 s, the average cross-correlation coefficient between the IFCB series of two consecutive days is 0.943, and the prediction accuracy of IFCB is 0.006 m. A triple-frequency PPP model that takes the IFCB into account is proposed. When using 3-h datasets, the positioning accuracy of triple-frequency PPP can be improved by 19, 13 and 21 % compared with the L1/L2-based PPP in the east, north and up directions, respectively.

A New Method of Ambiguity Resolution for Triple-Frequency GPS PPP

At present, reliable ambiguity resolution in GPS precise point positioning (PPP) can be achieved through the traditional model called “EWL-WL-NL”. In this paper, we proposed a new model of ambiguity resolution,“WL-WL-WL”, where making use of linear independence of coefficient vector of wide-lane combination. Firstly, using the Melbourne-Wubbena combination observable on L2 and L5, we could resolve extra-wide-lane ambiguity instantaneously. Then, the resolved unambiguous extra-wide-lane carrier-phase assists wide-lane ambiguity resolution (AR). Three wide-lane combinations whose coefficient vectors are linearly independent are chosen to compose one full-rank matrix so that the three narrow-lane ambiguity resolution can be achieved. As a result, with the triple-frequency signals, the correctness rate of narrow-lane ambiguity resolution achieves 90% within 60s, in contrast to only 63% within 180s in dual-frequency PPP. Therefore, we demonstrate that triple-frequency PPP has the potential to achieve ambiguity-fixed solutions within a few minutes and the efficiency of ambiguity resolution in triple-frequency PPP is higher.

BeiDou phase bias estimation and its application in precise point positioning with triple-frequency observable

At present, the BeiDou system (BDS) enables the practical application of triple-frequency observable in the Asia-Pacific region, of many possible benefits from the additional signal; this study focuses on exploiting the contribution of zero difference (ZD) ambiguity resolution (AR) to the precise point positioning (PPP). A general modeling strategy for multi-frequency PPP AR is presented, in which, the least squares ambiguity decorrelation adjustment (LAMBDA) method is employed in ambiguity fixing based on the full variance-covariance ambiguity matrix generated from the raw data processing model. Because of the reliable fixing of BDS L1 ambiguity faces more difficulty, the LAMBDA method with partial ambiguity fixing is proposed to enable the independent and instantaneous resolution of extra wide-lane (EWL) and wide-lane (WL). This mechanism of sequential ambiguity fixing is demonstrated for resolving ZD satellite phase bias and performing triple-frequency PPP AR with two reference station networks with a typical baseline of up to 400 and 800 km, respectively. Tests show that about $$90\,\%$$ of the EWL and WL phase bias of BDS has a consistency of better than 0.1 cycle, and this value decreases to $$<$$ 80 % for L1 phase bias for Experiment I, while all the solutions of Experiment II have a similar RMS of about 0.12 cycles. In addition, the repeatability of the daily mean phase bias agree to 0.093 cycles and 0.095 cycles for EWL and WL on average, which is much smaller than 0.20 cycles of L1. To assess the improvement of fixed PPP brought by applying the third frequency signal as well as the above phase bias, various ambiguity fixing strategy are considered in the numerical demonstration. It is shown that the impact of the additional signal is almost negligible when only float solution involved. It is also shown that by fixing EWL and WL together, as opposed to the single ambiguity fixing, will leads to an improvement in PPP accuracy by about $$20.6\,\%$$ on average. Attributed to the efficient resolution of EWL $$+$$ WL within about 2 min in Experiment I, the 0.5 m level positioning can be achieved in 10 min for both horizontal and vertical, compared to 50 min for horizontal and 30 min for vertical by the NONE/EWL/WL fixed solution. While, for Experiment II, the improvement in the convergence can only be seen for the horizontal as the TTFF takes about 40 min for EWL and WL to be resolved.

Triple-frequency GPS precise point positioning with rapid ambiguity resolution

At present, reliable ambiguity resolution in real-time GPS precise point positioning (PPP) can only be achieved after an initial observation period of a few tens of minutes. In this study, we propose a method where the incoming triple-frequency GPS signals are exploited to enable rapid convergences to ambiguity-fixed solutions in real-time PPP. Specifically, extra-wide-lane ambiguity resolution can be first achieved almost instantaneously with the Melbourne-Wubbena combination observable on L2 and L5. Then the resultant unambiguous extra-wide-lane carrier-phase is combined with the wide-lane carrier-phase on L1 and L2 to form an ionosphere-free observable with a wavelength of about 3.4 m. Although the noise of this observable is around 100 times the raw carrier-phase noise, its wide-lane ambiguity can still be resolved very efficiently, and the resultant ambiguity-fixed observable can assist much better than pseudorange in speeding up succeeding narrow-lane ambiguity resolution. To validate this method, we use an advanced hardware simulator to generate triple-frequency signals and a high-grade receiver to collect 1-Hz data. When the carrier-phase precisions on L1, L2 and L5 are as poor as 1.5, 6.3 and 1.5 mm, respectively, wide-lane ambiguity resolution can still reach a correctness rate of over 99 % within 20 s. As a result, the correctness rate of narrow-lane ambiguity resolution achieves 99 % within 65 s, in contrast to only 64 % within 150 s in dual-frequency PPP. In addition, we also simulate a multipath-contaminated data set and introduce new ambiguities for all satellites every 120 s. We find that when multipath effects are strong, ambiguity-fixed solutions are achieved at 78 % of all epochs in triple-frequency PPP whilst almost no ambiguities are resolved in dual-frequency PPP. Therefore, we demonstrate that triple-frequency PPP has the potential to achieve ambiguity-fixed solutions within a few minutes, or even shorter if raw carrier-phase precisions are around 1 mm. In either case, we conclude that the efficiency of ambiguity resolution in triple-frequency PPP is much higher than that in dual-frequency PPP.