Evaluation of BDS-2 and BDS-3 Satellite Atomic Clock Products and Their Effects on Positioning

The BeiDou Navigation Satellite System (BDS) has completed third phase construction and currently provides global services, with a mixed constellation of BDS-2 and BDS-3. The newly launched BDS-3 satellites are equipped with rubidium and passive hydrogen maser (PHM) atomic clocks. The performance of atomic clocks is one of the cores of satellite navigation system, which will affect the performance of positioning, navigation and timing (PNT). In this paper, we systematically analyze the characteristics of BDS-2 and BDS-3 atomic clocks, based on more than one year of precise satellite clock products and broadcast ephemeris. Firstly, the results of overlapping Allan variations demonstrate that BDS-3 Rb and PHM clocks improve better in stability than BDS-2 Rb clock and are comparable to GPS IIF Rb and Galileo PHM clocks. Accordingly, the STDs of BDS-3 broadcast satellite clock are better than GPS and BDS-2, which are at the same level with that of Galileo. Secondly, the inter-system bias (ISB) between BDS-2 and BDS-3 is analyzed by satellite clock datum comparison and precise point positioning (PPP). Surprisingly, the discrepancy between BDS-2 and BDS-3 satellite clock datum has a great difference between products that could reach up to about 10 ns for WHU satellite clock products and broadcast ephemeris. Moreover, the ISBs between BDS-2 and BDS-3 satellite clocks are quite stable over one-year periods. Thirdly, due to the improved stability of BDS-3 atomic clock, the 68% positioning accuracy is better than 0.65 m at 10 min for BDS-3 PPP, based on broadcast ephemeris. Besides, the non-negligible bias between BDS-2 and BDS-3 will greatly affect the BDS precise data processing. The accuracy of positioning is greatly improved when considering the ISB.

Assessment of Multi-Frequency PPP Ambiguity Resolution Using Galileo and BeiDou-3 Signals

From network RTK to PPP-RTK, it is highly expected that high-precision positioning within a few minutes can be achieved with a sparse reference network. In this study, we investigate a rapid multi-frequency PPP convergence strategy based on Galileo E1/E5a/E6 and BeiDou-3 B1C/B2a/B3I signals, whose unambiguous wide-lane observables can efficiently assist in speeding up narrow-lane ambiguity resolution. Furthermore, frequency-specific biases existing on the third-frequency observables have been observed to slow down multi-frequency PPP-AR convergence. In this study, we partially mitigated their effects by estimating a second satellite clock for the third frequency of signals. We validated this approach with one month of data collected from 22 stations. On average, it took about 18 min for PPP wide-lane ambiguity resolution (PPP-WAR) to converge, while 32 min were required for ambiguity-float PPP. Compared with dual-frequency PPP-AR, which needed nearly 12 min to converge, multi-frequency PPP-AR required 6 min only. Once there were more than 10 satellites involved in PPP, the convergence could be achieved within 3 min on average. Meanwhile, 81% and 62% of multi-frequency PPP-AR solutions converged successfully within 5 and 1 min, respectively. Finally, we carried out a vehicle-borne experiment to validate this approach in a kinematic environment. Owing to frequent cycle slips during the movement of vehicle, it took 14 min for B1C/B2a/B3I and E1/E5a/E6 PPP-AR to obtain reliable positions, and 19 min for those using the other signal combinations B1C/B2a/B2b and E1/E5a/E5b, owning to higher noise. Overall, these results are promising for achieving high-precision PPP positioning globally within a few minutes if multi-frequency biases can be handled well in the data processing.

Identification of BDS Satellite Clock Periodic Signals Based on Lomb-Scargle Power Spectrum and Continuous Wavelet Transform

Onboard satellite clocks are the basis of Global Navigation Satellite Systems (GNSS) operation, and their revolution periods are at the level of 2 per day (about 12 h) in the case of the Medium Earth Orbit (MEO) satellites. In this work, the authors analysed the entire BeiDou Navigation Satellite System (BDS) space segment (BDS-2 and BDS-3) in terms of the occurrence of periodic, repetitive signals in the clock products, and checked if they coincide with the orbital periods or their multiples. The Lomb-Scargle (L-S) power spectrum was used as a tool to determine the periods present in the BDS clock products, allowing for analyses based on incomplete input data; in this case, the incomplete data were the phase data with jumps and outliers removed. In addition, continuous wavelet transform (CWT) was used to produce a time−frequency representation showing the more complex behaviour of the satellite clock products. As shown in the case of geostationary and geosynchronous inclined orbit satellites, the main period was 23.935 h, while for the Medium Earth Orbit it was 12.887 h, with the BDS satellite orbital period being 12 h 53 m (12.883 h). Some effects connected with reference clock swapping are also visible in the power spectrum. The conducted analyses showed that the BDS-2 satellite clocks have much higher noise than the BDS-3 satellite clocks, meaning that the number of designated periods is greater, but their reliability is significantly lower. BDS-3 satellites have only been in operation for a very short time, thus this is the first analysis to include this type of data. Moreover, such a wide and complex analysis has not been carried out to date.

BDS Satellite Clock Prediction Considering Periodic Variations

The periodic noise exists in BeiDou navigation satellite system (BDS) clock offsets. As a commonly used satellite clock prediction model, the spectral analysis model (SAM) typically detects and identifies the periodic terms by the Fast Fourier transform (FFT) according to long-term clock offset series. The FFT makes an aggregate assessment in frequency domain but cannot characterize the periodic noise in a time domain. Due to space environment changes, temperature variations, and various disturbances, the periodic noise is time-varying, and the spectral peaks vary over time, which will affect the prediction accuracy of the SAM. In this paper, we investigate the periodic noise and its variations present in BDS clock offsets, and improve the clock prediction model by considering the periodic variations. The periodic noise and its variations over time are analyzed and quantified by short time Fourier transform (STFT). The results show that both the amplitude and frequency of the main periodic term in BDS clock offsets vary with time. To minimize the impact of periodic variations on clock prediction, a time frequency analysis model (TFAM) based on STFT is constructed, in which the periodic term can be quantified and compensated accurately. The experiment results show that both the fitting and prediction accuracy of TFAM are better than SAM. Compared with SAM, the average improvement of the prediction accuracy using TFAM of the 6 h, 12 h, 18 h and 24 h is in the range of 6.4% to 10% for the GNSS Research Center of Wuhan University (WHU) clock offsets, and 11.1% to 14.4% for the Geo Forschungs Zentrum (GFZ) clock offsets. For the satellites C06, C14, and C32 with marked periodic variations, the prediction accuracy is improved by 26.7%, 16.2%, and 16.3% for WHU clock offsets, and 29.8%, 16.0%, 21.0%, and 9.0% of C06, C14, C28, and C32 for GFZ clock offsets.

A Novel Long Short-Term Memory Predicted Algorithm for BDS Short-Term Satellite Clock Offsets

The satellite clocks carried on the BeiDou navigation System (BDS) are a self-manufactured hydrogen clock and improved rubidium clock, and their on-orbit performance and stabilities are not as efficient as GPS and Galileo satellite clocks caused of the orbital diversity of the BDS and the complexity of the space operating environment. Therefore, the existing BDS clock product cannot guarantee the high accuracy demand for precise point positioning in real-time scenes while the communication link is interrupted. To deal with this problem, we proposed a deep learning-based approach for BDS short-term satellite clock offset modeling which utilizes the superiority of Long Short-Term Memory (LSTM) derived from Recurrent Neural Networks (RNN) in time series modeling, and we call it QPLSTM. The ultrarapid predicted clock products provided by IGS (IGU-P) and four widely used prediction methods (the linear polynomial, quadratic polynomial, gray system (GM (1,1)), and Autoregressive Integrated Moving Average (ARIMA) model) are selected to compare with the QPLSTM. The results show that the prediction residual is lower than clock products of IGU-P during 6-hour forecasting and the QPLSM shows a greater performance than the mentioned four models. The average prediction accuracy has improved by approximately 79.6, 69.2, 80.4, and 77.1% and 68.3, 52.7, 66.5, and 69.8% during a 30 min and 1-hour forecasting. Thus, the QPLSTM can be considered as a new approach to acquire high-precision satellite clock offset prediction.

Comparisons of CODE and CNES/CLS GPS satellite bias products and applications in Sentinel-3 satellite precise orbit determination

To resolve undifferenced GNSS phase ambiguities, dedicated satellite products are needed, such as satellite orbits, clock offsets and biases. The International GNSS Service CNES/CLS analysis center provides satellite (HMW) Hatch-Melbourne-Wübbena bias and dedicated satellite clock products (including satellite phase bias), while the CODE analysis center provides satellite OSB (observable-specific-bias) and integer clock products. The CNES/CLS GPS satellite HMW bias products are determined by the Hatch-Melbourne-Wübbena (HMW) linear combination and aggregate both code (C1W, C2W) and phase (L1W, L2W) biases. By forming the HMW linear combination of CODE OSB corrections on the same signals, we compare CODE satellite HMW biases to those from CNES/CLS. The fractional part of GPS satellite HMW biases from both analysis centers are very close to each other, with a mean Root-Mean-Square (RMS) of differences of 0.01 wide-lane cycles. A direct comparison of satellite narrow-lane biases is not easily possible since satellite narrow-lane biases are correlated with satellite orbit and clock products, as well as with integer wide-lane ambiguities. Moreover, CNES/CLS provides no satellite narrow-lane biases but incorporates them into satellite clock offsets. Therefore, we compute differences of GPS satellite orbits, clock offsets, integer wide-lane ambiguities and narrow-lane biases (only for CODE products) between CODE and CNES/CLS products. The total difference of these terms for each satellite represents the difference of the narrow-lane bias by subtracting certain integer narrow-lane cycles. We call this total difference “narrow-lane” bias difference. We find that 3% of the narrow-lane biases from these two analysis centers during the experimental time period have differences larger than 0.05 narrow-lane cycles. In fact, this is mainly caused by one Block IIA satellite since satellite clock offsets of the IIA satellite cannot be well determined during eclipsing seasons. To show the application of both types of GPS products, we apply them for Sentinel-3 satellite orbit determination. The wide-lane fixing rates using both products are more than 98%, while the narrow-lane fixing rates are more than 95%. Ambiguity-fixed Sentinel-3 satellite orbits show clear improvement over float solutions. RMS of 6-h orbit overlaps improves by about a factor of two. Also, we observe similar improvements by comparing our Sentinel-3 orbit solutions to the external combined products. Standard deviation value of Satellite Laser Ranging residuals is reduced by more than 10% for Sentinel-3A and more than 15% for Sentinel-3B satellite by fixing ambiguities to integer values.

A Method to Accelerate the Convergence of Satellite Clock Offset Estimation Considering the Time-Varying Code Biases

Continuous and stable precision satellite clock offsets are an important guarantee for real-time precise point positioning (PPP). However, in real-time PPP, the estimation of a satellite clock is often interrupted for various reasons such as network fluctuations, which leads to a long time for clocks to converge again. Typically, code biases are assumed to stay constant over time in clock estimation according to the current literature. In this contribution, it is shown that this assumption reduces the convergence speed of estimation, and the satellite clocks are still unstable for several hours after convergence. For this reason, we study the influence of different code bias extraction schemes, that is, taking code biases as constants, extracting satellite code biases (SCBs), extracting receiver code biases (RCBs) and simultaneously extracting SCBs and RCBs, on satellite clock estimation. Results show that, the time-varying SCBs are the main factors leading to the instability of satellite clocks, and considering SCBs in the estimation can significantly accelerate the filter convergence and improve the stability of clocks. Then, the products generated by introducing SCBs in the clock estimation based on undifferenced observations are applied to PPP experiments. Compared with the original undifferenced model, clocks estimated using the new method can significantly accelerate the convergence speed of PPP and improve the positioning accuracy, which illustrates that our estimated clocks are effective and superior.

The CODE ambiguity-fixed clock and phase bias analysis products: generation, properties, and performance

The generation and use of GNSS analysis products that allow—particularly for the needs of single-receiver applications—precise point positioning with ambiguity resolution (PPP-AR) are becoming more and more popular. A general uncertainty concerns the question on how the necessary phase bias information should be provided to the PPP-AR user. Until now, each AR-enabling clock/bias representation method had its own practice to provide the necessary bias information. We have generalized the observable-specific signal bias (OSB) representation, as introduced in Villiger (J Geod 93:1487–1500, 2019) originally exclusively for pseudorange measurements, to carrier phase measurements. The existing common clock (CC) approach has been extended in a way that OSBs allowing for flexible signal and frequency handling between multiple GNSS become possible. Advantages of the proposed OSB-based PPP-AR approach are: GNSS biases can be provided in a consistent way for phase and code measurements and it is capable of multi-GNSS and suitable for standardization. This new, extended PPP-AR approach has been implemented by the Center for Orbit Determination in Europe (CODE). CODE clock products that adhere to the integer-cycle property have been submitted to the International GNSS Service (IGS) since mid of 2018 for three analysis lines: Rapid, Final, and MGEX (Multi-GNSS Extension). Ambiguity fixing is performed not only for GPS but also for Galileo. The integer-cycle property of between-satellite clock differences is of fundamental importance when comparing satellite clock estimates among various analysis lines, or at day boundaries. Both kinds of comparisons could be exploited at a very high level of consistency. Any retrieved comparison essentially indicated a standard deviation for between-satellite clocks from CODE of the order of 5 ps (1.5 mm in range). Finally, the integer-cycle property that may be recovered between the CODE Final clock and the accompanying bias product of consecutive daily sessions (using clock estimates additionally provided for the second midnight epoch) allows us to deduce GPS satellite clock and phase bias information that is consistent and continuous with respect to carrier phase observation data over two, three, or, in principle, yet more days. Phase-based clock densification from initially estimated integer-cycle-conform clock corrections at intervals of 300 s to 30 s (5 s in case of our Final clock product) is a matter of particular interest. Based on direct product comparisons and GRACE K-band ranging (KBR) data analysis, the quality of accordingly densified clock corrections could be confirmed to be on a level similar to that of “anchor” (300 s) clock corrections.