Satellite Clock Bias Research Articles

Grey modelling and real-time forecasting for the approximate non-homogeneous white exponential law BDS clock bias sequences

Precise forecasting of satellite clock bias is crucial for ensuring service quality and enhancing the efficiency of real-time precise point positioning (PPP).The grey model with many benefits is an excellent choice for predicting real-time clock bias. However, the absolute prediction error of a small number of satellites is very high in actual forecasting process. In order to address this issue, a non-homogeneous white exponential law grey model has been constructed specifically for predicting clock bias sequences with non-homogeneous class ratio approximating constants. Considering that any model is difficult to apply to different kinds of satellite clocks and clock bias signals, an adaptive selection strategy for forecast model is proposed to ensure more excellent prediction accuracy. Afterwards, a prediction scenario based on the observed products of the BeiDou satellite navigation system (BDS) is created to demonstrate the effectiveness of the approach described in this article. The outcomes of the method are then compared with those of a single grey model and the ultra-rapid predicted products. The outcomes demonstrate that this strategy’s prediction accuracy is less than 1 ns/day and that its prediction uncertainty is much decreased, which may improve data selection for real-time applications like real-time kinematics (RTK) and PPP.

An adaptive robust strategy based on hypothesis testing for satellite clock bias short-term forecasting

The study of short-term forecasting of satellite clock bias (SCB) is important to promote the development of real-time precise point positioning, and the Kalman filter model has certain advantages over other models in the single forecast model. However, the filtering performance will be degraded by the model bias, and a fading factor is constructed to control the effect brought by the model bias. Considering that the fading factor may cause the covariance matrix to be abnormally inflated or even invalid when the observed information is unreliable, the chi-square test is introduced as a test indicator. At the current epoch moment, the fading factor or robust estimation is conditionally applied based on the result of the hypothesis test. An improved IGGIII weight function is used by the robust estimation, which simplifies the solution of the parametric robust solution. BeiDou satellite navigation system-3 satellite clock bias data with different clock types were selected for the analysis of fitting accuracy and forecasting accuracy, and the experimental results showed that the proposed algorithm can well eliminate the anomalous perturbation phenomenon caused by model deviation and improve the fitting accuracy. The comparison of forecast accuracy between the proposed strategy and GM(1,1), quadratic polynomial, and autoregressive integrated moving average (ARIMA) models at 3, 6, and 12 h intervals reveals that the proposed strategy consistently demonstrates optimal forecast accuracy.

Enhancing satellite clock bias prediction in BDS with LSTM-attention model

Enhancing satellite clock bias prediction in BDS with LSTM-attention model

Improved SSA-Based GRU Neural Network for BDS-3 Satellite Clock Bias Forecasting.

Satellite clock error is a key factor affecting the positioning accuracy of a global navigation satellite system (GNSS). In this paper, we use a gated recurrent unit (GRU) neural network to construct a satellite clock bias forecasting model for the BDS-3 navigation system. In order to further improve the prediction accuracy and stability of the GRU, this paper proposes a satellite clock bias forecasting model, termed ITSSA-GRU, which combines the improved sparrow search algorithm (SSA) and the GRU, avoiding the problems of GRU's sensitivity to hyperparameters and its tendency to fall into local optimal solutions. The model improves the initialization population phase of the SSA by introducing iterative chaotic mapping and adopts an iterative update strategy based on t-step optimization to enhance the optimization ability of the SSA. Five models, namely, ITSSA-GRU, SSA-GRU, GRU, LSTM, and GM(1,1), are used to forecast the satellite clock bias data in three different types of orbits of the BDS-3 system: MEO, IGSO, and GEO. The experimental results show that, as compared with the other four models, the ITSSA-GRU model has a stronger generalization ability and forecasting effect in the clock bias forecasting of all three types of satellites. Therefore, the ITSSA-GRU model can provide a new means of improving the accuracy of navigation satellite clock bias forecasting to meet the needs of high-precision positioning.

Characterizing Periodic Variations of Atomic Frequency Standards via Their Frequency Stability Estimates.

The onboard atomic frequency standard (AFS) is a crucial element of Global Navigation Satellite System (GNSS) satellites. However, it is widely accepted that periodic variations can influence the onboard AFS. The presence of non-stationary random processes in AFS signals can lead to inaccurate separation of the periodic and stochastic components of satellite AFS clock data when using least squares and Fourier transform methods. In this paper, we characterize the periodic variations of AFS using Allan and Hadamard variances and demonstrate that the Allan and Hadamard variances of the periodics are independent of the variances of the stochastic component. The proposed model is tested against simulated and real clock data, revealing that our approach provides more precise characterization of periodic variations compared to the least squares method. Additionally, we observe that overfitting periodic variations can improve the precision of GPS clock bias prediction, as indicated by a comparison of fitting and prediction errors of satellite clock bias.

Research on modeling and predicting of BDS-3 satellite clock bias using the LSTM neural network model

Research on modeling and predicting of BDS-3 satellite clock bias using the LSTM neural network model

Performance of Galileo satellite products determined from multi-frequency measurements

Each Galileo satellite provides coherent navigation signals in four distinct frequency bands. International GNSS Service (IGS) analysis centers (ACs) typically determine Galileo satellite products based on the E1/E5a dual-frequency measurements due to the software limitation and the limited tracking capability of other signals in the early time. The goal of this contribution is to evaluate the quality of Galileo satellite products determined by using different dual-frequency (E1/E5a, E1/E5b, E1/E5, E1/E6) and multi-frequency (E1/E5a/E5b/E5/E6) measurements based on different sizes of ground networks. The performance of signal noise, the consistency of frequency-specific satellite phase center offsets and the stability of satellite phase biases are assessed in advance to confirm preconditions for multi-frequency processing. Orbit results from different dual-frequency measurements show that orbit precision determined from E1/E6 is clearly worse (about 35%) than that from other dual-frequency solutions. In view of a similar E1, E5a, E5b and E6 measurement quality, the degraded E1/E6 orbit performance is mainly attributed to the unfavorable noise amplification in the respective ionosphere-free linear combination. The advantage of using multi-frequency measurements over dual-frequency for precise orbit determination is clearly visible when using small networks. For instance, the ambiguity fixing rate is 80% for the multi-frequency solution while it is less than 40% for the dual-frequency solution if 150 s data sampling is employed in a 15-station network. Higher fixing rates result in better (more than 30%) satellite orbits and more robust satellite clock and phase bias products. In general, satellite phase bias products determined from a 20-station (or more) network are precise enough to conduct precise point positioning with ambiguity resolution (PPP-AR) applications. Multi-frequency kinematic PPP-AR solutions always show 5–10% precision improvement compared to those computed from dual-frequency observations.

Improved BDS-2/3 Satellite Ultra-Fast Clock Bias Prediction Based with the SSA-ELM Model

Ultra-fast satellite clock bias (SCB) products play an important role in real-time precise point positioning. Considering the low accuracy of ultra-fast SCB, which is unable to meet the requirements of precise point position, in this paper, we propose a sparrow search algorithm to optimize the extreme learning machine (SSA-ELM) algorithm in order to improve the performance of SCB prediction in the Beidou satellite navigation system (BDS). By using the sparrow search algorithm's strong global search and fast convergence ability, we further improve the prediction accuracy of SCB of the extreme learning machine. This study uses ultra-fast SCB data from the international GNSS monitoring assessment system (iGMAS) to perform experiments. First, the second difference method is used to evaluate the accuracy and stability of the used data, demonstrating that the accuracy between observed data (ISUO) and predicted data (ISUP) of the ultra-fast clock (ISU) products is optimal. Moreover, the accuracy and stability of the new rubidium (Rb-II) clock and hydrogen (PHM) clock onboard BDS-3 are superior to those of BDS-2, and the choice of different reference clocks affects the accuracy of SCB. Then, SSA-ELM, quadratic polynomial (QP), and a grey model (GM) are used for SCB prediction, and the results are compared with ISUP data. The results show that when predicting 3 and 6 h based on 12 h of SCB data, the SSA-ELM model improves the prediction model by ~60.42%, 5.46%, and 57.59% and 72.27%, 44.65%, and 62.96% as compared with the ISUP, QP, and GM models, respectively. When predicting 6 h based on 12 h of SCB data, the SSA-ELM model improves the prediction model by ~53.16% and 52.09% and by 40.66% and 46.38% compared to the QP and GM models, respectively. Finally, multiday data are used for 6 h SCB prediction. The results show that the SSA-ELM model improves the prediction model by more than 25% compared to the ISUP, QP, and GM models. In addition, the prediction accuracy of the BDS-3 satellite is better than that of the BDS-2 satellite.

Mind evolutionary algorithm optimization in the prediction of satellite clock bias using the back propagation neural network

Satellite clock bias is the key factor affecting the accuracy of the single point positioning of a global navigation satellite system. The traditional model back propagation (BP) neural network is prone to local optimum problems. This paper presents a prediction model and algorithm for the clock bias of the BP neural network based on the optimization of the mind evolutionary algorithm (MEA), which is used to optimize the initial weights and thresholds of the BP neural network. The accuracy of the comparison between clock bias data is verified with and without one-time difference processing. Compared with grey model (GM (1,1)) and BP neural network, this paper discusses the advantages and general applicability of this method from different constellation satellites, different atomic clock type satellites, and the amount of modeling data. The accuracy of the grey model (GM(1,1)), BP, and MEA-BP models for satellite clock bias prediction is analyzed and the root mean square error, range difference error, and the mean of the clock bias data compared. The results demonstrate that the prediction accuracy of the three satellites significantly increased after one-time difference processing and that they have good stability. The prediction accuracy of four sessions of 2 h, 3 h, 6 h, and 12 h obtained using the MEA-BP model was better than 0.74, 0.80, 1.12, and 0.87 ns, respectively. The MEA-BP model has a specific degree of improvement in the prediction accuracy of the different sessions. Additionally, the prediction accuracy of different models has a specific relationship with the length of the original modeling sequence, of which BP model is the most affected, and MEABP is relatively less affected by the length of the modeling sequence, indicating that the MEA-BP model has strong anti-interference ability.

Research on modeling and predicting of BDS3 satellite clock bias using the attention mechanism-based LSTM (AttLSTM) neural network model

In the Global Navigation Satellite System (GNSS), the satellite clock bias (SCB) plays an important role in the application of real-time precise point positioning (RT-PPP). Based on the operation of Beidou satellite global service, it is very important to establish a reliable Beidou SCB prediction model. In this research, an attention mechanism-based long short-term memory neural network (AttLSTM) model is applied to SCB prediction. The attention mechanism introduced in modelling can make the model pay less attention to useless information through weight allocation. In this paper, the BeiDou-3 Navigation Satellite System (BDS-3) satellite precision clock product provided by GFZ is used for clock prediction experiments. The proposed AttLSTM model, long short-term memory neural network (LSTM) model and quadratic polynomial (QP) model are compared and evaluated, and 12h and 24h SCB prediction experiments of BDS-3 satellite are set up. The results show that AttLSTM model can achieve high SCB prediction accuracy, and the averaged prediction accuracy of 12h and 24h can reach 1.41ns and 1.75ns. Compared with LSTM and QP models, the prediction accuracy of AttLSTM model is improved by 26.1%, 38.4% for 12h and 29.1%, 43.1% for 24h, respectively. Then, the clock bias predicted by the three models is applied to the static PPP positioning experiment, respectively. Through the analysis of the positioning results of 15 MGEX stations, the averaged positioning accuracy of AttLSTM model in the East, North and Up directions can reach 0.074m, 0.019m and 0.154m, respectively. Compared with LSTM and QP models, the positioning accuracy of AttLSTM model is improved by 42.5% and 44.4% in the East direction, 44.7% and 58.9% in the North direction, and 21.7% and 21.8% in the Up direction.

Analysis of Regional Satellite Clock Bias Characteristics Based on BeiDou System

With the continuous development of the Global Navigation Satellite System (GNSS), the calculation theory and strategy of the global Satellite Clock Bias (SCB) tends to be mature. However, in some eventualities with restricted conditions, the calculation and application of the global SCB are limited; hence, the application of regional SCB is derived. This paper focuses on the quality of regional SCB products in different regions, calculates three groups of regional SCB products, and analyzes their properties and application effects. We expand the double-differenced assessment method for SCB and extend satellite clock accuracy assessment to regional satellite clock products. Additionally, the Regional Effect Bias (REB) is introduced to analyze the influence of the relative position of satellite geometry on the SCB products due to the regional effects. The conclusions are as follows: (1) In low-latitude regions, SCB products have a high degree of completeness and a large number of satellite observations, which is conducive to expanding the positioning application range of regional SCB; (2) the low-latitude regions SCB will be affected by ionospheric activity, and the accuracy will be slightly lower than that of satellite clocks deviation in mid-latitudes; (3) in this paper, the REB in this area is in the level of 10−7. The experiment displays the result that the values of REB in low-latitude areas are larger, leading to fluctuated Precise Point Position (PPP) results. However, there are fewer stations in the mid-latitude regions, which will also affect the accuracy of PPP; (4) the accuracy of the positioning results of the regional satellite clock deviation in the Chinese region is higher than that of the global clock.

A new GM (1,1) model suitable for short‐term prediction of satellite clock bias

Due to the sensitivity of spaceborne atomic clock to many factors, the variation law of satellite clock bias (SCB) can be regarded as a grey system. The GM (1,1) model is a most classical and basic model of grey system, which has been successfully applied in SCB prediction. Moreover, many improved models have been proposed and widely used in various forecasts since GM (1,1) was generated. However, the prediction performance of these models is not obviously improved compared with the classical models in clock bias prediction. In view of this, a new GM (1,1) model has been come up with in this paper by optimising fitting model and initial condition. The new fitting model is obtained by differentiating time response function of winterisation, and the new initial condition is generated through one or more components of the original clock bias sequence. The authors employ GPS rapid and precise SCB provided by the International GNSS Service (IGS) for prediction experiments. The results show that the new GM (1,1) model is effective and feasible, and its prediction accuracy and stability are enormously better than that of the classical GM (1,1) model, especially for ultra-short-term prediction.

Integrated kinematic precise orbit determination and clock estimation for low Earth orbit satellites with onboard and regional ground observations

Many low Earth orbit (LEO) satellite constellations have been designed in recent years to provide global broadband Internet services. These constellations provide opportunities for LEO satellites to serve as navigation satellites by launching navigation signals while also being equipped with an onboard global navigation satellite system (GNSS) receiver. We propose a kinematic precise orbit determination (KPOD) and precise clock estimation (PCE) approach for LEO satellites by integrating regional ground observations and onboard observations of LEO satellites. By taking into account the LEO satellite clock bias, this approach can demonstrate the contributions of the ground and onboard observations of LEO satellites to the orbit and clock results, respectively. A composite LEO satellite constellation consisting of 168 satellites and observations from regional ground stations and LEO satellite onboard receivers are simulated considering the LEO satellite clock bias to verify the proposed approach. The results indicate that the convergence time of LEO satellite orbit determination can reach 9.38 min with the integrated KPOD (IKPOD) method, which is a reduction of 24.0% compared with the traditional KPOD method. However, the additional improvement in the LEO satellite orbit accuracy after convergence is very limited. The average root mean square (RMS) and standard deviation (STD) values of all LEO satellite clocks using the integrated PCE (IPCE) method with the participation of LEO satellite onboard observations can reach 0.27 ns and 0.15 ns, respectively. The improvements in the average RMS and STD are 42.6% and 60.5%, respectively, compared with the ground LEO PCE method. The convergence time and accuracy of LEO/GNSS precise point positioning can be improved by 44.6%, 48.3%, 26.7%, and 20.4% in the east, north, and up directions, respectively, using LEO satellite orbits and clocks from the IKPOD and IPCE methods compared with KPOD and PCE.

A Short-Term Forecasting Method of Inter-Frequency Clock Bias for GPS Block IIF Satellites

The time-varying biases within carrier phase observations are integrated into satellite clock offset parameters for precise clock estimation. Consequently, when the precise satellite clock bias is applied to the third frequency observation for precise point positioning (PPP), a new type of inter-frequency clock bias (IFCB) with satellite dependence should be noticed. If the IFCB is estimated together with the receiver coordinates, tropospheric wet delay, ambiguity and other parameters, it will increase the computational burden and lead to more time consumption. In order to solve this problem, the IFCB of GPS Block IIF satellites were estimated using 162 global uniformly distributed Multi-GNSS Experiment (MGEX) stations. By analyzing the time-varying characteristic of each satellite IFCB and combining the lag characteristics of the final ephemeris products, a modeling method of short-term IFCB prediction based on the epoch-by-epoch sliding Pearson autocorrelation function is proposed. The feasibility of this method was verified through the Student’s t-distribution, comparison with the measured IFCB, the posteriori residual of the third frequency carrier phase and the kinematic/static PPP solutions. The results showed that since the IFCB period was not a complete 24 h, the difference in the IFCBs time series on different days was increasingly significant with the passage of lag time, and the correlation constantly decreased. The peak-to-peak amplitudes of the IFCB difference reached 1.13, 3.44, 6.86 and 11.25 cm when the lag time was 1, 9, 19 and 29 days, respectively. In addition, based on the lag characteristic of final precise ephemerides released by the International GNSS Service (IGS) analysis centers, the prediction accuracy of the IFCB was evaluated with a time lag of 7 days. The root mean square of the posteriori residuals at the third-frequency observation decreased by approximately 51.3% compared to that without considering for IFCB correction. The triple-frequency uncombined PPP in the horizontal and vertical directions improved by approximately 33.2% and 17.2% for the static PPP solutions and 50.2% and 39.7% for the kinematic PPP solutions, respectively. In general, the accuracy and convergence time of the triple-frequency uncombined PPP were equivalently improved when the predicted IFCB and the measured IFCB were used.

Satellite Clock Batch Estimation Accuracy Analysis and Its Impacts on PPP

The ultra-rapid satellite clock product based on the satellite clock batch estimation is commonly used for high-precision and reliable precise point positioning (PPP) services. In order to clarify the effect of different ranging errors on the satellite clock batch estimation accuracy, the source of the satellite clock bias induced by the batch observation model is classified into the initial clock bias (ICB) and time-dependent bias (TDB). In addition to the effect of the ICB and TDB, the analytic relationship between the observation redundancy and the satellite clock batch estimation accuracy are derived and verified. The suitable number of stations is suggested to be 40 for the satellite clock batch estimation to achieve the counterbalance between the efficiency and saturable accuracy. For the PPP based on the batch-estimated satellite clock, the impacts of the ICB and TDB on PPP are clarified. The satellite clock batch estimation and PPP experiments are carried out to investigate the impacts of the ICB and TDB on the satellite clock batch estimation accuracy and the PPP performance. The ICB causes a significant bias for the batch-estimated satellite clock. The TDB is impacted by the assimilation ability of the batch-estimated satellite clock to the satellite orbit error. The convergence time and the positioning accuracy after the convergence of PPP are primarily affected by the ICB and TDB, respectively.

Improved GM (1,1) Model by Optimizing Initial Condition to Predict Satellite Clock Bias

The variation law of satellite clock bias (SCB) can be regarded as a grey system because the spaceborne atomic clock is very sensitive and vulnerable to many factors. GM (1,1) model is the core and foundation of the grey system, which has been highly valued and successfully applied in SCB prediction since its production. However, there are still some problems to be further studied such as the lack of stability of its prediction effect in practical application. In view of this, an improved GM (1,1) model by optimizing the initial condition has been proposed in this paper so as to increase the prediction performance. The new initial condition is obtained by the weighted combination of the latest and oldest components of the original clock bias sequence. And the weight values of these two components are acquired from a method of minimizing the sum of squares of fitting errors. We adopt GPS rapid precision SCB data provided by the International GNSS Service (IGS) for 15 mins, 30 mins, 1 h, 3 h, 6 h, 12 h, and 24 h prediction experiments. The results show that the improved GM (1,1) model is effective and feasible, and its prediction accuracy and stability are significantly better than those of the traditional GM (1,1) model, ARIMA model, and QP model, even for the SCB signal with obvious fluctuation.

Enhancing Short-Term Prediction of BDS-3 Satellite Clock Bias Based with BSO Optimized BP Neural Network

The satellite clock bias (SCB) prediction plays an important role in high-accuracy and real-time navigation and positioning. When predicting the SCB, the performance of the BP neural network is affected by the local optimum due to inaccurate initial parameters. Therefore, we propose an improved BP neural network based on the beetle swarm optimization (BSO-BP) algorithm to improve the performance of SCB prediction in third-generation Beidou satellite navigation system (BDS-3). The proposed model takes advantage of group learning strategy to optimize the initialization parameters of the BP neural network and obtains globally optimized parameters. In order to verify the proposed BSO-BP model, 15 BDS satellites are analyzed in terms of prediction accuracy and stability of SCB. The experimental results show that when predicting 1 hour SCB based on a 12 hours SCB data, the prediction accuracy of the BSO-BP model is the best, with an average accuracy of 0.064 ns. As compared with the LP, QP, and GM models, the average prediction accuracy of the proposed BSO-BP model increases by about 72.6%, 43.4%, and 86%, respectively. As the prediction time increases, the influence of the inaccurate initial parameters on SCB prediction gradually decreases, and the prediction accuracy improves. The proposed BSO-BP model has the best accuracy and stability when predicting the 1 h SCB based on the same data. The prediction stability of the proposed BSO-BP model improves by more than 36% as compared with LP, QP, and GM models. In addition, the prediction accuracies of PHM clock and Rb-II clock improved by more than 47%, as compared with that of the Rb clock. Therefore, the overall performance of the atomic clock based on BDS-3 is better than BDS-2. The positioning accuracy of the BSO-BP model can reach the centimeter level in east, north, and up directions.

GNSS observable-specific phase biases for all-frequency PPP ambiguity resolution

An unwritten rule to resolve GNSS ambiguities in precise point positioning (PPP-AR) is that users should follow faithfully the frequency choices and observable combinations mandated by satellite clock and phase bias providers. Switching to other frequencies of measurements requires that the satellite clocks be converted, albeit in a roundabout way, to agree with the new frequencies of code biases. Satellite phase biases, on the other hand, are prescribed conventionally as wide-lane and narrow-lane combinations, which prevents users from resolving other phase combinations in the case of multi-frequency observables. We therefore develop an approach to compute observable-specific phase biases (phase OSBs) in concert with the legacy, but ambiguity-fixed, satellite clocks to enable PPP-AR over any frequency choices and observable combinations at the user end, i.e., all-frequency PPP-AR. In particular, the phase OSBs on the baseline frequencies (e.g., L1/L2 for GPS and E1/E5a for Galileo) are estimated by decoupling the code OSBs pre-aligned with the satellite clocks; then satellite clocks are re-estimated by holding pre-resolved undifferenced ambiguities and phase OSBs on the baseline frequencies; finally, all third-frequency phase OSBs are determined by introducing the ambiguity-fixed satellite clocks above. We used a global network of multi-frequency GPS/Galileo data over a month to verify this approach. In dual-frequency PPP-AR using GPS L1/L2, L1/L5, Galileo E1/E5a, E1/E5b, E1/E5 and E1/E6 signals, over 95% of wide-lane and narrow-lane ambiguity residuals were within ±0.25 and ±0.15 cycles, respectively, after the code and phase OSB corrections on raw GNSS measurements. As a result, the ambiguity fixing rates reached around 95% in all PPP-AR tests, though it was only the satellite clocks aligned with the GPS L1/L2 and Galileo E1/E5a pseudorange that were applied throughout. We stress that the key to computing such phase OSBs for all-frequency PPP-AR is that the code OSBs have the same bias datum as that of the satellite clocks.

Short-term satellite clock bias forecast based on complementary ensemble empirical mode decomposition and quadratic polynomial

The nonlinear and nonstationary characteristics of satellite clock bias (SCB) have a harmful effect on the accuracy and stability of SCB forecast. To eliminate the influence of nonlinearity and non-stationarity, a hybrid forecast model was constructed that combines complementary ensemble empirical mode decomposition (CEEMD) and quadratic polynomial (QP), called CEEMD-QP. First, the SCB sequence is decomposed into several intrinsic mode function (IMF) components and one residual term by CEEMD. Second, permutation entropy (PE) and the correlation coefficient are used to quantitatively determine the IMF component with more noise and weak correlation with the original SCB signal. Finally, the SCB is reconstructed with the IMFs and residual components, and QP model is used to fit and forecast the clock bias. We adapt the observation part of ultra-rapid precise SCB data of GPS provided by IGS to forecast experiments. The results show that the CEEMD-QP method has obvious advantages of forecast accuracy and stability in short-term forecast.

A method for precisely predicting satellite clock bias based on robust fitting of ARMA models

A method for precisely predicting satellite clock bias based on robust fitting of ARMA models