A BDS satellite clock error prediction method based on GM-Elman model

Abstract

The application discloses a BDS satellite clock error prediction method of a GM-Elman model and belongs to the technical field of satellite navigation. The method comprises the following steps: S1, obtaining a raw time sequence of precise post-event clock errors of BDS satellites and clock error measured values corresponding to time instants of the sequence; S2, performing power function transformation on a GM(1, 1) prediction model, and obtaining clock error prediction values of future time instants through the transformed GM(1, 1) prediction model; S3, calculating residual error values and constructing a residual error sequence; S4, inputting the residual error sequence into an Elman neural network for training, and finally obtaining residual error prediction values of the future time instants; and S5, linearly adding the clock error prediction values of the future time instants and the residual error prediction values of the future time instants to obtain final clock error prediction values of the BDS satellites. By using the method, the BDS satellite clock error prediction precision and stability are effectively improved, the residual error compensation further reduces errors, and the method is suitable for BDS satellites of different orbits and on-board clocks.

Description

Technical Field

[0001] This invention relates to the field of satellite navigation technology, and in particular to a BDS satellite clock error prediction method based on the GM-Elman model. Background Technology

[0002] Accurate clock bias prediction for the BeiDou Navigation Satellite System (BDS) is crucial for achieving high-precision navigation, positioning, and timing. Currently, commonly used BDS satellite clock bias prediction methods include the GM(1,1) grey model, linear polynomial model, and quadratic polynomial model. While these methods meet basic prediction requirements, they still have several technical shortcomings. The prediction accuracy of the traditional GM(1,1) model is greatly affected by data quality, has limited adaptability to different satellite clocks, and is prone to large prediction errors. The long-term prediction accuracy of a single prediction model decreases significantly over time, resulting in insufficient overall stability of clock bias prediction. Although polynomial models are computationally simple, they have poor adaptability to BDS satellites with different orbit types such as GEO, IGSO, and MEO. Furthermore, existing models do not effectively compensate for or correct the residuals generated by clock bias prediction, further restricting the improvement of clock bias prediction accuracy and making it difficult to meet the practical application requirements of centimeter-level precise point positioning for BDS. Summary of the Invention

[0003] The purpose of this invention is to provide a BDS satellite clock error prediction method based on the GM-Elman model, thereby solving the aforementioned technical problems.

[0004] To achieve the above objectives, this invention provides a BDS satellite clock error prediction method based on the GM-Elman model, comprising the following steps: S1. Obtain the original time series of precise post-hoc clock errors of BDS satellite and the measured clock error values ​​at the corresponding times in the series; S2. Perform a power function transformation on the GM(1,1) forecast model, and obtain the clock difference forecast value for future times through the transformed GM(1,1) forecast model; S3. Calculate the residual value between the predicted clock difference value and the actual clock difference value at the corresponding time, and arrange the residual values ​​of consecutive time moments in chronological order to construct a residual sequence; S4. Input the residual sequence into the Elman neural network for training. After training, input the time parameters for future times to obtain the predicted residual values ​​for future times. S5. Linearly add the future time clock difference prediction value obtained in S2 to the future time residual prediction value obtained in S4 to obtain the final prediction value of BDS satellite clock difference.

[0005] Preferably, the original clock difference time series in S1 and the measured clock difference values ​​at the corresponding times in the series are derived from the precise post-hoc clock difference data of the BDS satellite. The original clock difference time series is denoted as: ; in, For time indexing, The length of the original time series data. for The original measured value of the clock difference at that time.

[0006] Preferably, the BDS satellites selected in S1 include GEO, IGSO, and MEO orbit types, and the onboard clocks carried by the satellites are rubidium atomic clocks or hydrogen atomic clocks.

[0007] Preferably, S2 specifically includes: S21, to Obtained by performing a power function transformation Transformation relationship and The formula is: ; in, for Clock difference sequence after power function transformation of time. It is a power exponent; S22, to The accumulation sequence generated by one accumulation is as follows: ; in, Indicates the order of Lagrange interpolation; S23. Solve the parameter sequence of the transformed GM(1,1) model using the least squares method. The formula is: ; in, To solve for the data matrix of the GM(1,1) model parameters after the power function transformation, This is a column vector of the original time series of precise clock errors. For development coefficient, This is the gray action quantity; S24, take the contents of S23 , Substituting into the time response function of the GM(1,1) model, the prediction model for the clock error sequence after cumulative subtraction is as follows: ; Then, the forecast data is obtained by restoring the forecast function, and the formula is: ; in, For the predicted time of satellite clock bias, , The number of moments for the forecast duration. for Predicted time difference value for Predicted values ​​of time-satellite clock bias using a power function transformation.

[0008] Preferably, the reconstructed prediction function in S24 represents the GM(1,1) model after power function transformation.

[0009] Preferably, in S3, when calculating the clock error residual value and constructing the residual sequence, for the first... At what time, its clock error residual value for: ; in, For the first Measured value of clock difference at time. For the first Clock error prediction values ​​of the GM(1,1) model after time power function transformation; Arrange the clock error residual values ​​at consecutive time points in chronological order to construct a clock error residual sequence. for: ; in, The length of the residual sequence is equal to the length of the original time series of the precise clock difference. Consistent.

[0010] Preferably, S4 specifically includes: S41. The Elman neural network consists of four layers: input layer, supporting layer, hidden layer, and output layer. Signals are input and transmitted through neurons in the input layer. Dynamic memory and local feedback are achieved through the supporting and hidden layers for network training. Finally, the output layer neurons linearly weight the signals and output them. The formulas are as follows: ; ; ; in, The connection weights from the receiving layer to the hidden layer. These are the connection weights from the input layer to the hidden layer. These are the connection weights from the hidden layer to the output layer. For the hidden layer transfer function, For the transfer function of the output layer neurons, for dimensional feedback state vector, for Hidden layer node vectors for 3D input layer node vector, for 3D output layer node vector; S42. Clock error residual sequence As input samples to the network, the sum of squared errors (SSE) function is used as the learning metric. Gradient descent backpropagation algorithm is employed to adjust the network weights until the model converges and training is complete. The formula for the SSE function is: ; in, Let be the sum of squared errors function. For the set of connection weights in an Elman neural network, for The actual output value of the network at any given time. for The target input value of the network at any given time; S43. Input the future time parameters into the trained Elman neural network, and output the future time parameters. Clock difference residual prediction value at time .

[0011] Preferably, in S5, the future... The formula for the final predicted value of the BDS satellite clock bias at time is: ; in, For the future The final predicted value of the BDS satellite clock bias at time [time]. For the future Clock error prediction values ​​of the GM(1,1) model after time power function transformation. For the future The predicted clock error residual value of the Elman neural network at time step.

[0012] Preferably, in S5, the original time series of BDS satellite precise clock difference in the previous 12 hours is used to construct the GM(1,1) model after power function transformation and complete the Elman neural network training. The BDS satellite clock difference value and clock difference residual value in the next 6 hours and 8 hours are predicted respectively. Then, the final predicted value of BDS satellite clock difference for the corresponding duration is obtained by linear addition.

[0013] Preferably, the root mean square error (RMS) is used as the evaluation index for clock error prediction accuracy. The formula for the root mean square error is: ; in, This is the measured value of the clock error. This is the predicted value of the clock difference. Number of samples participating in the evaluation Therefore, the BDS satellite clock error prediction method using the above-mentioned GM-Elman model has the following beneficial effects: 1. The GM-Elman combined model of the present invention effectively improves the accuracy and stability of the 6h and 8h short-term forecasts of BDS satellite clock bias, and the forecast accuracy is significantly improved compared with the traditional GM(1,1) polynomial model; 2. It has good adaptability to BDS satellites of different orbit types equipped with rubidium atomic clocks and hydrogen atomic clocks, and has a wide range of applications; 3. By training an Elman neural network using residual sequences and performing residual compensation, clock error information is fully utilized, effectively reducing forecast errors; 4. Using clock difference data from the previous 12 hours for modeling, clock difference forecasts for the next 6 hours and 8 hours can be quickly completed. The modeling and calculation process is simple and meets the actual engineering application needs of BDS.

[0014] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0015] Figure 1 The flowchart of the BDS satellite clock error prediction method based on the GM-Elman model of this invention is shown below. Figure 2 This is a schematic diagram of the Elman neural network model structure of the present invention; Figure 3 The graph shows the 6-hour forecast error variation of the four models of LPM, QPM, GM, and GM-Elman in this invention under different satellites, where (a) is PRN10 satellite, (b) is PRN19 satellite, (c) is PRN22 satellite, (d) is PRN26 satellite, (e) is PRN38 satellite, (f) is PRN40 satellite, (g) is PRN42 satellite, and (h) is PRN43 satellite. Figure 4 The graph shows the variation of 8-hour forecast error of the four models of LPM, QPM, GM and GM-Elman in this invention under different satellites, where (a) is PRN10 satellite, (b) is PRN19 satellite, (c) is PRN22 satellite, (d) is PRN26 satellite, (e) is PRN38 satellite, (f) is PRN40 satellite, (g) is PRN42 satellite and (h) is PRN43 satellite; Figure 5 This is a comparison chart showing the improvement rate of satellite clock error prediction accuracy of the satellite clock error prediction method of the present invention at 6-hour and 8-hour intervals. Detailed Implementation

[0016] To make the objectives, technical solutions, and advantages disclosed in the embodiments of the present invention clearer, the embodiments of the present invention will be further described in detail below with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are only used to explain the embodiments of the present invention and are not intended to limit the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of this application. Examples of the embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout.

[0017] It should be noted that the terms "comprising" and "having," and any variations thereof, are intended to cover non-exclusive inclusion, such as a process, method, system, product, or server that includes a series of steps or units, not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such process, method, product, or device.

[0018] The embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0019] like Figures 1-5 As shown, the present invention provides a BDS satellite clock error prediction method using the GM-Elman model, comprising the following steps: S1, obtaining the original time series of precise post-hoc clock errors of BDS satellites and the measured clock error values ​​at corresponding times; S2, performing a power function transformation on the GM(1,1) prediction model to obtain the predicted clock error values ​​for future times through the transformed GM(1,1) prediction model; S3, calculating the residual values ​​between the predicted clock error values ​​and the measured clock error values ​​at corresponding times, and arranging the residual values ​​of consecutive times in chronological order to construct a residual sequence; S4, inputting the residual sequence into an Elman neural network for training, and inputting future time parameters after training to obtain the predicted residual values ​​for future times; S5, linearly adding the predicted clock error values ​​for future times output by the GM(1,1) prediction model after the power function transformation in S2 with the predicted residual values ​​for future times output by the Elman neural network after training in S4 to obtain the final predicted value of the BDS satellite clock error.

[0020] Specifically, the original clock bias time series in S1 and the measured clock bias values ​​at the corresponding times in the series are derived from the precise post-hoc clock bias data of the BDS satellite. The original clock bias time series is denoted as: ; in, For time indexing, The length of the original time series data. for The original measured clock difference values ​​at each moment; to verify the prediction performance of the combined model, this invention used the precise post-hoc clock difference data of BDS satellites released by the GNSS Analysis Center of Wuhan University on August 5, 2024. The data sampling interval was 30 seconds. During this period, 34 Beidou satellites were in orbit. Their onboard clocks mainly included rubidium atomic clocks for geostationary orbit (GEO-Rb clocks), rubidium atomic clocks for inclined geosynchronous orbit (IGSO-Rb clocks), hydrogen atomic clocks for inclined geosynchronous orbit (IGSO-H clocks), rubidium atomic clocks for medium Earth orbit (MEO-Rb clocks), and hydrogen atomic clocks for medium Earth orbit (MEO-H clocks). Clock difference data from 8 satellites with different orbits, clock types, systems, and launch years were randomly selected for prediction experiments, specifically including BD-10 IGSO-5-Rb PRN10, BD-3 MEO-1-Rb PRN19, BD-3 MEO-4-Rb PRN22, and BD-3 The specific information of the selected satellites, namely MEO-11-Rb PRN26, BD-3IGSO-1-H PRN38, BD-3 IGSO-3-H PRN40, BD-3 MEO-20-Rb PRN42, and BD-3 MEO-21-Rb PRN43, is shown in Table 1. Relevant information about the selected satellites in Table 1

[0021] The changes in the 24-hour precise satellite clock bias time series of these eight satellites on August 5, 2025 are shown in Table 1. Among them, the clock bias time series of PRN10, PRN40, and PRN42 satellites show a negative monotonically decreasing trend, PRN19 shows a negative monotonically increasing trend, and the clock bias time series of PRN22, PRN26, PRN38, and PRN43 satellites show a positive monotonically increasing trend. The clock bias time series of all eight satellites have good linearity and are highly representative.

[0022] S2 specifically includes: S21, and... Obtained by performing a power function transformation Transformation relationship and The formula is: ; in, for Clock difference sequence after power function transformation of time. For power exponent; S22, for The accumulation sequence generated by one accumulation is as follows: ; in, S23. Denote the order of Lagrange interpolation; Solve the parameter sequence of the transformed GM(1,1) model using the least squares method. The formula is: ; in, To solve for the data matrix of the GM(1,1) model parameters after power function transformation, its core function is to provide basic model support for satellite clock error prediction by fitting the model parameters using the least squares method. This is a column vector of the original time series of precise clock errors. For development coefficient, The gray action quantity; S24, the amount in S23 , Substituting into the time response function of the GM(1,1) model, the prediction model for the clock error sequence after cumulative subtraction is as follows: ; Then, the forecast data is obtained by restoring the forecast function, and the formula is: ; in, For the predicted time of satellite clock bias, , The number of moments for the forecast duration. for Predicted time difference value for The predicted value of the satellite clock error is a power function transformation; and the restored prediction function in S24 represents the GM(1,1) model after the power function transformation.

[0023] When calculating the clock error residual value and constructing the residual sequence in S3, for the first... At what time, its clock error residual value for: ; in, For the first The measured clock error values ​​at each time, i.e., the precise clock error data released by the GNSS Analysis Center of Wuhan University. For the first Clock error prediction values ​​from the GM(1,1) model after power function transformation at each time step; clock error residual values ​​at consecutive time steps are arranged in chronological order to construct a clock error residual sequence. for: ; in, The length of the residual sequence is equal to the length of the original time series of the precise clock difference. Consistent.

[0024] S4 specifically includes: S41, the Elman neural network, which consists of a four-layer structure: input layer, supporting layer, hidden layer, and output layer. Signals are input and transmitted through neurons in the input layer. Dynamic memory and local feedback are achieved through the supporting and hidden layers for network training. Finally, the output layer neurons linearly weight the signals, with the following formulas: ; ; ; in, The connection weights from the receiving layer to the hidden layer. These are the connection weights from the input layer to the hidden layer. These are the connection weights from the hidden layer to the output layer. For the hidden layer transfer function, For the transfer function of the output layer neurons, for dimensional feedback state vector, for Hidden layer node vectors for 3D input layer node vector, for Dimensional output layer node vector; S42, convert clock error residual sequence As input samples to the network, the sum of squared errors (SSE) function is used as the learning metric. Gradient descent backpropagation algorithm is employed to adjust the network weights until the model converges and training is complete. The formula for the SSE function is: ; in, Let be the sum of squared errors function. For the set of connection weights in an Elman neural network, for The actual output value of the network at any given time. for The target input value for the time-series network; S43, input the future time parameters into the trained Elman neural network, and output the future time... Clock difference residual prediction value at time .

[0025] S5's vision of the future The formula for the final predicted value of the BDS satellite clock bias at time is: ; in, For the future The final predicted value of the BDS satellite clock bias at time [time]. For the future Clock error prediction values ​​of the GM(1,1) model after time power function transformation. For the future The clock error residual prediction value of the Elman neural network at each time point. In S5, the original time series of BDS satellite precise clock errors for the 12 hours prior to August 5, 2024, is used to construct a GM(1,1) model after power function transformation and complete the Elman neural network training. The clock error values ​​and clock error residual values ​​of BDS satellites for the next 6 hours and 8 hours are predicted respectively. The final predicted value of the BDS satellite clock error for the corresponding duration is obtained by linear summation. The root mean square error (RMS) is used as the evaluation index for clock error prediction accuracy. The formula for the root mean square error is: ; in, These are measured clock error values. Since the precision clock error data released by the GNSS Analysis Center of Wuhan University has an error of less than 0.1 nanoseconds, it is considered the true value. This is the predicted value of the clock difference. The number of samples participating in the evaluation. Experimental results show that when using the GM-Elman model of this invention for satellite clock bias data prediction, the RMS values ​​for the two satellites equipped with hydrogen atomic clocks reached 0.15 ns and 0.07 ns, respectively, when modeling and predicting satellite clock bias data for the next 6 hours based on the past 12 hours of satellite clock bias data. Compared with the LPM model, QPM model, and GM model, the RMS values ​​are significantly improved. Compared with the GM model, the RMS improvement rate of the GM-Elman model is 55.5% and 47.5%, respectively. For the six satellites equipped with rubidium atomic clocks, the RMS values ​​reached 2.19 ns, 0.04 ns, 0.07 ns, 0.09 ns, 0.20 ns, and 0.13 ns, respectively. Compared with the LPM model, QPM model, and GM model, the RMS values ​​are also significantly improved. Compared with the RMS value of the GM model, the RMS improvement rate of the GM-Elman model is 48.9%, 42.9%, 13.5%, 43.2%, 22.1%, and 39.6%, respectively, showing a significant improvement in RMS.

[0026] Meanwhile, when using satellite clock bias data from the past 12 hours to model and predict satellite clock bias data for the next 8 hours, the RMS values ​​for the two satellites equipped with hydrogen atomic clocks reached 0.15 ns and 0.17 ns, respectively. Compared with the LPM, QPM, and GM models, the RMS values ​​also showed a significant improvement. Compared with the GM model, the RMS improvement rate of the GM-Elman model was 52.9% and 39.9%, respectively. For the six satellites equipped with rubidium atomic clocks, the RMS values ​​reached 2.27 ns, 0.08 ns, 0.06 ns, 0.06 ns, 0.08 ns, 0.27 ns, and 0.09 ns, respectively. Compared with the LPM, QPM, and GM models, the RMS values ​​also showed a significant improvement. Compared with the RMS value of the GM model, the RMS improvement rate of the GM-Elman model was 41.9%, 44.5%, 37.8%, 47.5%, 52.9%, and 45.4%, respectively, showing a significant improvement in RMS values.

[0027] In summary, when using the GM-Elman model for 6-hour and 8-hour satellite clock bias data forecasts, the forecast error is significantly reduced compared to the other three models. The forecast error trend of the GM-Elman model is basically consistent with that of the GM model, but the forecast error value is significantly reduced. Furthermore, as the forecast duration increases, the absolute value of the forecast error for most satellites increases slowly and tends to stabilize. The 8-hour forecast error curve shows a more stable trend compared to the 6-hour forecast error curve. This indicates that the forecast accuracy of the GM-Elman model has been greatly improved, and its stability has been significantly enhanced.

[0028] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A BDS satellite clock error prediction method using the GM-Elman model, characterized in that, Includes the following steps: S1. Obtain the original time series of precise post-hoc clock errors of BDS satellite and the measured clock error values ​​at the corresponding times in the series; S2. Perform a power function transformation on the GM(1,1) forecast model, and obtain the clock difference forecast value for future times through the transformed GM(1,1) forecast model; S3. Calculate the residual value between the predicted clock difference value and the actual clock difference value at the corresponding time, and arrange the residual values ​​of consecutive time moments in chronological order to construct a residual sequence; S4. Input the residual sequence into the Elman neural network for training. After training, input the time parameters for future times to obtain the predicted residual values ​​for future times. S5. Linearly add the future time clock difference prediction value obtained in S2 to the future time residual prediction value obtained in S4 to obtain the final prediction value of BDS satellite clock difference.

2. The BDS satellite clock bias prediction method based on the GM-Elman model according to claim 1, characterized in that, The original clock bias time series in S1 and the corresponding measured clock bias values ​​at those times are derived from the precise post-hoc clock bias data from the BDS satellite. The original clock bias time series is denoted as: ； in, For time indexing, The length of the original time series data. for The original measured value of the clock difference at that time.

3. The BDS satellite clock bias prediction method based on the GM-Elman model according to claim 2, characterized in that, The BDS satellites selected in S1 include GEO, IGSO, and MEO orbit types, and the onboard clocks are either rubidium atomic clocks or hydrogen atomic clocks.

4. The BDS satellite clock error prediction method based on the GM-Elman model according to claim 3, characterized in that, S2 specifically includes: S21, to Obtained by performing a power function transformation Transformation relationship and The formula is: ； in, for Clock difference sequence after power function transformation of time. It is a power exponent; S22, to The accumulation sequence generated by one accumulation is as follows: ； in, Indicates the order of Lagrange interpolation; S23. Solve the parameter sequence of the transformed GM(1,1) model using the least squares method. The formula is: ； in, To solve for the data matrix of the GM(1,1) model parameters after the power function transformation, This is a column vector of the original time series of precise clock errors. For development coefficient, This is the gray action quantity; S24, take the contents of S23 , Substituting into the time response function of the GM(1,1) model, the prediction model for the clock error sequence after cumulative subtraction is as follows: ； Then, the forecast data is obtained by restoring the forecast function, and the formula is: ； in, For the predicted time of satellite clock bias, , The number of moments for the forecast duration. for Predicted time difference value for Predicted values ​​of time-satellite clock bias using a power function transformation.

5. The BDS satellite clock bias prediction method based on the GM-Elman model according to claim 4, characterized in that, The reconstructed prediction function in S24 represents the GM(1,1) model after power function transformation.

6. The BDS satellite clock bias prediction method based on the GM-Elman model according to claim 5, characterized in that, When calculating the clock error residual value and constructing the residual sequence in S3, for the first... At what time, its clock error residual value for: ； in, For the first Measured value of clock difference at time. For the first Clock error prediction values ​​of the GM(1,1) model after time power function transformation; Arrange the clock error residual values ​​at consecutive time points in chronological order to construct a clock error residual sequence. for: ； in, The length of the residual sequence is equal to the length of the original time series of the precise clock difference. Consistent.

7. The BDS satellite clock error prediction method based on the GM-Elman model according to claim 6, characterized in that, S4 specifically includes: S41. The Elman neural network consists of four layers: input layer, supporting layer, hidden layer, and output layer. Signals are input and transmitted through neurons in the input layer. Dynamic memory and local feedback are achieved through the supporting and hidden layers for network training. Finally, the output layer neurons linearly weight the signals and output them. The formulas are as follows: ； ； ； in, The connection weights from the receiving layer to the hidden layer. These are the connection weights from the input layer to the hidden layer. These are the connection weights from the hidden layer to the output layer. For hidden layer transfer functions, For the transfer function of the output layer neurons, for dimensional feedback state vector, for Hidden layer node vectors for 3D input layer node vector, for 3D output layer node vector; S42. Clock error residual sequence As input samples to the network, the sum of squared errors (SSE) function is used as the learning metric. Gradient descent backpropagation algorithm is employed to adjust the network weights until the model converges and training is complete. The formula for the SSE function is: ； in, Let be the sum of squared errors function. For the set of connection weights in an Elman neural network, for The actual output value of the network at any given time. for The target input value of the network at any given time; S43. Input the future time parameters into the trained Elman neural network, and output the future time parameters. Clock difference residual prediction value at time .

8. The BDS satellite clock bias prediction method based on the GM-Elman model according to claim 7, characterized in that, S5's vision of the future The formula for the final predicted value of the BDS satellite clock bias at time is: ； in, For the future The final predicted value of the BDS satellite clock bias at time [time]. For the future Clock error prediction values ​​of the GM(1,1) model after time power function transformation. For the future The predicted clock error residual value of the Elman neural network at time step.

9. The BDS satellite clock error prediction method based on the GM-Elman model according to claim 8, characterized in that, In S5, the original time series of BDS satellite precise clock bias from the previous 12 hours is used to construct a GM(1,1) model after power function transformation and Elman neural network training is completed. The BDS satellite clock bias value and clock bias residual value for the next 6 hours and 8 hours are predicted respectively. Then, the final predicted value of BDS satellite clock bias for the corresponding duration is obtained by linear addition.

10. A BDS satellite clock bias prediction method based on the GM-Elman model according to claim 9, characterized in that, The root mean square error (RMS) is used as an evaluation index for clock error prediction accuracy. The formula for the root mean square error is: ； in, This is the measured value of the clock error. This is the predicted value of the clock difference. The number of samples participating in the evaluation.

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