A clock error modeling and forecasting method for low earth orbit satellites

Abstract

The application belongs to the technical field of clock error prediction. The application provides a clock error modeling and prediction method for low-orbit satellites. In the embodiments of the present disclosure, based on original clock error data, the main period term characteristics in the satellite clock error sequence are detected, and a PQM model is established in combination with a quadratic polynomial. The trend term and the period term coefficient of the PQM model are estimated through the clock error change rate, the constant term is determined by using short-term clock error data close to the prediction end, and thus a high-precision short-term clock error prediction model is constructed, effectively improving the accuracy of clock error prediction, especially short-term prediction. By introducing the main period term and the quadratic drift term to construct the PQM model, and by deriving the PQM model, the influence of the constant term is eliminated, the estimation of the frequency initial deviation, the frequency drift rate and the period term amplitude coefficient is more focused on the dynamic change characteristics of the clock error, and the interference of the constant term drift on the period characteristic extraction is avoided.

Description

Technical Field

[0001] This disclosure relates to the field of clock error prediction technology, and in particular to a clock error modeling and prediction method for low-Earth orbit satellites. Background Technology

[0002] Low Earth Orbit (LEO) satellites, with their low orbital altitude, strong signal strength, and rapidly changing spatial geometry, offer the potential for rapid convergence in Precise Point Positioning (PPP), significantly shortening PPP convergence time compared to traditional medium and high Earth orbit (MEO) satellites. However, individual LEO satellites have limited coverage areas, necessitating the construction of large satellite constellations. In recent years, driven by global demand for broadband internet services, the deployment of LEO satellite constellations has grown rapidly. The US Starlink constellation and the UK OneWeb constellation have been completed, and China is accelerating its deployment, with the "Microspace" and "Future Mobility" constellations promoting the integrated development of LEO satellite navigation and communication.

[0003] Satellite time and frequency systems are the core of low-Earth orbit (LEO) intelligent satellite networks, and clock bias estimation and prediction are crucial for maintaining their high stability and accuracy. Compared to traditional Global Navigation Satellite Systems (GNSS), LEO satellite clock bias prediction is more challenging. Firstly, the crystal oscillators or miniature rubidium atomic clocks they carry are less powerful than the hydrogen or cesium atomic clocks used in GNSS satellites, and their clock bias characteristics differ significantly. Secondly, the complex operating environment of LEO satellites makes their clock biases susceptible to various interferences, exhibiting complex and nonlinear characteristics, further increasing the difficulty of clock bias modeling and prediction.

[0004] Satellite clock bias fitting and prediction models include linear models and quadratic polynomial models. Domestic and international scholars have achieved certain results in modeling low-Earth orbit (LEO) satellite clock bias. Wang et al. analyzed the prediction error of LEO satellite clocks, showing that its stability is related to multiple factors; Wang Jinqian et al. analyzed and predicted the clock bias characteristics of Sentinel-6A satellite, showing that the prediction accuracy decreases with time; Wu achieved a certain prediction accuracy using Sentinel-3B satellite data; Ge et al. analyzed the clock characteristics of the GRACE-FO mission satellites, verifying the effectiveness of the least squares harmonic estimation method in LEO satellite clock bias prediction. Although scholars at home and abroad have achieved a series of good research results and the accuracy of clock bias prediction in some reports has gradually improved, the clock bias variation characteristics of low-orbit satellites are very complex due to the constraints of the complex space environment and the design of different types of spaceborne atomic clocks. The problem of high-precision prediction has not yet been completely solved and deserves further detailed exploration and research by researchers.

[0005] Therefore, it is necessary to improve one or more of the problems existing in the above-mentioned related technical solutions.

[0006] It should be noted that this section is intended to provide background or context for the technical solutions of this disclosure as set forth in the claims. The description herein does not constitute an admission that it is prior art simply because it is included in this section. Summary of the Invention

[0007] The purpose of this disclosure is to provide a clock bias modeling and prediction method for low-Earth orbit satellites, thereby overcoming, to at least some extent, one or more problems caused by the limitations and defects of related technologies.

[0008] According to a first aspect of the present disclosure, a clock bias modeling and prediction method for low-Earth orbit satellites is provided, comprising: Obtain the raw clock bias data from the satellite and perform a detrending term to obtain the clock bias residual sequence; Fast Fourier Transform analysis was performed on the clock error residual sequence to extract the main periodic term; By introducing the principal periodic term into the quadratic polynomial model, a PQM model is constructed, and the derivative of the PQM model is obtained to obtain the RPQM model. Using the raw clock error data as input to the RPQM model, the optimal parameter vector to be estimated and the constant term are determined to obtain the prediction model; Predictive models are used to forecast future clock errors.

[0009] Furthermore, the expression for the PQM model is:

[0010] in, For satellites in The time difference For constant terms, This represents the relative frequency deviation. For frequency drift term, For random noise, , The number of clock differences selected. For the first The first amplitude coefficient of each principal periodic term For the first The second amplitude coefficient of each principal periodic term For the first The frequency of each principal periodic term , This represents the number of principal periodic terms in the model.

[0011] Furthermore, the expression for the RPQM model is:

[0012] in, This represents the rate of change of clock bias.

[0013] Furthermore, the steps of determining the optimal parameter vector and constant term to obtain the prediction model, using the original clock bias data as input to the RPQM model, include: Calculate the rate of change of clock bias based on the original clock bias data; The RPQM model is converted into matrix form and simplified. Based on the clock error rate of change, the least squares method is used to estimate the optimal parameters of the simplified RPQM model in order to determine the optimal parameter vector to be estimated in the PQM model; wherein, the optimal parameter vector to be estimated includes the optimal trend term coefficient and the optimal periodic term coefficient. Based on the PQM model that determines the optimal trend term coefficient and the optimal periodic term coefficient, the clock difference value at the starting point is predicted by using the average of several clock difference values ​​close to the forecast end, so as to obtain the constant term of the PQM model. The PQM model, which determines the optimal trend term coefficient, the optimal periodic term coefficient, and the constant term, is used as the prediction model.

[0014] Furthermore, the matrix form of the RPQM model is as follows:

[0015] The matrix form of the RPQM model is simplified as follows:

[0016] in, For the model matrix, The vector of parameters to be estimated , The clock error rate vector ; The optimal vector of parameters to be estimated is: .

[0017] According to a second aspect of the present disclosure, a computer-readable storage medium is provided, on which a computer program is stored, which, when executed by a processor, implements the steps of the clock bias modeling and prediction method for low-Earth orbit satellites described in any of the above embodiments.

[0018] According to a third aspect of the present disclosure, an electronic device is provided, comprising: Processor; and Memory for storing the executable instructions of the processor; The processor is configured to execute the steps of the clock bias modeling and prediction method for low-Earth orbit satellites described in any of the above embodiments by executing the executable instructions.

[0019] The technical solutions provided by the embodiments of this disclosure may include the following beneficial effects: In the embodiments of this disclosure, the clock bias modeling and prediction method for low-Earth orbit satellites described above achieves the following: First, based on the original clock bias data, the main periodicity feature in the satellite clock bias sequence is detected, and a PQM model is established by combining a quadratic polynomial. The coefficients of the trend and periodicity terms of the PQM model are estimated by the clock bias change rate, and the constant term is determined using short-term clock bias data close to the prediction end, thereby constructing a high-precision short-term clock bias prediction model and effectively improving the accuracy of clock bias prediction, especially short-term prediction. Second, by introducing the main periodicity term and the quadratic drift term to construct the PQM model and differentiating the PQM model, the influence of the constant term is eliminated. This makes the estimation of the initial frequency deviation, frequency drift rate, and periodicity term amplitude coefficient more focused on the dynamic change characteristics of the clock bias, avoiding interference from the constant term drift in the extraction of periodic features. Attached Figure Description

[0020] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with this disclosure and, together with the description, serve to explain the principles of this disclosure. It is obvious that the drawings described below are merely some embodiments of this disclosure, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort.

[0021] Figure 1 The diagram illustrates the steps of a clock bias modeling and prediction method for low-Earth orbit satellites in an exemplary embodiment of this disclosure. Figure 2 This illustrates the overall process of forecasting based on the rate of change of clock bias in an exemplary embodiment of this disclosure; Figure 3 This demonstrates that, in an exemplary embodiment of this disclosure, the GRACE-C satellite exhibits an approximately linear and monotonically decreasing trend in its daily clock bias sequence. Figure 4 This demonstrates that, in an exemplary embodiment of this disclosure, the GRACE-D satellite exhibits an approximately linear and monotonically decreasing trend in its daily clock bias sequence. Figure 5 This illustration shows the clock error residual sequence of the GRACE-C satellite in DOY 324, 2025, after removing the trend term, in an exemplary embodiment of this disclosure. Figure 6 This illustration shows the clock error residual sequence of the GRACE-D satellite in DOY 324, 2025, after removing the trend term, in an exemplary embodiment of this disclosure. Figure 7 The clock error FFT results of the GRACE-C satellite after removing the trend term in DOY 324, 2025, are shown in an exemplary embodiment of this disclosure. Figure 8 This illustrates the clock error FFT results of the GRACE-D satellite in DOY 324, 2025, after removing the trend term, in an exemplary embodiment of this disclosure. Figure 9 The first nine major periodic components extracted from the 2025 DOY 324 clock bias sequence by the GRACE-C satellite are shown in an exemplary embodiment of this disclosure. Figure 10 The first nine major periodic components extracted from the 2025 DOY 324 clock bias sequence by the GRACE-D satellite are shown in an exemplary embodiment of this disclosure. Figure 11 This illustration shows the scatter plot distribution of the periodic quadratic clock error rate of change model prediction accuracy of the GRACE-C satellite in DOY 324, 2025, over a period of 360 seconds, in an exemplary embodiment of this disclosure. Figure 12 This illustration shows the scatter plot distribution of the periodic quadratic clock error rate of change model prediction accuracy of the GRACE-D satellite in DOY 324, 2025, over a period of 360 seconds, in an exemplary embodiment of this disclosure. Figure 13 This illustration shows the scatter plot distribution of the periodic quadratic clock error rate of change model prediction accuracy of the GRACE-C satellite in DOY 324, 2025, within the range of 360s-900s in an exemplary embodiment of this disclosure. Figure 14 This illustration shows the scatter plot distribution of the periodic quadratic clock error rate of change model prediction accuracy of the GRACE-D satellite in DOY 324, 2025, within the range of 360s-900s in an exemplary embodiment of this disclosure. Figure 15 This diagram shows a summary of the prediction accuracy of the GRACE-C satellite's periodic quadratic clock error rate model for DOY 324 in 2025, in an exemplary embodiment of this disclosure. Figure 16 A summary graph showing the prediction accuracy of the GRACE-D satellite's periodic quadratic clock error rate model for DOY 324 in 2025, in an exemplary embodiment of this disclosure; Figure 17 This illustrates the forecast accuracy of the GRACE-C satellite based on various models for DOY 324 in 2025, in an exemplary embodiment of this disclosure. Figure 18 This illustrates the forecast accuracy of the GRACE-D satellite for DOY 324 in 2025 based on various models in an exemplary embodiment of this disclosure. Detailed Implementation

[0022] Example embodiments will now be described more fully with reference to the accompanying drawings. However, the example embodiments can be implemented in various forms and should not be construed as limited to the examples set forth herein; rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the concept of the example embodiments to those skilled in the art. The features, structures, or characteristics described may be combined in any suitable manner in one or more embodiments.

[0023] In addition, the accompanying drawings are only schematic illustrations of the embodiments of the present disclosure and are not necessarily drawn to scale. The same reference numerals in the drawings denote the same or similar parts, and thus their repeated description will be omitted. Some of the block diagrams shown in the drawings are functional entities and do not necessarily correspond to physically or logically independent entities.

[0024] A method for clock error modeling and prediction for low Earth orbit satellites is provided in this example embodiment. Referring to Figure 1 as shown, the method for clock error modeling and prediction for low Earth orbit satellites may include: Step S1: Obtain the original clock error data of the satellite and perform detrending to obtain the clock error residual sequence; Step S2: Perform fast Fourier transform analysis on the clock error residual sequence to extract the main periodic term; Step S3: Introduce the main periodic term into the quadratic polynomial model to construct the PQM model, and take the derivative of the PQM model to obtain the RPQM model; Step S4: Use the original clock error data as the input value of the RPQM model to determine the optimal parameter vector to be estimated and the constant term to obtain the prediction model; Step S5: Use the prediction model to predict the future clock error.

[0025] Through the above method for clock error modeling and prediction for low Earth orbit satellites, on the one hand, based on the original clock error data, the characteristics of the main periodic term in the satellite clock error sequence are detected, and the PQM model is established in combination with the quadratic polynomial. The trend term and the periodic term coefficient of the PQM model are estimated through the clock error rate of change, and the constant term is determined by using the short-term clock error data near the prediction end, so as to construct a high-precision short-term clock error prediction model, effectively improving the accuracy of clock error prediction, especially short-term prediction. On the other hand, by introducing the main periodic term and the quadratic drift term to construct the PQM model and taking the derivative of the PQM model, the influence of the constant term is eliminated, making the estimation of the initial frequency deviation, the frequency drift rate, and the amplitude coefficient of the periodic term more focused on the dynamic change characteristics of the clock error, avoiding the interference of the constant term drift on the extraction of the periodic characteristics.

[0026] Next, each step of the above method for clock error modeling and prediction for low Earth orbit satellites in this example embodiment will be described in more detail with reference to Figures 1 to 18 ​

[0027] In steps S1 and S2, the raw clock bias data of the satellite is acquired and detrended to obtain the clock bias residual sequence; the clock bias residual sequence is subjected to fast Fourier transform analysis to extract the main periodic term.

[0028] Specifically, after obtaining the raw clock bias data from the satellite, the trend term in the raw clock bias data is removed to obtain the clock bias residual sequence.

[0029] The clock error residual sequence after trend removal was analyzed by Fast Fourier Transform (FFT) to obtain the clock error spectral distribution results (i.e., the main periodic term).

[0030] In steps S3 to S5, the principal periodic term is introduced into the quadratic polynomial model to construct the PQM model, and the derivative of the PQM model is obtained to obtain the RPQM model; the original clock difference data is used as the input value of the RPQM model to determine the optimal parameter vector to be estimated and the constant term to obtain the prediction model; the prediction model is used to predict future clock differences.

[0031] Specifically, the Quadratic Polynomial Model (QPM) is a clock error prediction model built based on the physical characteristics of spaceborne atomic clocks, typically including the initial phase deviation of the satellite atomic clock. (i.e., constant term), initial frequency deviation and frequency drift rate Parameters are included. Compared to the linear model, the quadratic polynomial model incorporates a frequency drift term, thus effectively reflecting the long-term variation characteristics of atomic clocks. Especially for miniature rubidium atomic clocks or temperature-controlled crystal oscillators with good short-term stability but poor long-term stability, this model can more accurately describe the clock bias variation characteristics. Due to its simple structure and clear physical meaning, the quadratic polynomial model has been widely used in satellite clock bias modeling and prediction. In this model, it is assumed that the change of satellite clock bias over time can be approximated by a quadratic function, the mathematical expression of which is: (1) In the formula, For satellites in The time difference For initial phase deviation, This represents the initial frequency deviation. For frequency drift rate, For random noise, The model parameters can be estimated using the least squares method, and the solution formula is as follows: (2) In the formula, The vector of parameters to be estimated includes the initial phase deviation of the atomic clock. Initial frequency deviation and frequency drift rate ; The coefficient matrix, This is the clock error observation vector. Specifically, it can be represented as: (3) Quadratic polynomial models are suitable for scenarios where clock bias changes are relatively stable and can effectively reflect the long-term drift characteristics of onboard clocks. However, unlike GNSS satellites with orbital altitudes of approximately 20,000 km, LEO satellites orbit at altitudes of only a few hundred kilometers and operate at high speeds. Their environment is more complex and susceptible to periodic disturbances from various external factors such as geomagnetic field variations, thermal radiation effects, and atmospheric drag. Furthermore, the clock bias of LEO satellites is typically estimated based on GNSS signals, and the orbital periodicity of GNSS satellites introduces long-period variation components into the clock bias sequence. Therefore, relying solely on quadratic polynomial models is insufficient to fully reflect the dynamic characteristics of LEO satellite clock bias. To address this, this application introduces an additional periodic term into the traditional quadratic polynomial model to construct an improved periodic quadratic polynomial model. The mathematical expression of the PQM model is: (4) In the formula, For satellites in The time difference For initial phase deviation, This represents the initial frequency deviation. For frequency drift term, For random noise, , , and The first The amplitude coefficient and frequency of each periodic term, ,in This indicates the number of periodic terms in the model.

[0032] For periodic quadratic models, this application proposes a periodic quadratic model estimation method based on the clock difference rate of change.

[0033] This method effectively eliminates the constant term by modeling the clock error rate of change sequence. To address the drift problem and improve the accuracy of periodic feature extraction and model stability, a periodic quadratic clock error rate of change (RPQM) model is constructed, the mathematical expression of which is: (5) In the formula, Let $\frac{ ... The time difference is eliminated during the differentiation process, and the meanings of the remaining parameters are the same as in equation (4). Clock difference rate of change It can be calculated from clock difference observation data of adjacent epochs, and its value is approximately equal to The rate of change of clock difference at any given time. Specifically, it can be expressed as: (6) In the formula, and Representing time and time respectively Adjacent satellite clock bias data, The time interval between adjacent clock difference observation epochs is given. Using Equation 4 as the basic model for clock difference prediction, the frequency of the periodic term is first analyzed and its characteristics are fixed. Based on this, the model coefficients in Equation (5) are... As the parameter to be estimated, during the model parameter solution process, the existing clock difference data is used to perform optimal parameter estimation through the least squares method, thereby determining the prediction model for the periodic quadratic clock difference change rate. Assuming time... The rate of change of clock bias is expressed as The model can then be discretized as follows: (7) The above model can be simplified as follows: (8) in, For the model matrix, The vector of parameters to be estimated , The clock error rate vector .

[0034] Using the least squares method, the optimal estimates of the model parameters can be obtained by the following formula: (9) Based on the above estimation process, the obtained estimates of the clock bias rate of change can be used to solve for the corresponding model parameters within a selected time window. Among them, the constant term Predict the area near the starting point The clock error is estimated by extrapolation from individual clock error observations. The entire model is determined by combining the estimated parameter values. The overall process for forecasting based on the clock error rate of change is as follows: Figure 2 As shown.

[0035] In a specific embodiment, 1. Analysis of GRACE-FO satellite clock bias characteristics To verify the feasibility of the model proposed in this application and its effectiveness in clock bias prediction accuracy, onboard clock data from a low-Earth orbit satellite (Gravity Recovery and Climate Experiment Follow-On, GRACE-FO) was selected as the research object to conduct an analysis of the clock bias characteristics of the low-Earth orbit satellite. The GRACE-FO mission, jointly developed by the German Research Centre for Geosciences (GFZ) and the National Aeronautics and Space Administration (NASA), aims to obtain high-precision information on changes in the Earth's gravity field through precise measurement techniques. The GRACE-FO satellite system was launched and put into operation in 2018, employing a satellite chasing formation flight mode. It consists of two satellites, GRACE-C and GRACE-D, with an average inter-satellite distance of approximately 220 km. Both satellites carry a highly stable onboard clock system, the core component of which is an ultra-stable crystal oscillator (USO) operating at a frequency of 10 MHz, exhibiting excellent short-term frequency stability and low phase noise characteristics. The GRACE-C and GRACE-D have basically the same orbital altitude and main orbital parameters, providing good experimental conditions for comparative analysis of clock bias characteristics and joint modeling.

[0036] The data used in this application comes from Earthdata Search, the Earth Science Data Search and Access Portal developed by NASA. L1B-level CLK1B data products from the GRACE-C and GRACE-D satellites were obtained through this platform and used for subsequent analysis. The CLK1B product provides high temporal resolution onboard clock bias information and is an important data source for studying satellite clock performance and its dynamic characteristics in the GRACE-FO mission. CLK1B data is calculated based on the Single Point Positioning (SPP) method, estimating the deviation of the satellite receiver clock relative to GNSS system time using Global Navigation Satellite System (GNSS) observation data. To further improve the accuracy of the clock bias, precise orbital information and error correction models are incorporated into the CLK1B data generation process, resulting in clock bias data with high temporal stability and high accuracy. The product's time sampling interval is 10 seconds, which meets the accuracy requirements for clock bias variation characteristic analysis and model validation. This application selects GRACE-C and GRACE-D onboard clock data on day 324 of 2025 as the research sample. The data during this period is complete and without any abnormal jumps, which can well reflect the variation characteristics of the satellite clock under stable working conditions, providing reliable data support for the feasibility analysis of the subsequent model and the verification of the accuracy of clock error prediction.

[0037] like Figure 3 As shown, the GRACE-C satellite exhibits an approximately linear and monotonically decreasing trend in its daily clock bias sequence; for example... Figure 4 As shown, the clock bias sequence of the GRACE-D satellite exhibits an approximately linear and monotonically decreasing trend. The linear term in the clock bias sequence primarily reflects the inherent frequency deviation characteristics of the USO. The slopes of the two satellites are quite similar, indicating that the USOs carried by them have high consistency in terms of frequency constant deviation amplitude and stability.

[0038] Under longer timescales or high-precision modeling requirements, factors such as USO frequency drift and environmental disturbances can introduce certain nonlinear effects. While the amplitude of this effect is relatively small in the original clock error sequence, it is still significant in residual analysis and short-term forecast accuracy. Therefore, this application incorporates a quadratic term into the clock error modeling to describe clock error variations, thereby improving the model's ability to represent weak nonlinear characteristics. After removing the trend term from the clock error sequence, the result is as follows: Figure 5 and Figure 6 The clock error residual sequence is shown. Figure 5 The clock error residual sequence of GRACE-C satellite with trend term removed in DOY 324, 2025; Figure 6This is the clock error residual sequence of the GRACE-D satellite after removing the trend term in DOY 324, 2025. It can be seen that the residuals of the two satellites exhibit a slow fluctuation around zero mean, and the shape of the residuals is highly consistent, but there are still some differences in phase position and amplitude. Specifically, the peak-to-trough amplitude of the residuals of the GRACE satellite is controlled within ±20 ns, while the residual amplitude of the GRACE-D satellite is slightly larger, with peak-to-trough values ​​close to ±25 ns.

[0039] To further reveal the periodic variation characteristics of the onboard clock of the GRACE-FO satellite, this application performed Fast Fourier Transform (FFT) analysis on the clock error residual sequence after removing the trend. Figure 7 The clock error FFT results for the GRACE-C satellite after removing the trend term in DOY 324, 2025; Figure 8 The figure shows the clock error FFT results of the GRACE-D satellite after removing the trend term in DOY 324, 2025. As can be seen from the figure, the FFT results of both satellites exhibit several obvious discrete spectral peaks, with the spectral energy mainly concentrated at a few specific frequency positions, indicating that after removing the trend term, the GRACE-FO onboard clock has an analyzable periodic term.

[0040] Based on the amplitude of each frequency component, the first nine major periodic terms were extracted from the GRACE-C and GRACE-D satellite clock bias sequences, respectively, and their corresponding time-domain waveforms are shown below. Figure 9 and Figure 10 As shown. Among them, Figure 9 The first nine major periodic components extracted from the 2025 DOY 324 clock bias sequence of the GRACE-C satellite; Figure 10 These are the top nine major periodic components extracted from the DOY 324 clock bias sequence of the GRACE-D satellite in 2025. It can be seen that among the top nine major periodic components extracted by FFT, the periodic components around 12h, 8h, 6h, 4h, and 1.6h have relatively large amplitudes, exhibiting significant periodic characteristics. The remaining periodic components have relatively small amplitudes and have little impact on the overall clock bias variation.

[0041] The 12-hour periodic term may be related to the geometric repetition period of the GPS constellation, with GPS satellites having an orbital period of approximately 11 hours and 58 minutes. Because the GPS signals received by the GRACE-FO satellite exhibit a repeating geometric configuration within this period, it may cause systematic errors and clock disturbances, resulting in a significant semi-diurnal periodic component in the onboard clock. Furthermore, the GRACE-FO satellite experiences approximately three different variations in solar radiation incidence angle and thermal load cycles daily, which superimpose to produce an approximately 8-hour periodic term. The 6-hour and 4-hour periodic terms may originate from higher-order thermal control system responses, attitude adjustments, and changes in the GPS signal propagation path. Among the short-period components, the approximately 1.6-hour periodic term is the most prominent, highly consistent with the orbital period of the GRACE-FO satellite itself. GRACE-C and GRACE-D orbit at an altitude of approximately 490 km with an orbital period of approximately 96 minutes; this component reflects the direct modulation effect of orbital periodic disturbances on the clock system. In summary, the main periodic terms of the GRACE-FO satellite clock can be divided into two categories: one is the medium- and long-term periodic components caused by external signal sources (such as GPS geometry and propagation delay), and the other is closely related to the periodic changes in the satellite's own orbit and thermal environment.

[0042] 2. Performance analysis of the improved clock error prediction model Based on the above analysis, this application selects five main periodic terms in the clock bias modeling of GRACE-C and GRACE-D. First, the periodic structure of the GRACE-FO satellite clock is fixed, i.e., the number of periodic terms and their corresponding frequencies are fixed. The selected periodic components are all derived from the main spectral components extracted using Fast Fourier Transform (FFT) as described above. On this basis, a basic periodic quadratic model is constructed to ensure a systematic comparison of the clock bias prediction performance of different parameter estimation methods while maintaining a consistent periodic structure. In the model solution stage, a least-squares parameter estimation method based on the clock bias change rate is adopted, and a sliding time window technique is used to dynamically model and provide short-term forecasts for clock bias data on the same day. The sliding time window length is set from 30s to 900s, and parameter fitting and subsequent prediction are completed within each time window to obtain the model's response characteristics at different time scales.

[0043] Table 1. Forecasting strategies for GRACE-C and GRACE-D satellites

[0044] This application conducts a systematic analysis of the number of periodic terms, the length of the fitted arc segment, and the model accuracy under different forecast durations to determine the optimal forecasting strategy. During model construction, five main periodic terms are introduced for the GRACE-C and GRACE-D satellites. To avoid the overlapping effects between different time windows during the forecasting process, a rolling calculation method with the sliding window length consistent with the forecast step size is adopted. Multi-scale forecasting experiments ranging from 30s to 900s are conducted on the onboard clock data of the same day. Relevant parameter settings are shown in Table 1.

[0045] Figure 11 The scatter plot shows the prediction accuracy of the GRACE-C satellite's periodic quadratic clock error rate model for DOY 324 in 2025 over a 360s period. Figure 12 This is a scatter plot of the prediction accuracy of the GRACE-D satellite's periodic quadratic clock error rate model for DOY 324 in 2025, within the range of 30s-360s. From... Figure 11 and Figure 12 The scatter distribution results show that, under short-term prediction conditions within 360 seconds, the prediction residuals of the two satellites are highly concentrated, with the vast majority of errors being less than 0.1 ns. However, as the prediction duration gradually increases, such as... Figure 13 and Figure 14 As shown, the dispersion of the predicted residuals has increased, but overall it remains within 1 ns, without any systematic divergence. Figure 13 The scatter plot shows the prediction accuracy of the GRACE-C satellite's periodic quadratic clock error rate model for DOY 324 in 2025, within the range of 360s-900s. Figure 14 The image shows the scatter plot distribution of the GRACE-D satellite's periodic quadratic clock error rate of change model prediction accuracy for DOY324 in 2025, within the range of 360s to 900s. In comparison, the prediction results of the GRACE-C satellite are generally more stable, while GRACE-D shows a few outliers in certain time periods, but their proportion is extremely low and has a limited impact on the overall statistical characteristics.

[0046] To reduce the impact of extreme points on statistical results, Figure 15 and Figure 16 The root mean square (RMS) index of the absolute value of the residuals is used to summarize and evaluate the model accuracy under different prediction durations. Figure 15 A summary chart of the prediction accuracy of the GRACE-C satellite's periodic quadratic clock error rate model for DOY 324 in 2025; Figure 16This is a summary chart of the prediction accuracy of the GRACE-D satellite's periodic quadratic clock error rate model for DOY 324 in 2025. The results show that the prediction accuracy exhibits a smooth upward trend with increasing prediction duration. The predicted RMS of the GRACE-C satellite increased from approximately 0.02 ns at 30 s to approximately 0.44 ns at 900 s; while the predicted RMS of the GRACE-D satellite increased from approximately 0.01 ns to approximately 0.90 ns. The trends of the two satellites are basically consistent, further demonstrating that the proposed model has good generalization ability and stability under different payload conditions.

[0047] The results above show that the periodic quadratic model based on clock error rate of change (RPQM) exhibits high prediction accuracy for the GRACE-FO satellite clock within a 900s prediction range. Further analysis of the predictable duration under different accuracy constraints reveals that when prediction accuracy requirements are kept within 0.1ns and 0.2ns, respectively, the maximum predictable duration for the GRACE satellite is approximately 420s and 600s; while for GRACE-D, the maximum predictable durations are approximately 300s and 420s, respectively.

[0048] 3. Comparative Analysis of Clock Difference Prediction Accuracy To more intuitively demonstrate the advantages of this method, the forecasting performance of various clock error prediction models is compared and analyzed below under the same experimental conditions and statistical criteria. Figure 17 and Figure 18 The paper presents a comparison of the prediction accuracy of GRACE-C and GRACE-D satellites on clock bias data for day 324 of 2025 (DOY 324), using linear models, grey GM(1,1) models, quadratic polynomial models, and the periodic quadratic model based on the clock bias change rate proposed in this application. Figure 17 The forecast accuracy of the GRACE-C satellite for DOY 324 in 2025 based on various models; Figure 18 The forecast accuracy of the GRACE-D satellite for DOY324 in 2025 based on various models.

[0049] Comparison Method 1: Slide forecasts are made according to the forecast duration. A 30-second forecast time can be slided 2880 times. The forecast error of one arc segment generated by each forecast is statistically analyzed as a whole. Comparison Method 2: Perform RMS statistics for different points based on duration, and plot a curve for each method. from Figure 17 and Figure 18As can be seen, the overall trend of clock error prediction for GRACE-C and GRACE-D satellites using different models is positively correlated with the prediction time. With the extension of the prediction time, the differences in prediction accuracy among different models gradually become apparent and amplify. The linear model exhibits a faster rate of error growth, indicating that methods based solely on linear extrapolation are insufficient to effectively characterize the periodic disturbances contained in the clock error sequence. The quadratic polynomial model and the grey model show relatively gradual error growth trends in the short-to-medium-term prediction phase. In the GRACE-C satellite clock error prediction process, the method proposed in this application outperforms the other three methods within a short-term prediction range of 900s. For the GRACE-D satellite, the method proposed in this application consistently maintains the highest prediction accuracy when the prediction time does not exceed 540s.

[0050] By analyzing clock data from the GRACE-C and GRACE-D satellites during DOY 324 in 2025, the main periodic components in the onboard clocks were identified using the FFT method. The results show that, in addition to stable linear drift, significant multi-frequency periodic disturbances exist in the clock bias, mainly concentrated in the 12h, 8h, 6h, 4h, and 1.6h periods. Four main periodic components were introduced for the GRACE-C satellite, and five main periodic terms were introduced for the GRACE-D satellite. A short-term clock bias prediction experiment of 900s was conducted based on the proposed method, and the prediction results were compared with those of linear models, quadratic polynomial models, and grey models. Experimental results show that the proposed method has better prediction accuracy than other models within 900s for the GRACE-C satellite and within 540s for the GRACE-D satellite. Specifically, the prediction error is less than 0.1ns within the 7min and 5min prediction time ranges. In summary, the periodic quadratic model based on the clock error rate can effectively improve the accuracy and reliability of GRACE-FO satellite clock prediction, providing a useful reference for dynamic modeling and precise orbit determination of low-Earth orbit satellite time systems.

[0051] The aforementioned clock bias modeling and prediction method for low-Earth orbit satellites achieves two main objectives. First, based on raw clock bias data, the main periodicity term characteristics in the satellite clock bias sequence are detected, and a Phase Quantitative Model (PQM) is established using a quadratic polynomial. The coefficients of the trend and periodicity terms in the PQM model are estimated using the clock bias change rate, and the constant term is determined using short-term clock bias data close to the prediction endpoint. This constructs a high-precision short-term clock bias prediction model, effectively improving the accuracy of clock bias prediction, especially short-term prediction. Second, by introducing the main periodicity term and a quadratic drift term to construct the PQM model and differentiating the PQM model, the influence of the constant term is eliminated. This allows the estimation of initial frequency deviation, frequency drift rate, and periodicity term amplitude coefficients to focus more on the dynamic changes in clock bias, avoiding interference from constant term drift in the extraction of periodic features.

[0052] In exemplary embodiments of this disclosure, a computer-readable storage medium is also provided, on which a computer program is stored, which, when executed by a processor, can implement the steps of the clock bias modeling and prediction method for low-Earth orbit satellites described in any of the above embodiments. In some possible implementations, various aspects of this application can also be implemented as a program product comprising program code, which, when run on a terminal device, causes the terminal device to perform the steps of the various exemplary embodiments of this application described in the section on clock bias modeling and prediction methods for low-Earth orbit satellites described above.

[0053] In exemplary embodiments of this disclosure, an electronic device is also provided, which may include a processor and a memory for storing executable instructions of the processor. The processor is configured to perform the steps of the clock bias modeling and prediction method for low-Earth orbit satellites described in any of the above embodiments by executing the executable instructions.

[0054] Those skilled in the art will understand that various aspects of this application can be implemented as a system, method, or program product. Therefore, various aspects of this application can be specifically implemented in the following forms: a completely hardware implementation, a completely software implementation (including firmware, microcode, etc.), or a combination of hardware and software implementations, collectively referred to herein as a "circuit," "module," or "system."

[0055] Other embodiments of this disclosure will readily occur to those skilled in the art upon consideration of the specification and practice of the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of this disclosure that follow the general principles of this disclosure and include common knowledge or customary techniques in the art not disclosed herein. The specification and examples are to be considered exemplary only, and the true scope and spirit of this disclosure are indicated by the appended claims.

Claims

1. A clock bias modeling and prediction method for low-Earth orbit satellites, characterized in that, include: Obtain the raw clock bias data from the satellite and perform a detrending term to obtain the clock bias residual sequence; Fast Fourier Transform analysis was performed on the clock error residual sequence to extract the main periodic term; By introducing the principal periodic term into the quadratic polynomial model, a PQM model is constructed, and the derivative of the PQM model is obtained to obtain the RPQM model. Using the raw clock error data as input to the RPQM model, the optimal vector of parameters to be estimated and the constant term are determined to obtain the prediction model; Predictive models are used to forecast future clock errors.

2. The clock bias modeling and prediction method for low-Earth orbit satellites according to claim 1, characterized in that, The expression for the PQM model is: in, For satellites in The time difference For constant terms, This represents the relative frequency deviation. For frequency drift term, For random noise, , The number of clock differences selected. For the first The first amplitude coefficient of each principal periodic term For the first The second amplitude coefficient of each principal periodic term For the first The frequency of each principal periodic term , This represents the number of principal periodic terms in the model.

3. The clock bias modeling and prediction method for low-Earth orbit satellites according to claim 2, characterized in that, The expression for the RPQM model is: in, This represents the rate of change of clock bias.

4. The clock bias modeling and prediction method for low-Earth orbit satellites according to claim 3, characterized in that, The steps involved in determining the optimal parameter vector and constant terms to obtain the prediction model, using the raw clock bias data as input to the RPQM model, include: Calculate the rate of change of clock bias based on the original clock bias data; The RPQM model is converted into matrix form and simplified. Based on the clock error rate of change, the least squares method is used to estimate the optimal parameters of the simplified RPQM model in order to determine the optimal parameter vector to be estimated in the PQM model; wherein, the optimal parameter vector to be estimated includes the optimal trend term coefficient and the optimal periodic term coefficient. Based on the PQM model that determines the optimal trend term coefficient and the optimal periodic term coefficient, the clock difference value at the starting point is predicted by using the average of several clock difference values ​​close to the forecast end, so as to obtain the constant term of the PQM model. The PQM model, which determines the optimal trend term coefficient, the optimal periodic term coefficient, and the constant term, is used as the prediction model.

5. The clock bias modeling and prediction method for low-Earth orbit satellites according to claim 4, characterized in that, The matrix form of the RPQM model is: The matrix form of the RPQM model is simplified as follows: in, For the model matrix, The vector of parameters to be estimated , The clock error rate vector ; The optimal vector of parameters to be estimated is: 。 6. A computer-readable storage medium having a computer program stored thereon, characterized in that, When executed by a processor, the program implements the steps of the clock bias modeling and prediction method for low-Earth orbit satellites as described in any one of claims 1 to 5.

7. An electronic device, characterized in that, include: processor; as well as Memory for storing the executable instructions of the processor; The processor is configured to execute the steps of the clock bias modeling and prediction method for low-Earth orbit satellites according to any one of claims 1 to 5 by executing the executable instructions.

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