A new method for extracting geocentric motion modes based on successive variational modal decomposition

Abstract

The present invention discloses a new method for extracting geocentric motion modes based on successive variational mode decomposition, relates to the technical field of geocentric motion time series signal processing, solves the problem that the existing IMF extraction criteria are insufficient in GCM time series analysis based on SVMD, and the key points of the technical solution specifically include the following steps: decomposing the geocentric motion GCM time series by a successive variational mode decomposition (SVMD) method to decompose each modal IMF; based on each modal IMF decomposed in step a, extracting the effective IMF from each modal IMF according to the proposed dual-threshold progressive extraction criteria, the effect is that the present invention can better extract the effective IMF by using the dual-threshold progressive extraction criteria, thereby providing a basis for subsequent analysis of GCM periodic signals.

Description

Technical Field

[0001] The present invention relates to the technical field of geocentric motion time series signal processing, and in particular to a new method for geocentric motion mode extraction based on successive variational mode decomposition. Background Art

[0002] Geocentre Motion (GCM) is caused by mass migration of factors such as the atmosphere, ocean, and land water, and the GCM time series reflects the changes in the geocentre motion, which contains periodic signals driven by multiple factors. In order to extract the periodic signals in the GCM time series, it is necessary to use the signal decomposition method. SVMD (Successive Variational Mode Decomposition, SVMD) is an improved method of Variational Mode Decomposition (VMD). In the SVMD decomposition process, only one The algorithm will be no more than Under the condition of , the modal IMF (Intrinsic Mode Function, IMF) is decomposed without fixing the penalty factor. Therefore, It is an important parameter that needs to be set according to the signal characteristics before SVMD decomposition. In order to extract the periodic signal in the GCM time series more precisely, it is necessary to set a larger However, for noisy GCM time series, the larger This may result in an IMF bandwidth that is too narrow, resulting in excessive IMFs and the appearance of noise-dominated IMFs. Therefore, considering the unique properties of geocentric motion time series data, it is extremely important to explore appropriate IMF extraction criteria.

[0003] Currently, when extracting IMFs from SVMD decomposition, relevant research has proposed appropriate extraction criteria based on data characteristics. In processing ship radiated noise signals, to extract useful signals, relevant research has introduced a dual-threshold criterion based on the correlation coefficient, thereby extracting effective IMFs from the SVMD decomposition while eliminating noise-dominated modes. To address this issue, some research has also used criteria based on amplitude-aware permutation entropy, inverse permutation entropy, and correlation coefficients to extract effective modes. In ultrasonic detection signal processing, relevant research has significantly improved the signal-to-noise ratio of ultrasonic detection signals by removing noise modes using a single-threshold criterion based on the correlation coefficient.

[0004] Although these criteria have achieved some success in their respective application areas, relying solely on the existing extraction criteria for IMFs derived from SVMD decomposition in GCM time series analysis suffers from the following three limitations: 1. Excessive information loss: Using a single-threshold correlation coefficient criterion can lead to high thresholds due to the high correlation coefficients of individual IMFs, which can easily result in significant information loss, affecting the accuracy of subsequent data analysis and processing. 2. Failure to effectively extract key signals: Because annual and semiannual signals are important signals in GCMs, and the correlation coefficient of the semiannual signal is too small, using a single-threshold or dual-threshold correlation coefficient criterion can mistakenly identify the semiannual cycle as a noise-dominated IMF, leading to the inadvertent rejection of key signals. 3. Complex threshold determination: In practice, extraction criteria based on amplitude-aware permutation entropy, inverse permutation entropy, and correlation coefficients require extensive experimentation to determine the threshold, making the threshold determination process relatively complex.

[0005] In summary, in order to overcome the shortcomings of the existing IMF extraction criteria in GCM time series analysis based on SVMD, it is particularly important to propose an IMF extraction criterion that is theoretically more suitable for GCM analysis. Summary of the Invention

[0006] The purpose of the present invention is to provide a new method for extracting geocentric motion modes based on successive variational modal decomposition to solve the problems raised in the above background technology.

[0007] To achieve the above object, the present invention provides the following technical solutions:

[0008] A new method for extracting geocentric motion modes based on successive variational mode decomposition (SVMD) is proposed. Based on the IMF decomposed by SVMD (Successive Variational Mode Decomposition), valid IMFs are extracted using a newly proposed dual-threshold progressive extraction criterion and analyzed. The method specifically includes the following steps:

[0009] Step a: Setup , use SVMD to decompose the GCM time series and obtain each IMF;

[0010] Step b: Calculate the correlation coefficient between each IMF decomposed by SVMD and the original GCM time series, and calculate the standard deviation of the correlation coefficient as the first threshold. Extract the IMFs greater than this threshold, and take the IMFs corresponding to the annual signal and the semi-annual signal as the effective IMFs. Then, calculate the kurtosis of the remaining IMFs, calculate their average value and use it as the second threshold. Extract the IMFs with kurtosis less than this threshold as the effective IMFs.

[0011] The effective IMFs extracted in step c are compared with those extracted by the single-threshold and double-threshold correlation coefficient criteria, and further compared with the period decomposed by singular spectrum analysis (SSA) to verify the advantages and applicability of the double-threshold progressive extraction criterion.

[0012] As a further technical solution of the present invention, the extraction process of different extraction criteria is as follows:

[0013] The single threshold criterion of correlation coefficient is as follows: the correlation coefficient of each mode of the original signal decomposed by SVMD is calculated, and the single threshold is calculated based on the maximum and average values of the correlation coefficients;

[0014] ① Use SVMD to decompose the original GCM time series;

[0015] ②Calculate the correlation coefficient of each IMF , as shown in formula (1).

[0016] (1) Where, is the original GCM time series, is the average of the original signal, is the decomposed mode, is the mode average, Indicates time, N Indicates the number of data.

[0017] ③Calculate the single threshold of the correlation coefficient , as shown in formula (2).

[0018] (2)

[0019] Where, is the maximum value among all IMF correlation coefficients, is the average of the IMF correlation coefficients.

[0020] For correlation coefficients greater than the threshold The IMF is considered to be an effective IMF. When , it is considered to be the noise-dominated IMF. At the same time, the noise-dominated IMF is removed and the effective IMF is extracted.

[0021] When using the correlation coefficient single threshold criterion to process the IMFs of the GCM time series decomposed by SVMD, since the GCM time series contains complex multi-scale periodic information, after SVMD decomposition, some periodic signals corresponding to the IMF have a high degree of consistency with the original series characteristics, and the correlation coefficient is much higher than other IMFs, which will raise the threshold. , which results in most IMFs with low correlation coefficients being misjudged as noise-dominated modes and eliminated.

[0022] The double-threshold criterion of correlation coefficient is as follows: the correlation coefficient of each mode of the original signal decomposed by SVMD is calculated, and the high threshold and low threshold are calculated based on the maximum and average values of the correlation coefficients;

[0023] ① Use SVMD to decompose the original GCM time series;

[0024] ②Calculate the correlation coefficient of each IMF , as shown in formula (1).

[0025] ③Calculate the high threshold of the correlation coefficient and low threshold , as shown in formula (3) and formula (4).

[0026] (3)

[0027] (4)

[0028] Where, is the maximum value among all IMF correlation coefficients, is the average of the IMF correlation coefficients.

[0029] For correlation coefficients greater than the threshold and The IMF is considered to be an effective IMF. When , it is considered to be the noise-dominated IMF. At the same time, the noise-dominated IMF is removed and the effective IMF is extracted.

[0030] When using the double-threshold criterion for correlation coefficient to process the IMFs of the GCM time series decomposed by SVMD, the double-threshold setting can retain most modes. However, for major cycles such as the semi-annual term, the correlation coefficient between the corresponding IMF and the original series may be too low, making it unable to pass the low threshold, resulting in the major cycle being eliminated as a noise-dominated mode.

[0031] As a further technical solution of the present invention, a new extraction criterion is proposed based on the characteristics of GCM data: a double-threshold progressive extraction method. The extraction process of the double-threshold progressive extraction criterion is as follows:

[0032] ① Use SVMD to decompose the original GCM time series;

[0033] SVMD decomposes the original signal into two parts, as shown in formula (5):

[0034] (5)

[0035] Where, is the original GCM time series, is the first of all the modes decomposed m modal, Except for the m All modes except the one.

[0036] When decomposing, four principles should be followed to form a constraint problem, as shown in formula (6):

[0037] (6)

[0038] in, It is the criterion for controlling each mode to be more compact at its center frequency. is the control residual signal Hedi m The criterion for minimizing the spectral overlap between modes is, Is to avoid the decomposition of m The mode was decomposed before m- 1 criterion for modal repetition, It's balance 、 and Parameters, is the original GCM time series, is the first of all the modes decomposed m modal, Except for the m All modes except the one.

[0039] Then, Formula (6) is transformed into an unconstrained problem and converted into frequency form, and the optimal mode is obtained by iteration.

[0040] ②Calculate the correlation coefficient of each IMF , as shown in formula (1), and calculate the standard deviation of each correlation coefficient , as shown in formula (7).

[0041] (1) In the formula, is the original GCM time series, is the average of the original signal, is the decomposed mode, is the mode average, Indicates time, N Indicates the number of data.

[0042] (7)

[0043] in, is the number of decomposition modes, is the average of the correlation coefficients, is the correlation coefficient of each IMF.

[0044] ③Extract the correlation coefficient greater than Modes filtered by this threshold represent modes highly correlated with the original time series. However, if the above steps are used solely as extraction criteria, if the correlation coefficient of a mode (such as the anniversary term) is much higher than the mean, the calculated standard deviation may be too large, resulting in the inadvertent elimination of some valid signals. Therefore, further processing of the remaining IMFs is essential. Furthermore, since the anniversary and semiannual signals are important periodic signals in GCM time series, step 4 is necessary.

[0045] ④ Regardless of whether the IMF corresponding to the annual signal or the semi-annual signal is higher than the threshold , all of them are retained as valid signals.

[0046] ⑤Calculate the kurtosis of the residual IMF Kurt , as in formula (8), and calculate the average value of the residual IMF kurtosis.

[0047] (8)

[0048] in, is the standard deviation of the series, is the mean of the series, Represents a sequence.

[0049] ⑥ Extract the IMF that is smaller than the average kurtosis as the effective IMF.

[0050] When applying extraction criteria to the IMFs of GCM time series decomposed by SVMD, the single- and dual-threshold correlation coefficient criteria mentioned above tend to overlook important but low-correlation semiannual cycles, leading to their misjudgment and removal. Furthermore, the single-threshold correlation coefficient criteria present another problem: because the correlations of individual IMFs are much greater than those of others, the threshold is too high, resulting in excessive IMF removal. The proposed criteria take these characteristics into account, not only extracting these cycles individually but also setting the standard deviation of the correlation coefficient and the mean kurtosis of the remaining IMFs to retain more IMFs. This overcomes the shortcomings of the single- and dual-threshold correlation coefficient criteria when applied to GCM data analysis.

[0051] As a further technical solution of the present invention: a dual-threshold progressive extraction criterion, that is, combining the prominent features of the GCM time series, using the standard deviation of the correlation coefficient as the first step extraction criterion, and the average value of the kurtosis as the second step extraction criterion.

[0052] Compared with the prior art, the present invention has the following beneficial effects:

[0053] Based on the characteristics of the GCM time series, the present invention proposes an extraction criterion adapted thereto, which solves the defects of the existing extraction criteria in the analysis of the GCM time series, including the two aspects of main feature extraction and information retention. In terms of main feature extraction, since the correlation coefficient double threshold criterion uses the threshold calculated by the correlation coefficient as the judgment basis, important periods in the GCM with too low correlation coefficients, such as the semi-annual term, will be mistakenly eliminated as noise-dominated modes. This extraction criterion fully considers the importance of the annual signal and the semi-annual signal in the GCM, and performs separate extraction for their corresponding IMFs. In terms of information retention, the IMFs decomposed from the GCM time series based on SVMD are extracted using the threshold determined by the existing single-threshold extraction criterion for correlation coefficient. Since the correlation between some IMFs and the original GCM series is much greater than that of the other IMFs, the threshold is raised, resulting in excessive signal loss. To this end, we use the standard deviation of the correlation coefficient as the first threshold, and take into account that using only the standard deviation of the correlation coefficient will also lead to the above-mentioned problems, and because there is a positive correlation between kurtosis and noise, when the noise intensity increases, the kurtosis value increases, so the kurtosis average is introduced as the second threshold, which can more accurately screen the remaining IMFs according to the noise characteristics. At the same time, compared with the criteria of using amplitude-aware permutation entropy, inverse permutation entropy and correlation coefficient, determining the threshold is relatively simple. Therefore, the present invention can better extract effective IMFs by using the dual-threshold progressive extraction criterion, providing a basis for subsequent analysis of GCM periodic signals. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 For the technology roadmap.

[0055] Figure 2 is a plot of the relationship between kurtosis and noise.

[0056] Figure 3 It is the extraction process with the standard deviation of the correlation coefficient as the threshold.

[0057] Figure 4 is the extraction process with the mean value of kurtosis as the threshold.

[0058] Figure 5 Figure 2 is the IMF energy proportion diagram based on different extraction criteria.

[0059] Figure 6 Comparison diagram of the main cycles decomposed by SVMD and SSA. DETAILED DESCRIPTION

[0060] The following will provide a clear and complete description of the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of them. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall within the scope of protection of the present invention.

[0061] The present invention mainly includes:

[0062] ① A dual-threshold progressive extraction criterion suitable for GCM time series has been developed. Existing extraction criteria have limitations when applied to GCM time series. Therefore, based on the significant characteristics of GCMs, this invention uses the standard deviation of the correlation coefficient and the average value of the kurtosis as dual thresholds. The standard deviation of the correlation coefficient is used as the first threshold, enabling the selection of IMFs with high correlation with the original series. The average value of the kurtosis is used as the second threshold, leveraging the positive correlation between kurtosis and noise to more efficiently extract valid IMFs.

[0063] ② The SVMD method was applied to the GCM time series decomposition for the first time, providing a new data processing perspective for GCM analysis and helping to more accurately analyze the periodic signals in GCM.

[0064] The project aims to develop a new method for extracting geocentric motion modes based on successive variational mode decomposition (SVMD) to accurately extract valid IMFs. This method aims to address three major issues with traditional extraction methods in GCM data analysis: excessive information loss, inability to effectively extract important GCM signal signatures, and complex threshold determination.

[0065] See also Figure 1 The present invention discloses a new method for extracting geocentric motion modes based on successive variational mode decomposition.

[0066] First, using the monthly GCM data derived from satellite laser ranging (SLR) data from the Center for Space Research (CSR) at the University of Texas (hereinafter referred to as CSR-SLR monthly GCM data) as an example, the correlation coefficients of the IMFs decomposed by SVMD were calculated.

[0067] Then, the standard deviation of the correlation coefficient is calculated as the first threshold, and valid IMFs are extracted based on values greater than this threshold.

[0068] Secondly, regardless of whether the IMF corresponding to the anniversary item or the semi-annual item passes the first threshold, it is proposed as a valid IMF;

[0069] Finally, the kurtosis average of the remaining IMFs is calculated and used as the second threshold, and valid IMFs are extracted based on the value less than this threshold. In order to analyze the relationship between kurtosis and noise, simulated data is used to conduct experiments and the kurtosis of the annual signal after adding white noise of different intensities is calculated ( Figure 2 At the same time, in order to verify the adaptability of the dual-threshold progressive extraction criterion proposed in this invention, the sample sequences of different periodic compositions in GCM (Table 1) were extracted, and the standard deviation of the correlation coefficient was used as the first threshold in the extraction process ( Figure 3 ) and the extraction process using the mean value of kurtosis as the second threshold ( Figure 4 Two studies were conducted based on the measured data of CSR-SLR monthly GCM: (1) The IMF energy proportion under different extraction criteria was calculated ( Figure 5 ); (2) The effective IMF extracted by the dual threshold progressive extraction criterion was subjected to fast Fourier transform to obtain the spectrum diagram, which was compared with the main period decomposed by SSA ( Figure 6 ) as well as the period and amplitude of the effective IMF decomposed by SVMD (Table 2) and the period and amplitude of the main cycle decomposed by SSA (Table 3).

[0070] like Figure 2 As shown in Figure 1, the kurtosis of the anniversary signal changes when white noise of different intensities is added. It can be seen that as the intensity of white noise increases, the kurtosis of the sequence also increases accordingly. This means that the more noise a sequence contains, the greater its kurtosis.

[0071] As shown in Table 1, using the partial periodic signals decomposed from the CSR-SLR monthly solution, the CSR-SLR bimonthly solution, and the GRACE-OBP-GFZ (GCM time series calculated by the German Research Centre for Geosciences (GFZ) using the Ocean Bottom Pressure (OBP) model and the Gravity Recovery and Climate Experiment (GRACE) satellite data) as examples, we added white noise of varying intensities to generate nine different combinations of sample sequences. Among them: a ( X 1), a ( X 2), a ( X 3) represents the monthly solution data from CSR-SLR X A sample sequence composed of components, b ( Y 1), b ( Y2), b ( Y 3) Indicates the CSR-SLR bimonthly solution data Y The sample sequence composed of components and c ( Z 1), c ( Z 2), c ( Z 3) Represents GRACE-OBP-GFZ data Z A sample sequence consisting of components.

[0072]

[0073] Figure 3 and Figure 4 The points in the pink box represent the periodic signals decomposed by the SVMD method in the sample sequence. Figure 3 It can be seen that the effective signal is not completely extracted after the first step of screening, and the correlation coefficient of the pure noise mode is very small, close to 0, which shows that the correlation coefficient of the mode dominated by noise is generally small. b ( Y 2) and b ( Y 3) It can be seen that only two periodic signals were extracted from the three periodic signals, and the 183-day (half-year) periodic signal was not decomposed, but only the 190-day signal with a similar frequency was extracted. Therefore, the present invention speculates that b ( Y 2) and b ( Y 3) The half-year periodic signals of these two sample series are replaced by the 190-day signal dominated by noise. To verify this conjecture, this paper calculated and found that the noise energy ratio of the 190-day signal is b ( Y 2) 0.9% of the sample sequence and b ( Y 3) 1.5% of the sample sequence. When the standard deviation of the white noise is reduced to 0.7, the 183-day signal can be decomposed. At this time, the noise energy proportion of the 183-day signal is 0.6%, which is less than the noise energy proportion of the 190-day signal. This verifies the correctness of the speculation and also shows that the SVMD decomposition method is more inclined to decompose the mode with the highest energy near the center frequency.

[0074] Before screening the remaining modes in the second step, the dual-threshold progressive extraction criterion proposed in the present invention has already extracted the annual and semi-annual modes separately. Therefore, modes with greater correlation as well as the annual and semi-annual modes have been extracted as valid modes. Figure 4It can be seen that the effective signals in the sample sequence are all successfully extracted, while some modes dominated by noise are removed. This shows that the dual threshold progressive extraction criterion can effectively extract the effective signals in the sequence. It is worth noting that Figure 3 and Figure 4 Some sample sequences in a ( X 2) a ( X 3) c ( Z 1) c ( Z 2) and c ( Z 3)) 4 known cycles were added, but observation Figure 3 and Figure 4 It can be seen that only three periods are proposed. This is because the first step of the dual-threshold progressive extraction criterion treats annual and semiannual periods as valid periods and extracts them regardless of whether they meet the first step extraction criteria. Therefore, the dual-threshold progressive extraction criterion, combining the prominent characteristics of GCM time series, uses the standard deviation of the correlation coefficient as the first step extraction criterion and the mean kurtosis as the second step extraction criterion, verifying the effectiveness of the dual-threshold progressive extraction criterion.

[0075] Figure 5 The energy proportion of each IMF obtained by SVMD decomposition of the CSR-SLR monthly solution GCM time series and the comparison results of the three extraction criteria are shown. Figure 6 The comparison of the effective IMF extracted by SVMD decomposition and the main period decomposition by SSA based on the dual threshold progressive extraction criterion proposed in this invention is shown. Table 2 and Table 3 show the period and amplitude of these signals. Figure 5 The red ellipse in the figure represents the IMF extracted by the single-threshold criterion of the correlation coefficient, the blue box represents the IMF extracted by the double-threshold criterion of the correlation coefficient, and the pink box represents the IMF extracted by the criterion proposed in this invention. Figure 5 It can be seen that among the three components, although the IMF extracted by the correlation coefficient double threshold criterion retains more than 75% of the sequence information, it fails to extract the prominent features in the GCM, such as X and Z The semiannual cycle of direction was not identified, in which Y The semiannual cycle in the component cannot be effectively decomposed. The single threshold criterion of correlation coefficient discards too many IMFs, resulting in serious information loss. Therefore, these two reference criteria have great defects in extracting IMFs from GCM time series. Figure 5 、 Figure 6As can be seen from Tables 2 and 3, the proposed extraction criterion retains a relatively large number of modes, and the key GCM cycles, namely the annual and semiannual periods, are also effectively preserved. This indicates that the proposed dual-threshold progressive extraction criterion effectively preserves key information, ensuring the integrity of critical information and demonstrating improved applicability.

[0076]

[0077] SSA is widely used and effective in time series processing. Table 3 shows the period and amplitude of the main periods decomposed using SSA. The bolded components in Tables 2 and 3 represent the common signals extracted by both SVMD and SSA. Further analysis shows that 82% of the main periods decomposed by SSA match the main periods extracted by SVMD combined with the dual-threshold progressive extraction criterion. Furthermore, the main mode amplitudes decomposed by SVMD are highly consistent with the main component amplitudes decomposed by SSA, with the difference within 0.4 mm. This demonstrates the feasibility of using the SVMD method combined with the dual-threshold progressive extraction criterion in GCM time series analysis.

[0078] From Table 2, we can find that SVMD can decompose the periodic signal of 570 days (close to 560 days) in all three directions, while SSA can only decompose the periodic signal of 570 days (close to 560 days) in all three directions. Y and Z The periodic signal is separated from the components, but X The components cannot be well separated from the 570-day (close to 560-day) periodic signal. Related research shows that the 570-day (close to 560-day) periodic signal is caused by the repetitiveness of the sun's altitude angle above the LAGEOS (Laser Geodynamics Satellite, LAGEOS) 1 satellite orbital plane, which is the astrodynamic effect of the nodal annual cycle. In order to verify the SSA in X Whether the 570-day periodic signal can be effectively separated from the original GCM time series X Component removal in Table 3 X After the main period of the component is detected, SSA decomposition is performed again. The results show that SSA can well separate the 570-day submillimeter oscillation period signal, and its amplitude is consistent with that of SVMD in X The amplitude of the 570-day signal decomposed from the components is consistent. This further shows that compared with the SSA method, the SVMD method combined with the double-threshold progressive extraction criterion can more effectively decompose the main periodic signal on the three components. Figure 6 It can be seen that the periodic signal extracted by SVMD combined with the dual-threshold progressive extraction criterion is finer than that of SSA.

[0079]

[0080] In summary, the dual-threshold progressive extraction criterion proposed in this paper combines the prominent characteristics of GCM time series, uses the standard deviation of the correlation coefficient as the first threshold, and the mean value of the kurtosis as the second threshold. Through dual-threshold extraction, it can effectively retain the key annual and semi-annual period modes while avoiding excessive information loss, greatly improving the accuracy and reliability of GCM time series analysis, and fully demonstrating the effectiveness and advantages of this criterion in GCM time series analysis.

[0081] The following is an explanation of some terms in this industry:

[0082] 1. Geocentre Motion (GCM): The Earth's center of mass is the mass center of the Earth system, which consists of the solid Earth, oceans, surface water, cryosphere, and atmosphere. Geocentre motion refers to the displacement of the Earth's center of mass (CM) relative to the geometric center of the solid Earth surface (CF), which is mainly affected by changes in the mass load of the atmosphere, oceans, and land water bodies. Using the SVMD method combined with the dual-threshold progressive extraction criterion can help better reveal the GCM's X 、 Y and Z The time-varying characteristics of the direction of the Earth's mass can provide a better insight into the dynamic process of the Earth's mass redistribution.

[0083] 2. Correlation coefficient: The correlation coefficient can reflect the correlation between two sequences. The closer the correlation coefficient is to 1, the higher the correlation is. Conversely, the closer the correlation coefficient is to 0, the lower the correlation is.

[0084] 3. Kurtosis: A quantitative statistical indicator that measures the distribution characteristics of signal random variables.

[0085] 4. Successive Variational Mode Decomposition (SVMD): A signal decomposition method that decomposes a complex signal into a series of modal functions, each of which represents a specific frequency component in the signal.

[0086] 5. Maximum penalty factor: It is an important parameter that needs to be set before using the successive variational mode decomposition signal.

[0087] 6. Variational Mode Decomposition (VMD): This method is a signal decomposition method that iteratively searches for the optimal solution of the variational model to determine the frequency center and bandwidth of each component, effectively separating the signal components.

[0088] 7. Gravity Recovery and Climate Experiment (GRACE): The GRACE gravity satellite provides direct observational means for continuous monitoring of the time-varying Earth gravity field, which can be used to quantify mass redistribution, including terrestrial water storage, polar ice sheets, mountain glaciers, and ocean water mass.

[0089] 8. Satellite Laser Ranging (SLR): This technology uses laser pulses emitted by a ground-based satellite laser ranging system to track and observe artificial Earth satellites equipped with laser-reflecting prisms. SLR satellites have a simple structure and are sensitive to GCMs.

[0090] 9. Single-threshold criterion for correlation coefficient: A threshold is calculated using the maximum and average values of the correlation coefficient. The modes of SVMD decomposition are extracted based on this threshold. Modes greater than this threshold are considered to be signal IMFs containing noise, and modes less than this threshold are considered to be noise-dominated modes.

[0091] 10. Double-threshold criterion for correlation coefficient: Two thresholds, a high threshold and a low threshold, are calculated using the maximum and average values of the correlation coefficient. The modes of SVMD decomposition are extracted based on the high and low thresholds. Modes greater than the high threshold are considered to be signal modes, modes less than the high threshold and greater than the low threshold are considered to be signal modes containing noise, and modes less than the low threshold are considered to be noise-dominated modes.

[0092] 11. Fast Fourier transform: It is a method of transforming signals from the time domain to the frequency domain for analysis.

[0093] 12. Singular Spectrum Analysis (SSA): It is a time series analysis method that aims to extract meaningful components such as trends, cyclical fluctuations and noise from complex time series.

[0094] 13. LAGEOS (Laser Geodynamics Satellite): This laser geodynamics satellite provides a permanent reference point for precise geodynamic measurements of crustal motion, regional strain, fault motion, polar wander, Earth rotation changes, solid Earth tides, and other geodynamic parameters, as well as for the calculation of related dynamic parameters. Furthermore, the high-precision LAGEOS ranging data provided by the satellite's laser ranging system enables the determination of the position of any point on Earth through multi-point positioning and orbital dynamics models.

[0095] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above and that the invention can be embodied in other specific forms without departing from the spirit or essential characteristics of the invention. Therefore, the embodiments should be considered in all respects as illustrative and non-restrictive, and the scope of the invention is defined by the appended claims, not the foregoing description, and all variations within the meaning and range of equivalents of the claims are intended to be included therein. Any reference sign in a claim should not be construed as limiting the claim to which it relates.

[0096] In addition, it should be understood that although this specification is described in terms of implementation methods, not every implementation method contains only one independent technical solution. This narrative method of the specification is only for the sake of clarity. Those skilled in the art should regard the specification as a whole. The technical solutions in each embodiment can also be appropriately combined to form other implementation methods that can be understood by those skilled in the art.

Claims

1. A new method for extracting geocentric motion modes based on successive variational modal decomposition, characterized by: The specific steps include: a: The GCM time series of geocentric motion is decomposed by the successive variational mode decomposition (SVMD) method to obtain the IMF of each mode; b: Based on the modal IMFs decomposed in step a, the effective IMFs in each modal IMF are extracted according to the proposed double-threshold progressive extraction criterion; The dual threshold progressive extraction criterion: ①Calculate the correlation coefficient of each modal IMF , as shown in formula (1), and calculate the standard deviation of each correlation coefficient , as shown in formula (7): （1） Where, is the original GCM time series, is the average of the original signal, is the decomposed mode, is the mode average, Indicates time, N Indicates the number of data; （7） Where, is the number of decomposition modes, is the average of the correlation coefficients, is the correlation coefficient of each modal IMF; ②Extract the correlation coefficient greater than The modes filtered by this threshold represent the modes that are highly correlated with the original time series; ③ Regardless of whether the modal IMF corresponding to the annual signal or the semi-annual signal is higher than the threshold , all of them are retained as valid signals; ④Calculate the kurtosis of the residual mode IMF , as shown in formula (8), and calculate the average value of the residual mode IMF kurtosis: （8） Where, is the standard deviation of the series, is the mean of the series, Represents a sequence; ⑤ Extract the modal IMF with a value less than the average kurtosis as the effective IMF; Step c analyzes the effective IMF extracted in step b.

2. The new method for extracting geocentric motion modes based on successive variational modal decomposition according to claim 1 is characterized in that: Step a includes the following sub-steps: a1. Set the maximum penalty factor based on the characteristics of the GCM time series ; a2.In Under the condition of , the GCM time series is decomposed using the successive variational mode decomposition (SVMD) method to obtain the IMF of each mode.

3. The new method for extracting geocentric motion modes based on successive variational modal decomposition according to claim 1 is characterized in that: Step b includes the following sub-steps: b1. Calculate the correlation coefficient between each modal IMF and the original GCM time series in a2; b2. Calculate the standard deviation of the correlation coefficient between each modal IMF and the original GCM time series in a2 , which is used as the threshold for initial extraction; b3. Extract the correlation coefficient greater than Modal IMF of b4. Check whether the anniversary signal and the semi-annual signal exist in the modal IMF extracted in b3. If not, extract the modal IMF corresponding to the anniversary signal and the semi-annual signal; b5. Calculate the kurtosis of the residual mode IMF; b6. Calculate the average value of the residual mode IMF kurtosis and use it as the threshold for secondary extraction; b7. Based on the remaining modal IMFs, extract the modal IMFs with a kurtosis value less than the mean value.

4. The new method for extracting geocentric motion modes based on successive variational modal decomposition according to claim 1 is characterized in that: Step c includes the following sub-steps: c1. Compare and analyze the effective IMF extracted by the single-threshold and double-threshold criteria of the correlation coefficient; c2. Compare and analyze the main periods decomposed by SSA to verify the advantages and applicability of the extraction method.

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