Novel geocentric motion mode extraction method based on successive variational mode decomposition

Abstract

The invention discloses a new geocentric motion mode extraction method based on successive variational mode decomposition, relates to the technical field of geocentric motion time sequence signal processing, and solves the problem that an existing IMF extraction criterion is insufficient in GCM time sequence analysis based on SVMD. According to the technical scheme, the method specifically comprises the following steps that a GCM time sequence is decomposed through a successive variational mode decomposition (SVMD) method, and IMFs of all modes are obtained through decomposition; and on the basis of each modal IMF decomposed in the step a), according to the proposed double-threshold progressive extraction criterion, the effective IMF in each modal IMF is extracted, and the effect is that the effective IMF can be better extracted by using the double-threshold progressive extraction criterion, and a basis is provided 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 specifically to a new method for extracting geocentric motion modes based on successive variational mode decomposition. Background Art

[0002] Geocentre Motion (GCM) is caused by the mass migration of factors such as the atmosphere, ocean, and terrestrial water, and the GCM time series reflects the changes in geocentric motion, which contains periodic signals driven by multiple factors. In order to extract the periodic signals in the GCM time series, a signal decomposition method needs to be used. SVMD (Successive Variational Mode Decomposition) is an improved method of Variational Mode Decomposition (VMD). During the SVMD decomposition process, only one α needs to be set. max . The algorithm will decompose the mode IMF (Intrinsic Mode Function) under the condition of not exceeding α max , without the need to fix the penalty factor. Therefore, α max is an important parameter that needs to be set according to the signal characteristics before performing SVMD decomposition. To extract the periodic signals in the GCM time series more precisely, a larger α needs to be set through experiments. max . However, for the noisy GCM time series, a larger α max may cause the IMF bandwidth to be too narrow, resulting in too many IMFs and the appearance of noise-dominated IMFs. Therefore, considering the unique properties presented by the geocentric motion time series data, it becomes extremely crucial to explore the IMF extraction criteria adapted to it.

[0003] Currently, when extracting the IMFs of SVMD decomposition, relevant research has proposed corresponding extraction criteria based on data characteristics. When processing ship radiated noise signals, in order to extract the useful signals therein, relevant research has introduced the correlation coefficient double-threshold criterion to extract the effective IMFs of SVMD decomposition and eliminate the noise-dominated modes at the same time; for this problem, some research has also adopted criteria such as amplitude-aware permutation entropy, inverse permutation entropy, and correlation coefficient to extract effective modes; in ultrasonic detection signal processing, relevant research has used the correlation coefficient single-threshold criterion to remove noise modes, significantly improving the signal-to-noise ratio of ultrasonic detection signals.

[0004] Although these guidelines have achieved certain results in their respective application fields, in the time series analysis of GCM, for the IMFs obtained by SVMD decomposition, there are the following three major limitations in relying solely on the existing extraction guidelines mentioned above: 1. Excessive information loss: When using the single threshold criterion of correlation coefficient, due to the relatively high correlation coefficient of individual IMFs, the threshold is prone to be too high, resulting in a large amount of information loss, thus affecting the accuracy of subsequent data analysis and processing; 2. The main signals cannot be effectively extracted: Since the annual and semi-annual cycle signals are important signals in GCM, and the correlation coefficient of the semi-annual cycle signal is too small, using the single threshold or double threshold criterion of correlation coefficient will misidentify the semi-annual cycle as an IMF dominated by noise, resulting in the key signals being erroneously excluded. 3. The determination of the threshold is complex: In actual operation, for the extraction guidelines based on amplitude-aware permutation entropy, inverse permutation entropy, and correlation coefficient, a large number of experiments are required to determine the threshold, and the threshold determination process is relatively complex.

[0005] In summary, to overcome the deficiencies of the existing IMF extraction guidelines in the time series analysis of GCM based on SVMD, it is particularly important to propose an IMF extraction guideline 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 the geocentric motion mode based on successive variational mode decomposition to solve the problems proposed in the above background technology.

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

[0008] A new method for extracting the geocentric motion mode based on successive variational mode decomposition. Based on the IMFs decomposed by SVMD (Successive Variational Mode Decomposition, SVMD), according to the newly proposed double-threshold progressive extraction criterion, the effective IMFs are extracted, and the extracted effective IMFs are analyzed. Specifically, it includes the following steps:

[0009] Step a) Set α max , and use SVMD to decompose the GCM time series to 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, and extract the IMFs greater than this threshold. At the same time, the IMFs corresponding to the annual signal and the semi-annual signal are taken as effective IMFs together; then, calculate the kurtosis of the remaining IMFs, calculate its average value and take it as the second threshold, and extract the IMFs with kurtosis less than this threshold as effective IMFs;

[0011] Compare and analyze the effective IMFs extracted in step c) with the single-threshold and double-threshold criteria of the correlation coefficient, and further compare with the periods decomposed by the 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 processes of different extraction criteria are as follows:

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

[0014] ① Decompose the original GCM time series using SVMD;

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

[0016]

[0017] In formula (1), f(t) is the original GCM time series, is the average of the original signal, imf i (t) is the decomposed mode, is the average of the mode, t represents time, and N represents the number of data.

[0018] ③ Calculate the single-threshold C of the correlation coefficient g , as shown in formula (2).

[0019]

[0020] In the formula, CC max is the maximum value among the correlation coefficients of each IMF, and CC avg is the average value of the correlation coefficients of each IMF.

[0021] For the IMF with a correlation coefficient greater than the threshold C g , it is considered an effective IMF. When the correlation coefficient is less than C g , it is considered an IMF dominated by noise. At the same time, remove the IMF dominated by noise and extract the effective IMF.

[0022] When applying the single-threshold criterion of the correlation coefficient to process each IMF 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 coincidence with the characteristics of the original sequence, and the correlation coefficient is much higher than that of other IMFs, which will raise the threshold C g, resulting in most IMFs with low correlation coefficients being misjudged as noise-dominated modes and being excluded.

[0023] The double-threshold criterion for the correlation coefficient is as follows: Calculate the correlation coefficients of each mode of the original signal decomposed by SVMD, and calculate the high threshold and the low threshold based on the maximum value and the average value of the correlation coefficients.

[0024] ① Decompose the original GCM time series using SVMD;

[0025] ② Calculate the correlation coefficient r of each IMF i , as shown in Equation (1).

[0026] ③ Calculate the high threshold P 1 and the low threshold P 2 , as shown in Equation (3) and Equation (4).

[0027]

[0028]

[0029] In the formula, CC max is the maximum value among the correlation coefficients of each IMF, and CC avg is the average value of the correlation coefficients of each IMF.

[0030] For the IMF with a correlation coefficient greater than the thresholds P 1 and P 2 , it is considered an effective IMF. When the correlation coefficient is less than P 2 , it is considered a noise-dominated IMF. At the same time, remove the noise-dominated IMF and extract the effective IMF.

[0031] When applying the double-threshold criterion of the correlation coefficient to process each IMF of the GCM time series decomposed by SVMD, although the setting of the double threshold can retain most modes, for a main period such as the semi-annual term, the correlation coefficient between the corresponding IMF and the original sequence may be too low, making it unable to pass the low threshold, resulting in the exclusion of this main period as a noise-dominated mode.

[0032] As a further technical solution of the present invention: A new extraction criterion is proposed according to the characteristics of GCM data: the double-threshold progressive extraction method. The extraction process of the double-threshold progressive extraction criterion is as follows:

[0033] ① Decompose the original GCM time series using SVMD;

[0034] SVMD decomposes the original signal into two parts, as shown in Equation (5):

[0035] f(t) = u m (t) + f r (t) (5)

[0036] In the formula, f(t) is the original GCM time series, and u m (t) is the m-th mode among all the decomposed modes, and f r (t) is all the modes except the m-th mode.

[0037] Four criteria are followed during decomposition to form a constrained problem, as shown in formula (6):

[0038]

[0039] Among them, J 1 is the criterion to control each mode to be more compact at its central frequency, and J 2 is the criterion to minimize the spectral overlap between the residual signal f r (t) and the m-th mode, and J 3 is the criterion to avoid the m-th mode decomposed being a repetition of the previous m - 1 decomposed modes. α is the parameter to balance J 1 , J 2 and J 3 . f(t) is the original GCM time series, and u m (t) is the m-th mode among all the decomposed modes, and f r (t) is all the modes except the m-th mode.

[0040] Then, formula (6) is transformed into an unconstrained problem and converted into the frequency form, and iteration is performed to find the optimal mode.

[0041] ② Calculate the correlation coefficient r i of each IMF, as shown in formula (1), and calculate the standard deviation ξ of each correlation coefficient, as shown in formula (7).

[0042]

[0043] In the formula, f(t) is the original GCM time series, is the average of the original signal, and imf i (t) is the decomposed mode, is the average of the mode, t represents time, and N represents the number of data.

[0044]

[0045] Among them, M is the number of decomposed modes, is the average of the correlation coefficients, and r i are the correlation coefficients of each IMF.

[0046] ③Extract the modes with correlation coefficients greater than ξ. The modes selected by this threshold represent the modes highly correlated with the original time series. However, relying solely on the above steps as the extraction criterion, if the correlation coefficient of a certain mode (such as the annual term) is much higher than the mean value, the calculated standard deviation may be too large, resulting in some valid signals being erroneously excluded. Therefore, it is very necessary to further process the remaining IMFs. At the same time, since the annual and semi-annual signals are important periodic signals in the GCM time series, step ④ is required.

[0047] ④Regardless of whether the IMFs corresponding to the annual signal or the semi-annual signal are higher than the threshold ξ, they are all retained as valid signals.

[0048] ⑤Calculate the kurtosis Kurt of the remaining IMFs as shown in formula (8), and calculate the average value of the kurtosis of the remaining IMFs.

[0049]

[0050] where σ is the standard deviation of the sequence, μ is the mean value of the sequence, and x represents the sequence.

[0051] ⑥Extract the IMFs less than the average kurtosis value as the valid IMFs.

[0052] When applying the extraction criterion to each IMF of the GCM time series decomposed by SVMD, since the GCM time series contains an important but low-correlation semi-annual cycle, the above single-threshold and double-threshold criteria of correlation coefficients are likely to ignore this cycle and lead to misjudgment and exclusion. In addition, there is another problem with the single-threshold criterion of correlation coefficients. Since the correlation of individual IMFs is much greater than that of other IMFs, the threshold is too high, resulting in too many IMFs being excluded. The criterion of the present invention fully considers these characteristics. It not only extracts this cycle separately, but also sets the standard deviation of the correlation coefficient and the average value of the kurtosis of the remaining IMFs to retain more IMFs, solving the defects of the single-threshold and double-threshold criteria of correlation coefficients applied to GCM data analysis.

[0053] As a further technical solution of the present invention: a double-threshold progressive extraction criterion, that is, a scheme that combines the prominent features of the GCM time series, uses the standard deviation of the correlation coefficient as the first extraction criterion, and the average value of the kurtosis as the second extraction criterion.

[0054] Compared with the prior art, the beneficial effects of the present invention are:

[0055] Based on the characteristics of the GCM time series, the present invention proposes an appropriate extraction criterion, which solves the defects of the existing extraction criteria in the analysis of GCM time series, including two aspects: main feature extraction and information retention. In terms of main feature extraction, since the double-threshold criterion of correlation coefficient uses the threshold calculated by the correlation coefficient as the judgment basis, important periods with too low correlation coefficients in GCM, such as the semi-annual term, will be mis-eliminated as modes dominated by noise. This extraction criterion fully considers the importance of the annual signal and the semi-annual signal in GCM, and extracts them separately for the corresponding IMFs. In terms of information retention, for each IMF decomposed from the GCM time series based on SVMD, the threshold determined by the existing single-threshold extraction criterion of correlation coefficient is used to extract the decomposed IMF. Since the correlation of some IMFs with the original GCM sequence is much greater than that of the rest of the IMFs, the threshold is raised, resulting in excessive signal loss. Therefore, we use the standard deviation of the correlation coefficient as the first threshold, and considering that only using the standard deviation of the correlation coefficient will also cause the above problems, and since there is a positive correlation between kurtosis and noise, when the noise intensity increases, the kurtosis value rises, so the average kurtosis 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 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 using the double-threshold progressive extraction criterion, providing a basis for subsequent analysis of GCM periodic signals. Description of the Drawings

[0056] Figure 1 is the technical roadmap.

[0057] Figure 2 is the relationship diagram between kurtosis and noise.

[0058] Figure 3 is the extraction process with the standard deviation of the correlation coefficient as the threshold.

[0059] Figure 4 is the extraction process with the average kurtosis as the threshold.

[0060] Figure 5 is the IMF energy proportion diagram based on different extraction criteria.

[0061] Figure 6 is the comparison diagram of the main periods decomposed by SVMD and SSA. Detailed Implementation Modes

[0062] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts belong to the scope of protection of the present invention.

[0063] The content of the present invention mainly includes:

[0064] ① An extraction criterion applicable to GCM time series is invented: the double-threshold progressive extraction criterion. There are certain limitations in the existing extraction criteria applied to GCM time series. Therefore, based on the significant features of GCM, the standard deviation of the correlation coefficient and the average value of kurtosis are used as double thresholds in the present invention. The standard deviation of the correlation coefficient is used as the first threshold, which can screen out the IMFs with high correlation with the original sequence. The average value of kurtosis is used as the second threshold, which can utilize the positive correlation between kurtosis and noise to more efficiently extract the effective IMFs.

[0065] ② The SVMD method is first applied to the decomposition of GCM time series, providing a new perspective for data processing in analyzing GCM and helping to more accurately analyze the periodic signals in GCM.

[0066] The research purpose of this project is to develop a new method for extracting the geocentric motion mode based on successive variational mode decomposition (SVMD) to achieve the accurate extraction of effective IMFs. This method aims to solve three major problems existing in the traditional extraction criteria in GCM data analysis: excessive information loss, inability to effectively extract the important feature signals of GCM, and complex threshold determination process.

[0067] Please refer to Figure 1 , the present invention discloses a new method for extracting the geocentric motion mode based on successive variational mode decomposition,

[0068] First, taking the monthly solution GCM data calculated by the Center for Space Research (CSR) at the University of Texas at Austin, USA, based on Satellite Laser Ranging (SLR) data (abbreviated as CSR-SLR monthly solution GCM data in the present invention) as an example, calculate the correlation coefficients of each IMF decomposed by SVMD;

[0069] Then, calculate the standard deviation of the correlation coefficient as the first threshold, and extract the effective IMFs based on being greater than this threshold;

[0070] Secondly, whether the IMFs corresponding to the annual term or the semi-annual term pass the first threshold or not, they are all proposed as effective IMFs;

[0071] Finally, calculate the average kurtosis of the remaining IMFs and use it as the second threshold, and extract the effective IMFs based on those less than this threshold. To analyze the relationship between kurtosis and noise, simulation data was used for experiments, and the kurtosis of the annual signal after adding white noise of different intensities was calculated ( Figure 2 ). At the same time, to verify the adaptability of the dual-threshold progressive extraction criterion proposed in the present invention, the sample sequences (Table 1) with different cycles in the GCM were extracted, with the standard deviation of the correlation coefficient as the extraction process of the first threshold ( Figure 3 ) and the average kurtosis as the extraction process of the second threshold ( Figure 4 ). Two studies were carried out based on the measured data of the CSR-SLR monthly solution GCM: (1) Calculate the proportion of IMF energy under different extraction criteria ( Figure 5 ); (2) Perform a fast Fourier transform on the effective IMFs extracted by the dual-threshold progressive extraction criterion to obtain a spectrogram, and compare it with the main periods decomposed by SSA ( Figure 6 ) and the periods and amplitudes of the effective IMFs decomposed by SVMD (Table 2) and the periods and amplitudes of the main periods decomposed by SSA (Table 3).

[0072] As Figure 2 shown, it shows the kurtosis change of the annual signal with the addition of white noise of different intensities. It can be seen that as the intensity of white noise increases, the kurtosis of the sequence also increases accordingly. This shows that the more noise contained in the sequence, the greater its kurtosis.

[0073] As shown in Table 1, taking the partial periodic signals decomposed by CSR-SLR monthly solution, CSR-SLR bimonthly solution, and GRACE-OBP-GFZ (GCM time series solved by the German Research Centre for Geosciences, GFZ, combining Ocean Bottom Pressure, OBP, and Gravity Recovery and Climate Experiment, GRACE) data as examples, white noise of different intensities was added to generate 9 different combinations of sample sequences. Among them: a(X1), a(X2), a(X3) represent the sample sequences composed of the X component of the CSR-SLR monthly solution data, b(Y1), b(Y2), b(Y3) represent the sample sequences composed of the Y component of the CSR-SLR bimonthly solution data, and c(Z1), c(Z2), c(Z3) represent the sample sequences composed of the Z component of the GRACE-OBP-GFZ data.

[0074] Table 1 Sample sequences (d represents days)

[0075]

[0076] Figure 3 and Figure 4 The points in the pink boxes indicate the periodic signals decomposed by the SVMD method in the sample sequence. From Figure 3 it can be seen that after the first-step screening, the effective signals are not completely extracted, and the correlation coefficients of the pure noise modes are very small, close to 0, indicating that the correlation coefficients of the modes dominated by noise are generally small. At the same time, from b(Y2) and b(Y3), it can be seen that only two of the three periodic signals are extracted, and the 183-day (half-year) periodic signal is not decomposed, but only the 190-day signal with a similar frequency is extracted. Therefore, the present invention speculates that the half-year periodic signals of the two groups of sample sequences b(Y2) and b(Y3) are replaced by the 190-day signal dominated by noise. To verify this speculation, it is calculated in this paper that the noise energy ratios of the 190-day signal are 0.9% of the b(Y2) sample sequence and 1.5% of the b(Y3) sample sequence respectively. When the standard deviation of white noise is reduced to 0.7, the 183-day signal can be decomposed. At this time, the noise energy ratio of the 183-day signal is 0.6%, which is less than the noise energy ratio of the 190-day signal, verifying the correctness of the speculation and indicating that the SVMD decomposition method is more inclined to decompose the mode with the largest energy near the central frequency.

[0077] Before screening the remaining modes in the second step, the double-threshold progressive extraction criterion proposed by the present invention has separately extracted the annual and half-year periodic modes. Therefore, the modes with relatively large correlations and the annual and half-year periodic modes have been extracted as effective modes. Figure 4 It can be seen that the effective signals in the sample sequence are all successfully extracted, and at the same time, some modes dominated by noise are removed. This shows that the double-threshold progressive extraction criterion can effectively extract the effective signals in the sequence. It should be noted that Figure 3 and Figure 4 4 known periods are added to some sample sequences (a(X2), a(X3), c(Z1), c(Z2) and c(Z3)) in Figure 3 and Figure 4 it can be found that only 3 periods are extracted. This is because in the first-step criterion extraction process of the double-threshold progressive extraction criterion, whether the annual period and the half-year period meet the first-step extraction criterion or not, they will be regarded as effective periods and extracted. Therefore, the double-threshold progressive extraction criterion: 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 kurtosis as the second-step extraction criterion, verifies the effectiveness of the double-threshold progressive extraction criterion.

[0078] Figure 5It shows the energy proportion of each IMF obtained by decomposing the CSR - SLR monthly solution GCM time series using SVMD, as well as the comparison results of three extraction criteria. Figure 6 It shows the comparison between the effective IMFs decomposed and extracted using SVMD based on the dual - threshold progressive extraction criterion proposed in the present invention and the main periods decomposed by SSA. Tables 2 and 3 show the periods and amplitudes of these signals. Among them, Figure 5 the red ellipses in represent the IMFs extracted by the single - threshold criterion of correlation coefficient, the blue boxes represent the IMFs extracted by the dual - threshold criterion of correlation coefficient, and the pink boxes represent the IMFs extracted by the criterion proposed in the present invention. From Figure 5 it can be seen that among the three components, although the IMF extracted by the dual - threshold criterion of correlation coefficient retains more than 75% of the sequence information, it fails to extract the prominent features in the GCM. For example, the semi - annual periods in the X and Z directions cannot be identified, and among them, the semi - annual period in the Y component cannot be effectively decomposed. The single - threshold criterion of correlation coefficient discards too many IMFs, resulting in serious information loss. Therefore, these two borrowed criteria have great defects in the extraction of GCM time series IMFs. And from Figure 5 、 Figure 6 Tables 2 and 3, it can be seen that using the extraction criterion of the present invention retains more modes, and the main periods in the GCM, namely the annual and semi - annual periods, are effectively retained. This means that the dual - threshold progressive extraction criterion proposed in the present invention can effectively retain the main information, ensure the integrity of the key information, and demonstrate better applicability.

[0079] Table 2 Effective IMFs Decomposed by SVMD

[0080]

[0081]

[0082] SSA is widely used and has good effects when dealing with time series. Table 3 shows the periods and amplitudes of the main periods decomposed by SSA. The bold parts in Tables 2 and 3 represent the common signals extracted by the two methods of SVMD and SSA. Further analysis shows that among the main periods decomposed by SSA, 82% of the signals can match the main periods extracted by combining the dual - threshold progressive extraction criterion with SVMD. At the same time, the amplitudes of the main modes decomposed by SVMD are highly consistent with the amplitudes of the main components decomposed by SSA, and the difference is controlled within 0.4 mm. This indicates that it is feasible to apply the SVMD method combined with the dual - threshold progressive extraction criterion in the analysis of GCM time series.

[0083] As can be seen from Table 2, SVMD can decompose periodic signals with a period of 570 days (close to 560 days) in all three directions, while SSA can only separate such periodic signals in the Y and Z components, but the X component cannot well separate the periodic signals with a period of 570 days (close to 560 days). Related research shows that the periodic signal with a period of 570 days (close to 560 days) is the annual intersection celestial dynamics effect caused by the repeatability of the solar altitude angle above the orbital plane of LAGEOS (Laser Geodynamics Satellite) 1 satellite. To verify whether SSA can effectively separate the periodic signal with a period of 570 days in the X component, after removing the main periods shown in the X component of Table 3 from the X component of the original GCM time series, SSA decomposition was performed again. The results show that SSA can better separate the submillimeter oscillation periodic signal with a period of 570 days, and its amplitude is consistent with the amplitude of the 570-day signal decomposed by SVMD in the X component. 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 signals in the three components. At the same time, from Figure 6 it can be seen that the periodic signals extracted by SVMD combined with the double-threshold progressive extraction criterion are finer than those of SSA.

[0084] Table 3 Main periods decomposed by SSA

[0085]

[0086] In summary, the double-threshold progressive extraction criterion proposed in the present invention: combining the prominent features of the GCM time series, using the standard deviation of the correlation coefficient as the first threshold and the average value of kurtosis as the second threshold, through the extraction of double thresholds, can effectively retain the annual and semi-annual key periodic modes, while avoiding excessive loss of information, 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.

[0087] The following are some explanations of technical terms in this industry:

[0088] 1. Geocentre Motion (GCM): The geocenter of the Earth is the mass center of the Earth system composed of the solid Earth, oceans, surface water, cryosphere, and atmosphere. Geocentre motion refers to the displacement of the geocenter (CM) of the Earth relative to the geometric center (CF) of the solid Earth's surface, mainly affected by changes in the mass loads of the atmosphere, oceans, and terrestrial water bodies. Using the SVMD method combined with the double-threshold progressive extraction criterion helps to better reveal the time-varying characteristics of GCM in the X, Y, and Z directions, thus better understanding the dynamic process of the Earth's mass redistribution.

[0089] 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. Conversely, the closer the correlation coefficient is to 0, the lower the correlation.

[0090] 3. Kurtosis: A quantitative statistical index used to measure the distribution characteristics of signal random variables.

[0091] 4. Successive Variational Mode Decomposition (SVMD): A method for signal decomposition. It can decompose a complex signal into a series of mode functions, and each mode function represents a specific frequency component in the signal.

[0092] 5. Maximum penalty factor: An important parameter that needs to be set before using successive variational mode decomposition for signals.

[0093] 6. Variational Mode Decomposition (VMD): A method for signal decomposition. In the process of obtaining decomposition components, this method determines the frequency center and bandwidth of each component by iteratively searching for the optimal solution of the variational model, so as to effectively separate the components of the signal.

[0094] 7. Gravity Recovery and Climate Experiment (GRACE) satellite: The GRACE gravity satellite provides a direct observation means for continuously monitoring the time-varying Earth's gravity field, and the time-varying Earth's gravity field can be used to quantify mass redistribution such as terrestrial water storage, polar ice sheets, mountain glaciers, and seawater mass.

[0095] 8. Satellite Laser Ranging (SLR): It uses the laser pulses emitted by the satellite laser ranging system installed on the ground to track and observe artificial Earth satellites equipped with laser retroreflectors. The SLR satellite has a simple structure and is sensitive to GCM.

[0096] 9. Correlation coefficient single-threshold criterion: Calculate a threshold using the maximum value and average value of the correlation coefficient. Based on this threshold, extract each mode decomposed by SVMD. Modes greater than this threshold are considered signal IMFs containing noise, and modes less than this threshold are considered noise-dominated modes.

[0097] 10. Correlation coefficient double-threshold criterion: Two thresholds, a high threshold and a low threshold, are calculated using the maximum value and the average value of the correlation coefficient. Based on the high and low thresholds, each mode decomposed by SVMD is extracted. Modes greater than the high threshold are considered signal modes, modes less than the high threshold and greater than the low threshold are considered signal modes containing noise, and modes less than the low threshold are considered noise-dominated modes.

[0098] 11. Fast Fourier Transform: It is a method for analyzing signals by transforming them from the time domain to the frequency domain.

[0099] 12. Singular Spectrum Analysis (SSA): It is a time series analysis method aimed at extracting meaningful components from complex time series, such as trends, periodic fluctuations, and noise.

[0100] 13. LAGEOS (Laser Geodynamics Satellite) satellite: A laser geodynamics satellite that provides a permanent reference point for precise geodynamic measurements of crustal movement, regional strain, fault movement, polar motion, changes in the Earth's rotation, solid Earth tides, etc., and for solving related dynamic parameters. In addition, due to the high-precision LAGEOS ranging data provided by the satellite laser ranging system, the positions of points on the Earth can be determined through multi-point positioning and orbit dynamics models.

[0101] For those skilled in the art, it is obvious that the present invention is not limited to the details of the above exemplary embodiments, and can be implemented in other specific forms without departing from the spirit or basic characteristics of the present invention. Therefore, from any point of view, the embodiments should be regarded as exemplary and non-restrictive. The scope of the present invention is defined by the appended claims rather than the above description. Therefore, all changes falling within the meaning and scope of the equivalent elements of the claims are intended to be encompassed within the present invention. Any reference signs in the claims should not be regarded as limiting the claims involved.

[0102] In addition, it should be understood that although this specification is described according to embodiments, not every embodiment only contains an independent technical solution. This narrative way of the specification is only for clarity. Those skilled in the art should regard the specification as a whole, and the technical solutions in each embodiment can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.

Claims

1. A new method for extracting geocentric motion modes based on successive variational mode decomposition, characterized in that: The specific steps include: a) Decompose the GCM time series of geocentric motion by successive variational mode decomposition (SVMD) method to obtain the IMF of each mode; b) Based on the modal IMFs decomposed in 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: ① Calculate the correlation coefficient r of each modal IMF i , as shown in formula (1), and calculate the standard deviation ξ of each correlation coefficient, as shown in formula (7): Where f(t) is the original GCM time series, f is the average of the original signal, and imf is i (t) is the decomposed mode, is the average of the modes, t represents time, and N represents the number of data; Where M is the number of decomposed modes, r is the average of the correlation coefficients, and r i is the correlation coefficient of each modal IMF; ② Extract the modes with correlation coefficient greater than ξ. The modes selected 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 ξ, it will be retained as a valid signal; ④ Calculate the kurtosis Kurt of the residual mode IMF, as shown in formula (8), and calculate the average value of the kurtosis of the residual mode IMF: In the formula, σ is the standard deviation of the sequence, μ is the mean of the sequence, and x represents the sequence; ⑤ Extract the modal IMF that is less than the average kurtosis as the effective IMF; c) Analyze the effective IMF extracted in b).

2. According to claim 1, a new method for extracting geocentric motion modes based on successive variational mode decomposition is characterized in that: Step a includes the following sub-steps: a1. According to the characteristics of the GCM time series, set the maximum penalty factor α max ; a2. In α max 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. According to claim 1, a new method for extracting geocentric motion modes based on successive variational mode decomposition 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 ξ of the correlation coefficient between each modal IMF and the original GCM time series in a2, and use it as the threshold for initial extraction; b3. Extract the modal IMF with correlation coefficient greater than ξ; 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 that are less than the mean kurtosis.

4. The new method for extracting geocentric motion modes based on successive variational mode 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 correlation coefficient single threshold and double threshold criteria; c2. The main periods are decomposed by SSA and compared with those by Singular Spectrum Analysis (SSA) to verify the advantages and applicability of the extraction method.

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