GNSS deformation monitoring denoising method based on improved CEEMDAN and fuzzy permutation entropy

Abstract

The invention discloses a GNSS deformation monitoring denoising method based on improved CEEMDAN and fuzzy permutation entropy, and relates to the technical field of mode decomposition denoising. Comprising the following steps: data acquisition: acquiring original GNSS deformation monitoring data which comprises position quantities in N, E and U directions, comes from a plurality of GNSS observation stations and comprises a plurality of interference noises; and CEEMDAN decomposition: performing complete ensemble empirical mode decomposition on the original GNSS deformation monitoring data to obtain a plurality of intrinsic mode function IMF components. According to the method, the CEEMDAN is improved to optimize the noise intensity coefficient and the number of decomposition times, so that the modal aliasing phenomenon of a traditional CEEMDAN method is effectively reduced, and IMF component redundancy is reduced; the complexity and the noise ratio of each IMF component are quantified by combining fuzzy permutation entropy, high and low frequency modality accurate classification is realized by matching with a dynamic threshold mechanism, and the problems that a traditional method lacks an effective noise recognition mechanism and depends on experience to set a classification standard are solved.

Description

Technical Field

[0001] This invention relates to the field of modal decomposition-based denoising technology, specifically a GNSS deformation monitoring denoising method based on improved CEEMDAN and fuzzy permutation entropy. Background Technology

[0002] With the rapid development of urbanization and infrastructure construction, geological disaster monitoring, slope stability analysis, and large-scale structural health management have placed higher demands on the accuracy and real-time performance of GNSS deformation monitoring. GNSS technology, due to its advantages of all-weather operation, wide coverage, and high precision, has been widely applied in the field of deformation monitoring. However, in actual observation processes, GNSS displacement data is often affected by various factors, including ionospheric delay, multipath effects, receiver noise, and ambient temperature fluctuations. These interferences result in a large amount of noise in the original observation signal, severely impacting the reliability of the data and subsequent analysis results.

[0003] Currently, denoising of GNSS deformation monitoring data commonly employs signal decomposition methods such as Empirical Mode Decomposition (EMD), Ensemble Empirical Mode Decomposition (EEMD), and Complete Ensemble Empirical Mode Decomposition (CEEMDAN). These methods can decompose complex non-stationary and nonlinear signals into multiple intrinsic mode functions (IMFs) of different frequencies, which helps to distinguish between signals and noise to some extent. However, the traditional CEEMDAN method still faces several challenges in practical GNSS data processing. For example, mode aliasing is prone to occur during mode decomposition, making it difficult to clearly distinguish between useful and noise signals; the number of decomposed IMF components is redundant, making it difficult to effectively determine which should be retained and which should be removed, often relying on subjective settings based on empirical rules; furthermore, current methods typically lack built-in noise identification mechanisms and parameter adaptive functions, resulting in denoising results that depend on initial settings, leading to poor stability and versatility. Fuzzy Permutation Entropy (FPE), as a nonlinear dynamic index for measuring signal complexity, has been applied in various signal anomaly detection and feature recognition scenarios in recent years. It can be used to analyze the complexity of IMF components, thereby providing a quantitative basis for signal mode classification. However, current technologies have not effectively integrated the FPE index with mode decomposition methods such as CEEMDAN, and there is still a lack of an integrated mode classification and denoising method based on complexity discrimination. At the same time, current denoising methods usually do not introduce error evaluation and closed-loop feedback mechanisms for denoising results, making it difficult to dynamically optimize processing parameters. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this invention provides a GNSS deformation monitoring denoising method based on an improved CEEMDAN and fuzzy permutation entropy. This method solves the problems in existing technologies, such as the easy occurrence of mode aliasing and IMF component redundancy in traditional CEEMDAN, the lack of an effective noise identification mechanism and closed-loop feedback, and the reliance on empirical settings for denoising results.

[0005] To achieve the above objectives, the present invention provides the following technical solution: a GNSS deformation monitoring denoising method based on improved CEEMDAN and fuzzy permutation entropy, the method comprising: S1. Data Acquisition: Acquire raw GNSS deformation monitoring data, which includes positional quantities in the N, E, and U directions. The data comes from multiple GNSS observation stations and contains various types of interference noise. S2, CEEMDAN decomposition: The original GNSS deformation monitoring data is subjected to complete ensemble empirical mode decomposition to obtain multiple intrinsic mode function (IMF) components. The decomposition process separates the low-frequency trend term and high-frequency noise component in the data by optimizing the noise intensity coefficient and the number of decompositions. S3. Fuzzy permutation entropy calculation: Calculate the fuzzy permutation entropy value for each IMF component. The entropy value is used to quantify the complexity and noise ratio of the component. S4. High and low frequency mode classification: Based on the fuzzy permutation entropy value, a dynamic threshold mechanism is used to divide the IMF components into high frequency mode components and low frequency mode components. The high frequency mode contains noise signals, while the low frequency mode retains the original signal trend components. S5. Bayesian wavelet denoising: Bayesian wavelet denoising is performed on high-frequency modal components. The denoising process combines wavelet analysis and Bayesian theory to remove noise. S6. Signal Reconstruction and Error Assessment: The high-frequency mode components and low-frequency mode components after Bayesian wavelet denoising are superimposed and reconstructed to obtain the purified GNSS deformation monitoring signal. The denoising effect is assessed by mean square error and signal-to-noise ratio. S7. Feedback and optimization: If the denoising effect evaluation result does not reach the preset performance threshold, the number of components of the complete set empirical mode decomposition, the fuzzy permutation entropy discrimination criterion, or the threshold function parameters of Bayesian wavelet denoising are adjusted according to the evaluation result.

[0006] Furthermore, the CEEMDAN decomposition includes: Based on the additive noise model, the original GNSS deformation monitoring data is perturbed multiple times by adding Gaussian white noise to generate the multiple IMF components. The IMF components obtained from each perturbation decomposition are subjected to multiple iterations of averaging to obtain the final stable IMF component results. IMF components are arranged from high to low frequency and include high-frequency disturbance terms, signal trend terms, and mixed components. In the process of complete set empirical mode decomposition, a noise amplitude factor and a stopping criterion are introduced to control the decomposition depth.

[0007] Furthermore, the calculation of the fuzzy permutation entropy includes: The IMF components are divided into time windows, and a fixed-dimensional delay vector sequence is extracted from each window; The similarity between time-delay vectors is calculated using fuzzy membership functions to generate the fuzzy permutation entropy value; A higher fuzzy permutation entropy value indicates a stronger signal complexity, while a lower fuzzy permutation entropy value indicates that the components have a trend. The high-frequency modal components and the low-frequency modal components can be distinguished by setting a critical range for fuzzy permutation entropy or by using a clustering algorithm to determine an adaptive threshold.

[0008] Furthermore, the high- and low-frequency mode classification includes: Statistical analysis was performed on the fuzzy permutation entropy values ​​of all IMF components, including the extraction of mean, variance, and range parameters. The threshold function is constructed based on statistical analysis results and is adaptively adjusted for different datasets; Components above the threshold are defined as high-frequency modal components, and components below the threshold are defined as low-frequency modal components. The dynamic classification algorithm supports a sliding window mechanism, which adjusts the judgment criteria in real time according to changes in the input signal; The classification results are used to determine which modes are included in the Bayesian wavelet denoising process.

[0009] Furthermore, the Bayesian wavelet denoising process includes: High-frequency modal components undergo discrete wavelet transform to convert the time-domain signal into a wavelet coefficient sequence; Wavelet coefficients are denoised using soft or hard thresholding based on the Bayesian thresholding function; The Bayesian threshold function is constructed based on the maximum a posteriori probability estimation and optimized by combining the signal prior and the statistical characteristics of noise. The wavelet basis can be any of the Daubechies, Symlet, or Coiflet wavelet basis, and the number of wavelet decomposition layers is dynamically set according to the spectral energy distribution. The wavelet coefficients after Bayesian wavelet denoising are subjected to inverse wavelet transform to recover the time-domain denoised signal.

[0010] Furthermore, the signal reconstruction, error assessment, and feedback tuning include: The low-frequency modal components are superimposed and reconstructed with the high-frequency modal components after Bayesian wavelet denoising to obtain the purified GNSS deformation monitoring signal. The error between the purified GNSS deformation monitoring signal and the original GNSS deformation monitoring data is calculated by mean square error, and the denoising effect is judged by combining the signal-to-noise ratio, correlation coefficient, and maximum error evaluation index. If the denoising effect evaluation result does not reach the preset performance threshold, the system will automatically start the feedback tuning mechanism and dynamically adjust the number of components of the complete set empirical mode decomposition, the fuzzy permutation entropy discrimination standard, or the threshold function parameters of Bayesian wavelet denoising according to the evaluation result. The feedback and optimization mechanism supports point-by-point optimization at the GNSS station level.

[0011] Furthermore, the optimization adjustment of the noise intensity coefficient and the number of decompositions during the complete set empirical mode decomposition process includes: Preset the initial range of noise intensity coefficient and number of decompositions; Based on the error assessment results, an iterative search is used to adjust the noise intensity coefficient and the number of decompositions. The adjustment process aims to reduce mode aliasing and reduce redundant components. The optimization and adjustment process is performed each time feedback is received.

[0012] Furthermore, the dynamic threshold mechanism based on fuzzy permutation entropy values ​​includes: The critical range of fuzzy permutation entropy was determined by statistical analysis of the FPE distribution of historical GNSS deformation monitoring data; The adaptive threshold uses K-means clustering or Gaussian mixture model to cluster the FPE values ​​of IMF components to identify noise component clusters and signal component clusters, and determines the threshold for separating high-frequency mode components and low-frequency mode components based on the clustering results. The dynamic threshold mechanism is adjusted individually for data processing at different GNSS observation stations or in different time periods.

[0013] Furthermore, the construction and optimization of the Bayesian threshold function includes: The Bayesian threshold function is constructed using the maximum a posteriori probability estimation method, combining the prior probability distribution of wavelet coefficients with the statistical characteristics of noise. The prior probability distribution of the wavelet coefficients is modeled using a generalized Gaussian distribution or a Laplace distribution. The statistical characteristics of the noise are obtained by estimating the noise in the high-frequency modal components, and the estimation method adopts the median absolute deviation method. The optimization of the Bayesian threshold function parameters is achieved by minimizing the estimation risk during the iterative denoising process. The risk assessment combines the spectral characteristics of the original signal and the noise.

[0014] Furthermore, the triggering conditions and parameter adjustments for the feedback optimization are further refined, including: When the mean square error is higher than the preset upper limit of error or the signal-to-noise ratio is lower than the preset lower limit of signal-to-noise ratio, the feedback tuning mechanism is triggered. The range of dynamically adjusted parameters is limited to a preset effective range. The parameters include the additional noise intensity coefficient of CEEMDAN decomposition, the delay vector dimension of fuzzy permutation entropy, and the number of decomposition layers for Bayesian wavelet denoising. Optimize point by point at the GNSS station level, and adjust the corresponding CEEMDAN decomposition parameters or Bayesian wavelet denoising parameters individually based on the denoising effect evaluation results of each station. The feedback optimization mechanism is executed at the end of each denoising processing cycle.

[0015] Compared with the prior art, the beneficial effects of the present invention are as follows: This invention improves the noise intensity coefficient and decomposition number of CEEMDAN, effectively reducing mode aliasing and IMF component redundancy in the traditional CEEMDAN method. It combines fuzzy permutation entropy quantification of the complexity and noise proportion of each IMF component with a dynamic threshold mechanism to achieve accurate high- and low-frequency mode classification, addressing the lack of an effective noise identification mechanism and reliance on empirically set classification criteria in traditional methods. For high-frequency modes, Bayesian wavelet denoising is employed, combining wavelet analysis and Bayesian theory to retain effective information in high-frequency signals while removing noise. A feedback tuning mechanism is introduced to dynamically adjust the number of decomposition components, fuzzy permutation entropy discrimination criteria, and Bayesian wavelet denoising parameters based on error assessment results, overcoming the shortcomings of traditional methods such as lack of closed-loop optimization and reliance on initial settings for denoising results. Overall, this significantly improves the denoising accuracy, stability, and adaptability of GNSS deformation monitoring data, ensuring data reliability in subsequent applications such as geological disaster early warning and structural health management. Attached Figure Description

[0016] Figure 1 This is a flowchart of the method of the present invention; Figure 2 This is a flowchart of the CEEMDAN decomposition and optimization process of the present invention; Figure 3 This is a flowchart of the feedback optimization mechanism of the present invention. Detailed Implementation

[0017] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0018] Please see Figure 1-3This invention provides a GNSS deformation monitoring denoising method based on improved CEEMDAN and fuzzy permutation entropy, the method comprising: S1. Data Acquisition: Acquire raw GNSS deformation monitoring data, which includes positional quantities in the N, E, and U directions. The data comes from multiple GNSS observation stations and contains various types of interference noise. S2, CEEMDAN decomposition: The original GNSS deformation monitoring data is subjected to complete ensemble empirical mode decomposition to obtain multiple intrinsic mode function (IMF) components. The decomposition process separates the low-frequency trend term and high-frequency noise component in the data by optimizing the noise intensity coefficient and the number of decompositions. S3. Fuzzy permutation entropy calculation: Calculate the fuzzy permutation entropy value for each IMF component. The entropy value is used to quantify the complexity and noise ratio of the component. S4. High and low frequency mode classification: Based on the fuzzy permutation entropy value, a dynamic threshold mechanism is used to divide the IMF components into high frequency mode components and low frequency mode components. The high frequency mode contains noise signals, while the low frequency mode retains the original signal trend components. S5. Bayesian wavelet denoising: Bayesian wavelet denoising is performed on high-frequency modal components. The denoising process combines wavelet analysis and Bayesian theory to remove noise. S6. Signal Reconstruction and Error Assessment: The high-frequency mode components and low-frequency mode components after Bayesian wavelet denoising are superimposed and reconstructed to obtain the purified GNSS deformation monitoring signal. The denoising effect is assessed by mean square error and signal-to-noise ratio. S7. Feedback and optimization: If the denoising effect evaluation result does not reach the preset performance threshold, the number of components of the complete set empirical mode decomposition, the fuzzy permutation entropy discrimination criterion, or the threshold function parameters of Bayesian wavelet denoising are adjusted according to the evaluation result.

[0019] Specifically, multiple GNSS observation stations deployed in a geological disaster monitoring area were selected to acquire raw deformation monitoring data in the N, E, and U directions over a continuous period of time for each station. The data included ionospheric delay, multipath effects, receiver noise, and noise components caused by environmental interference. CEEMDAN decomposition was performed on the raw data. Initial ranges for the noise intensity coefficients and the number of decomposition iterations were preset. Based on the error assessment results, these two parameters were adjusted through iterative search. The goal during the process was to reduce mode mixing and redundant IMF components, ultimately obtaining multiple IMF components and achieving effective separation of low-frequency trend terms from high-frequency noise components in the data.

[0020] For each IMF component, a fixed-length time window is defined. Within each window, a fixed-dimensional sequence of delayed vectors is extracted. The similarity between these delayed vectors is calculated using a fuzzy membership function. Finally, the fuzzy permutation entropy value for each IMF component is calculated using a formula. The fuzzy membership function formula is as follows: , In the formula and These are the j-th and k-th delay vectors, respectively. Let be the Euclidean distance between the two vectors. The fuzziness coefficient is determined based on the data standard deviation; the formula for calculating the fuzzy permutation entropy is: , In the formula The fuzzy probability of the i-th delay vector arrangement is obtained by normalizing the membership degree.

[0021] Based on the fuzzy permutation entropy values ​​of each IMF component, a dynamic thresholding mechanism is used to separate high-frequency mode components (HFMCs) from low-frequency mode components (LMCs). HFMCs primarily contain noise signals, while LMCs retain the trend components of the original signal. Bayesian wavelet denoising is performed on the HFMCs. First, a discrete wavelet transform is used to convert the time-domain signal into a wavelet coefficient sequence. The wavelet basis is the Daubechies series, and the decomposition level is dynamically set according to the spectral energy distribution of the HFMCs. Then, a Bayesian thresholding function is constructed based on the maximum a posteriori probability estimation to perform soft thresholding on the wavelet coefficients. The soft thresholding formula is: , In the formula These are the original wavelet coefficients. For Bayesian threshold, The sign function is used; finally, the inverse wavelet transform is performed on the processed wavelet coefficients to recover the time-domain denoised signal.

[0022] The denoised high-frequency mode components are superimposed with the low-frequency mode components to reconstruct the purified GNSS deformation monitoring signal. The denoising effect is evaluated using mean square error and signal-to-noise ratio. If the evaluation result does not reach the preset performance threshold, the number of CEEMDAN decomposition components, the fuzzy permutation entropy discrimination criterion, or the threshold function parameters for Bayesian wavelet denoising are adjusted based on the evaluation results. This implementation effectively solves the problems of mode aliasing, reliance on empirical parameter settings, and lack of feedback optimization in traditional methods, improving denoising stability and accuracy.

[0023] In this embodiment, the CEEMDAN decomposition includes: Based on the additive noise model, multiple IMF components are generated by adding Gaussian white noise to the original GNSS deformation monitoring data to achieve multiple data perturbations. The IMF components obtained from each perturbation decomposition are subjected to multiple iterations of averaging to obtain the final stable IMF component results. IMF components are arranged from high to low frequency and include high-frequency disturbance terms, signal trend terms, and mixed components. In the process of complete set empirical mode decomposition, a noise amplitude factor and a stopping criterion are introduced to control the decomposition depth.

[0024] Specifically, when performing CEEMDAN decomposition on the acquired raw GNSS deformation monitoring data, a disturbance signal is first constructed based on the additive noise model. The formula for the disturbance signal is: , In the formula The signal after the disturbance. This is the original GNSS deformation monitoring signal. It is Gaussian white noise and satisfies , The noise variance is determined based on the fluctuation characteristics of the original data. Multiple perturbations are achieved by adding different types of Gaussian white noise. After each perturbation, empirical mode decomposition (IMF) is performed on the signal to obtain the corresponding IMF components.

[0025] For each IMF component obtained from perturbation decomposition, perform iterative averaging. The iterative calculation formula for the i-th IMF component is: , In the formula The intermediate result of the i-th iteration is given by N, which is the length of the time window. Through multiple iterations, the IMF components tend to stabilize, and finally a stable set of IMF components is obtained.

[0026] The obtained IMF components are arranged from high to low frequency, with the high-frequency components corresponding to disturbance noise in the data, the low-frequency components corresponding to the trend components of the original signal, and the intermediate components being mixed components. A noise amplitude factor is introduced during the decomposition process to control the intensity of the added noise at each step, and a stopping criterion is set to control the decomposition depth. The stopping criterion formula is: , In the formula, M represents the number of IMF components obtained from the current decomposition. The maximum value among all current IMF components, when Decomposition stops when the value is below a preset threshold. This implementation method effectively avoids pseudo-modal phenomena and improves the decomposition quality and stability of IMF components.

[0027] In this embodiment, the calculation of fuzzy permutation entropy includes: The IMF components are divided into time windows, and a fixed-dimensional delay vector sequence is extracted from each window; The similarity between time-delayed vectors is calculated using fuzzy membership functions to generate fuzzy permutation entropy values. A higher fuzzy permutation entropy value indicates a stronger signal complexity, while a lower fuzzy permutation entropy value indicates that the components have a trend. High-frequency modal components can be distinguished from low-frequency modal components by setting a critical range for fuzzy permutation entropy or by using a clustering algorithm to determine an adaptive threshold.

[0028] Specifically, for each IMF component obtained from CEEMDAN decomposition, it is divided into multiple time windows of fixed length. Within each time window, a sequence of delayed vectors of fixed dimensions is extracted. The expression for the j-th delayed vector is: In the formula, m is the embedding dimension. The delay time is determined based on the time resolution and signal characteristics of the IMF components. This represents the value of the IMF component at the corresponding time.

[0029] The similarity between any two delay vectors is calculated using fuzzy membership functions. The formula for the fuzzy membership function is as follows: , In the formula Let r be the Euclidean distance between the j-th and k-th delay vectors, and r be the ambiguity coefficient. , Let be the sample standard deviation of the current IMF component. Based on the similarity of all delay vector pairs, calculate the fuzzy probability of each delay vector permutation. , In the formula, Z is the normalization factor. Let be the membership degree of the j-th delay vector in the i-th permutation.

[0030] Through formula The fuzzy permutation entropy value of each IMF component is calculated, where N is the number of all possible permutations of delay vectors. Based on the actual data characteristics, a critical range for the fuzzy permutation entropy is set, or a K-means clustering algorithm is used to cluster the fuzzy permutation entropy values ​​of all IMF components to identify noise component clusters and signal component clusters, thereby determining the threshold for distinguishing high-frequency mode components from low-frequency mode components. This implementation method can accurately quantify the complexity and noise ratio of IMF components, providing a reliable basis for subsequent modal classification.

[0031] In this embodiment, high- and low-frequency mode classification includes: Statistical analysis was performed on the fuzzy permutation entropy values ​​of all IMF components, including the extraction of mean, variance, and range parameters. The threshold function is constructed based on statistical analysis results and is adaptively adjusted for different datasets; Components above the threshold are defined as high-frequency modal components, and components below the threshold are defined as low-frequency modal components. The dynamic classification algorithm supports a sliding window mechanism, which adjusts the judgment criteria in real time according to changes in the input signal; The classification results are used to determine which modes are included in the Bayesian wavelet denoising process.

[0032] Specifically, the fuzzy permutation entropy values ​​of all IMF components obtained from CEEMDAN decomposition are collected, and statistical analysis is performed on these entropy values ​​to extract the mean. ,variance Statistical parameters such as mean and range, among which the mean is... Reflects the average level and variance of the entropy of the fuzzy permutation of all IMF components. It reflects the degree of dispersion of the entropy value.

[0033] A threshold function is constructed based on the statistical analysis results. The expression for the threshold function is as follows: , In the formula, k is an adjustment factor, adjusted according to the noise intensity and signal trend characteristics of GNSS monitoring data to ensure that the threshold can effectively distinguish between high-frequency noise and low-frequency signals. The fuzzy permutation entropy value of each IMF component is then compared with the threshold. In comparison, the entropy value is higher than The IMF component is defined as a high-frequency modal component with an entropy value lower than 1. The IMF component is defined as the low-frequency modal component.

[0034] A sliding window mechanism is employed during the classification process to update the statistical parameters of the fuzzy permutation entropy value in real time as the input signal changes, thereby dynamically adjusting the threshold. This ensures that the classification criteria can adapt to the time-varying characteristics of the signal. Based on the classification results, the high-frequency mode components that need to undergo Bayesian wavelet denoising are determined. This implementation achieves adaptive classification of IMF components, avoiding classification errors caused by traditional subjective threshold setting, and improving classification accuracy.

[0035] In this embodiment, the Bayesian wavelet denoising process includes: High-frequency modal components undergo discrete wavelet transform to convert the time-domain signal into a wavelet coefficient sequence; Wavelet coefficients are denoised using soft or hard thresholding based on the Bayesian thresholding function; The Bayesian threshold function is constructed based on the maximum a posteriori probability estimate and optimized by combining the signal prior and the statistical characteristics of noise. The wavelet basis can be any of the Daubechies, Symlet, or Coiflet wavelet basis, and the number of wavelet decomposition layers is dynamically set according to the spectral energy distribution. The wavelet coefficients after Bayesian wavelet denoising are subjected to inverse wavelet transform to recover the time-domain denoised signal.

[0036] Specifically, a discrete wavelet transform is performed on the high-frequency modal components obtained from the partitioning, converting the time-domain signal into a wavelet coefficient sequence. The discrete wavelet transform formula is as follows: , In the formula These are wavelet coefficients. For scale parameters, The wavelet basis functions are selected using the Daubechies wavelet basis, and the order of the wavelet basis is determined based on the spectral characteristics of the high-frequency modal components.

[0037] A Bayesian threshold function is constructed based on maximum a posteriori probability estimation, assuming wavelet coefficients It follows a Laplace prior distribution, and its probability density function is: , In the formula For the scaling parameter, the Bayesian threshold By maximizing the posterior probability and combining it with the statistical characteristics of noise in high-frequency modal components, the median absolute deviation method is used to estimate the noise standard deviation. , In the formula, MAD is the median absolute deviation of the wavelet coefficients.

[0038] The wavelet coefficients are processed using a soft thresholding method, and the processing formula is as follows: , In the formula These are the denoised wavelet coefficients. Based on the spectral energy distribution of the high-frequency modal components, the number of wavelet decomposition levels is dynamically set to ensure coverage of the main frequency components of the signal. An inverse wavelet transform is performed on the denoised wavelet coefficients; the inverse transform formula is as follows: This method restores the time-domain denoised signal. It effectively preserves the valid signal components in the high-frequency modal components while removing noise, thus improving denoising accuracy.

[0039] In this embodiment, signal reconstruction, error assessment, and feedback tuning include: The low-frequency modal components are superimposed and reconstructed with the high-frequency modal components after Bayesian wavelet denoising to obtain the purified GNSS deformation monitoring signal. The error between the purified GNSS deformation monitoring signal and the original GNSS deformation monitoring data is calculated by mean square error, and the denoising effect is judged by combining the signal-to-noise ratio, correlation coefficient, and maximum error evaluation index. If the denoising effect evaluation result does not reach the preset performance threshold, the system will automatically start the feedback tuning mechanism and dynamically adjust the number of components of the complete set empirical mode decomposition, the fuzzy permutation entropy discrimination standard, or the threshold function parameters of Bayesian wavelet denoising according to the evaluation result. The feedback and optimization mechanism supports point-by-point optimization at the GNSS station level.

[0040] Specifically, the low-frequency modal components are superimposed with the high-frequency modal components (after Bayesian wavelet denoising) at corresponding time points to reconstruct the purified GNSS deformation monitoring signal. The reconstruction formula is as follows: , In the formula For low-frequency modal components, These are the high-frequency modal components after denoising.

[0041] The denoising effect is evaluated using mean squared error, signal-to-noise ratio, correlation coefficient, and maximum error. The formula for mean squared error is: , In the formula For the raw GNSS data at time The value of , To reconstruct the signal at time... The value of N is the number of data samples; The signal-to-noise ratio formula is: .

[0042] If the evaluation results fail to meet the preset performance threshold, a feedback optimization mechanism is automatically activated, adjusting parameters based on the deviation direction of the evaluation indicators: if the mean square error is too high, the number of components in the CEEMDAN decomposition is increased; if the signal-to-noise ratio is too low, the threshold function parameters for Bayesian wavelet denoising are adjusted; if the classification accuracy is insufficient, the discrimination criterion for fuzzy permutation entropy is corrected. Simultaneously, point-by-point optimization at the GNSS station level is supported, with corresponding parameters adjusted individually for the denoising effect of each station. This implementation method ensures that the denoising effect meets the requirements and improves the consistency of multi-station data.

[0043] In this embodiment, the optimization adjustment of the noise intensity coefficient and the number of decompositions during the complete set empirical mode decomposition process includes: Preset the initial range of noise intensity coefficient and number of decompositions; Based on the error assessment results, an iterative search is adopted to adjust the noise intensity coefficient and the number of decompositions. The adjustment process aims to reduce mode aliasing and reduce redundant components. The optimization and adjustment process is performed each time feedback is received.

[0044] Specifically, before CEEMDAN decomposition, based on the sampling frequency and noise characteristics of the GNSS monitoring data, the initial range of the noise intensity coefficient and the initial range of the number of decompositions are preset. The initial range of the noise intensity coefficient is usually 0.1-0.4, and the initial range of the number of decompositions is usually 100-200 times.

[0045] Based on the error assessment results after signal reconstruction, an iterative search method is used to adjust the noise intensity coefficient and the number of decompositions: The number of decompositions is fixed, and the noise intensity coefficient is gradually adjusted. The modal aliasing degree of the IMF components after each adjustment is calculated, and the noise intensity coefficient with the lowest modal aliasing degree is selected. This noise intensity coefficient is then fixed, and the number of decompositions is gradually adjusted. The proportion of redundant IMF components after each adjustment is calculated, and the decomposition number with the lowest redundancy proportion is selected. The adjustment process consistently aims to reduce modal aliasing and redundant components, ensuring that the IMF components can effectively separate low-frequency trends from high-frequency noise.

[0046] Each time the feedback tuning mechanism is activated, the above optimization and adjustment process is repeated, updating the noise intensity coefficient and the number of decompositions based on the latest error assessment results, so that CEEMDAN decomposition can adapt to changes in data characteristics. This implementation method can continuously improve the quality of CEEMDAN decomposition, providing high-quality IMF components for subsequent denoising processing.

[0047] In this embodiment, the dynamic threshold mechanism based on the fuzzy permutation entropy value includes: The critical range of fuzzy permutation entropy was determined by statistical analysis of the FPE distribution of historical GNSS deformation monitoring data; The adaptive threshold uses K-means clustering or Gaussian mixture model to cluster the FPE values ​​of IMF components to identify noise component clusters and signal component clusters, and determines the threshold for separating high-frequency mode components and low-frequency mode components based on the clustering results. The dynamic threshold mechanism is adjusted individually during data processing at different GNSS observation stations or in different time periods.

[0048] Specifically, historical GNSS deformation monitoring data is collected, CEEMDAN decomposition and fuzzy permutation entropy calculation are performed on the historical data, the fuzzy permutation entropy distribution characteristics of all IMF components in the historical data are statistically analyzed, and the critical range of fuzzy permutation entropy is determined. This range reflects the boundary characteristics of fuzzy permutation entropy between noise components and signal components in the historical data.

[0049] For the fuzzy arrangement of the entropy values ​​of the IMF components in the current data, K-means clustering algorithm or Gaussian mixture model is used for cluster analysis to divide the entropy values ​​into noise component clusters and signal component clusters. The midpoint of the cluster center of the two clusters is used as the adaptive threshold for separating the high-frequency mode components and the low-frequency mode components.

[0050] For data from different GNSS observation stations or different time periods, the fuzzy permutation entropy distribution of data from each station and time period is statistically analyzed. The critical range or clustering parameters are then adjusted individually to ensure that the threshold adapts to the differences in observation environments at different stations and the noise variations over different time periods. This implementation method enables the dynamic threshold to have good adaptability and improves the accuracy of modal classification in different scenarios.

[0051] In this embodiment, the construction and optimization of the Bayesian threshold function includes: The Bayesian threshold function is constructed using the maximum a posteriori probability estimation method, combining the prior probability distribution of wavelet coefficients with the statistical properties of noise. The prior probability distribution of wavelet coefficients is modeled using a generalized Gaussian distribution or a Laplace distribution. The statistical characteristics of the noise were obtained by estimating the noise in the high-frequency modal components using the median absolute deviation method. The optimization of the Bayesian threshold function parameters is achieved by minimizing the estimation risk during the iterative denoising process. The risk assessment combines the spectral characteristics of the original signal and the noise.

[0052] Specifically, the Bayesian threshold function is constructed using the maximum a posteriori probability estimation method. It is assumed that the wavelet coefficients follow a generalized Gaussian distribution, and its probability density function is: , In the formula For shape parameters, For scale parameters, Let gamma be the function. Combining the prior distribution of the signal and the statistical characteristics of noise, the posterior probability distribution of the wavelet coefficients is derived, and the Bayesian threshold is obtained by maximizing the posterior probability.

[0053] Noise in high-frequency modal components is estimated, and the standard deviation of noise is calculated using the median absolute deviation method. , In the formula, MAD represents the median absolute deviation of the high-frequency wavelet coefficients after discrete wavelet transform of the high-frequency modal components. During the iterative denoising process, the Bayesian threshold function parameters are optimized with the goal of minimizing the estimation risk. The estimation risk is combined with the spectral characteristics of the original signal and noise, and the parameters are adjusted by comparing the deviation between the reconstructed signal and the ideal signal. This implementation allows the Bayesian threshold to adapt to different noise intensities, improving the stability of the denoising effect.

[0054] In this embodiment, the triggering conditions and parameter adjustments for feedback tuning include: When the mean square error is higher than the preset upper limit of error or the signal-to-noise ratio is lower than the preset lower limit of signal-to-noise ratio, the feedback optimization mechanism is triggered. The range of dynamically adjusted parameters is limited to a preset effective range. The parameters include the additional noise intensity coefficient of CEEMDAN decomposition, the delay vector dimension of fuzzy permutation entropy, and the number of decomposition layers for Bayesian wavelet denoising. Optimize point by point at the GNSS station level, and adjust the corresponding CEEMDAN decomposition parameters or Bayesian wavelet denoising parameters individually based on the denoising effect evaluation results of each station. The feedback optimization mechanism is executed at the end of each noise reduction processing cycle.

[0055] Specifically, the preset upper limit of mean square error and the lower limit of signal-to-noise ratio serve as the triggering conditions for the feedback optimization mechanism. When the mean square error of the reconstructed signal is higher than the preset upper limit, or the signal-to-noise ratio is lower than the preset lower limit, the feedback optimization mechanism is automatically triggered.

[0056] The range of dynamically adjusted parameters is clearly defined: the additional noise intensity coefficient of CEEMDAN decomposition is limited to 0.1-0.5, the delay vector dimension of fuzzy permutation entropy is limited to 3-7, and the decomposition level of Bayesian wavelet denoising is limited to 3-6. This ensures that the parameter adjustments are within the effective range and avoids processing failure due to abnormal parameters.

[0057] Optimization is performed point-by-point at the GNSS station level. For each station, the denoising performance evaluation results are used to individually adjust the corresponding CEEMDAN decomposition parameters, fuzzy permutation entropy discrimination criteria, or Bayesian wavelet denoising parameters. For example, if a station exhibits significant multipath effects, the noise intensity coefficient of the CEEMDAN decomposition is appropriately increased. After each denoising processing cycle, a feedback tuning mechanism is implemented to promptly correct parameter deviations. This implementation method ensures that the denoising performance of data from each station consistently meets preset requirements, improving the overall processing reliability.

[0058] In summary, this invention effectively reduces mode aliasing and IMF component redundancy in the traditional CEEMDAN method by improving the noise intensity coefficient and decomposition number of CEEMDAN optimizations. It combines fuzzy permutation entropy quantification of the complexity and noise proportion of each IMF component with a dynamic threshold mechanism to achieve accurate high- and low-frequency mode classification, addressing the lack of an effective noise identification mechanism and reliance on empirically set classification standards in traditional methods. For high-frequency modes, Bayesian wavelet denoising is employed, combining wavelet analysis and Bayesian theory to retain effective information in high-frequency signals while removing noise. A feedback tuning mechanism is introduced to dynamically adjust the number of decomposition components, fuzzy permutation entropy discrimination criteria, and Bayesian wavelet denoising parameters based on error evaluation results, overcoming the shortcomings of traditional methods that lack closed-loop optimization and rely on initial settings for denoising results. Overall, this significantly improves the denoising accuracy, stability, and adaptability of GNSS deformation monitoring data, ensuring data reliability in subsequent applications such as geological disaster early warning and structural health management.

[0059] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article, or apparatus.

[0060] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A GNSS deformation monitoring denoising method based on improved CEEMDAN and fuzzy permutation entropy, characterized in that, The method comprises: S1, data acquisition: acquiring original GNSS deformation monitoring data, the data containing position quantities in N, E and U directions, derived from multiple GNSS observation sites and containing various interference noises; S2, CEEMDAN decomposition: performing complete ensemble empirical mode decomposition on the original GNSS deformation monitoring data to obtain multiple intrinsic mode function (IMF) components, the decomposition process being controlled by optimizing noise intensity coefficient and decomposition times to separate low-frequency trend items and high-frequency noise parts in the data; S3, fuzzy permutation entropy calculation: calculating the fuzzy permutation entropy value of each IMF component, the entropy value being used to quantify the complexity and noise proportion of the component; S4, high and low frequency mode classification: based on the fuzzy permutation entropy value, the IMF components are divided into high frequency mode components and low frequency mode components by using a dynamic threshold mechanism, the high frequency mode containing noise signals and the low frequency mode retaining the trend component of the original signal; S5, Bayesian wavelet denoising: performing Bayesian wavelet denoising processing on the high frequency mode components, the denoising processing combining wavelet analysis and Bayesian theory to remove noises; S6, signal reconstruction and error evaluation: superimposing and reconstructing the high frequency mode components and the low frequency mode components processed by the Bayesian wavelet denoising to obtain the purified GNSS deformation monitoring signal, and evaluating the denoising effect by mean square error and signal-to-noise ratio; S7, feedback optimization: if the denoising effect evaluation result does not reach the preset performance threshold, adjusting the number of components of the complete ensemble empirical mode decomposition, the fuzzy permutation entropy discrimination standard or the threshold function parameters of the Bayesian wavelet denoising according to the evaluation result.

2. The GNSS deformation monitoring denoising method based on improved CEEMDAN and fuzzy permutation entropy according to claim 1, characterized in that, The CEEMDAN decomposition comprises: Based on an additive noise model, the original GNSS deformation monitoring data is disturbed multiple times by adding Gaussian white noise to the data to generate the multiple IMF components; The IMF components obtained by each disturbance decomposition are processed by multiple iteration and mean value calculation to obtain the final stable IMF component results; The IMF components are arranged from high to low frequency and contain high frequency disturbance items, signal trend items and mixed components; A noise amplitude factor and a stop criterion are introduced in the complete ensemble empirical mode decomposition process to control the decomposition depth.

3. The GNSS deformation monitoring denoising method based on improved CEEMDAN and fuzzy permutation entropy according to claim 1, characterized in that, The fuzzy permutation entropy calculation comprises: The IMF components are divided into time windows, and a fixed dimension delay vector sequence is extracted in each window; The similarity between the delay vectors is calculated by using a fuzzy membership function to generate the fuzzy permutation entropy value; The higher the fuzzy permutation entropy value, the stronger the signal complexity, and the lower the fuzzy permutation entropy value, the more the component has a trend; An adaptive threshold is determined by setting a fuzzy permutation entropy critical range or using a clustering algorithm to distinguish the high frequency mode components and the low frequency mode components.

4. The GNSS deformation monitoring denoising method based on improved CEEMDAN and fuzzy permutation entropy according to claim 1, characterized in that, The high and low frequency mode classification comprises: Statistical analysis is performed on the fuzzy permutation entropy values of all the IMF components, including mean value, variance and range parameter extraction; A threshold function is constructed based on the statistical analysis results and is adaptively adjusted for different data sets; The components higher than the threshold are defined as high frequency mode components, and the components lower than the threshold are defined as low frequency mode components; A dynamic classification algorithm supports a sliding window mechanism to real-time correct the judgment standard according to the change of the input signal. The classification result is used to determine which modalities enter the Bayesian wavelet denoising process.

5. The GNSS deformation monitoring denoising method based on improved CEEMDAN and fuzzy permutation entropy according to claim 1, characterized in that, The Bayesian wavelet denoising process comprises: The high-frequency modality component performs a discrete wavelet transform to convert a time-domain signal into a wavelet coefficient sequence; The wavelet coefficient is subjected to soft threshold or hard threshold denoising according to a Bayesian threshold function; The Bayesian threshold function is constructed according to a maximum a posteriori probability estimation and is optimized in combination with signal prior and noise statistical characteristics; The wavelet basis can be any one of a Daubechies, Symlet, or Coiflet wavelet basis, and the wavelet decomposition level is dynamically set according to the spectral energy distribution; The wavelet coefficient after the Bayesian wavelet denoising process is subjected to an inverse wavelet transform to restore a time-domain denoised signal.

6. The GNSS deformation monitoring denoising method based on improved CEEMDAN and fuzzy permutation entropy according to claim 1, characterized in that, The signal reconstruction and error evaluation and feedback optimization comprise: The low-frequency modality component and the high-frequency modality component after the Bayesian wavelet denoising process are superimposed to reconstruct a purified GNSS deformation monitoring signal; The error between the purified GNSS deformation monitoring signal and the original GNSS deformation monitoring data is calculated by a mean square error, and the denoising effect is judged in combination with a signal-to-noise ratio, a correlation coefficient, and a maximum error evaluation index; If the denoising effect evaluation result does not reach a preset performance threshold, a feedback optimization mechanism is automatically started, and the number of components of the complete set empirical mode decomposition, the fuzzy permutation entropy discrimination standard, or the threshold function parameters of the Bayesian wavelet denoising are dynamically adjusted according to the evaluation result; The feedback optimization mechanism supports point-by-point optimization at the GNSS station site level.

7. The GNSS deformation monitoring denoising method based on improved CEEMDAN and fuzzy permutation entropy according to claim 2, characterized in that, In the complete set empirical mode decomposition process, the optimization adjustment of the noise intensity coefficient and the decomposition number comprises: An initial range of the noise intensity coefficient and the decomposition number is preset; Based on the error evaluation result, the noise intensity coefficient and the decomposition number are adjusted by an iterative search, and the adjustment process aims to reduce modal aliasing and reduce redundant components; The optimization adjustment process is performed at each feedback optimization.

8. The GNSS deformation monitoring denoising method based on improved CEEMDAN and fuzzy permutation entropy according to claim 1, characterized in that, The dynamic threshold mechanism based on the fuzzy permutation entropy value comprises: A fuzzy permutation entropy critical range is determined by statistical analysis of the FPE distribution of historical GNSS deformation monitoring data; An adaptive threshold is determined by clustering analysis of the FPE values of the IMF components by a K-means clustering algorithm or a Gaussian mixture model to identify noise component clusters and signal component clusters, and the threshold value for separating high-frequency modality components and low-frequency modality components is determined according to the clustering result; The dynamic threshold mechanism is individually adjusted in different GNSS observation station data processing or different time period data processing.

9. The GNSS deformation monitoring denoising method based on improved CEEMDAN and fuzzy permutation entropy according to claim 5, characterized in that, The construction and optimization of the Bayesian threshold function comprise: The Bayesian threshold function is constructed by a maximum a posteriori probability estimation method in combination with the prior probability distribution of the wavelet coefficient and the statistical characteristics of the noise; The prior probability distribution of the wavelet coefficient is modeled by a generalized Gaussian distribution or a Laplace distribution; The statistical characteristics of the noise are obtained by noise estimation in the high-frequency modality component, and the estimation method uses a median absolute deviation method; The optimization of the Bayesian threshold function parameters is realized by minimizing the estimation risk in the iterative denoising process, and the risk evaluation combines the spectral characteristics of the original signal and the noise.

10. The GNSS deformation monitoring denoising method based on improved CEEMDAN and fuzzy permutation entropy according to claim 1, characterized in that, The trigger condition and parameter adjustment refinement of the feedback optimization comprise: The feedback tuning mechanism is triggered when the mean square error is higher than a preset upper error limit or the signal-to-noise ratio is lower than a preset lower signal-to-noise ratio limit; The dynamically adjusted parameter range is limited in a preset effective interval, and the parameters include an additional noise intensity coefficient of CEEMDAN decomposition, a delay vector dimension of fuzzy permutation entropy, and a decomposition layer number of Bayesian wavelet denoising; The GNSS station point level is optimized point by point, and the corresponding CEEMDAN decomposition parameters or Bayesian wavelet denoising parameters are adjusted according to the denoising effect evaluation result of each station point; The feedback tuning mechanism is executed at the end of each denoising processing period.

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