Method for extracting beidou monitoring displacement trend based on third-order smooth prior analysis algorithm

Abstract

The application discloses a Beidou monitoring displacement trend extraction method based on a third-order smooth prior analysis algorithm, which comprises the following steps: the cut-off frequency of the SPA algorithm under different regularization parameters is counted; when the SPA algorithm is used to extract the coordinate sequence trend, the cut-off frequency is specified according to the difference of the required filtering cut-off frequency in different application scenarios, namely, the regularization parameter used by the algorithm is determined, so that the sequence trend is extracted; based on the time-frequency characteristic analysis of the smooth prior analysis (SPA), the third-order differential operator matrix is used for filtering, the third-order smooth prior analysis algorithm (SPA-L3) is constructed, and the high-precision extraction of the GNSS deformation monitoring displacement trend is realized. The application can provide a high-precision displacement trend extraction method by combining the third-order smooth prior analysis (SPA-L3) for time-frequency characteristic analysis. The method can adaptively select the regularization parameter according to different application scenarios, so as to flexibly adjust the filtering cut-off frequency, and ensure that the optimal trend extraction result is obtained under various monitoring environments.

Description

Technical Field

[0001] This invention belongs to the field of GNSS (Global Navigation Satellite System) positioning and navigation technology, and relates to GNSS structural health monitoring technology, specifically to a method for extracting displacement trends in BeiDou monitoring based on a third-order smoothing prior analysis algorithm. Background Technology

[0002] GNSS deformation monitoring technology has become an important tool widely used in geological engineering, urban construction, and infrastructure monitoring in recent years. Its principle involves receiving satellite signals using the Global Navigation Satellite System (GNSS) and employing high-precision positioning technology to monitor the location information of monitoring points over a long period. GNSS deformation monitoring collects location information from multiple monitoring points within a target area through continuous observation, and combines differential techniques (such as RTK or PPP) to remove the influence of atmospheric and orbital errors, thereby achieving centimeter-level or millimeter-level deformation accuracy. Real-time acquisition and analysis of deformation data can effectively detect the deformation trend of the ground or structures, providing reliable data for disaster early warning, structural safety assessment, and geological exploration. While the accuracy and efficiency of deformation monitoring methods have greatly improved with the continuous advancement of GNSS technology, challenges remain when monitoring large areas and complex terrains, including insufficient data acquisition density and complex error sources.

[0003] To address these challenges, GNSS-based deformation monitoring methods have gradually incorporated intelligent and automated data processing and analysis technologies in recent years. By integrating big data analytics and machine learning, large volumes of monitoring data can be intelligently analyzed to automatically identify and predict deformation trends. Simultaneously, leveraging wireless sensor networks, IoT technology, and cloud computing platforms, GNSS deformation monitoring enables remote data acquisition and real-time transmission, significantly improving monitoring efficiency and coverage. Furthermore, for large-scale monitoring projects, distributed computing and multi-source data fusion technologies can efficiently process massive amounts of data and generate accurate deformation information in real time. The integration of these technologies has propelled GNSS deformation monitoring technology towards higher precision, higher efficiency, and lower cost, achieving significant application results in disaster monitoring, urban infrastructure construction, and geological disaster early warning. Summary of the Invention

[0004] To address the aforementioned issues, this invention discloses a displacement trend extraction method based on a third-order smoothing prior analysis algorithm for BeiDou monitoring. It delves into the related problems of displacement trend extraction from deformation monitoring sequences, extracting dynamic displacement information and various vibration characteristics of the monitored object from the original GNSS deformation monitoring coordinate sequence. This provides a high-precision and high-stability displacement trend extraction method for structural health monitoring technology.

[0005] To achieve the above objectives, the technical solution of the present invention is as follows:

[0006] The method for extracting displacement trends from BeiDou monitoring based on a third-order smoothing prior analysis algorithm includes the following steps:

[0007] Step 1: Statistically determine the cutoff frequency of the SPA algorithm under different regularization parameters, and determine the correspondence between the cutoff frequency and the regularization parameter. Then, the regularization parameter used by the algorithm can be determined according to the required filtering cutoff frequency.

[0008] Step 2: Select the differential operator matrix in the smoothing prior analysis algorithm, and perform comprehensive analysis based on the amplitude-frequency response corresponding to different differential orders to determine the order of the differential operator matrix and thus perform SPA filtering;

[0009] Step 3. Adjust the regularization parameters and the determined differential operator matrix according to the filter cutoff frequency, calculate the sequence trend term and determine the displacement extreme points to obtain trend extraction results with different smoothness levels.

[0010] The specific steps are as follows:

[0011] Step 1. Statistically determine the cutoff frequencies of the SPA algorithm under different regularization parameters, and establish the correspondence between the cutoff frequencies and regularization parameters. Subsequently, the regularization parameters used in the algorithm can be determined based on the required filtering cutoff frequency.

[0012]

[0013] Based on equation (1), different regularization parameters are used to generate the transformation matrix, and the amplitude-frequency response of different transformation matrices when filtering 1Hz data is obtained and the cutoff frequency is statistically analyzed.

[0014] Calculation of amplitude-frequency response and determination of cutoff frequency:

[0015] (1) Construct a d-order differential matrix D based on the smoothed prior analysis order d. d The transformation matrix is ​​generated using the regularization parameter λ. Where I is the identity matrix.

[0016] (2) To P d Input a unit pulse signal δ(n) and obtain the response signal.

[0017] y(n)=D d ·δ(n) (2)

[0018] Performing a discrete Fourier transform on y(n) yields the frequency domain response Y(w), which in turn gives the normalized amplitude-frequency response 20log. 10(|Y(w)| / |Y(0)|) and amplitude-frequency response curves, and determine the cutoff frequency f when the amplitude drops to -3dB in the Nyquist frequency range (0~0.5Hz). cutoff .

[0019] (3) By iterating through different λ values ​​and recording the corresponding f values cutoff A quantitative relationship diagram between the two is established, and λ is selected accordingly to meet the filtering requirements of the target frequency band.

[0020] Analysis shows that the amplitude-frequency response of the transformation matrix can be approximated as a low-pass filter, which retains the low-frequency components in the sequence while significantly filtering out the higher-frequency components; and the cutoff frequency (where the amplitude-frequency response drops to -3dB) decreases as the regularization parameter increases.

[0021] The cutoff frequencies of the SPA algorithm under different regularization parameters are statistically analyzed. When using the SPA algorithm to extract trends from coordinate sequences, the cutoff frequency can be specified according to the differences in the required filtering cutoff frequencies in different application scenarios, thereby determining the regularization parameters used by the algorithm and thus extracting the sequence trends.

[0022] Step 2. Select the differential operator matrix in the smoothing prior analysis algorithm, and perform comprehensive analysis based on the amplitude-frequency response corresponding to different differential orders to determine the order of the differential operator matrix and then perform SPA filtering.

[0023] In the SPA algorithm, the transformation matrix P transforms the original sequence into a trend sequence. d The calculation formula is as shown in (1).

[0024] Wherein, the differential operator matrix D d The order d is generally chosen to be d=2, that is, the second-order differential operator matrix D2 is used, and the specific matrix is ​​shown in equation (3). The order d of the differential operator matrix is ​​changed to d=1 and d=3 to generate the first-order differential operator matrix D1 and the third-order differential operator D3, as shown in equations (4) and (5).

[0025]

[0026] Transformation matrices P1, P2, and P3 are generated using D1, D2, and D3 respectively. The regularization parameter λ of the three transformation matrices is adjusted so that the cutoff frequency f of the three transformation matrices is equal. cutoff All are the same (1.11 × 10⁻⁶). -3 (Hz), compare the amplitude-frequency response of the three transformation matrices.

[0027] Analysis shows that, for higher frequency components, the amplitude-frequency response of the SPA algorithm's transformation matrix is ​​roughly the same when using different derivative orders d; however, when analyzing the ultra-low frequency range (0–0.0025 Hz) near the cutoff frequency, with a derivative order d = 3, the amplitude-frequency response is significantly different for frequencies below the cutoff frequency f. cutoff The highest amplitude is found in the portion of the amplitude response where the amplitude is greater than -3dB; meanwhile, for frequencies higher than the cutoff frequency f... cutoff The portion of the amplitude response with an amplitude less than -3dB exhibits the strongest attenuation effect. This characteristic indicates that when using the third-order differential operator matrix D3 for SPA filtering, it can filter out sequence components above the cutoff frequency to the greatest extent while achieving the best retention effect for components below the cutoff frequency. Considering the ultra-low frequency characteristics of the trend term in the GNSS coordinate sequence, SPA filtering using the third-order differential operator matrix D3 has a superior filtering effect compared to the traditional second-order differential operator matrix D2.

[0028] Step 3. Adjust the regularization parameters and the determined differential operator matrix according to the filter cutoff frequency, calculate the sequence trend term and determine the displacement extreme points to obtain trend extraction results with different smoothness levels.

[0029] Based on the above analysis, compared to the traditional SPA algorithm which uses second-order differential operators, using third-order differential operators to generate the transformation matrix P3 achieves better trend extraction and periodic term filtering results. Furthermore, the filtering cutoff frequency f required by the algorithm's application scenario can be adjusted accordingly. cutoff Choose an appropriate regularization parameter λ to achieve the best results.

[0030] The complete process for extracting trend items with different levels of smoothness is as follows:

[0031] Suppose the original signal X consists of two parts:

[0032] X = X s +X t (6)

[0033] Where X s X represents the stationary periodic term in the signal. t The trend term in the signal can be represented by a linear observation model as follows:

[0034] X t =Hθ+v (7)

[0035] Where H is the observation matrix, θ is the regression parameter, and v is the observation error.

[0036] At this point, the optimal solution for estimating the regression parameter θ is obtained using regularized least squares. Then it can be passed Identify the trend term in the original signal:

[0037]

[0038] To satisfy equation (8), the differential term ||D| must be made equal to 0. d (Hθ)|| approaches zero, thus allowing estimation of the predicted trend Hθ based on prior information. The solution to equation (8) can be expressed as:

[0039]

[0040] The observation matrix H can be obtained by analyzing the characteristics of the original signal X. To facilitate analysis and simplify the selection of the observation matrix H, let it be the identity matrix I. Then equation (10) can be transformed into:

[0041]

[0042] Based on the above analysis, a third-order differential operator is used for filtering, specifically a third-order differential matrix D3; at this point, the algorithm extracts the sequence trend term. for:

[0043]

[0044] By adjusting the regularization parameter λ according to the previously determined cutoff frequency, trend extraction results with different degrees of smoothness can be obtained. After obtaining the displacement trend sequence through filtering, the first derivative of the trend sequence is first performed to obtain the rate of change sequence of the trend sequence. In the rate of change sequence, the points where the sign of the rate of change changes (i.e., from positive to negative, or from negative to positive), as well as the start and end points of the sequence, are the displacement extreme points. Then, by performing difference operations based on the time and coordinate values ​​of each displacement extreme point, the time period, displacement amplitude, and displacement velocity of each displacement can be obtained.

[0045] The beneficial effects of this invention are:

[0046] This invention achieves high-precision extraction of displacement trends from GNSS deformation monitoring through an innovative design of the third-order smooth prior analysis (SPA-L3) algorithm. Based on the amplitude-frequency response optimization of the third-order differential operator, this method significantly suppresses high-frequency noise while preserving low-frequency deformation characteristics, effectively improving anti-interference capability compared to traditional low-order algorithms. By establishing a quantitative mapping relationship between regularization parameters and filter cutoff frequencies, it supports adaptive parameter configuration according to the needs of scenarios such as geological disaster early warning and structural health monitoring, achieving precise control of the cutoff frequency within a certain range. The algorithm integrates a time-frequency analysis framework with efficient computational strategies, improving data processing efficiency while maintaining millimeter-level accuracy. It meets the real-time processing needs of large-scale continuous monitoring data such as slope displacement and bridge vibration, providing a standardized solution for displacement trend extraction in complex environments. Attached Figure Description

[0047] Figure 1 This is a flowchart of the implementation of the BeiDou monitoring displacement trend extraction method based on the third-order smoothing prior analysis algorithm described in this invention.

[0048] Figure 2 This is a flowchart of a step change identification method based on total variation filtering;

[0049] Figure 3 This is a flowchart of a method for comprehensively expressing displacement trend monitoring information;

[0050] Figure 4 This is a flowchart of the multi-parameter adaptive variational mode decomposition algorithm;

[0051] Figure 5 This is a graph showing the results of extracting extreme points of displacement trends;

[0052] Figure 6 This is a graph showing the extracted displacement trend results of a measured coordinate sequence in a complex mountainous area. Detailed Implementation

[0053] The present invention will be further illustrated below with reference to the accompanying drawings and specific embodiments. It should be understood that the following specific embodiments are for illustrative purposes only and are not intended to limit the scope of the invention.

[0054] As shown in the figure, the specific steps of the BeiDou monitoring displacement trend extraction method based on the third-order smoothing prior analysis algorithm described in this invention are as follows:

[0055] Step 1. Statistically determine the cutoff frequencies of the SPA algorithm under different regularization parameters, and establish the correspondence between the cutoff frequencies and regularization parameters. Subsequently, the regularization parameters used in the algorithm can be determined based on the required filtering cutoff frequency.

[0056]

[0057] Calculation of amplitude-frequency response and determination of cutoff frequency:

[0058] (1) Construct a d-order differential matrix D based on the smoothed prior analysis order d. d The transformation matrix is ​​generated using the regularization parameter λ. Where I is the identity matrix.

[0059] (2) To P d Input a unit pulse signal δ(n) and obtain the response signal.

[0060] y(n)=D d ·δ(n) (2)

[0061] Performing a discrete Fourier transform on y(n) yields the frequency domain response Y(w), which in turn gives the normalized amplitude-frequency response 20log. 10(|Y(w)| / |Y(0)|) and amplitude-frequency response curves, and determine the cutoff frequency f when the amplitude drops to -3dB in the Nyquist frequency range (0~0.5Hz). cutoff .

[0062] (3) By iterating through different λ values ​​and recording the corresponding f values cutoff A quantitative relationship diagram between the two is established, and λ is selected accordingly to meet the filtering requirements of the target frequency band.

[0063] According to equation (1), the transformation matrix of the SPA algorithm is generated using different regularization parameters, and the amplitude-frequency response of different transformation matrices when filtering 1Hz data is obtained as follows: Figure 2 As shown. Analysis reveals that the amplitude-frequency response of the transformation matrix can be approximated as a low-pass filter, which retains low-frequency components in the sequence while significantly filtering out higher-frequency components; and the cutoff frequency (where the amplitude-frequency response drops to -3dB) decreases as the regularization parameter increases.

[0064] The cutoff frequencies of the SPA algorithm under different regularization parameters are as follows: Figure 3 As shown, when using the SPA algorithm to extract trends from coordinate sequences, the cutoff frequency can be specified according to the differences in the required filtering cutoff frequency in different application scenarios, thereby determining the regularization parameters used by the algorithm and thus extracting the sequence trend.

[0065] Step 2. Select the differential operator matrix in the smoothing prior analysis algorithm, and perform comprehensive analysis based on the amplitude-frequency response corresponding to different differential orders to determine the order of the differential operator matrix and then perform SPA filtering.

[0066] In the SPA algorithm, the transformation matrix P transforms the original sequence into a trend sequence. d The calculation formula is as shown in (1).

[0067] Wherein, the differential operator matrix D d The order d is generally chosen to be d=2, that is, the second-order differential operator matrix D2 is used, and the specific matrix is ​​shown in equation (3). The order d of the differential operator matrix is ​​changed to d=1 and d=3 to generate the first-order differential operator matrix D1 and the third-order differential operator D3, as shown in equations (4) and (5).

[0068]

[0069] Transformation matrices P1, P2, and P3 are generated using D1, D2, and D3 respectively. The regularization parameter λ of the three transformation matrices is adjusted so that the cutoff frequency f of the three transformation matrices is equal. cutoff All are the same (1.11 × 10⁻⁶). -3 (Hz), compare the amplitude-frequency response of the three transformation matrices, such as Figure 4 As shown.

[0070] Depend on Figure 4 -(a) It can be seen that, when using different differential orders d, the amplitude-frequency response of the SPA algorithm transformation matrix is ​​roughly the same for the higher frequency part; however, when analyzing the ultra-low frequency range (0~0.0025Hz) near the cutoff frequency, such as... Figure 4 -(b) It can be seen that when the differential order d = 3, for frequencies below the cutoff frequency f cutoff The highest amplitude is found in the portion of the amplitude response where the amplitude is greater than -3dB; meanwhile, for frequencies higher than the cutoff frequency f... cutoff The portion of the amplitude response with an amplitude less than -3dB exhibits the strongest attenuation effect. This characteristic indicates that when using the third-order differential operator matrix D3 for SPA filtering, it can filter out sequence components above the cutoff frequency to the greatest extent while achieving the best retention effect for components below the cutoff frequency. Considering the ultra-low frequency characteristics of the trend term in the GNSS coordinate sequence, SPA filtering using the third-order differential operator matrix D3 has a superior filtering effect compared to the traditional second-order differential operator matrix D2.

[0071] Step 3. Adjust the regularization parameters and the determined differential operator matrix according to the filter cutoff frequency, calculate the sequence trend term and determine the displacement extreme points to obtain trend extraction results with different smoothness levels.

[0072] Based on the above analysis, compared to the traditional SPA algorithm which uses second-order differential operators, using third-order differential operators to generate the transformation matrix P3 achieves better trend extraction and periodic term filtering results. Furthermore, the filtering cutoff frequency f required by the algorithm's application scenario can be adjusted accordingly. cutoff Choose an appropriate regularization parameter λ to achieve the best results.

[0073] The complete process for extracting trend items with different levels of smoothness is as follows:

[0074] Suppose the original signal X consists of two parts:

[0075] X = X s +X t (6)

[0076] Where X s X represents the stationary periodic term in the signal. t The trend term in the signal can be represented by a linear observation model as follows:

[0077] X t =Hθ+v (7)

[0078] Where H is the observation matrix, θ is the regression parameter, and v is the observation error.

[0079] At this point, the optimal solution for estimating the regression parameter θ is obtained using regularized least squares. Then it can be passed Identify the trend term in the original signal:

[0080]

[0081] To satisfy equation (8), the differential term ||D| must be made equal to 0. d (Hθ)|| approaches zero, thus allowing estimation of the predicted trend Hθ based on prior information. The solution to equation (8) can be expressed as:

[0082]

[0083] The observation matrix H can be obtained by analyzing the characteristics of the original signal X. To facilitate analysis and simplify the selection of the observation matrix H, let it be the identity matrix I. Then equation (10) can be transformed into:

[0084]

[0085] Based on the above analysis, a third-order differential operator is used for filtering, specifically a third-order differential matrix D3; at this point, the algorithm extracts the sequence trend term. for:

[0086]

[0087] By adjusting the regularization parameter λ according to the previously determined cutoff frequency, trend extraction results with different smoothness levels can be obtained. After obtaining the displacement trend sequence through filtering, the first derivative of the trend sequence is first performed to obtain the rate of change sequence. The points where the sign of the rate of change changes (i.e., from positive to negative, or from negative to positive), as well as the start and end points of the sequence, are the displacement extreme points. Then, by performing a difference operation based on the time and coordinate values ​​of each displacement extreme point, the time period, displacement amplitude, and displacement velocity of each displacement can be obtained. The displacement extreme points of the trend sequence extracted by this method are shown in the attached figure. Figure 5 As shown.

[0088] Appendix Figure 6 The trend sequence extraction results obtained by this algorithm show that the trend extraction results of the SPA-L3 algorithm are basically consistent with the actual trend.

[0089] It should be noted that the above content merely illustrates the technical concept of the present invention and should not be construed as limiting the scope of protection of the present invention. For those skilled in the art, various improvements and modifications can be made without departing from the principle of the present invention, and all such improvements and modifications fall within the scope of protection of the claims of the present invention.

Claims

1. A method for extracting displacement trends from BeiDou monitoring based on a third-order smoothing prior analysis algorithm, characterized in that: Step 1: Statistically determine the cutoff frequency of the SPA algorithm under different regularization parameters, and determine the correspondence between the cutoff frequency and the regularization parameter. Then, determine the regularization parameter to be used in the algorithm based on the required filtering cutoff frequency. The selection of the regularization parameter requires determining the correspondence between the cutoff frequency and the regularization parameter. Based on equation (1), different regularization parameters are used to generate the transformation matrix, and the amplitude-frequency response of different transformation matrices when filtering 1Hz data is obtained and the cutoff frequency is statistically analyzed. (1)； The methods for calculating the amplitude-frequency response and determining the cutoff frequency are as follows: (1) Based on the order of smooth prior analysis , build 1st order differential matrix ; Using regularization parameters Generate transformation matrix , where I is the identity matrix; (2) To Input unit pulse signal Obtain response signal = (2)； right The frequency domain response is obtained by performing a discrete Fourier transform. Thus, the normalized amplitude-frequency response is obtained. The amplitude-frequency response curves were obtained, and the cutoff frequency at which the amplitude drops to -3dB was determined within the Nyquist frequency range of 0–0.5Hz. ; (3) By traversing different Value and record the corresponding Establish a quantitative relationship diagram between the two, and select accordingly. To meet the filtering requirements of the target frequency band; Step 2: Select the differential operator matrix in the smoothing prior analysis algorithm, and perform comprehensive analysis based on the amplitude-frequency response corresponding to different differential orders to determine the order of the differential operator matrix and thus perform SPA filtering; Step 3: Adjust the regularization parameters and the determined differential operator matrix according to the filter cutoff frequency, calculate the sequence trend term and determine the displacement extreme points to obtain trend extraction results with different smoothness levels.

2. The method for extracting displacement trends from BeiDou monitoring based on a third-order smoothing prior analysis algorithm as described in claim 1, characterized in that, The order of the differential operator matrix in the smoothing prior analysis algorithm described in step 2 is obtained according to the following formulas: In the SPA algorithm, the transformation matrix converts the original sequence into a trend sequence. The calculation formula is as follows (1); Among them, the differential operator matrix order Select as That is, using the second-order differential operator matrix The specific matrix is ​​shown in equation (3); the order of the differential operator matrix is... Change to and Generate a first-order differential operator matrix With third-order differential operators See equations (4) and (5); (3)； (4)； (5)。 3. The method for extracting displacement trends from BeiDou monitoring based on a third-order smoothing prior analysis algorithm as described in claim 1, characterized in that, The complete process for extracting trend items with different smoothness levels as described in step 3 is as follows: Assume the original signal Composed of two parts composition: (6)； in This refers to the stationary periodic term in the signal; The trend term in the signal is represented by a linear observation model as follows: (7)； in For the observation matrix, For regression parameters, This is the observation error; At this point, regularized least squares method is used to estimate the regression parameters. optimal solution And then through Identify the trend term in the original signal: (8)； To satisfy equation (8), the differential term must be made Approaching zero, predicting trends based on prior information. The estimate; the solution to equation (8) is expressed as: (9)； (10)； Observation matrix By analyzing the original signal The characteristics were obtained to facilitate analysis and simplify the observation matrix. The choice of is used to convert it into an identity matrix. Then equation (10) is transformed into: (11)； Based on the above analysis, a third-order differential operator is used for filtering, that is, a third-order differential matrix is ​​used. ; At this point, the algorithm extracts the sequence trend term. for: (12)； Adjust the regularization parameters according to the cutoff frequency determined in step 1. This yields trend extraction results with varying degrees of smoothness. After obtaining the displacement trend sequence through filtering, the first derivative of the trend sequence is performed to obtain the rate of change sequence of the trend sequence. The points where the sign of the rate of change changes, as well as the start and end points of the sequence, are the displacement extreme points. Then, the difference operation is performed based on the time and coordinate values ​​of each displacement extreme point to obtain the time period, displacement amplitude, and displacement velocity of each displacement.

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