Novel noise HV spectrum ratio imaging method based on polarization

By using polarization analysis technology to separate noise wave field components in the micro-movement HV spectrum ratio method, calculate the HV spectrum ratio and imaging, the problem of multi-peak and insignificant peaks is solved, and more efficient and accurate underground structure analysis is achieved.

CN119596388AActive Publication Date: 2025-03-11CHINESE ACAD OF GEOLOGICAL SCI

Patent Information

Application Number
CN202411785340.X
Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
Filing Date
2024-12-06
Publication Date
2025-03-11
Estimated Expiration
2044-12-06

AI Technical Summary

Technical Problem

The existing micro-moving HV spectrum ratio methods are prone to problems of multi-peak and insignificant peaks when the noise HV spectrum ratio wavefield is mixed, resulting in the inability to accurately extract excellent frequencies.

Method used

Using a new noise HV spectrum ratio imaging method based on polarization, the linear bulk wave and nonlinear plane wave components are separated by polarization analysis technology, the respective HV spectrum ratio is calculated, and the profile imaging of the frequency domain and the depth domain is performed.

Benefits of technology

It significantly improves the accuracy and efficiency of underground structure analysis, optimizes the data processing flow, automatically generates frequency and depth profiles, improves the simplicity of operation and the reliability of results, and can more accurately analyze and identify the wave impedance interface and velocity structure of the underground.

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Abstract

The invention discloses a novel noise HV spectrum ratio imaging method based on polarization. The method comprises the following steps: collecting background noise data in a predetermined monitoring area; preprocessing the acquired background noise data, wherein the preprocessing comprises data format conversion, data splicing, mean value removal, trend removal, waveform pinching and band-pass filtering processing; preprocessed data are processed by adopting a polarization analysis technology, a wave field is separated by setting a threshold value and analyzing a power spectrum, and a background noise area is divided into a linear body wave component and a nonlinear surface wave component. According to the polarization-based noise HV spectrum ratio imaging method, the precision and efficiency of underground structure analysis are remarkably improved through the precise separation of the wave field and the automatic processing flow of the HV spectrum ratio, and by means of polarization analysis optimized data processing, the automatically generated frequency and depth profile and the automatically picked excellent frequency and peak value, the imaging precision and efficiency of the underground structure are greatly improved. And the analysis of the underground velocity structure and the wave impedance interface is more accurate.
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Description

Technical Field

[0001] The invention relates to a novel forming method, in particular to a novel polarization-based noise HV spectrum ratio imaging method. Background Art

[0002] The micromotion HV spectrum ratio method is based on the three-component micromotion information of a single station on the earth's surface. By appropriately averaging the horizontal component, the spectrum of the horizontal component and the vertical component is obtained. This spectrum ratio is a function of frequency and is called the HV spectrum ratio or HVSR (Horizontal-to-Vertical Spectral Ratio). Even though there is controversy over the micromotion source component, the HV spectrum ratio method believes that the peak frequency of the HV spectrum ratio curve corresponds to the resonance frequency of the site S wave, or the dominant frequency, through which the site's amplification factor and sediment thickness can be obtained. Due to the work of Nakamura (1989, 2000), the micromotion HV spectrum ratio method has been promoted in the field of engineering seismic, also known as the Nakamura method, but Nakamura himself prefers to call it the QTS (Quasi-Transfer Spectra) method (Nakamura, 2019). Since the micromotion HV spectrum ratio method was proposed, some experiments (Lermo and Chávez-García, 1993; Gitterman et al., 1996; Seekins et al., 1996; et al., 1997) showed that the HV spectrum ratio method can be used to identify the fundamental resonant frequency of the sedimentary layer, and the good correspondence between the peak frequency of the micro-motion HV spectrum ratio and the fundamental resonant frequency of the SH wave of the sedimentary layer was confirmed by the one-dimensional model. These studies used randomly distributed surface sources to simulate micro-motion waveforms. When the sedimentary layer and the bedrock showed a significant impedance ratio, the HV spectrum ratio produced a peak near the fundamental S-wave resonance frequency. For a single low-velocity layer on a half-space, Malischewsky and Scherbaum (2004) theoretically proved that the higher the impedance ratio between the low-velocity layer and the half-space, the better the correlation between the peak frequency of the micro-motion HV spectrum ratio and the fundamental resonant frequency of the S wave of the layer. The HV spectrum ratio method uses the HV peak to estimate the resonant frequency and is suitable for quickly studying site characteristics. Compared with the method based on seismic records, the micro-HV spectrum ratio method has the advantages of being simple, fast, economical, and less restricted by the site. Although there are still some controversies in its theoretical explanation, it has been widely used in the field of engineering earthquakes. Especially in Japan and other places. The HV spectrum ratio method is also widely used in micro-zoning, that is, to divide, identify and map areas of earthquake intensity, liquefaction or landslide possibility.

[0003] The focus of the controversy over the micromotion HV spectrum ratio method is which energy component in the micromotion record is dominant. One view is that the main energy component of the micromotion is body waves (Nakamura, 1989, 2000, 2009; Herak, 2008), especially the vertically incident SH waves. Under this assumption, the peak of the HV spectrum ratio curve corresponds to the resonance frequency of the overlying soft soil layer, so the HV spectrum ratio curve can be used to estimate the site amplification effect and sediment thickness. Another view is that the main energy component of the micromotion is Rayleigh waves, especially the fundamental mode Rayleigh waves. Under this assumption, the HV spectrum ratio curve is mainly affected by the ellipticity of the fundamental Rayleigh wave, and theoretically there is no direct correlation with the actual site amplification effect. In a clarification article by Nakamura (2000), this view does not seem to be agreed with. It is believed that the similarity between the shape of the HV spectral ratio curve and the fundamental Rayleigh wave ellipticity curve does not mean that the HV spectral ratio curve corresponds to the Rayleigh wave fundamental mode ellipticity curve. Regardless of the degree of influence of the Rayleigh wave, the HV spectral ratio curve (QTS) can give the resonant frequency of the SH wave to a certain extent, thereby inferring the dominant frequency and site amplification factor of the site.

[0004] In addition, the contributions of different seismic phases to the microvibration HV spectrum ratio curve are different in the frequency bands outside the peak amplitude frequency. Some authors believe that surface waves play a dominant role in the microvibration wave field, at least in the range above the fundamental resonance frequency of the sediment layer (Lunedei & Albarello, 2009; Albarello & Lunedei, 2011). Bonnefoy-Claudet et al. (2006b) used sources in the bedrock to simulate the microvibration HV spectrum ratio curve and found that the peak of the HV spectrum ratio curve was mainly caused by S-wave resonance. However, since the wave field around the array is mainly generated by human activities, it is closer to surface waves, and the actual HV spectrum ratio may be mainly affected by the fundamental mode Rayleigh wave. In the research on the theoretical basis of HV spectrum ratio, the most commonly used method is to model the HV spectrum ratio curve by using the ellipticity of Rayleigh waves and use the HV spectrum ratio curve to invert the medium structure. Due to the controversy over the micro-vibration waveform components, some noise source models are currently being developed, such as the DSS (Distributed Surface Sources) and DFA (Diffuse Field Approach) models.

[0005] Therefore, based on the views of different researchers and the development of seismic interferometry theory, there are two obvious branches in the research on micromotion HV spectrum ratio (Lunedei & Malischewsky, 2015): The first branch is to identify the Rayleigh wave components in the micromotion records, study the Rayleigh wave ellipticity (or ellipticity ZH amplitude ratio), and use the Rayleigh wave ellipticity or the combined Rayleigh wave dispersion curve to image the medium structure. This method is not limited to seismic background noise, but also used for natural earthquake events, mainly in the field of regional or shallow surface seismic tomography. The theoretical basis of this direction is the Rayleigh wave theory in horizontally layered media. In order to distinguish it from the micromotion HV spectrum ratio, the more specific term "Rayleigh wave ZH amplitude ratio" is generally used. The second branch is to study the wave type components in the micromotion records, develop different micromotion source models, consider the contribution of different wave type components, and explain the measured HV spectrum ratio curve. For example, the micromotion source model based on the dominant S wave, the diffuse background noise model, etc. This direction has gradually developed with the development of seismic interferometry theory. However, as Lunedei & Malischewsky (2015) pointed out, developing a theoretical model to explain the HV spectral ratio curve requires that the model is an actual model of the background noise wave field and should first be consistent with the findings of the actual background noise wave field, rather than just providing a possible way to generate and explain the HV spectral ratio curve.

[0006] Polarization analysis (Zebker et al., 2010) refers to a method of obtaining underground medium information by analyzing the polarization characteristics of seismic waveforms in seismic signal processing. Polarization analysis is a technology used in seismic signal processing, which can help us understand the physical properties of underground media (Yilmaz et al, 2011; Dong et al, 2021). In HV spectral ratio imaging, polarization analysis can be used to optimize data processing and improve imaging accuracy. The application of polarization analysis in HV spectral ratio imaging mainly includes the following aspects:

[0007] (1) Improve signal-to-noise ratio: Using polarization analysis, we can determine the polarization direction of the seismic signal and partially filter out the non-polarized signal, thereby improving the signal-to-noise ratio of the data. This enables HV spectral imaging to better capture the reflection information of the underground medium and improve the imaging quality.

[0008] (2) Identification of abnormal media: Abnormal underground media may cause differences in polarization signals in different directions. Using polarization analysis, we can detect these anomalies and provide more information for HV spectral ratio imaging.

[0009] (3) Determine physical properties: Seismic waves undergo polarization rotation when passing through a medium. We can use polarization analysis to infer the physical properties of the underground medium, such as compressibility, shear, and density. This helps interpret HV spectral ratio imaging results and improves our understanding of underground structures.

[0010] Polarization analysis is widely used in seismology, mainly including the following aspects:

[0011] (1) Earthquake location: Earthquake location refers to the process of determining the depth and location of an earthquake source. Polarization analysis can be used to determine the depth and location of an earthquake source. Since earthquake signals undergo polarization changes during propagation, sources at different locations and depths will result in different polarization characteristics. This characteristic can be used to locate an earthquake.

[0012] (2) Analysis of underground medium structure: Polarization analysis can be used to study the physical properties and structure of underground media, such as changes in groundwater levels, cracks and faults in rocks, etc. When seismic waves pass through the underground, they are affected by the structure of the medium, and different structures will result in different polarization characteristics. By analyzing the polarization characteristics, the structure and physical properties of the underground medium can be inferred.

[0013] (3) Seismic imaging: Seismic imaging refers to the process of using seismic data to construct a three-dimensional model of the underground medium. Polarization analysis can be used for seismic imaging. Polarization features can be extracted during seismic data processing to determine the physical properties and structure of the underground medium, and then to construct a three-dimensional model of the underground medium.

[0014] (4) Rock mechanics research: Polarization analysis can be used to study the mechanical properties of rocks, such as the stress state of rocks, cracks and faults in rocks, etc. When seismic waves pass through rocks, polarization changes occur. By analyzing the polarization characteristics, the mechanical properties and structure of rocks can be inferred.

[0015] (5) Sediment research: Polarization analysis can be used to study the physical properties of sediments, such as thickness, density, and porosity. Since the physical properties of sediments are related to the waveform polarization characteristics, the physical properties and structure of sediments can be inferred by analyzing the polarization characteristics.

[0016] The existing background noise HV spectrum ratio method is a method of calculating the Fourier spectrum ratio of the horizontal and vertical components of the data recorded by the three-component seismic instrument. It is generally recognized that the peak frequency of the HV spectrum ratio curve is consistent with or close to the resonance frequency in the loose sediment layer, and studies have shown that when the wave impedance interface ratio is large, the near-surface noise source will produce a sharp peak. Therefore, the HV spectrum ratio curve can reflect the velocity structure of the underground, and judge whether there is a strong wave impedance interface underground and calculate the interface depth based on whether there is a peak. However, this method still has some shortcomings. At present, the research on HV spectrum ratio is based on three directions, namely body waves, surface waves and mixed wave fields. When we cannot further judge the wave field that plays a decisive role in the HV spectrum ratio curve, there will be problems such as multiple peaks, inability to judge the peaks, and inability to extract the superior frequencies. Summary of the invention

[0017] In order to solve the above problems, the present invention provides a novel noise HV spectrum ratio imaging method based on polarization, which solves the problem of multiple peaks and unclear peaks when obtaining the HV spectrum ratio due to mixed noise HV spectrum ratio wave fields.

[0018] The present invention is achieved through the following technical solution: a novel polarization-based noise HV spectrum ratio imaging method, comprising the following specific steps:

[0019] Step 1: Collect background noise data in a predetermined monitoring area;

[0020] Step 2: preprocessing the collected background noise data, including data format conversion, data splicing, de-averaging, de-trending, waveform pinching out, and bandpass filtering;

[0021] Step 3: Use polarization analysis technology to process the preprocessed data, separate the wave field by setting thresholds and analyzing power spectra, and distinguish the background noise into linear body wave components and nonlinear surface wave components;

[0022] Step 4: Perform HV spectrum ratio analysis on the separated linear body waves and nonlinear surface waves, and calculate their respective HV spectrum ratios;

[0023] Step 5: Use the obtained HV spectrum ratio to perform profile imaging in the frequency domain according to the relative positions of the stations;

[0024] Step 6: Use the obtained HV spectrum ratio to perform profile imaging in the depth domain according to the relative positions of the stations.

[0025] As a preferred technical solution, in step 2, the bandpass filter is adjusted for a specific frequency range to optimize the response to the specific frequency and enhance the signal-to-noise ratio of the target frequency.

[0026] As a preferred technical solution, in step three, the polarization analysis further includes using a multi-dimensional data processing technology based on time window and frequency selection to ensure the accuracy and efficiency of wave field separation.

[0027] As a preferred technical solution, in step 4, the HV spectrum ratio analysis includes applying an adaptive algorithm to reduce noise in the analysis and optimize the accuracy of the HV spectrum ratio curve.

[0028] As a preferred technical solution, the frequency domain and depth domain profile imaging steps include using high-resolution image processing technology to clearly reveal the velocity structure and wave impedance interface under the surface.

[0029] The data processing system of the present invention comprises:

[0030] Data acquisition module, used to collect background noise in the monitoring area;

[0031] Data processing module, equipped with software for data preprocessing, polarization analysis, HV spectrum ratio calculation and profile imaging;

[0032] The display module is used to display the results of profile imaging and provide an interactive operation interface.

[0033] As a preferred technical solution, the data processing module includes at least one microprocessor and a related storage device, and the storage device stores instructions and algorithms for executing the above steps.

[0034] The beneficial effects of the present invention are as follows: the present invention significantly improves the accuracy and efficiency of underground structure analysis by adopting a polarization-based noise HV spectrum ratio imaging method, optimizes the data processing flow, allows automatic generation of frequency and depth profiles, and further improves the simplicity of operation and the reliability of results;

[0035] By automatically picking up the outstanding frequencies and peaks of the HV spectrum ratio, the present invention can more accurately analyze and identify the underground wave impedance interface and velocity structure, which is of great significance for the prevention of geological disasters and the exploration of resources. Automated processing not only reduces human errors, but also speeds up data processing, making the analysis of seismic data more efficient and accurate.

[0036] The present invention utilizes polarization analysis to significantly optimize the processing and analysis of seismic data, improves imaging accuracy, enhances the application value of data, and brings innovative technological advances to the fields of seismology and geological exploration. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0038] Figure 1 This is a comparison diagram of the numerical simulation waveform before and after polarization processing of the present invention;

[0039] Figure 2 This is a comparison diagram of the vector end diagrams before and after polarization of the numerical simulation waveform of the present invention;

[0040] Figure 3 This is a comparison diagram of the actual station waveform before and after polarization processing of the present invention;

[0041] Figure 4 This is a comparison diagram of a single HV spectrum ratio curve before and after polarization processing of the present invention;

[0042] Figure 5 It is a frequency domain profile diagram of the HV spectrum ratio of the present invention;

[0043] Figure 6 It is a depth domain profile diagram of the HV spectrum ratio of the present invention. DETAILED DESCRIPTION

[0044] All features disclosed in this specification, or steps in all methods or processes disclosed, except mutually exclusive features and / or steps, can be combined in any manner.

[0045] Any feature disclosed in this specification (including any additional claims, abstract and drawings), unless otherwise stated, may be replaced by other equivalent or alternative features with similar purposes. That is, unless otherwise stated, each feature is only an example of a series of equivalent or similar features.

[0046] A novel noise HV spectrum ratio imaging method based on polarization of the present invention is based on the idea of ​​wavefield separation, utilizes the polarization characteristics of the wavefield, adopts a polarization method to perform wavefield separation on the background noise, utilizes different thresholds and wavefield quality control, separates the background noise of the whole wavefield into relatively pure body waves and surface waves, and then further calculates the HV spectrum ratio value, obtains the HV spectrum ratio value under different wavefield conditions, and thus improves the imaging resolution and imaging quality, and includes the following specific steps:

[0047] Step 1: Collect background noise data in a predetermined monitoring area;

[0048] Step 2: preprocessing the collected background noise data, including data format conversion, data splicing, de-averaging, de-trending, waveform pinching out, and bandpass filtering;

[0049] Step 3: Use polarization analysis technology to process the preprocessed data, separate the wave field by setting thresholds and analyzing power spectra, and distinguish the background noise into linear body wave components and nonlinear surface wave components;

[0050] Step 4: Perform HV spectrum ratio analysis on the separated linear body waves and nonlinear surface waves, and calculate their respective HV spectrum ratios;

[0051] Step 5: Use the obtained HV spectrum ratio to perform profile imaging in the frequency domain according to the relative positions of the stations;

[0052] Step 6: Use the obtained HV spectrum ratio to perform profile imaging in the depth domain according to the relative positions of the stations.

[0053] Wherein, in step 2, the bandpass filter is adjusted for a specific frequency range to optimize the response to the specific frequency and enhance the signal-to-noise ratio of the target frequency.

[0054] In step three, the polarization analysis further includes using a multi-dimensional data processing technology based on time window and frequency selection to ensure the accuracy and efficiency of wave field separation.

[0055] In step 4, the HV spectrum ratio analysis includes applying an adaptive algorithm to reduce noise in the analysis and optimize the accuracy of the HV spectrum ratio curve.

[0056] The frequency domain and depth domain profile imaging steps include using high-resolution image processing technology to clearly reveal the velocity structure and wave impedance interface under the surface.

[0057] The data processing system of the present invention comprises:

[0058] Data acquisition module, used to collect background noise in the monitoring area;

[0059] Data processing module, equipped with software for data preprocessing, polarization analysis, HV spectrum ratio calculation and profile imaging;

[0060] The display module is used to display the results of profile imaging and provide an interactive operation interface.

[0061] The data processing module includes at least one microprocessor and a related storage device, in which instructions and algorithms for executing the above steps are stored.

[0062] In order to study the feasibility of polarization analysis in HV spectral ratio imaging, we first conducted numerical simulations to verify the feasibility of polarization analysis in wave field separation. Then we used the actual deployed stations to perform polarization processing, and finally analyzed the results by comparing the results before and after processing. The actual data station spacing is 5m, the number of stations is 94, the effective acquisition time is 4 hours, and the sampling frequency is 1000Hz.

[0063] Figure 1 It is a three-component numerical simulation of a single station, with a total of 1200 sample points, of which 1-600 are linear motion (body waves) and 601-1200 are plane elliptical motion (surface waves). Figure 1 Figures a and b are the waveform comparisons before and after processing. We can see that when the threshold T0 is 0.8, we can separate the two wave fields very well.

[0064] Figure 2 This is the comparison of the waveform and the vector diagram before and after the numerical simulation waveform is polarized. Figure 2 a is the original numerical simulation waveform, b is the vector diagram corresponding to the original numerical simulation waveform, through Figure 2 From a and b in the figure, we can see that the original data is divided into the surface wave component of the inverse radial ellipse and the body wave component of linear motion; Figure 2 c and d in the figure are the surface wave components after polarization processing. Figure 2 The e and f in the figure are the body wave components after polarization processing; by comparison, it can be seen that polarization can well separate surface waves from body waves from the perspective of wave field.

[0065] Figure 3 This is a comparison of the actual three-component short-period station waveform before and after polarization. Figure 3 Figures a and b are the waveform comparisons before and after polarization processing. We can see that when the threshold R0 is 0.2, we can separate the body wave and the surface wave well. Figure 3 Figures c and d are continuous waveforms over a longer period of time. The polarized waveforms have clearer relative amplitudes and clearer energy.

[0066] Figure 4 This is a comparison of the single HV spectrum ratio curve before and after polarization. The blue and red are the comparison of the single HV spectrum ratio curve before and after polarization treatment. We can see that the HV spectrum ratio curve after polarization treatment has a larger ratio than before treatment, more peaks, clearer curve changes, and carries more medium information. This is very beneficial for subsequent imaging, inversion, extraction of superior frequencies, estimation of sediment thickness and other data processing.

[0067] Figure 5Figures a and b are the waveform and variable area HV spectrum ratio profile in the frequency domain, respectively, where 0-20Hz and 20-100Hz are displayed at equal intervals. We can clearly see that there is a clear phase axis in the 0-20Hz range, which extends from west to east and from deep to shallow. There is an obvious convex structure between 80 and 90 channels, which is speculated to be the uplift of underground rock formations, which is related to the migration of groundwater and stratum movement. There is a continuous energy axis in the 20-100Hz range, with a frequency range of 50-70Hz. According to the empirical formula, the depth range is roughly about 20-30cm, which should be the bluestone and asphalt layers on the surface.

[0068] Figure 6 The empirical formula for HV spectrum ratio frequency domain is Depth = 108f -1.551 From the depth domain profile obtained by depth conversion, we can see that there are continuous positrons in the depth range of 5-15m, among which there are three positrons in tracks 1-40, one continuous positron in tracks 40-70, and a shallower continuous positron in tracks 70-90. The entire structure has obvious faults between tracks 38-44 and 70-75, and it is speculated that hidden faults have occurred at these two locations.

[0069] The above is only a specific embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or substitutions that are not conceived through creative work should be included in the protection scope of the present invention. Therefore, the protection scope of the present invention should be based on the protection scope defined in the claims.

Claims

1. A novel polarization-based noise HV spectrum ratio imaging method, characterized in that: The specific steps include: Step 1: Collect background noise data in a predetermined monitoring area; Step 2: preprocessing the collected background noise data, including data format conversion, data splicing, de-averaging, de-trending, waveform pinching out, and bandpass filtering; Step 3: Use polarization analysis technology to process the preprocessed data, separate the wave field by setting thresholds and analyzing power spectra, and distinguish the background noise into linear body wave components and nonlinear surface wave components; Step 4: Perform HV spectrum ratio analysis on the separated linear body waves and nonlinear surface waves, and calculate their respective HV spectrum ratios; Step 5: Use the obtained HV spectrum ratio to perform profile imaging in the frequency domain according to the relative positions of the stations; Step 6: Use the obtained HV spectrum ratio to perform profile imaging in the depth domain according to the relative positions of the stations.

2. The novel polarization-based noise HV spectrum ratio imaging method according to claim 1, characterized in that: In step 2, the bandpass filter is adjusted for a specific frequency range to optimize the response to the specific frequency and enhance the signal-to-noise ratio of the target frequency.

3. The novel polarization-based noise HV spectrum ratio imaging method according to claim 1, characterized in that: In step three, the polarization analysis further includes using a multi-dimensional data processing technology based on time window and frequency selection to ensure the accuracy and efficiency of wave field separation.

4. The novel polarization-based noise HV spectrum ratio imaging method according to claim 1, characterized in that: In step 4, the HV spectrum ratio analysis includes applying an adaptive algorithm to reduce noise in the analysis and optimize the accuracy of the HV spectrum ratio curve.

5. The novel polarization-based noise HV spectrum ratio imaging method according to claim 1, characterized in that: The frequency domain and depth domain profile imaging steps include using high-resolution image processing technology to clearly reveal the velocity structure and wave impedance interface under the surface.

6. A data processing system for executing the method according to any one of claims 1 to 5, characterized in that: The system includes: Data acquisition module, used to collect background noise in the monitoring area; Data processing module, equipped with software for data preprocessing, polarization analysis, HV spectrum ratio calculation and profile imaging; The display module is used to display the results of profile imaging and provide an interactive operation interface.

7. The data processing system according to claim 6, characterized in that: The data processing module includes at least one microprocessor and a related storage device, and the storage device stores instructions and algorithms for executing the above steps.

Citation Information

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