Self-adaptive background noise imaging method based on CMP gather strategy

By employing the CMP gather strategy and adaptive stacking technology, the imaging resolution and signal-to-noise ratio problems of background noise imaging technology when the station distribution is sparse or the ray path density is insufficient are solved, thus achieving high-precision imaging of underground structures.

CN121784827APending Publication Date: 2026-04-03CHINESE ACAD OF GEOLOGICAL SCI
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Patent Information

Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
Filing Date
2026-01-14
Publication Date
2026-04-03

AI Technical Summary

Technical Problem

Existing background noise imaging techniques suffer from low imaging resolution in areas where the distribution of stations is not dense enough or the ray path density is insufficient, and linear superposition does not significantly improve the signal-to-noise ratio.

Method used

An adaptive background noise imaging method based on CMP gather strategy is adopted. After deploying seismic arrays in the target area and performing cross-correlation calculations in time segments, the adaptive CMP stacking strategy is used to change the aperture and select the aperture with the strongest dispersion energy and the best continuity for stacking, thereby improving the signal-to-noise ratio and imaging resolution.

Benefits of technology

It improves the signal-to-noise ratio and enhances imaging resolution, especially in areas with insufficient noise time or insufficient ray path density, achieving high-precision imaging of underground structures.

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Abstract

The invention relates to the technical field of background noise processing, and discloses an adaptive background noise imaging method based on a CMP gather strategy, comprising the following specific steps: S1, arranging a seismic array in a target area, and collecting continuous background noise data; s2, preprocessing the background noise data, including data format conversion, data splicing, de-trending, de-averaging, band-pass filtering and spectrum whitening; s3, segmenting the preprocessed data according to time, performing cross-correlation calculation on the data among different stations to obtain a plurality of cross-correlation functions, and performing linear superposition on the cross-correlation functions of a plurality of time periods; according to the adaptive background noise imaging method based on the CMP gather strategy, noise cross-correlation functions are superposed according to the CMP gather superposition principle, so that the purpose of improving the signal-to-noise ratio of signals is achieved, and in addition, an adaptive strategy is applied in the superposition process; the purpose of increasing the number of common center points and improving the signal-to-noise ratio is achieved by changing the station apertures participating in superposition.
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Description

Technical Field

[0001] This invention relates to the field of background noise processing technology, specifically to an adaptive background noise imaging method based on the CMP gather strategy. Background Technology

[0002] Background noise refers to random noise originating from unknown locations, sources, and times in the subsurface. Background noise imaging technology utilizes background noise data received by seismic stations, cross-correlates noise data from different stations at the same time to obtain empirical Green's functions between stations, extracts surface wave dispersion energy from the cross-correlation function, and finally uses the dispersion information to invert and obtain the shear wave velocity structure of the study area. The core theory of this method is to calculate the seismic signal with one station as a virtual source and the other as the receiver by cross-correlation calculation of background noise data from two stations. This signal differs from the Green's functions between stations only in amplitude, and this has been proven through theories such as mode equipartition, time inversion symmetry, and steady-phase approximation.

[0003] Bensen et al. (2007) proposed a background noise imaging process that has been widely accepted by subsequent scholars and has been widely applied in various fields. In regional-scale background noise imaging, Yao et al. (2006) used a temporary seismic network in the southeastern Tibetan Plateau to extract broadband Rayleigh wave phase velocity dispersion curves through background noise tomography and the traditional dual-station method, and used this to invert the velocity structure of the crust and upper mantle in the southeastern Tibetan Plateau. Fang et al. (2010) applied background noise imaging technology to noise data recorded by seismic arrays in North China and obtained Rayleigh wave group velocity distribution maps for 4-30s in North China. In global-scale imaging studies, Yang et al. (2007) selected 125 broadband seismic stations from the global seismic network. Using noise data recorded by these stations, they obtained broadband group velocity dispersion curves using the frequency-wavenumber method, and finally obtained imaging results with higher resolution than traditional seismic imaging. Bensen et al. (2008) used two years of noise data recorded by 203 seismic stations in North America and applied background noise imaging technology to obtain the distribution maps of 8-25s period Lafite and Rayleigh wave group velocities in the study area. Due to its source-independent nature, background noise imaging has also been widely used in monitoring subsurface media, including monitoring earthquakes, faults, and volcanic activity in the Earth's crust.

[0004] In exploration geophysics, the term "stacking" refers to a method of combining multiple geophysical measurements to generate a single output measurement. Assuming the signal is coherent while the noise is incoherent or random, using multiple geophysical measurements instead of a single measurement can reduce the noise level in the final output. Among the stacking techniques commonly used in geophysical measurements, the most widely applied method is stacking seismic reflection data traces acquired at a Common Middle Point (CMP). CMP stacking, proposed by Mayne in 1962, has since become the foundation of seismic reflection data acquisition and processing. The concept of CMP stacking is relatively simple: the distance between the seismic source and the detector is systematically varied so that the same subsurface point is probed multiple times using different source-detector offsets. The result is a CMP gather in which multiple traces image the same middle point, but with different ray paths. The acquired seismic traces are then corrected for normal time difference, followed by summation and normalization to obtain the stacked traces. The superposition process is based on only one assumption: the signal is coherent, while noise is generally random, and under appropriate conditions, superposition will eliminate several types of noise, with the most effective elimination of random noise. Mathematically, the signal-to-noise ratio will improve. The number of stacked traces or coverage times is N. The significant success of CMP stacking technology lies in its ability to separate noise from the signal at the same frequency, whereas almost all other seismic data processing either removes both signal and noise simultaneously or simply modifies the amplitudes of the signal and noise without changing the scale. Shortly after the advent of CMP stacking, many studies began to point out that direct averaging stacking was not the optimal method for all seismic data. Several alternative stacking techniques have been proposed for seismic data acquired in different environments and for seismic data with certain special characteristics. These alternative stacking techniques include Nth root stacking, random stacking, and single-trace iterative stacking.

[0005] Because CMP stacking has a significant effect on improving the signal-to-noise ratio, in recent years people have begun to apply CMP stacking technology to the processing of passive source seismic data. For example, Pang et al. (2022) combined the traditional two-stage method with CMP stacking to suppress the signal-to-noise ratio caused by passive source seismic data. The spurious energy generated by the ambiguity and the broadening of the bandwidth of high-frequency passive surface wave imaging demonstrate the reliability of CMP stacking application in passive source seismic data processing.

[0006] Existing background noise imaging techniques approximate the Green's function between two stations by using the cross-correlation function of noise between them. Currently, the main approach is to improve the signal-to-noise ratio (SNR) by truncating the original data into many segments, cross-correlating them, and then linearly superimposing the results. However, when the original data length is insufficient or the study area is relatively quiet, linear superposition does not significantly improve the SNR. Furthermore, current background noise imaging techniques also suffer from reduced resolution in areas with insufficient station density and ray path density. Summary of the Invention

[0007] To address the problems mentioned in the background art, the present invention provides the following technical solution: an adaptive background noise imaging method based on CMP gather strategy, comprising the following specific steps: S1. Deploy seismic arrays in the target area and collect continuous background noise data; S2. Preprocess the background noise data, including data format conversion, data splicing, detrending, mean removal, bandpass filtering, and spectral whitening; S3. Divide the preprocessed data into time segments, perform cross-correlation calculations on the data between different stations to obtain multiple cross-correlation functions, and linearly superimpose the cross-correlation functions of multiple time periods. S4. With the common center point of the seismic trace remaining unchanged, the CMP stacking aperture is changed so that the range of CMP stacking is changed from small to large. The cross-correlation function after stacking is used for dispersion extraction, and the aperture with the strongest dispersion energy and the best continuity is selected as the optimal CMP aperture. S5. The stations participating in the cross-correlation are respectively used as virtual sources and receivers. An adaptive strategy is adopted during the superposition process to adaptively superimpose the cross-correlation functions at different locations until the optimal CMP aperture is reached. S6. Surface wave dispersion measurement is performed using the superimposed cross-correlation function.

[0008] Furthermore, S3 specifically includes: The preprocessed background noise data is divided into data segments with a duration of one day according to time and saved in sac format. Cross-correlation calculations are performed on the noise data of different stations in the same time period. The cross-correlation calculation involves fixing the data from one station and moving the data from another station along the time axis by one sampling point each time. This moving time is called the delay value. The corresponding points of the overlapping parts of the two signals are multiplied and accumulated to obtain the correlation value under the delay value. By traversing all delay values, multiple cross-correlation functions are obtained. The cross-correlation functions of each data segment are linearly superimposed, which means adding the corresponding points of each cross-correlation function and then dividing by the number of data segments.

[0009] Furthermore, the method for frequency dispersion extraction in S4 is specifically as follows: The dispersion extraction method selected is the dual-stage method. The image conversion algorithm of the dual-stage method is used to measure and extract the dispersion curve of Rayleigh surface waves. The dispersion results of the cross-correlation function of different CMP stacking apertures are compared. The result with the strongest dispersion energy, continuous and smooth change of dispersion curve and less interference is selected as the corresponding aperture as the optimal CMP stacking aperture.

[0010] Furthermore, the implementation of the image conversion algorithm first requires narrowband filtering of the cross-correlation function to filter out signal components with different periods. The filtering results are arranged to form a time-period image with the period on the horizontal axis and time on the vertical axis. The ratio of the distance between the two stations corresponding to the cross-correlation function to time is used as the phase velocity. Then, the time-period image is converted into a velocity-period image, and finally the dispersion result corresponding to the cross-correlation function is obtained.

[0011] Furthermore, S5 specifically includes: The stations participating in the cross-correlation function are respectively used as virtual sources and receivers. The common center point gathers composed of cross-correlation functions are selected based on whether the midpoints between the virtual sources and receivers correspond to the cross-correlation function are consistent, and then superimposed. The points corresponding to the cross-correlation functions participating in the superposition are added together and then divided by the number of channels participating in the superposition. An adaptive strategy is adopted during the superposition process, that is, the CMP aperture is gradually expanded from the edge of the array to the center of the array according to the spatial distribution of the array. The aperture is adjusted according to the principle of achieving the maximum possible number of superposition channels. Adaptive CMP superposition is performed on the cross-correlation functions at different positions until the optimal CMP aperture is reached.

[0012] Compared with the prior art, the present invention has the following beneficial effects: This adaptive background noise imaging method based on the CMP gather strategy improves the signal-to-noise ratio (SNR) by superimposing noise cross-correlation functions according to the CMP gather stacking principle. Furthermore, an adaptive strategy is applied during the stacking process to increase the number of common midpoints by changing the apertures of the participating stations, thus further enhancing the SNR. When the noise time is insufficient and the work area is relatively quiet, the cross-correlation function SNR is low. Superimposing data from different channels in the CMP gather suppresses random signals and improves the SNR. When the ray paths are not dense enough, the final imaging resolution will decrease. An adaptive strategy is used during CMP stacking to simultaneously obtain common midpoint data located at both the station location and the middle of the station, thereby improving the imaging resolution. This invention applies adaptive CMP stacking technology to both simulated and actual data, improving the quality of cross-correlation function dispersion extraction and laying the foundation for high-precision imaging of underground structures. Attached Figure Description

[0013] Figure 1This is a diagram showing the spatial distribution of noise sources and the distribution of stations in the model of this invention; Figure 2 This is a schematic diagram illustrating the variation of the CMP stacked aperture at the common center point in this invention. Figure 3 This is a superimposed dispersion energy diagram of CMP gathers with different apertures according to the present invention; Figure 4 This is a schematic diagram of the CMP adaptive aperture variation of the present invention; Figure 5 This is the CMP adaptive aperture dispersion group velocity energy diagram of the present invention; Figure 6 This is a distribution map of the stations in the target area of ​​this invention; Figure 7 This is a comparison diagram of waveforms and dispersion before and after preprocessing of the original data in this invention; Figure 8 This is a CMP superposition profile of the cross-correlation function of the present invention; Figure 9 The present invention provides an adaptive aperture superposition dispersion energy map and a CMP superposition aperture dispersion energy map with a constant common center point. Figure 10 This is a distribution diagram of the number of stacking times of the CMP gather in this invention. Detailed Implementation

[0014] 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.

[0015] An example of the adaptive background noise imaging method based on the CMP gather strategy is as follows: Please see Figures 1-10 An adaptive background noise imaging method based on the CMP gather strategy includes the following specific steps: S1. Deploy seismic arrays in the target area and collect continuous background noise data; S2. Preprocess the background noise data, including data format conversion, data splicing, detrending, mean removal, bandpass filtering, and spectral whitening; S3. Divide the preprocessed data into time segments, perform cross-correlation calculations on the data from different stations to obtain multiple cross-correlation functions, and then linearly superimpose the cross-correlation functions from multiple time segments, specifically as follows: The preprocessed background noise data is divided into one-day data segments according to time and saved in SAC format. Cross-correlation calculation is performed on the noise data of different stations for the same time period. The cross-correlation calculation involves fixing the data of one station and moving the data of another station along the time axis by one sampling point each time. This time value of movement is called the delay value. The corresponding points of the overlapping parts of the two signals are multiplied and accumulated to obtain the correlation value under the delay value. By traversing all delay values, multiple cross-correlation functions are obtained. The cross-correlation functions of each data segment are linearly superimposed, that is, the corresponding points of each cross-correlation function are added together and then divided by the number of data segments.

[0016] It should be noted that when the signal-to-noise ratio of the cross-correlation function is not high, a phase-weighted superposition method can be considered to improve the signal-to-noise ratio by enhancing the superposition weight of the phase-consistent part of the cross-correlation function.

[0017] S4. With the common center point of the seismic trace remaining unchanged, the CMP stacking aperture is changed so that the range of CMP stacking is changed from small to large. The cross-correlation function after stacking is used for dispersion extraction, and the aperture with the strongest dispersion energy and the best continuity is selected as the optimal CMP aperture. Among them, the dispersion extraction method selected is the dual-stage method. The image conversion algorithm of the dual-stage method is used to measure and extract the dispersion curve of Rayleigh surface wave. The dispersion results of the cross-correlation function of different CMP stacked apertures are compared, and the result with the strongest dispersion energy, continuous and smooth change of dispersion curve and less interference is selected as the corresponding aperture as the optimal CMP stacked aperture.

[0018] When selecting the aperture, the main basis is the dispersion energy map of the superposition results. The point with the strongest energy in each period of the map is extracted to form a dispersion curve. The aperture corresponding to the curve with the highest smoothness is selected as the optimal aperture.

[0019] It should also be noted that the implementation of the image conversion algorithm first requires narrowband filtering of the cross-correlation function (setting a center period every 0.1s, with each center period corresponding to a filter with a width of 0.04s) to filter out signal components of different periods. The filtering results are arranged to form a time-period image with the period on the horizontal axis and time on the vertical axis. The ratio of the distance between the two stations corresponding to the cross-correlation function to time is used as the phase velocity. Then, the time-period image is converted into a velocity-period image, and finally the dispersion result corresponding to the cross-correlation function is obtained.

[0020] S5. The stations participating in the cross-correlation process act as virtual sources and receivers, respectively. During the superposition process, an adaptive strategy is adopted to adaptively superimpose the cross-correlation functions at different locations until the optimal CMP aperture is reached. Specifically: The stations participating in the cross-correlation function are used as virtual sources and receivers, respectively. The common center point gather of the cross-correlation function is selected based on whether the midpoints between the virtual source and receiver are consistent, and then the gathers are superimposed. The points corresponding to the cross-correlation functions participating in the superposition are added together and then divided by the number of channels participating in the superposition. An adaptive strategy is adopted during the superposition process, that is, the CMP aperture is gradually expanded from the edge of the array to the center of the array according to the spatial distribution of the array. The aperture is adjusted according to the principle of achieving the maximum possible number of channels superimposed. Adaptive CMP superposition is performed on the cross-correlation functions at different positions until the optimal CMP aperture is reached.

[0021] S6. Surface wave dispersion measurement is performed using the superimposed cross-correlation function.

[0022] The superposition effect of the above method is verified by setting up a two-layer velocity model, such as... Figure 1 As shown in Figure a, 10,000 Rayleigh and Lafite wave sources are randomly set as noise sources within the model, such as... Figure 1 As shown in Figure b, a linear array of 50 stations was deployed, spaced 4m apart. The parameters for the forward modeling are shown in Table 1. Noise cross-correlation between stations was performed using the waveform data recorded by the linear array to obtain 1225 cross-correlation functions.

[0023] Table 1 Forward Simulation Parameter Table Forward parameters Parameter values Model stratigraphic number 2 Model shear wave velocity First level: 0.25km / s Second level: 1km / s Model P-wave velocity First level: 1.35km / s Second level: 2km / s Model density <![CDATA[First layer: 1.9 g / cm 3 Second layer 2.5 g / cm 3 > Stations 50 Station spacing 4m Distance of the epicenter from the array center 1-1.5km Amplitude range 0-1 Center frequency range 0.5-25Hz Simulation time 3600s Number of earthquake focal points 10000 earthquake source type Rayleigh / Lafbo Superposition of CMP gathers based on the cross-correlation function of simulated data, such as Figure 2 As shown in the figure, the aperture of the CMP gather gradually increases while the common center point position remains unchanged. Dispersion energy is extracted using the superposition results of different apertures with the same common center point, and the superposition effects are compared. Figure 3 As shown, Figure 3 for Figure 2 The five CMP aperture cross-correlation function dispersion extraction results shown are illustrated below. b1-b5 represent the extraction results for apertures ranging from 1 to 5 times the station spacing. Figure 4 As shown, with the increase of CMP aperture, the dispersion energy becomes more concentrated, and the dispersion extraction results are improved.

[0024] To further leverage the advantages of CMP overlay, this invention employs an adaptive strategy during the simulation data overlay process, such as... Figure 4 As shown, a smaller CMP aperture is used during the stacking process near the edge channels, while a larger stacking aperture is selected in the middle of the array. This increases the number of common subsurface points, thereby improving imaging resolution. Figure 5 As shown, Figure 5 This represents the dispersion extraction results of five apertures during adaptive stacking, where... Figure 5 d.1-d.5 and Figure 4The values ​​c.1-c.5 correspond to these values.

[0025] To verify the feasibility of the above-described method of the present invention, the following specific implementation was carried out: Using a seismic survey line approximately 104 km long deployed in the field, comprising 260 short-period seismometers spaced 400 m apart, three-component seismic records for 30 consecutive days were obtained. The station distribution is as follows: Figure 6 As shown.

[0026] First, the raw data collected in the field (i.e., background noise data) is converted to the commonly used SAC format using software such as mseed2sac. Then, the raw data collected in different time periods is merged into a continuous data set, so that it can be divided into equal-length data segments according to time. Figure 7 As shown, the segmented data undergoes preprocessing, where... Figure 7 Figure (a) shows the original data, which has been processed by detrending and mean removal to obtain the final data. Figure 7 The results shown in Figure (b) are then filtered to obtain data in the effective frequency band. Figure 7 The results in (c) show that sharp spikes in the data were removed after filtering. Finally, spectral whitening was performed on the noisy data to broaden its spectrum, and the results are as follows. Figure 7 As shown in (d) in the figure, the effective frequency band spectrum is enhanced.

[0027] The preprocessed background noise data is cross-correlation calculated, such as... Figure 8 As shown, a clear surface wave signal is visible. The asymmetry of the surface wave signal on the time axis is caused by the uneven spatial distribution of the noise source.

[0028] With the common center point of the seismic traces unchanged, the CMP stacking aperture is changed to vary the range of CMP stacking from small to large. Based on the dual-stage method, dispersion extraction is performed using the cross-correlation function after stacking, and the aperture with the strongest dispersion energy and the best continuity is selected as the optimal CMP aperture.

[0029] like Figure 9 As shown, Figure 9 In the table, a.1-a.5 represent the dispersion results of the cross-correlation function of the CMP stacking aperture from the edge of the array to the center of the array with 5, 6, 7, 8, and 9 times the station spacing, respectively, which reflect the adaptive stacking strategy. Figure 9 Figures b.1-b.5 represent the dispersion extraction results of CMP stacking apertures with station spacing of 5, 10, 15, 20, and 25 times the common center point, respectively, while keeping the common center point unchanged. The figures show that higher quality dispersion extraction results can be obtained when the stacking aperture is sufficiently large. According to... Figure 9The dispersion results show that when the CMP aperture is 25 times the station spacing, the dispersion energy is continuous and there is less interference. At this time, the dispersion extraction effect is the best, and 25 times the station spacing can be used as the optimal superposition aperture.

[0030] After obtaining the optimal stacking aperture, adaptive CMP stacking is performed on the entire array, such as... Figure 10 As shown, the CMP stacking aperture is smaller at both ends of the array, resulting in a smaller number of cross-correlation functions participating in the stacking, while it is larger in the middle of the array.

[0031] 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. An adaptive background noise imaging method based on CMP gather strategy, characterized in that, The specific steps include the following: S1. Deploy seismic arrays in the target area and collect continuous background noise data; S2. Preprocess the background noise data, including data format conversion, data splicing, detrending, mean removal, bandpass filtering, and spectral whitening; S3. Divide the preprocessed data into time segments, perform cross-correlation calculations on the data between different stations to obtain multiple cross-correlation functions, and linearly superimpose the cross-correlation functions of multiple time periods. S4. With the common center point of the seismic trace remaining unchanged, the CMP stacking aperture is changed so that the range of CMP stacking is changed from small to large. The cross-correlation function after stacking is used for dispersion extraction, and the aperture with the strongest dispersion energy and the best continuity is selected as the optimal CMP aperture. S5. The stations participating in the cross-correlation are respectively used as virtual sources and receivers. An adaptive strategy is adopted during the superposition process to adaptively superimpose the cross-correlation functions at different locations until the optimal CMP aperture is reached. S6. Surface wave dispersion measurement is performed using the superimposed cross-correlation function.

2. The adaptive background noise imaging method based on CMP gather strategy according to claim 1, characterized in that: Specifically, S3 is: The preprocessed background noise data is divided into data segments with a duration of one day according to time and saved in sac format. Cross-correlation calculations are performed on the noise data of different stations in the same time period. The cross-correlation calculation involves fixing the data from one station and moving the data from another station along the time axis by one sampling point each time. This moving time is called the delay value. The corresponding points of the overlapping parts of the two signals are multiplied and accumulated to obtain the correlation value under the delay value. By traversing all delay values, multiple cross-correlation functions are obtained. The cross-correlation functions of each data segment are linearly superimposed, which means adding the corresponding points of each cross-correlation function and then dividing by the number of data segments.

3. The adaptive background noise imaging method based on CMP gather strategy according to claim 1, characterized in that: The method for extracting frequency dispersion in S4 is as follows: The dispersion extraction method selected is the dual-stage method. The image conversion algorithm of the dual-stage method is used to measure and extract the dispersion curve of Rayleigh surface waves. The dispersion results of the cross-correlation function of different CMP stacking apertures are compared. The result with the strongest dispersion energy, continuous and smooth change of dispersion curve and less interference is selected as the corresponding aperture as the optimal CMP stacking aperture.

4. The adaptive background noise imaging method based on CMP gather strategy according to claim 3, characterized in that: The implementation of the image conversion algorithm first requires narrowband filtering of the cross-correlation function (setting a center period every 0.1s, with each center period corresponding to a filter with a width of 0.04s) to filter out signal components of different periods. The filtering results are arranged to form a time-period image with the period on the horizontal axis and time on the vertical axis. The ratio of the distance between the two stations corresponding to the cross-correlation function to time is used as the phase velocity. Then, the time-period image is converted into a velocity-period image, and finally the dispersion result corresponding to the cross-correlation function is obtained.

5. The adaptive background noise imaging method based on CMP gather strategy according to claim 1, characterized in that: Specifically, S5 is: The stations participating in the cross-correlation function are respectively used as virtual sources and receivers. The common center point gathers composed of cross-correlation functions are selected based on whether the midpoints between the virtual sources and receivers correspond to the cross-correlation function are consistent, and then superimposed. The points corresponding to the cross-correlation functions participating in the superposition are added together and then divided by the number of channels participating in the superposition. An adaptive strategy is adopted during the superposition process, that is, the CMP aperture is gradually expanded from the edge of the array to the center of the array according to the spatial distribution of the array. The aperture is adjusted according to the principle of achieving the maximum possible number of superposition channels. Adaptive CMP superposition is performed on the cross-correlation functions at different positions until the optimal CMP aperture is reached.