Anterior ancient life buried hill interior fault identification method based on compressed sensing dip angle transformation
Through the method based on compression-sensing inclination transformation, the problem of weak signals being disturbed by noise in deep seismic data is solved, and the clear identification and signal extraction of the deep seismic data are achieved, which improves the quality of deep seismic data and the accuracy of target recognition.
Patent Information
- Application Number
- CN202310857594.7
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2023-07-13
- Publication Date
- 2025-01-14
AI Technical Summary
In deep and hidden oil and gas reservoir exploration, weak signals in seismic data are interfered with by random noise, resulting in difficulty in signal processing and interpretation. The prior art is not ideal for noise suppression when amplifying weak signals, and compression perception is in the compression reconstruction of seismic signal. There are problems such as cumbersome processing, difficulty in designing observation matrix, and slow reconstruction.
Using a method based on compression-sensing inclination transformation, we can obtain perceived weak signals by reading seismic data, setting the number of samples of compressed sensing coefficients, constructing an initial random matrix for subsampling, performing inclination transformation, extracting the main inclination vector distribution matrix, and repeating processing to obtain perceived weak signals. , realize the identification of the inside faults of the Qianshan Mountain.
This method can quickly build a sparse matrix, clearly identify the inner fault of the latent mountain, effectively extract weak lithologic interface reflections and fault reflection waves, maintain the relative strength of the signal, highlight the relationship between the inner fault system of the latent mountain, and improve the quality of deep seismic data and Target recognition accuracy.
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Abstract
Description
Technical Field
[0001] The invention relates to the technical field of oil and gas exploration, and in particular to a method for identifying internal faults of a Pre-Paleozoic buried hill based on compressed sensing dip angle transformation. Background Art
[0002] With the continuous improvement of seismic acquisition technology, the exploration depth is getting deeper and deeper, and the seismic information contained in seismic data is becoming richer. In the exploration of deep and concealed oil and gas reservoirs, due to the limitations of seismic data acquisition conditions and the influence of complex geological conditions and other factors, weak signals will be generated in the original seismic data, that is, there is a serious random noise in the profile that interferes with the effective signal, which will bring difficulties to the processing and interpretation of seismic data. It is necessary to study the enhancement method of deep seismic weak signals to improve the quality of deep seismic data, which is also the basic work for the accurate identification of exploration target earthquakes.
[0003] Currently, traditional methods include entropy spectrum method, stochastic resonance method, nonlinear superposition method, etc., but these methods are not ideal in suppressing random noise while amplifying weak signals.
[0004] Compressed sensing, also known as compressed sampling or sparse sampling, is a technique for finding sparse solutions to underdetermined linear systems. Compressed sensing is used in electronic engineering, especially signal processing, to acquire and reconstruct sparse or compressible signals. This method uses the sparse characteristics of the signal to restore the original desired signal from fewer measured values compared to the Nyquist theorem.
[0005] The core ideas of compressed sensing theory mainly include two points. The first is the sparse structure of the signal. The traditional Shannon signal representation method only exploits the minimum prior information of the sampled signal, that is, the bandwidth of the signal. However, many widely concerned signals in real life have some structural characteristics. Relative to the freedom of bandwidth information, these structural characteristics are determined by a smaller part of the freedom of the signal. In other words, with little information loss, this signal can be represented by a small number of digital codes. So, in this sense, this signal is a sparse signal (or an approximately sparse signal, a compressible signal). Another point is the irrelevant characteristic. The acquisition of useful information of sparse signals can be completed by compressing the signal into smaller sample data through a non-adaptive sampling method. Theoretically, it is proved that the sampling method of compressed sensing is just a simple operation of correlating the signal with a set of certain waveforms. These waveforms are required to be irrelevant to the sparse space where the signal is located.
[0006] The compressed sensing method discards the redundant information in the current signal sampling. It directly transforms the continuous-time signal to obtain compressed samples, and then uses optimization methods in digital signal processing to process the compressed samples. The optimization algorithm required to recover the signal here is often an underdetermined linear inverse problem with known signal sparseness.
[0007] However, the current compressed sensing in seismic signal compression and reconstruction has some shortcomings, such as cumbersome initial sparse processing process, difficult observation matrix design, and slow compression and reconstruction speed.
[0008] The original meaning of gradient is a vector (vector), which means that the directional derivative of a function at this point has the maximum value along this direction, that is, the function changes fastest along this direction (the direction of this gradient) at this point, and the rate of change is the largest (the modulus of the gradient). By calculating the three-dimensional gradient, the transformed inclination data volume can be obtained, and this transformation is called inclination transformation.
[0009] After the dip transformation, the signal has "sparseness", that is, there are only a small number of large value points where the fault exists (dip domain), and the values of most other points are close to 0. This algorithm highlights the fault information more than the conventional Fourier transform, wavelet transform, etc., so that the faults developed in the buried hill can be quickly and effectively identified.
[0010] In the Chinese patent application with application number: CN201610284067.1, a method and device for reverse time migration processing of seismic data are involved. The method includes: constructing a single frequency Helmholtz matrix of the seismic data according to the medium information, and calculating the inverse matrix of the Helmholtz matrix using the selected sparse matrix inversion algorithm; performing Fourier transform on the single shot data to obtain the corresponding frequency domain single shot data; multiplying the inverse matrix of Helmholtz with the source wavelet of the single shot data to obtain the forward wave field, and multiplying the inverse matrix of Helmholtz with the frequency domain single shot data to obtain the reverse wave field; multiplying the forward wave field with the reverse wave field to obtain the real part data to obtain the single-frequency single shot reverse time migration image data. The various embodiments of the invention are used to solve the dispersion problem when implementing reverse time migration processing, and different media can share the same reverse time migration implementation method, which has good stability, reduces the difficulty of operation, and greatly improves the efficiency of reverse time migration processing.
[0011] In the Chinese patent application with application number: CN201810636774.1, a method and system for processing seismic data are involved. The method includes: preprocessing pre-stack data; constructing an interpolation operator P for selecting seismic traces; constructing a spatial differential operator C for performing differential operations on data along the direction of the event axis and a temporal differential operator Lt for performing differential operations on data along the time direction; constructing an f function with edge-preserving and structure-preserving or smoothing effects; constructing an inversion objective function based on the obtained interpolation operator P, spatial differential operator C, temporal differential operator Lt, and f function; solving the constructed inversion objective function through an optimization algorithm. The invention uses a spatial structural operator to maintain the details of the geological structure, and uses an interpolation operator to interpolate the missing seismic traces, which has played a good effect in the processing of different types of pre-stack seismic data with low signal-to-noise ratio.
[0012] In the Chinese patent application with application number: CN201811094492.X, a fault-constrained tomographic inversion method and inversion system are involved, which includes: step 1: calculating the underground geological dip and fault attributes of the offset profile based on the seismic offset image; step 2: constructing a set of structural constraint pre-conditioning tomographic equations based on the underground geological dip and fault attributes; step 3: solving the set of structural constraint pre-conditioning tomographic equations to obtain the underground velocity model and the offset profile. The fault-constrained tomographic inversion method of the invention realizes the automation of the entire tomographic inversion iterative process, does not require manual intervention, reduces manual workload, shortens the processing cycle, and improves work efficiency; by introducing underground geological dip and fault attributes to constrain the velocity model, the inverted velocity model has a higher accuracy in complex structural areas, especially fault-developed areas, and meets the requirements of subsequent offset imaging.
[0013] The above existing technologies are greatly different from the present invention and fail to solve the technical problem we want to solve. Therefore, we have invented a new method for identifying the inner faults of the Pre-Paleozoic buried hill based on dip angle transformation and compressed sensing. Summary of the invention
[0014] The purpose of the present invention is to provide a method for identifying internal faults of Pre-Paleozoic buried hills based on compressed sensing dip transformation, which can realize accurate identification of underground hidden geological bodies by adaptively enhancing weak seismic signals.
[0015] The purpose of the present invention can be achieved by the following technical measures: a method for identifying internal faults of a pre-Paleozoic buried hill based on compressed sensing dip angle transformation, the method for identifying internal faults of a pre-Paleozoic buried hill based on compressed sensing dip angle transformation comprises:
[0016] Step 1, read the seismic data file;
[0017] Step 2, setting the sampling times K of the compressed sensing coefficient;
[0018] Step 3: Construct an initial random matrix and perform subsampling on the seismic record traces.
[0019] Step 4: Perform dip transformation to obtain the sparse coefficient vector CSQ in the dip domain, and extract the main dip vector distribution matrix CSQ. * ;
[0020] Step 5: For the K CSQs * , take their average to obtain CSQ ** , perform inverse dip transformation on it to obtain the perceived weak signal S * .
[0021] The object of the present invention can also be achieved by the following technical measures:
[0022] In Step 1, read the seismic data file, select the weak signal area S where the buried hill interior to be processed is located, which consists of a series of seismic traces S j , j = 1, 2, 3... J.
[0023] In Step 3, perform subsampling on the J seismic record traces S in the selected area with different sparsities.
[0024] In Step 3, when performing subsampling on S, the length of the seismic trace S j is M, randomly generate a random vector C with only N 1s and the rest 0s, with a length of M, N << M, i , i = 1, 2, 3,..., J, to form a matrix Φ, that is:
[0025] C i = (0, 1, 0, 0, 1,... 0, 1, 0... ) T
[0026] C i is a column vector with a length of M, randomly containing N 1s and M - N 0s;
[0027] Φ = {C1 C2 C3... C J-1 C J}
[0028] Φ is a random matrix composed of J column vectors C i .
[0029] In Step 3, process the seismic record trace S, multiply the corresponding components of C i with S i to obtain a new vector CS i , i = 1, 2, 3... J, and all the processed data form a matrix CS, that is:
[0030] CS = S · Φ.
[0031] In step 4, the CS is subjected to an inclination transformation to obtain an inclination domain sparse coefficient vector CSQ. The inclination transformation formula is as follows:
[0032] Constructing the tilt transformation matrix The inclination angle is transformed into: CSQ = Q T CS Q.
[0033] In step 4, for the inclination domain sparse coefficients CSQ, only the 5% with the largest absolute value is retained, and the rest are set to 0, and the main inclination vector distribution matrix CSQ is obtained. * .
[0034] In step 4, repeat steps 3-4 K times to obtain K CSQs. * .
[0035] In step 5, for the K CSQs obtained K times, * , taking the average value to get CSQ ** , perform an inverse tilt transformation to obtain the perceived weak signal S * :
[0036] S * =[QQ T +βI] -1 QCQ ** Q T [QQ T ] -1
[0037] Where β is the damping coefficient, which ranges from 0 to 0.05, and I is the unit matrix.
[0038] The method for identifying internal faults of pre-Paleozoic buried hills based on compressed sensing dip angle transformation further includes, after step 5, outputting new seismic data S * .
[0039] The Pre-Paleozoic buried-hill internal fault dip transform domain compressed sensing identification method of the present invention can quickly construct a sparse matrix, and after processing, the internal fault of the buried-hill is clearly broken.
[0040] Aiming at the dip characteristics and compressible sensing characteristics of the internal signals of the buried hill, the present invention proposes a dip transformation weak signal extraction method under the framework of compressed sensing, and realizes the fault signal recognition. The weak reflections corresponding to the lithologic interface of the stratum and the weak reflections corresponding to the faults in the seismic signal inside the buried hill appear on the dip vector corresponding to the dip transformation domain after compressed sensing sparse sampling. In particular, combined with the characteristic of compressed sensing inverse transformation that only restores signals but not noise, the present invention effectively extracts the weak lithologic interface reflection waves and weak fault reflection waves inside the buried hill. The present invention not only obtains the internal signals of the buried hill, but also maintains the relative strength of the internal signals of the buried hill, and highlights the relationship between the internal fault systems of the buried hill, which is not available in the current petroleum industry. BRIEF DESCRIPTION OF THE DRAWINGS
[0041] Figure 1 It is a flow chart of a specific embodiment of the method for identifying inner faults of Pre-Paleozoic buried hills based on compressed sensing dip transformation of the present invention;
[0042] Figure 2 is a schematic diagram of an original seismic profile in a specific embodiment of the present invention;
[0043] Figure 3 A schematic diagram of a processed seismic profile provided in a specific embodiment of the present invention;
[0044] Figure 4 A schematic diagram of an original seismic profile in another specific embodiment of the present invention;
[0045] Figure 5 It is a schematic diagram of a processed seismic profile provided in another specific embodiment of the present invention. DETAILED DESCRIPTION
[0046] It should be noted that the following detailed descriptions are exemplary and are intended to provide further explanation of the present invention. Unless otherwise specified, all technical and scientific terms used herein have the same meanings as those commonly understood by those skilled in the art to which the present invention belongs.
[0047] It should be noted that the terms used herein are only for describing specific embodiments and are not intended to limit exemplary embodiments according to the present invention. As used herein, unless the context clearly indicates otherwise, the singular form is also intended to include the plural form. In addition, it should be understood that when the terms "comprising" and / or "including" are used in this specification, it indicates the presence of features, steps, operations and / or combinations thereof.
[0048] The method for identifying the inner fault of the Pre-Paleozoic buried hill based on compressed sensing dip angle transformation of the present invention comprises the following steps:
[0049] Step 1: Read the seismic data file;
[0050] Step 2: Set a certain number of sampling times K for the compressive sensing coefficient;
[0051] Step 3: Subsample the J seismic record traces S in the selected area with different sparsities;
[0052] Step 4: Perform dip transformation on CS to obtain the dip-domain sparse coefficient vector CSQ;
[0053] Repeat steps 3 - 4 for K times.
[0054] Step 5: For the K CSQs obtained in K times * , take their average to obtain CSQ ** , perform inverse dip transformation on it to obtain the perceived weak signal S * ;
[0055] S6: Output the new seismic data S * .
[0056] The following are several specific embodiments of applying the present invention
[0057] Embodiment 1
[0058] As Figure 1 shown, Figure 1 is the flow chart of the method for identifying the internal faults of pre-Paleozoic buried hills based on compressive sensing dip transformation of the present invention. The method for identifying the internal faults of pre-Paleozoic buried hills based on compressive sensing dip transformation includes:
[0059] S1: Read the seismic data file, select the weak signal area S where the internal buried hill to be processed is located, which is composed of a series of seismic traces S j (j = 1, 2, 3…J);
[0060] S2: Set a certain number of sampling times K for the compressive sensing coefficient;
[0061] S3: Subsample the J seismic record traces S in the selected area with different sparsities:
[0062] The seismic trace S j has a length of M. Randomly generate a random vector C i (i = 1, 2, 3,…J) with only N 1s and the rest 0s, and a length of M (N << M), and form a matrix Φ, that is:
[0063] C i =(0, 1, 0, 0, 1,…0, 1, 0…) T
[0064] C i is a column vector with a length of M, randomly containing N 1s and M - N 0s;
[0065] Φ={C1 C2 C3…C J-1 C J}
[0066] Φ is J column vector C i A random matrix.
[0067] Process the seismic record S using C i With S i Multiply the corresponding components to get the new vector CS i (i=1, 2, 3...J), all processed data form a matrix CS, that is:
[0068] CS=S·Φ
[0069] S4: Perform inclination transformation on CS to obtain the inclination domain sparse coefficient vector CSQ. The inclination transformation formula is as follows:
[0070] Constructing the tilt transformation matrix The inclination angle is transformed into: CSQ = Q T Q
[0071] For the inclination domain sparse coefficients CSQ, only the 5% with the largest absolute value is retained, and the rest are set to 0 to obtain the main inclination vector distribution matrix CSQ * .
[0072] Repeat steps 3-4 K times.
[0073] S5: For K CSQs obtained K times * , taking the average value to get CSQ ** , perform an inverse tilt transformation to obtain the perceived weak signal S * :
[0074] S * =[QQ T +βI] -1 QCQ ** Q T [QQ T ] -1
[0075] Where β is the damping coefficient, which ranges from 0 to 0.05, and I is the unit matrix.
[0076] S6: Output new seismic data S*.
[0077] Example 2
[0078] In the specific embodiment 2 of the present invention, the method for identifying the inner fault of the pre-Paleozoic buried hill based on dip angle transformation and compressed sensing includes the following steps:
[0079] S1: Read the seismic data file and select the weak signal area S where the buried hill interior to be processed is located. As shown in Figure 2 , it consists of a series of seismic traces S j (j = 1, 2, 3…J); This figure is the "input seismic data" in Figure 1 , and the white line in Figure 2 is the top of the buried hill. Below the white line is the signal of the buried hill interior, which is chaotic and the reflection signals of the buried hill interior lithologic interface and faults cannot be identified;
[0080] S2: Set a certain compressive sensing coefficient sampling number K = 180;
[0081] S3: Subsample the J seismic traces S in the selected area with different sparsities. For the seismic trace S
[0082] , the length is M = 2500. Randomly generate a random vector C j with only N = 150 1s and the rest 0s, and the length is M = 2500 (N << M) (i = 1, 2, 3,…J), and form a matrix Φ, that is: i
[0083] C i =(0, 1, 0, 0, 1,…0, 1, 0…) T
[0084] C i is a column vector with length M, randomly containing N 1s and M - N 0s;
[0085] Φ={C1 C2 C3…C J-1 C J}
[0086] Φ is a random matrix composed of J column vectors C i . Process the seismic trace S, multiply the corresponding components of C
[0087] with S i to obtain a new vector CS i (i = 1, 2, 3…J). All the processed data forms a matrix CS, that is: i
[0088] CS = S·Φ
[0089] S4. Perform dip transformation on CS to obtain the dip domain sparse coefficient vector CSQ. The dip transformation formula is as follows:
[0090] Construct a dip transformation matrix The dip transformation is: CSQ = Q T CS Q
[0091] For the dip domain sparse coefficient CSQ, only keep the 5% with the largest absolute value, and set the rest to 0 to obtain the main dip vector distribution matrix CSQ*.
[0092] Repeat steps 3 - 4 for K times.
[0093] S5. For the K CSQs obtained in K times * , take their average to get CSQ ** , perform an inverse dip transformation on it to obtain the perceived weak signal S * :
[0094] S * = [QQ T + βI] -1 QCSQ ** Q T [QQ T -1
[0095] In the formula, β is the damping coefficient, with a value of 0.05, and I is the identity matrix.
[0096] S6. Output the new seismic data S * , as Figure 3 shown, this figure is Figure 1 the "new seismic data" in Figure 2 , which is the seismic data corresponding to the application of the present invention and can identify the buried hill interior fault. The figure clearly shows the lithological interface reflection and fault signals.
[0097] Embodiment 3
[0098] In the specific Embodiment 3 of applying the present invention, the method for identifying the buried hill interior fault in the pre - Paleozoic based on dip transformation and compressive sensing includes the following steps:
[0099] S1: Read the seismic data file, and select the weak signal area S where the buried hill interior to be processed is located. As Figure 4 shown, it is composed of a series of seismic traces S j (j = 1, 2, 3…J);
[0100] S2: Set a certain compressive sensing coefficient sampling number K = 50;
[0101] S3: Use different sparsities to perform sub - sampling on the J seismic traces S in the selected area:
[0102] The length of the seismic trace S j is M = 1300. Randomly generate a random vector C i (i = 1, 2, 3,…J) with only N = 130 1s and the rest 0s, and a length of M = 1300 (N << M) to form a matrix Φ, that is:
[0103] C i =(0,1,0,1,0,0,…0,1,0,1,1…) T
[0104] C i is a column vector of length M, containing N random 1s and MN random 0s;
[0105] Φ={C1 C2 C3…C J-1 C J}
[0106] Φ is J column vector C i A random matrix.
[0107] Process the seismic record S using C i With S i Multiply the corresponding components to get the new vector CS i (i=1, 2, 3...J), all processed data form a matrix CS, that is:
[0108] CS=S·Φ
[0109] S4. Perform inclination transformation on CS to obtain the inclination domain sparse coefficient vector CSQ. The inclination transformation formula is as follows:
[0110] Constructing the tilt transformation matrix The inclination angle is transformed into: CSQ = Q T Q
[0111] For the inclination domain sparse coefficients CSQ, only the 5% with the largest absolute value is retained, and the rest are set to 0 to obtain the main inclination vector distribution matrix CSQ * .
[0112] Repeat steps 3-4 K times.
[0113] S5. For K CSQs obtained K times * , taking the average value to get CSQ ** , perform an inverse tilt transformation to obtain the perceived weak signal S * :
[0114] S * =[QQ T +βI] -1 QCQ ** Q T [QQ T ] -1
[0115] Where β is the damping coefficient, which is 0.05, and I is the unit matrix.
[0116] S6. Output new seismic data S * ,like Figure 5 As shown in the figure, the lithology interface reflection and the internal reflection of the buried hill are clearly displayed.
[0117] Finally, it should be noted that the above is only a preferred embodiment of the present invention and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art can still modify the technical solutions recorded in the aforementioned embodiments or replace some of the technical features therein with equivalents. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.
[0118] Except for the technical features described in the specification, all other technical features are known technologies to those skilled in the art.
Claims
1. A method for identifying internal faults in pre-Paleozoic buried hills based on compressed sensing dip transformation, characterized in that: The method for identifying internal faults of the Pre-Paleozoic buried hill based on compressed sensing dip angle transformation includes: Step 1, read the seismic data file; Step 2, setting the sampling times K of the compressed sensing coefficient; Step 3, construct an initial random matrix and perform sub-sampling on the seismic traces; Step 4: Perform inclination transformation to obtain the inclination domain sparse coefficient vector CSQ, and extract the main inclination vector distribution matrix CSQ * ; Step 5: For K CSQs * , taking the average value to get CSQ ** , perform an inverse tilt transformation to obtain the perceived weak signal S * .
2. The method for identifying internal faults of Pre-Paleozoic buried hills based on compressed sensing dip transformation according to claim 1 is characterized in that: In step 1, read the seismic data file, select the weak signal area S where the buried hill is located to be processed, and select a series of seismic traces S j , j=1,2,3…J.
3. The method for identifying internal faults of Pre-Paleozoic buried hills based on compressed sensing dip transformation according to claim 1 is characterized in that: In step 3, different sparsities are used to subsample S of the J seismic traces S in the selected area.
4. The method for identifying internal faults of Pre-Paleozoic buried hills based on compressed sensing dip transformation according to claim 3 is characterized in that: In step 3, when subsampling S, the seismic trace S j has a length of M, and a random vector C of length M is randomly generated, with only N 1s and the rest 0s, where N << M i , i = 1, 2, 3, … J, to form a matrix Φ, that is: C i =(0,1,0,0,1,…0,1,0…) T C i is a column vector of length M, containing N random 1s and MN random 0s; Φ={C1 C2 C3…C J-1 C J } Φ is J column vector C i A random matrix.
5. The method for identifying internal faults of Pre-Paleozoic buried hills based on compressed sensing dip transformation according to claim 4 is characterized in that: In step 3, the seismic trace S is processed using C i With S i Multiply the corresponding components to get the new vector CS i , i = 1, 2, 3...J, all processed data form a matrix CS, that is: CS=S·Φ.
6. The method for identifying internal faults of Pre-Paleozoic buried hills based on compressed sensing dip transformation according to claim 5 is characterized in that: In step 4, the CS is subjected to an inclination transformation to obtain an inclination domain sparse coefficient vector CSQ. The inclination transformation formula is as follows: Constructing the tilt transformation matrix The inclination angle is transformed into: CSQ = Q T CS Q.
7. The method for identifying internal faults of Pre-Paleozoic buried hills based on compressed sensing dip transformation according to claim 6 is characterized in that: In step 4, for the inclination domain sparse coefficients CSQ, only the 5% with the largest absolute value is retained, and the rest are set to 0, and the main inclination vector distribution matrix CSQ is obtained. * .
8. The method for identifying internal faults of Pre-Paleozoic buried hills based on compressed sensing dip transformation according to claim 7 is characterized in that: In step 4, repeat steps 3-4 K times to obtain K CSQs. * .
9. The method for identifying internal faults of Pre-Paleozoic buried hills based on compressed sensing dip transformation according to claim 8, characterized in that: In step 5, for the K CSQs obtained K times, * , taking the average value to get CSQ ** , perform an inverse tilt transformation to obtain the perceived weak signal S * : S * =[QQ T +βI] -1 QCSQ ** Q T [QQ T ] -1 Where β is the damping coefficient, which ranges from 0 to 0.05, and I is the unit matrix.
10. The method for identifying internal faults of Pre-Paleozoic buried hills based on compressed sensing dip transformation according to claim 1, characterized in that: The method for identifying internal faults of pre-Paleozoic buried hills based on compressed sensing dip angle transformation further includes, after step 5, outputting new seismic data S * .
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