CN117055107B - Seismic interpolation method based on interaction of Framelet transformation and Lp pseudo-norms - Google Patents
Seismic interpolation method based on interaction of Framelet transformation and Lp pseudo-norms Download PDFInfo
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- G01V1/28—Processing seismic data, e.g. for interpretation or for event detection
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- G01V1/307—Analysis for determining seismic attributes, e.g. amplitude, instantaneous phase or frequency, reflection strength or polarity
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Abstract
The invention discloses a method based on the Framelet transformation and l p A pseudo-norm interaction seismic interpolation method relates to the field of oil and gas exploration and seismic data processing, and solves the problem that details of seismic data are ignored in the existing sparse transform-based method, and further adverse effects are caused on processing and interpretation of high-resolution data. The invention includes constructing a Framelet transform function F; performing the Framelet transformation on the seismic data, and performing the l on the seismic data after the Framelet transformation p Pseudo-norm constraint, constructing an objective function of seismic interpolation; determining and updating update formulas of the seismic data, the Lagrangian multiplier and the dual variables; and determining a final seismic data interpolation result according to the relation among the seismic data, the Lagrangian multiplier and the dual variables. The method is used for seismic data interpolation.
Description
Technical Field
The invention relates to the field of oil and gas exploration and seismic data processing, in particular to a method based on the Framelet transformation and the seismic data processingA pseudo-norm interaction seismic interpolation method.
Background
In seismic exploration, data acquisition is the most basic step. The quality of the acquired data affects the understanding of the subsurface structure. However, due to the limitation of the acquisition conditions, the problem of missing traces cannot be ignored, especially in areas with complex geological conditions, such as the Qiang pond basin. Incomplete seismic data will impair subsequent steps such as amplitude and offset analysis and interpretation of results. Therefore, seismic data reconstruction is necessary.
The seismic data interpolation method based on the sparse transformation is rapid in development. This approach does not require a priori information required for wave equation based approaches, and is relatively simple to calculate compared to predictive filtering based approaches. This approach assumes that the seismic data is sparse in a certain region, and missing traces break the sparsity. They reconstruct the dataThe problem is considered an optimization problem with sparse constraints. Constraint is generally associated withThe norms are related. Common transformations include Fourier transforms, radon transforms, curvelet transforms, wavelet transforms, seilet transforms, streamlet transforms, and shearlet transforms. These transformations provide a good tool for analyzing sparsity of seismic data. While these transformations have been widely used in practical processing, they tend to ignore details of the seismic data. Such loss of detail may adversely affect the processing and interpretation of the high resolution data.
Disclosure of Invention
The invention aims at: the invention provides a method based on the Framelet transformation sumThe seismic interpolation method based on pseudo-norm interaction solves the problem that details of seismic data are ignored in the existing sparse transform-based method, and further adverse effects are generated on processing and interpretation of high-resolution data, and the interpolation effect is improved.
The technical scheme adopted by the invention for achieving the purpose is as follows:
step 1: building a Framelet transform function；
Step 2: performing the Framelet transformation on the seismic data, and performing the Framelet transformation on the seismic dataPseudo-norm constraint, constructing an objective function of seismic interpolation;
step 3: determining seismic dataLagrangian multiplier term->And dual variables>Updating formulas of the three and +.>Lagrangian multiplier term->And dual variables>Updating;
processing the seismic data update sub-problem by using the cost Ma Yinli to finish the process ofIs updated according to the update of (a);
updating sub-problem of Lagrangian multiplier sub-item to complete pairing according to generalized contraction methodIs updated according to the update of (a);
the update sub-problem of the pair-pair variable is paired according to the alternate direction multiplier methodIs updated according to the update of (a);
step 4: and determining a final seismic data interpolation result according to the relation among the seismic data, the Lagrangian multiplier and the dual variables.
Step 1 comprises the following steps:
step 1.1: order theRepresenting a scale function->And->Representing two wavelet functions, < >>Representing scale factors, scale functionsThe specific definitions of the numbers and wavelet functions are as follows equation (1) and equation (2):
（1）
（2）
wherein the method comprises the steps of，/>And->Low-pass and high-pass filters for scale functions and wavelet functions, < >>Meaning 0,1,2;
step 1.2: from equation (1) and equation (2), the equation for the Framelet transform can be expressed as:
is a signal>And->A low frequency part and a high frequency part of the scale function and the wavelet function, respectively, < >>Representation->sub-Framelet transform +_>And->The specific expression of (2) is as follows:
（4）
（5）
step 2 comprises the following steps:
step 2.1: the relationship between the complete seismic data and the observations is represented by the following equation (6):
（6）
in the method, in the process of the invention,representing the complete seismic data, which is also the result expected by interpolation,/for example>Representing the observation result->Representing a sampling matrix, which may be a diagonal matrix;
step 2.2: by usingThe pseudo-norm is used as a main body of a reconstruction method, sparse constraint is carried out on the seismic data after the Framelet transformation, and an objective function of seismic interpolation is constructed, wherein the objective function is shown in the following formula (7):
（7）
wherein the method comprises the steps ofIs Lagrangian factor, +.>For the Framelet transform +_>，/>Representation->Norms (F/F)>Representation ofPseudo-norms;
step 2.3: solving equation (7) using the alternate direction multiplier method, equation (7) becomes whereinRepresentation ofThe Lagrangian multiplier term:
（8）
equation (8) is further expressed as:
（9）
is a dual variable, ++>Is->After that, the formula (9) is decomposed into three sub-problems as the following formulas (10) - (12), -, respectively>Is a dual parameter, its value is defined by itself;
（10）
（11）
（12）
step 3 comprises the following steps:
step 3.1: equation (10) is typicalThe norm problem, let equation (13) equal 0, can be applied to +.>Updating is performed as shown in the following formula (14):
（13）
（14）
wherein the method comprises the steps ofIs unit momentArray (S)>The method is a transpose of the Framelet transformation, so that the method is simplified, and is convenient to solve by using a conjugate gradient method;
step 3.2: solving the formula (11) by adopting a generalized contraction methodIs updated by the following formula:
（15）
step 3.3: according to the alternate direction multiplier method, can obtainThe updated formula of (c) should be:
（16）
step 4 comprises the following steps:
step 4.1: judgingIf true, let seismic data +.>Lagrangian multiplier term->Dual variable->The method comprises the steps of carrying out a first treatment on the surface of the If not, output ++>As a final interpolation result.
The invention has the following beneficial effects because the technical scheme is adopted:
the invention adopts the Framelet transformation to groundTransforming the seismic data by adoptingThe pseudo-norm is used as a main body of a reconstruction method, sparse constraint is carried out on the seismic data after the Framelet transformation, and a method based on the Framelet transformation and ++is provided by combining an alternate direction multiplier method>A pseudo-norm interaction seismic interpolation method. The method solves the problem that details of the seismic data are ignored in the existing sparse transform-based method, and further adverse effects are generated on processing and interpretation of high-resolution data, and achieves the effect of improving the signal-to-noise ratio of seismic data reconstruction.
Drawings
FIG. 1 shows a frame-based transform sum according to the present inventionA flow chart of a pseudo-norm interaction seismic interpolation method;
FIG. 2 is a seismic section of raw seismic data of the invention
FIG. 3 is a seismic section of the invention with a random loss of 40% of the seismic data;
FIG. 4 is an interpolation of a seismic section of the invention with a random loss of 40% of the seismic data;
fig. 5 is a residual of the interpolation result of the present invention.
Detailed Description
Hereinafter, embodiments of the present invention will be described in detail. While the invention will be described and illustrated in conjunction with certain specific embodiments, it will be understood that it is not intended to limit the invention to these embodiments alone. On the contrary, the invention is intended to cover modifications and equivalent arrangements included within the scope of the appended claims.
In addition, numerous specific details are set forth in the following description in order to provide a better illustration of the invention. It will be understood by those skilled in the art that the present invention may be practiced without these specific details.
The invention is described in detail below in connection with fig. 1-5.
The invention solves the technical problems that: the method solves the problem that details of the seismic data are ignored in the existing sparse transform-based method, and further adverse effects are generated on processing and interpretation of high-resolution data, and achieves the effect of improving the signal-to-noise ratio of seismic data reconstruction.
Technical means: based on Framelet transformation sumA pseudo-norm interaction seismic interpolation method comprises the following steps of
Step 1: building a Framelet transform function；
Step 2: performing the Framelet transformation on the seismic data, and performing the Framelet transformation on the seismic dataPseudo-norm constraint, constructing an objective function of seismic interpolation;
step 3: determining seismic dataLagrangian multiplier term->And dual variables>Updating formulas of the three and +.>Lagrangian multiplier term->And dual variables>Updating;
step 4: and determining a final seismic data interpolation result according to the relation among the seismic data, the Lagrangian multiplier and the dual variables.
Step 1 comprises the following steps:
step 1.1: order theRepresenting a scale function->And->Representing two wavelet functions, < >>The specific definitions of scale factors, scale functions and wavelet functions are represented by the following formulas (1) and (2):
（1）
（2）
wherein the method comprises the steps of，/>And->Low-pass and high-pass filters for scale functions and wavelet functions, < >>Meaning 0,1,2;
step 1.2: from equation (1) and equation (2), the equation for the Framelet transform can be expressed as:
is a signal>And->A low frequency part and a high frequency part of the scale function and the wavelet function, respectively, < >>Representation->sub-Framelet transform +_>And->The specific expression of (2) is as follows:
（4）
（5）
step 2 comprises the following steps:
step 2.1: the relationship between the complete seismic data and the observations is represented by the following equation (6):
（6）
in the method, in the process of the invention,representing the complete seismic data, which is also the result expected by interpolation,/for example>Representing the observation result->Representing a sampling matrix, which may be a diagonal matrix;
step 2.2: by usingThe pseudo-norm is used as a main body of a reconstruction method, sparse constraint is carried out on the seismic data after the Framelet transformation, and an objective function of seismic interpolation is constructed, wherein the objective function is shown in the following formula (7):
（7）
wherein the method comprises the steps ofIs Lagrangian factor, +.>For the Framelet transform +_>，/>Representation->Norms (F/F)>Representation ofPseudo-norms;
step 2.3: solving equation (7) using the alternate direction multiplier method, equation (7) becomes whereinRepresentation ofThe Lagrangian multiplier term:
（8）
equation (8) is further expressed as:
（9）
is a dual variable, ++>Is->After that, the formula (9) is decomposed into three sub-problems as the following formulas (10) - (12), -, respectively>Is a dual parameter, its value is defined by itself;
（10）
（11）
（12）
step 3 comprises the following steps:
step 3.1: equation (10) is typicalThe norm problem, let equation (13) equal 0, can be applied to +.>Updating is performed as shown in the following formula (14):
（13）
（14）
wherein the method comprises the steps ofIs a unitary matrix->The method is a transpose of the Framelet transformation, so that the method is simplified, and is convenient to solve by using a conjugate gradient method;
step 3.2: solving the formula (11) by adopting a generalized contraction methodIs updated by the following formula:
（15）
step 3.3: according to the alternate direction multiplier method, can obtainThe updated formula of (c) should be:
（16）
step 4 comprises the following steps:
step 4.1: judgingIf true, let seismic data +.>Lagrangian multiplier term->Dual variable->The method comprises the steps of carrying out a first treatment on the surface of the If not, output ++>As a final interpolation result.
The technical effects are as follows: the invention adopts the Framelet transformation to transform the seismic data, adopts the following steps ofThe pseudo-norm is used as a main body of a reconstruction method, sparse constraint is carried out on the seismic data after the Framelet transformation, and a method based on the Framelet transformation and ++is provided by combining an alternate direction multiplier method>A pseudo-norm interaction seismic interpolation method. The method solves the problem that details of the seismic data are ignored in the existing sparse transform-based method, and further adverse effects are generated on processing and interpretation of high-resolution data, and achieves the effect of improving the signal-to-noise ratio of seismic data reconstruction.
Example 1
As shown in fig. 1-5, the seismic data interpolation is as follows:
step 1.1: order theRepresenting a scale function->And->Representing two wavelet functions, < >>The specific definitions of scale factors, scale functions and wavelet functions are represented by the following formulas (1) and (2):
（1）
（2）
wherein the method comprises the steps of，/>And->Low-pass and high-pass filters for scale functions and wavelet functions, < >>Meaning 0,1,2;
step 1.2: from equation (1) and equation (2), the equation for the Framelet transform can be expressed as:
is a signal>And->A low frequency part and a high frequency part of the scale function and the wavelet function, respectively, < >>Representation->sub-Framelet transform +_>And->The specific expression of (2) is as follows:
（4）
（5）
step 2.1: the relationship between the complete seismic data and the observations is represented by the following equation (6):
（6）
in the method, in the process of the invention,representing the complete seismic data, which is also the result expected by interpolation,/for example>Representing the observation result->Representing a sampling matrix, which may be a diagonal matrix;
step 2.2: by usingThe pseudo-norm is used as a main body of a reconstruction method, sparse constraint is carried out on the seismic data after the Framelet transformation, and an objective function of seismic interpolation is constructed, wherein the objective function is shown in the following formula (7):
（7）
wherein the method comprises the steps ofIs Lagrangian factor, +.>For the Framelet transform +_>，/>Representation->Norms (F/F)>Representation ofPseudo-norms;
step 2.3: solving equation (7) using the alternate direction multiplier method, equation (7) becomes whereinRepresentation ofThe Lagrangian multiplier term:
（8）
equation (8) is further expressed as:
（9）
is a dual variable, ++>Is->After that, the formula (9) is decomposed into three sub-problems as the following formulas (10) - (12), -, respectively>Is a dual parameter, its value is defined by itself;
（10）
（11）
（12）
step 3.1: equation (10) is typicalThe norm problem, let equation (13) equal 0, can be applied to +.>Updating is performed as shown in the following formula (14):
（13）
（14）
wherein the method comprises the steps ofIs a single sheetBit matrix,/->The method is a transpose of the Framelet transformation, so that the method is simplified, and is convenient to solve by using a conjugate gradient method;
step 3.2: solving the formula (11) by adopting a generalized contraction methodIs updated by the following formula:
（15）
step 3.3: according to the alternate direction multiplier method, can obtainThe updated formula of (c) should be:
（16）
step 4.1: judgingIf true, let seismic data +.>Lagrangian multiplier term->Dual variable->The method comprises the steps of carrying out a first treatment on the surface of the If not, output ++>As a final interpolation result.
And (3) effect analysis: as shown in fig. 2-5, the interpolated result preferably reconstructs missing seismic data, which is closer to the original seismic data, thus proving the correctness of the method. The invention utilizes the Framelet transformation andthe pseudo-norm interaction method performs interpolation processing on the seismic data. The invention adopts the Framlet transformation to transform the seismic data, adopts +.>The pseudo-norm is used as a main body of a reconstruction method, sparse constraint is carried out on the seismic data after the Framelet transformation, and a method based on the Framelet transformation and ++is provided by combining an alternate direction multiplier method>A pseudo-norm interaction seismic interpolation method. The method solves the problem that details of the seismic data are ignored in the existing sparse transform-based method, and further adverse effects are generated on processing and interpretation of high-resolution data, and achieves the effect of improving the signal-to-noise ratio of seismic data reconstruction.
Claims (5)
1. Based on Framelet transformation sumA method of pseudo-norm interaction seismic interpolation comprising the steps of:
step 1: building a Framelet transform function；
Step 2: performing the Framelet transformation on the seismic data, and performing the Framelet transformation on the seismic dataThe pseudo-norm constraint is adopted, an objective function of seismic interpolation is constructed, and the objective function is decomposed into three sub-problems, namely a seismic data updating sub-problem, a Lagrangian multiplier updating sub-problem and a dual variable updating sub-problem;
step 3: determining seismic dataLagrangian multiplier term->And dual variables>Updating formulas of the three and +.>Lagrangian multiplier term->And dual variables>Updating;
processing the seismic data update sub-problem by using the cost Ma Yinli to finish the process ofIs updated according to the update of (a);
updating sub-problem of Lagrangian multiplier sub-item to complete pairing according to generalized contraction methodIs updated according to the update of (a);
the update sub-problem of the dual variable is completed according to the alternate direction multiplication methodIs updated according to the update of (a);
step 4: and determining a final seismic data interpolation result according to the relation among the updated seismic data, the Lagrangian multiplier and the dual variables.
2. A Framelet transform sum based method according to claim 1Pseudo-norm interaction groundThe vibration interpolation method is characterized in that: the step 1 comprises the following steps:
step 1.1: order theRepresenting a scale function->And->Representing two wavelet functions, < >>The specific definitions of scale factors, scale functions and wavelet functions are represented by the following formulas (1) and (2):
（1）
（2）
wherein the method comprises the steps of，/>And->Low-pass and high-pass filters for scale functions and wavelet functions, < >>Meaning 0,1,2;
step 1.2: from equation (1) and equation (2), the equation for the Framelet transform can be expressed as:
（3）
is a signal>And->A low frequency part and a high frequency part of the scale function and the wavelet function, respectively, < >>Representation->sub-Framelet transform +_>And->The specific expression of (2) is as follows:
（4）
（5）。
3. a Framelet transform sum based method according to claim 1The seismic interpolation method of pseudo-norm interaction is characterized in that: the step 2 comprises the following steps:
step 2.1: the relationship between the complete seismic data and the observations is represented by the following equation (6):
（6）
in the method, in the process of the invention,representing the complete seismic data, which is also the result expected by interpolation,/for example>Representing the observation result->Representing a sampling matrix, the sampling matrix being a diagonal matrix;
step 2.2: by usingThe pseudo-norm is used as a main body of a reconstruction method, sparse constraint is carried out on the seismic data after the Framelet transformation, and an objective function of seismic interpolation is constructed, wherein the objective function is shown in the following formula (7):
（7）
wherein the method comprises the steps ofIs Lagrangian factor, +.>For the Framelet transform +_>，/>Representation->Norms (F/F)>Representation->Pseudo-norms;
step 2.3: solving equation (7) using the alternate direction multiplier method, equation (7) becomes whereinRepresentation->The Lagrangian multiplier term:
（8）
equation (8) is further expressed as:
（9）
is a dual variable, ++>Is->After that, the formula (9) is decomposed into three sub-problems as the following formulas (10) - (12), -, respectively>Is a dual parameter, its value is defined by itself;
（10）
（11）
（12）。
4. a Framelet transform sum based method according to claim 3The seismic interpolation method of pseudo-norm interaction is characterized in that: the step 3 comprises the following steps:
step 3.1: equation (10) is typicalThe norm problem, let equation (13) equal 0, can be applied to +.>Updating is performed as shown in the following formula (14):
（13）
（14）
the formula (13) expands the formula (10), and derives m from the expanded formula, and specifically explains the updating step of m;
wherein the method comprises the steps ofIs a unitary matrix->The method is a transpose of the Framelet transformation, so that the formula (10) is simplified, and the method is convenient to solve by using a conjugate gradient method;
step 3.2: solving the formula (11) by adopting a generalized contraction methodIs updated by the following formula:
（15）
step 3.3: according to the alternate direction multiplier method, can obtainThe updated formula of (c) should be:
（16）。
5. a Framelet transform sum based method according to claim 1The seismic interpolation method of pseudo-norm interaction is characterized in that: the step 4 comprises the following steps:
step 4.1: judgingIf true, let seismic data +.>Pulling upGelang Ri multiplier sub-term->Dual variable->The method comprises the steps of carrying out a first treatment on the surface of the If not, output ++>As a final interpolation result.
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|---|---|---|---|---|
| US4594693A (en) * | 1983-11-04 | 1986-06-10 | Mobil Oil Corporation | Seismic trace interpolation using f-k filtering |
| CN110261912A (en) * | 2019-07-23 | 2019-09-20 | 河北地质大学 | The interpolation and denoising method and system of a kind of seismic data |
| CN112526599A (en) * | 2019-09-17 | 2021-03-19 | 中国石油化工股份有限公司 | Wavelet phase estimation method and system based on weighted L1 norm sparsity criterion |
| CN115469364A (en) * | 2022-10-21 | 2022-12-13 | 成都理工大学 | Earthquake denoising method and system based on stable Framelet transformation |
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| EP2784551A3 (en) * | 2013-03-26 | 2015-10-28 | CGG Services SA | System and method for interpolating seismic data by matching pursuit in fourier transform |
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| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US4594693A (en) * | 1983-11-04 | 1986-06-10 | Mobil Oil Corporation | Seismic trace interpolation using f-k filtering |
| CN110261912A (en) * | 2019-07-23 | 2019-09-20 | 河北地质大学 | The interpolation and denoising method and system of a kind of seismic data |
| CN112526599A (en) * | 2019-09-17 | 2021-03-19 | 中国石油化工股份有限公司 | Wavelet phase estimation method and system based on weighted L1 norm sparsity criterion |
| CN115469364A (en) * | 2022-10-21 | 2022-12-13 | 成都理工大学 | Earthquake denoising method and system based on stable Framelet transformation |
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