Multi-component seismic data reconstruction method and device
By combining Z-transform and vector adaptive prediction filter, the problem of neglecting component relationships in multi-component seismic data reconstruction is solved, achieving high-precision data reconstruction and ensuring the integrity of the seismic wavefield.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- CHINA NAT PETROLEUM CORP
- Filing Date
- 2024-10-25
- Publication Date
- 2026-04-28
AI Technical Summary
Existing multi-component seismic data interpolation methods ignore the relationships between components, leading to inaccurate reconstructed data and even disrupting the complete structure of the seismic wavefield.
The local dip angle of multi-component seismic data is obtained by Z-transform, a vector adaptive prediction filter is constructed, prediction filtering interpolation is performed using this filter, and inverse Z-transform is performed to finally reconstruct the multi-component seismic data.
Identifying potential correlations between components enables high-precision reconstruction of multi-component irregular missing seismic signal data, providing a solid data foundation.
Smart Images
Figure CN121934149A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of seismic signal processing technology, specifically to a method for reconstructing multi-component seismic data, a device for reconstructing multi-component seismic data, an electronic device, and a machine-readable storage medium. Background Technology
[0002] To meet the needs of complex oil and gas exploration, seismic exploration technology has gradually developed into the multi-component exploration stage. Multi-component seismic exploration simultaneously records two horizontal components and one vertical component of the wavefield, yielding richer information about the subsurface medium and providing more accurate results for identifying lithology, porosity, fractures, and gas-bearing properties. Multi-component exploration technology has broad application prospects in solving complex lithological problems, studying stratigraphic anisotropy, and identifying fractured oil and gas reservoirs.
[0003] In the field of seismic data interpolation and reconstruction, multi-component seismic exploration, like single-component exploration, also faces the problem of regularized reconstruction of seismic data. However, current interpolation methods for multi-component seismic data mainly focus on processing each component separately, decomposing the vector field signal into several scalar field signals, ignoring the relationship between the components in the multi-component vector data, and may even destroy the complete structure of the seismic wave field, resulting in inaccurate reconstructed data. Summary of the Invention
[0004] The purpose of this invention is to provide a method and apparatus for reconstructing multi-component seismic data, so as to at least solve the problem that the current interpolation methods for multi-component seismic data mainly focus on processing each component separately, decomposing the vector field signal into several scalar field signals, ignoring the relationship between the components in the multi-component vector data, and even destroying the complete structure of the seismic wave field, resulting in inaccurate reconstructed data.
[0005] To achieve the above objectives, a first aspect of the present invention provides a method for reconstructing multi-component seismic data, the method comprising: Acquire multi-component seismic data to be reconstructed; The multi-component seismic data is subjected to Z-transform to obtain the transformed multi-component seismic data; Based on the transformed multi-component seismic data, the local dip angle of each component is obtained; Based on the local tilt angle of each component, a vector adaptive prediction filter is constructed; The transformed multi-component seismic data is subjected to predictive filtering and interpolation using a vector adaptive predictive filter to obtain the reconstructed vector wavefield. The reconstructed vector wavefield is subjected to an inverse Z-transform to obtain the reconstructed multi-component seismic data.
[0006] Optionally, based on the transformed multi-component seismic data, the local dip angle of each component is obtained, including: Based on the transformed multi-component seismic data, a prediction operator is constructed using local plane wave theory; Based on the prediction operator, a linear iterative optimization method is applied to calculate the local tilt angle of each component as it varies with time and space.
[0007] Optionally, a vector adaptive prediction filter is constructed based on the local tilt angle of each component, including: Based on the local tilt angle of each component, the plane wave decomposition filter for each component is obtained; A vector adaptive prediction filter is constructed based on the plane wave decomposition filter of each component.
[0008] Optionally, the expression for the plane wave decomposition filter is: ; in, For components Plane wave decomposition filter; For components Plane wave decomposition filter; For components Plane wave decomposition filter; For components The local tilt angle; For components The local tilt angle; For components The local tilt angle; For components The wave field; For components The wave field; For components The wave field; For unit space delay operators; This is a unit time delay operator.
[0009] Optionally, the expression for the vector adaptive prediction filter is: ; in, It is a vector adaptive prediction filter; , , , , , , , , All are weight coefficients of the vector adaptive prediction filter; For components Plane wave decomposition filter; For components Plane wave decomposition filter; For components A plane wave decomposition filter.
[0010] Optionally, after constructing the vector adaptive prediction filter, the method further includes: An objective function is constructed based on the coefficients of the vector adaptive prediction filter and the shaping regularization constraint. The objective function is solved using the conjugate gradient method, and the coefficients of the adaptive prediction filter are iteratively updated until a preset number of iterations are reached. The coefficients corresponding to the minimum solution of the objective function are then taken as the optimal coefficients of the vector adaptive prediction filter.
[0011] Optionally, the expression for the objective function is: ; in, For mask operators; This is the transformed multi-component seismic data; These are the coefficients of the vector adaptive prediction filter; This refers to the missing seismic data in the transformed multi-component seismic data. To impose shaping regularization constraints on the coefficients of the vector adaptive prediction filter; is the regularization coefficient.
[0012] Optionally, a vector adaptive prediction filter is used to perform predictive filtering interpolation on the transformed multi-component seismic data to obtain the reconstructed vector wavefield, including: The reconstructed vector wave field is obtained using the following formula: ; in, For components The reconstructed vector wave field; For components The reconstructed vector wave field; For components The reconstructed vector wave field; , , , , , , , , All are weight coefficients of the vector adaptive prediction filter; For components Plane wave decomposition filter; For components Plane wave decomposition filter; For components Plane wave decomposition filter; For components The wave field; For components The wave field; For components The wave field; For unit space delay operators; This is a unit time delay operator.
[0013] A second aspect of the present invention provides an apparatus for reconstructing multi-component seismic data, the apparatus comprising: The data acquisition module is used to acquire multi-component seismic data to be reconstructed; The first transformation module is used to perform Z-transform on the multi-component seismic data to obtain transformed multi-component seismic data. The local dip angle determination module is used to obtain the local dip angle of each component based on the transformed multi-component seismic data. The filter building module is used to build vector adaptive prediction filters based on the local tilt angle of each component; The data reconstruction module is used to perform predictive filtering and interpolation on the transformed multi-component seismic data using a vector adaptive prediction filter to obtain the reconstructed vector wavefield. The second transformation module is used to perform an inverse Z-transform on the reconstructed vector wavefield to obtain the reconstructed multi-component seismic data.
[0014] A third aspect of the present invention provides an electronic device including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the above-described method for reconstructing multi-component seismic data.
[0015] On the other hand, the present invention provides a machine-readable storage medium storing instructions for causing a machine to perform the reconstruction method of multi-component seismic data as described in any of the preceding claims.
[0016] This technical solution can identify the potential correlations and differences between components during the reconstruction of multi-component data, and generate high-precision reconstruction results of multi-component irregular missing seismic signal data, providing good data for subsequent multi-component signal processing.
[0017] Other features and advantages of the embodiments of the present invention will be described in detail in the following detailed description section. Attached Figure Description
[0018] The accompanying drawings are provided to further illustrate embodiments of the present invention and form part of the specification. They are used together with the following detailed description to explain the embodiments of the present invention, but do not constitute a limitation thereof. In the drawings: Figure 1 This is a flowchart of the multi-component seismic data reconstruction method provided by the present invention; Figure 2 This is a structural diagram of the multi-component seismic data reconstruction device provided by the present invention; Figure 3 This is a schematic diagram of the complete seismic data provided by the present invention; Figure 4 This is a schematic diagram of earthquake data with 60% irregular random missing data provided by the present invention; Figure 5 This is a schematic diagram of the reconstructed seismic data provided by the present invention.
[0019] Explanation of reference numerals in the attached figures 10 - Data acquisition module; 20 - First transformation module; 30 - Local tilt angle determination module; 40 - Filter construction module; 50 - Data reconstruction module; 60 - Second transformation module. Detailed Implementation
[0020] The specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings. It should be understood that the specific embodiments described herein are for illustration and explanation only and are not intended to limit the present invention.
[0021] Figure 1 This is a flowchart of the multi-component seismic data reconstruction method provided by the present invention; Figure 2 This is a structural diagram of the multi-component seismic data reconstruction device provided by the present invention; Figure 3 This is a schematic diagram of the complete seismic data provided by the present invention; Figure 4 This is a schematic diagram of earthquake data with 60% irregular random missing data provided by the present invention; Figure 5 This is a schematic diagram of the reconstructed seismic data provided by the present invention.
[0022] like Figure 1 As shown, an embodiment of the present invention provides a method for reconstructing multi-component seismic data, the method comprising: Step 1: Obtain the multi-component seismic data to be reconstructed; Step 2: Perform Z-transform on the multi-component seismic data to obtain the transformed multi-component seismic data; Step 3: Based on the transformed multi-component seismic data, obtain the local dip angle of each component; Step 4: Construct a vector adaptive prediction filter based on the local tilt angle of each component; Step 5: Use a vector adaptive prediction filter to perform predictive filtering interpolation on the transformed multi-component seismic data to obtain the reconstructed vector wavefield; Step 6: Perform an inverse Z-transform on the reconstructed vector wavefield to obtain the reconstructed multi-component seismic data.
[0023] Specifically, in this embodiment, the multi-component seismic data to be reconstructed is the common-detector point data received by a multi-component detector with irregular time-domain missing data, including two horizontal components. , and a vertical component ; Specifically, the multi-component seismic data is subjected to a Z-transform to obtain transformed multi-component seismic data, which is based on the horizontal component. , and vertical components Perform Z-transform separately, specifically for the horizontal component. Taking Z-transform as an example: Based on the theory of local plane waves, with spatial and temporal sampling intervals of... and In a sampling system, the slope of the discrete space is defined as... As a unit, represented as , It is the local slope of a continuous space. Because... p Regardless of the sampling interval, it can be directly used for irregular datasets; the time delay between two adjacent locations is the slope. We can obtain:
[0024] Will When the transformation is applied to time and space orientation, the above equation becomes:
[0025] in, For unit time delay operators, Represents the unit space delay operator. for Z-transform; Similarly, sampling is performed on the horizontal component. The method of performing Z-transform yields Z-transform results ,as well as Z-transform results .
[0026] Furthermore, based on the transformed multi-component seismic data, the local dip angle of each component is obtained, including: Based on the transformed multi-component seismic data, a prediction operator is constructed using local plane wave theory; Based on the prediction operator, a linear iterative optimization method is applied to calculate the local tilt angle of each component as it varies with time and space.
[0027] Specifically, this includes constructing a tilt-dependent prediction operator using local plane wave theory, and applying a linear iterative optimization method to calculate the two horizontal components separately. , and a vertical component The local tilt angle that varies with time and space.
[0028] Furthermore, based on the local tilt angle of each component, a vector adaptive prediction filter is constructed, including: Based on the local tilt angle of each component, the plane wave decomposition filter for each component is obtained; A vector adaptive prediction filter is constructed based on the plane wave decomposition filter of each component.
[0029] Among them, the horizontal component Z-transform results Calculate local tilt angle plane wave decomposition filter For example: The plane wave decomposition equation has a nonlinear equation regarding the local slope:
[0030] It can also be written in vector form:
[0031] in Wavefield of known data The approximate equality implies minimizing the left-hand side. By analytically linearizing the equation, linear iterative optimization methods can be applied to solve the linear system.
[0032] in, For the slope increment, This is the initial value for slope estimation. It is a derivative of the equation concerning of Convolution filter coefficients. Initial slope after solving the system. By increasing This update allows us to solve the linear problem once again.
[0033] Similarly, the horizontal component is thus obtained. and vertical components Inclination angle and and plane wave decomposition filter and .
[0034] Specifically, the expression for the plane wave decomposition filter is:
[0035] in, For components Plane wave decomposition filter; For components Plane wave decomposition filter; For components Plane wave decomposition filter; For components The local tilt angle; For components The local tilt angle; For components The local tilt angle; For components The wave field; For components The wave field; For components The wave field; For unit space delay operators; This is a unit time delay operator.
[0036] Furthermore, for multi-component signals, we define the M-order non-stationary non-causal predictive filter as follows:
[0037] In the formula, It is space The vector at point M is a three-component vector. Here, M and i are the number and index of the surrounding record channels, respectively. , and Let x and y represent two horizontal components and one vertical component, respectively, while x and y represent spatial axes. For a three-component signal, the local tilt angles of different components are calculated separately. Using these as constraints, a multi-component vector filter is constructed. The prediction filter coefficients can be represented by the convolution of a 3×3 coefficient matrix and a diagonal matrix related to the tilt angle, yielding the expression for the vector adaptive prediction filter:
[0038] in, It is a vector adaptive prediction filter; , , , , , , , , All are weight coefficients of the vector adaptive prediction filter; For components Plane wave decomposition filter; For components Plane wave decomposition filter; For components A plane wave decomposition filter.
[0039] Furthermore, since the coefficients of the vector adaptive prediction filter affect the reconstruction results of seismic data, in this embodiment, after constructing the vector adaptive prediction filter based on the local dip angle of each component, the method further includes: An objective function is constructed based on the coefficients of the vector adaptive prediction filter and the shaping regularization constraint. The objective function is solved using the conjugate gradient method, and the coefficients of the adaptive prediction filter are iteratively updated until a preset number of iterations are reached. The coefficients corresponding to the minimum solution of the objective function are then taken as the optimal coefficients of the vector adaptive prediction filter. After obtaining the optimal coefficients of the vector adaptive prediction filter in the above manner, data reconstruction can be performed, which can effectively ensure the accuracy of the reconstructed seismic data.
[0040] Specifically, the expression for the objective function is:
[0041] in, For mask operators; This is the transformed multi-component seismic data; These are the coefficients of the vector adaptive prediction filter; This refers to the missing seismic data in the transformed multi-component seismic data. To impose shaping regularization constraints on the coefficients of the vector adaptive prediction filter; is the regularization coefficient.
[0042] In this embodiment, the filter coefficients are updated by minimizing the difference between the known portion of the missing seismic data in the transformed multi-component seismic data and the interpolated data in each iteration. In this iteration process, the missing seismic data in the transformed multi-component seismic data can be preset known seismic data. In order to obtain the coefficients of the optimal vector adaptive prediction filter, the coefficients are updated in each iteration.
[0043] Among them, for the mask operator For seismic data interpolation with non-steady spatial variations, when seismic data is missing, shaping regularization can make the coefficients of the adaptive prediction filter change spatially. Maintain smoothness. Therefore, filter coefficients can be estimated from the coefficients of adjacent channels. This can be achieved by constructing a chosen mask operator matrix. This is achieved by using a matrix where the missing data position is 1 and the known data position is 0. The adaptive prediction filter coefficient matrix and... Multiplication yields a mask operator that can recognize autoregressive equations and solve boundary problems. Taking one-dimensional data as an example, take... for The filter coefficients are 3. Let... Given a matrix where the filter coefficients are 1 in one position and 0 in all other positions, we can obtain:
[0044] in, Each value represents how many inputs are missing. A value of 0 indicates that the regression equation was used to estimate the filter coefficients. Let 1 represent a value of 0, and 0 represent all other values.
[0045] Furthermore, a vector adaptive prediction filter is used to perform predictive filtering interpolation on the transformed multi-component seismic data to obtain the reconstructed vector wavefield, including: The reconstructed vector wave field is obtained using the following formula:
[0046] in, For components The reconstructed vector wave field; For components The reconstructed vector wave field; For components The reconstructed vector wave field; , , , , , , , , All are weight coefficients of the vector adaptive prediction filter; For components Plane wave decomposition filter; For components Plane wave decomposition filter; For components Plane wave decomposition filter; For components The wave field; For components The wave field; For components The wave field; For unit space delay operators; This is a unit time delay operator.
[0047] More specifically, in the process of obtaining the reconstructed vector wavefield, the missing data can be interpolated by solving the linear inversion problem using the multi-component prediction filter coefficients. Seismic data interpolation can be achieved by solving the following least-squares problem:
[0048] Simultaneously satisfy
[0049] in, It is transformed multi-component seismic data that needs to be reconstructed. The coefficients of the vector adaptive prediction filter; This refers to the missing seismic data in the transformed multi-component seismic data. These are known seismic traces (obtainable from collected seismic data); It is the mask matrix of the valid equation.
[0050] Through the above method, the accuracy of the interpolation results gradually increases with the iteration process, and the method for solving the inverse problem is more accurate than the direct interpolation method in existing technologies. like Figure 2 As shown, embodiments of the present invention also provide a reconstruction apparatus for multi-component seismic data, the apparatus comprising: Data acquisition module 10 is used to acquire multi-component seismic data to be reconstructed; The first transformation module 20 is used to perform Z-transform on the multi-component seismic data to obtain transformed multi-component seismic data. The local dip angle determination module 30 is used to obtain the local dip angle of each component based on the transformed multi-component seismic data. The filter construction module 40 is used to construct a vector adaptive prediction filter based on the local tilt angle of each component; Data reconstruction module 50 is used to perform predictive filtering interpolation on the transformed multi-component seismic data using a vector adaptive prediction filter to obtain the reconstructed vector wavefield. The second transformation module 60 is used to perform an inverse Z-transform on the reconstructed vector wave field to obtain reconstructed multi-component seismic data.
[0051] This invention also provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the above-described method for reconstructing multi-component seismic data.
[0052] This invention also provides a readable storage medium storing instructions for causing a machine to execute the above-described method for reconstructing multi-component seismic data.
[0053] In one specific implementation, such as Figure 3 The diagram shows a complete set of seismic data, representing (left) Z-component seismic data and (right) X-component seismic data. The seismic data described above was processed by randomly removing 60% of the data, resulting in the following... Figure 4 The earthquake data shown is used to reconstruct the earthquake data using this scheme, and the final result is as follows. Figure 5 The prediction filtering reconstruction results are shown below. (Through, as shown...) Figure 3 and Figure 5 The comparison shows that after reconstructing the seismic data using the vector adaptive prediction filter of this scheme, the complete seismic data and the reconstructed seismic data are quite similar. Therefore, the reconstruction method of multi-component seismic data in this scheme has high data reconstruction accuracy.
[0054] Those skilled in the art will understand that all or part of the steps in the methods of the above embodiments can be implemented by a program instructing related hardware. This program is stored in a storage medium and includes several instructions to cause a microcontroller, chip, or processor to execute all or part of the steps of the methods described in the various embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as a USB flash drive, a portable hard drive, a read-only memory (ROM), a random access memory (RAM), a magnetic disk, or an optical disk.
[0055] Those skilled in the art will clearly understand that, for the sake of convenience and brevity, the above-described division of functional units and modules is merely an example. In practical applications, the above functions can be assigned to different functional units and modules as needed, that is, the internal structure of the device can be divided into different functional units or modules to complete all or part of the functions described above. The functional units and modules in the embodiments can be integrated into one processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit. The integrated unit can be implemented in hardware or as a software functional unit. Furthermore, the specific names of the functional units and modules are only for easy distinction and are not intended to limit the scope of protection of this invention.
[0056] The optional embodiments of the present invention have been described in detail above with reference to the accompanying drawings. However, the embodiments of the present invention are not limited to the specific details described above. Within the scope of the technical concept of the embodiments of the present invention, various simple modifications can be made to the technical solutions of the embodiments of the present invention, and these simple modifications all fall within the protection scope of the embodiments of the present invention. It should also be noted that the various specific technical features described in the above specific embodiments can be combined in any suitable manner without contradiction. To avoid unnecessary repetition, the embodiments of the present invention will not further describe the various possible combinations.
[0057] Furthermore, various different embodiments of the present invention can be combined in any way, as long as they do not violate the spirit of the embodiments of the present invention, they should also be regarded as the content disclosed by the embodiments of the present invention.
Claims
1. A method for reconstructing multi-component seismic data, characterized in that, The method includes: Acquire multi-component seismic data to be reconstructed; The multi-component seismic data is subjected to Z-transform to obtain the transformed multi-component seismic data; Based on the transformed multi-component seismic data, the local dip angle of each component is obtained; Based on the local tilt angle of each component, a vector adaptive prediction filter is constructed; The transformed multi-component seismic data is subjected to predictive filtering and interpolation using a vector adaptive predictive filter to obtain the reconstructed vector wavefield. The reconstructed vector wavefield is subjected to an inverse Z-transform to obtain the reconstructed multi-component seismic data.
2. The method for reconstructing multi-component seismic data according to claim 1, characterized in that, Based on the transformed multi-component seismic data, the local dip angle of each component is obtained, including: Based on the transformed multi-component seismic data, a prediction operator is constructed using local plane wave theory; Based on the prediction operator, a linear iterative optimization method is applied to calculate the local tilt angle of each component as it varies with time and space.
3. The method for reconstructing multi-component seismic data according to claim 1, characterized in that, Based on the local tilt angle of each component, a vector adaptive prediction filter is constructed, including: Based on the local tilt angle of each component, the plane wave decomposition filter for each component is obtained; A vector adaptive prediction filter is constructed based on the plane wave decomposition filter of each component.
4. The method for reconstructing multi-component seismic data according to claim 3, characterized in that, The expression for the plane wave decomposition filter is: ； in, For components Plane wave decomposition filter; For components Plane wave decomposition filter; For components Plane wave decomposition filter; For components The local tilt angle; For components The local tilt angle; For components The local tilt angle; For components The wave field; For components The wave field; For components The wave field; For unit space delay operators; This is a unit time delay operator.
5. The method for reconstructing multi-component seismic data according to claim 1, characterized in that, The expression for the vector adaptive prediction filter is: ； in, It is a vector adaptive prediction filter; , , , , , , , , All are weight coefficients of the vector adaptive prediction filter; For components Plane wave decomposition filter; For components Plane wave decomposition filter; For components Plane wave decomposition filter.
6. The method for reconstructing multi-component seismic data according to claim 1, characterized in that, After constructing the vector adaptive prediction filter, the method further includes: An objective function is constructed based on the coefficients of the vector adaptive prediction filter and the shaping regularization constraint. The objective function is solved using the conjugate gradient method, and the coefficients of the adaptive prediction filter are iteratively updated until a preset number of iterations are reached. The coefficients corresponding to the minimum solution of the objective function are then taken as the optimal coefficients of the vector adaptive prediction filter.
7. The method for reconstructing multi-component seismic data according to claim 6, characterized in that, The expression for the objective function is: ； in, For mask operators; This is the transformed multi-component seismic data; These are the coefficients of the vector adaptive prediction filter; This refers to the missing seismic data in the transformed multi-component seismic data. To impose shaping regularization constraints on the coefficients of the vector adaptive prediction filter; is the regularization coefficient.
8. The method for reconstructing multi-component seismic data according to claim 1, characterized in that, The transformed multi-component seismic data is subjected to predictive filtering and interpolation using a vector adaptive prediction filter to obtain the reconstructed vector wavefield, including: The reconstructed vector wave field is obtained using the following formula: ； in, For components The reconstructed vector wave field; For components The reconstructed vector wave field; For components The reconstructed vector wave field; , , , , , , , , All are weight coefficients of the vector adaptive prediction filter; For components Plane wave decomposition filter; For components Plane wave decomposition filter; For components Plane wave decomposition filter; For components The wave field; For components The wave field; For components The wave field; For unit space delay operators; This is a unit time delay operator.
9. A reconstruction device for multi-component seismic data, characterized in that, The device includes: The data acquisition module is used to acquire multi-component seismic data to be reconstructed; The first transformation module is used to perform Z-transform on the multi-component seismic data to obtain transformed multi-component seismic data. The local dip angle determination module is used to obtain the local dip angle of each component based on the transformed multi-component seismic data. The filter building module is used to build vector adaptive prediction filters based on the local tilt angle of each component; The data reconstruction module is used to perform predictive filtering and interpolation on the transformed multi-component seismic data using a vector adaptive prediction filter to obtain the reconstructed vector wavefield. The second transformation module is used to perform an inverse Z-transform on the reconstructed vector wavefield to obtain the reconstructed multi-component seismic data.
10. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the method for reconstructing multi-component seismic data as described in any one of claims 1-8.
11. A readable storage medium storing instructions for causing a machine to perform a method for reconstructing multi-component seismic data as described in any one of claims 1-8.