CN110865409B - A seismic wave impedance inversion method based on low-rank regularization of wave impedance - Google Patents

A seismic wave impedance inversion method based on low-rank regularization of wave impedance Download PDF

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CN110865409B
CN110865409B CN201911210757.2A CN201911210757A CN110865409B CN 110865409 B CN110865409 B CN 110865409B CN 201911210757 A CN201911210757 A CN 201911210757A CN 110865409 B CN110865409 B CN 110865409B
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陈雷平
李曙
贺达江
丁黎明
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Linyi Gaoxin Tap Water Co ltd
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Abstract

本发明公开了一种基于波阻抗低秩正则化的地震波阻抗反演方法。本发明直接针对地震波阻抗建立低秩正则化项，并在此基础上建立具有波阻抗低秩约束的目标函数，最后巧妙地利用奇异值分解进行求解，利用波阻抗的低秩性进行地震反演，充分利用了地震波阻抗数据的低秩性先验信息且提高了地震波阻抗反演的准确性。

Figure 201911210757

The invention discloses a seismic wave impedance inversion method based on wave impedance low-rank regularization. The invention directly establishes a low-rank regularization term for the seismic wave impedance, and establishes an objective function with a low-rank constraint of the wave impedance on this basis, and finally uses the singular value decomposition to solve the problem cleverly, and uses the low-rank property of the wave impedance for seismic inversion. , making full use of the low-rank prior information of seismic impedance data and improving the accuracy of seismic impedance inversion.

Figure 201911210757

Description

Seismic wave impedance inversion method based on wave impedance low-rank regularization
Technical Field
The invention belongs to the field of seismic exploration, and particularly relates to a seismic wave impedance inversion method based on wave impedance low-rank regularization.
Background
Seismic wave impedance is an important earth medium parameter that is numerically equal to the product of the density of the subsurface rock and the velocity of the seismic waves as they propagate in the rock. It is closely related to the lithology, porosity, etc. of the rock. Therefore, how to effectively and accurately obtain the seismic wave impedance is always an important research content in seismic exploration. The seismic wave impedance inversion technology reversely deduces the wave impedance of the underground medium by using prior information according to data such as earthquake, well logging and the like obtained by observation, and is the most main way for obtaining the seismic wave impedance at present. With the development of seismic acquisition technology and signal processing technology, the quality of seismic data is higher and higher. The quality of the seismic inversion results is more and more obviously limited by the utilization of prior information. How to mine and effectively use prior information becomes a hot problem in the current seismic inversion field.
The low rank nature of the data is an important a priori information that has been widely focused and used in recent years. Such as: a large number of literature reports based on data low-rank prior exist in the fields of voice and image signal denoising, seismic data denoising/reconstruction and the like. In the field of seismic exploration, the low-rank characteristic is mainly used for the aspects of seismic data denoising, missing seismic channel completion, seismic data reconstruction and the like. In these studies, the low rank property of seismic data was utilized. Namely: the matrix formed by combining the seismic data of different sub-blocks with similar waveforms in the seismic data has the characteristic of low rank. When the low rank property is used, the denoising/reconstruction effect of the data is obviously improved compared with the traditional method. Thus, these methods experimentally demonstrate the advantage of low rank as a priori information for use in seismic data processing. It should be noted that these seismic data processing methods only utilize the low rank property of seismic data in the data domain. The output results after they are processed are still only seismic data.
Since seismic data has low rank properties between different sub-blocks, seismic data is closely related to the nature and structure of the subsurface rock formations. The former is a signalized representation of the latter. If there is similarity between different sub-blocks of the former, the corresponding position of the latter should also have such similarity. In this case, the matrix formed by the wave resistances of the sub-blocks, such as wave impedance, density, poisson ratio, and the like, has low rank. Low rank is a non-localized a priori information. In electromagnetic parametric inversion, the idea of using low rank approximation is mentioned in the article "A low-rank approximation for large-scale 3D controlled-source electromagnetic Gauss-Newton inversion" published in Geophysics journal by Manual Amaya et al (2016). However, their idea is to use its low rank approximation to replace the Hessian matrix in the inversion process to reduce the computational complexity, and not to directly utilize the low rank of the model parameters to be inverted. At present, no literature report is available on the utilization of the low rank property of wave impedance, and the wave impedance is obtained by using a seismic inversion method according to seismic data.
The existing seismic wave impedance inversion method mainly utilizes the seismic data or the localized prior information of the wave impedance. For example, the sparsity prior information utilized by sparse inversion is a localized prior information. It is a utilization of sparsity of local regions of seismic data or wave impedance under the L0, L1, or Lp (0< p <1) norm, lacking the utilization of a priori information between different sub-regions. In recent years, the seismic wave impedance inversion method based on total variation regularization which is concerned with uses gradient sparsity prior information of wave impedance, but the prior information is also localized, and non-localized prior information is not used.
In sum, the low rank characteristic has outstanding advantages in the fields of signal/data denoising, reconstruction and the like as important non-local prior information, so that the low rank characteristic plays an important role in seismic data processing. The seismic wave impedance should also have low rank. However, none of the existing seismic wave impedance inversion methods utilize this a priori information.
Disclosure of Invention
In order to solve the problems, the invention discloses a seismic wave impedance inversion method based on wave impedance low-rank regularization. The method directly establishes the low-rank regularization item aiming at the seismic wave impedance, establishes the target function with the low-rank constraint of the wave impedance on the basis, finally solves the problem by skillfully utilizing singular value decomposition, performs seismic inversion by utilizing the low-rank property of the wave impedance, fully utilizes the low-rank property of seismic wave impedance data and improves the accuracy of seismic wave impedance inversion.
In order to achieve the purpose, the technical scheme of the invention is as follows:
a seismic wave impedance inversion method based on wave impedance low-rank regularization comprises the following steps:
step one, inputting seismic data and a wave impedance initial model, and constructing an initial low-rank approximate matrix of wave impedance: constructing an initial low-rank approximate matrix by adopting a similarity search-based method;
step two, establishing a wave impedance low-rank regularization target function
Let the whole wave impedance profile be divided into K sub-data blocks with overlap, in which the ith sub-data block is denoted as diThen the wave impedance is adjustedThe optimal low rank approximation problem is expressed as:
Figure BDA0002297996710000021
Φiis a similarity measure and selection operator with diIs a reference block for selecting the sum d from the seismic wave impedance miQ sub-data blocks with the smallest euclidean distance of (d) are arranged into a matrix of size P × Q, P representing sub-data block diLength of (d); oiObtaining an optimal low-rank approximate matrix to be solved; | | non-woven hairFExpressing the Frobenius norm of the matrix, and rank () is the operation of solving the rank of the matrix; λ represents a low-rank constrained regularization parameter;
on the basis of the formula (1), constructing an objective function of a wave impedance low-rank regularization seismic wave impedance inversion problem:
Figure BDA0002297996710000022
g represents a forward operator during seismic wave impedance inversion, and S represents seismic data obtained through observation; eta is the regularization coefficient of the wave impedance low-rank regularization term used for balancing the data fidelity term
Figure BDA0002297996710000023
And the magnitude of the role played by the low-rank regularization term in inversion, wherein the larger the value is, the larger the role played by the low-rank regularization term is;
step three, solving the objective function
For the objective function of the formula (2), the solution is carried out in two steps, firstly, the optimal low rank approximation is solved, namely, the optimal value of the formula (1) is solved, and the optimal low rank approximation of the formula (1)
Figure BDA0002297996710000031
Writing into:
Figure BDA0002297996710000032
solving equation (3) using singular value decomposition and hard threshold shrinkage, let ΦiThe singular value of m is decomposed into:
Φim＝Αdiag(ω)ΒH (4)
a and B respectively represent two unitary matrixes generated after singular value decomposition; omega is a singular value vector obtained by singular value decomposition; diag (ω) represents a diagonal matrix spanned by singular value vectors ω obtained by singular value decomposition, and the element on the diagonal is ω; h represents a conjugate operator; optimal low rank approximation
Figure BDA0002297996710000033
The solution of (a) is written as:
Figure BDA0002297996710000034
where λ represents the threshold used in hard threshold contraction, HTλ() Represents a hard threshold function defined by the formula:
Figure BDA0002297996710000035
where Γ represents the input HTλ() Data in the function;
in finding OiAfter the optimal solution of (3), O is fixed in the objective function of equation (2)iThe update formula of the wave impedance inversion result is obtained as follows:
Figure BDA0002297996710000036
equation (7) is a least squares problem, and equation (7) is formulated with respect to
Figure BDA0002297996710000037
The equation of (c):
Figure BDA0002297996710000038
wherein G is*And phii *Are G and phi respectivelyiUsing conjugate gradient method to quickly solve to obtain inversion result
Figure BDA0002297996710000039
Step four, inversion results are obtained
Figure BDA00022979967100000310
And restoring the original wave impedance model into a 2-dimensional section or a 3-dimensional data volume according to a process opposite to the process of constructing the long column vector m, namely completing the whole inversion process.
In a further improvement, η is determined by a cross-validation method, namely: adjusting according to the difference between the inversion result and the true value, and selecting a value which enables the inversion error to be minimum, namely a value of eta; the seismic data are seismic signals recorded by a geophone.
In a further improvement, in the first step, the step of constructing an initial low-rank approximation matrix of wave impedance comprises the following steps: arranging an initial wave impedance model into a long column vector m according to columns; then determining a sub-data block d as a reference, and assuming that the length of the sub-data block d is P; sequentially selecting Q sub-data blocks with the minimum Euclidean distance from d from m; arranging the Q sub-data blocks into a matrix, and completing construction of an initial low-rank approximate matrix; the initial wave impedance is a 2-dimensional matrix or a 3-dimensional data volume.
Drawings
FIG. 1 is a flow chart of the construction of an initial low rank approximation matrix; in the figure, (a) represents a wave impedance profile, (b) represents similar blocks with similar euclidean distances, and (c) an initial low rank approximation matrix;
FIG. 2 is a flow chart of the present invention;
FIG. 3 is a plot of a portion of data taken from the Marmousi2 model as a true model of wave impedance;
FIG. 4 is an initial model of wave impedance;
FIG. 5 is seismic data for inversion;
FIG. 6 is a Rake wavelet with a dominant frequency of 40 Hz;
fig. 7 is a diagram of inversion results.
Detailed Description
The invention is further explained with reference to the drawings and the embodiments.
Example 1
In order to solve the problems that the existing seismic wave impedance inversion method does not utilize non-local prior information of wave impedance, cannot grasp the essential characteristics of the seismic wave impedance from the global angle and the like, the seismic wave impedance inversion method based on the low-rank regularization of the wave impedance is provided. The core of the method is as follows: the method comprises the steps of directly establishing a low-rank regularization term aiming at seismic wave impedance, establishing a target function with low-rank wave impedance constraint on the basis, and finally solving by skillfully utilizing singular value decomposition. At present, no research report for seismic inversion by using low rank property of wave impedance exists.
(1) Constructing a low rank approximation matrix of wave impedance
And constructing a low-rank approximate matrix by adopting a similarity search-based method. The specific implementation method comprises the following steps: the initial wave impedance model is first arranged column by column into a long column vector m. Then, a sub-data block d is determined as a reference, assuming that it is of length P. And sequentially selecting Q sub-data blocks with the minimum Euclidean distance from d from m. Finally, the Q sub-data blocks are arranged into a matrix, and the obtained matrix is a low-rank matrix. The construction process is shown in figure 1. It should be noted that, in order to observe the similarity between sub-data blocks, the wave impedance in (a) in fig. 1 is not vectorized in advance.
(2) Establishing a wave impedance low-rank regularization target function
Suppose that the entire wave impedance profile can be divided into K sub-data blocks overlapping each other, where the ith sub-data block is denoted as diThen, the wave impedance optimal low rank approximation problem can be expressed as:
Figure BDA0002297996710000051
Φiis a similarity measure and selection operator which uses diIs a reference block for selecting the sum d from the seismic wave impedance miQ sub-data blocks having the smallest euclidean distance of (a) are arranged into a matrix of size P × Q. O isiAnd (4) obtaining an optimal low-rank approximate matrix to be solved. | | non-woven hairFThe Frobenius norm, rank () of the matrix is the operation to find the rank of the matrix. λ represents a low rank constrained regularization parameter. On the basis, an objective function of the wave impedance low-rank regularization seismic wave impedance inversion problem can be constructed:
Figure BDA0002297996710000052
wherein G represents a forward operator during seismic wave impedance inversion, and S represents seismic data obtained by observation. Eta is the regularization coefficient of the wave impedance low-rank regularization term used for balancing the data fidelity term
Figure BDA0002297996710000053
And the magnitude of the role played by the low-rank regularization term in the inversion, wherein the larger the value of the low-rank regularization term is, the larger the role played by the low-rank regularization term is.
(3) Solving of an objective function
For the objective function constructed in the previous step, the solution can be carried out in two steps. The first step is to solve for the optimal low rank approximation, i.e. to solve for the optimal value of:
Figure BDA0002297996710000054
the optimized expression for this equation can be written as:
Figure BDA0002297996710000055
the problem is solved using singular value decomposition and hard threshold shrinkage. Let phiiThe singular value of m is decomposed into:
Φim＝Αdiag(ω)ΒH
wherein, A and B respectively represent two unitary matrixes generated after singular value decomposition. And omega is a singular value vector obtained by singular value decomposition. diag (ω) denotes a diagonal matrix spanned by a singular value vector ω obtained by singular value decomposition, and the element on the diagonal is ω. In view of this, the optimal low rank can be approximated
Figure BDA0002297996710000056
The solution of (a) is written as:
Figure BDA0002297996710000057
where λ represents the threshold used in hard threshold contraction, HTλ() Represents a hard threshold function defined by the formula:
Figure BDA0002297996710000058
where Γ represents the input HTλ() Data in the function.
In finding OiAfter the optimal solution of (3), O is fixed in the objective functioniThe update formula of the wave impedance inversion result can be obtained as follows:
Figure BDA0002297996710000061
this is a least squares problem which can be collated as to
Figure BDA0002297996710000062
The equation of (c):
Figure BDA0002297996710000063
wherein G is*And phii *Are G and phi respectivelyiThe companion operator of (c). This problem can be solvedTo solve quickly using the conjugate gradient method.
Finally, inversion results are obtained
Figure BDA0002297996710000064
And restoring the original wave impedance model into a 2-dimensional section or a 3-dimensional data volume according to a process opposite to the process of constructing the long column vector m, namely completing the whole inversion process.
The specific calculation example of the invention is as follows: part of the data from the Marmousi2 model as a true model of wave impedance is shown in fig. 3, where there are 452 sample points in the depth direction and 512 samples in the distance direction. The initial model of wave impedance is a gaussian low-pass filter of the real model as shown in fig. 4; the seismic data used for inversion is obtained by obtaining a reflection coefficient by using a wave impedance real model, then performing convolution on the reflection coefficient and a Rake wavelet with a main frequency of 40Hz, and adding 20% of white Gaussian noise as shown in FIG. 5. The wavelet is a Rake wavelet with a dominant frequency of 40Hz, as shown in FIG. 6; the results are shown in FIG. 7.
It can be seen that the wave impedance inversion result obtained by the method is very similar to the real wave impedance model, and the boundary surface of the stratum can be clearly depicted in the inversion result.
While embodiments of the invention have been disclosed above, it is not limited to the applications set forth in the description and embodiments, which are fully applicable to various fields of endeavor for which the invention is intended, and further modifications may readily be effected therein by those skilled in the art, without departing from the general concept defined by the claims and their equivalents, which are to be limited not to the specific details shown and described herein.

Claims (3)

1.一种基于波阻抗低秩正则化的地震波阻抗反演方法，其特征在于，包括如下步骤：1. a seismic wave impedance inversion method based on wave impedance low rank regularization, is characterized in that, comprises the steps: 步骤一、输入地震数据、初始波阻抗模型，构建波阻抗模型的初始低秩近似矩阵：采用基于相似性搜索的方法构建初始低秩近似矩阵；Step 1: Input the seismic data and the initial wave impedance model, and construct the initial low-rank approximate matrix of the wave impedance model: the method based on similarity search is used to construct the initial low-rank approximate matrix; 步骤二、建立波阻抗低秩正则化目标函数Step 2. Establish low-rank regularization objective function of wave impedance 设整个波阻抗剖面可以分成K个相互之间有交叠的子数据块，其中的第i个子数据块记为di，则将波阻抗最优低秩近似问题表述为：Assuming that the entire wave impedance profile can be divided into K sub-data blocks that overlap with each other, and the i-th sub-data block is denoted as d i , the optimal low-rank approximation problem of wave impedance can be expressed as:
Figure FDA0003159529690000011
Figure FDA0003159529690000011
 Φi是一个相似性度量与选择算子，以子数据块di为参考块，用于从地震波阻抗m中选择出与di的欧氏距离最小的Q个子数据块，并排列成一个大小为P×Q的矩阵，P表示子数据块di的长度；Oi为待求的最优低秩近似矩阵；|| ||F表示矩阵的Frobenius范数，rank()为求矩阵的秩的运算；λ表示低秩约束的正则化参数；Φ i is a similarity measure and selection operator, with sub-data block d i as a reference block, it is used to select Q sub-data blocks with the smallest Euclidean distance from d i from the seismic wave impedance m, and arrange them into a size is a P×Q matrix, P represents the length of the sub-data block d i ; O i is the optimal low-rank approximate matrix to be found; || || F represents the Frobenius norm of the matrix, and rank() is the rank of the matrix. operation; λ represents the regularization parameter of the low-rank constraint; 在式(1)的基础上，构建波阻抗低秩正则化地震波阻抗反演问题的目标函数：On the basis of Equation (1), the objective function of the low-rank normalized seismic wave impedance inversion problem is constructed:
Figure FDA0003159529690000012
Figure FDA0003159529690000012
 其中，G表示地震波阻抗反演时的正演算子，S表示观测得到的地震数据；η为波阻抗低秩正则项的正则化系数，用于平衡数据保真项
Figure FDA0003159529690000013
和低秩正则项在反演中所起的作用的大小，值越大说明低秩正则项起的作用越大；Among them, G represents the forward operator in seismic wave impedance inversion, S represents the observed seismic data; η is the regularization coefficient of the low-rank regularization term of the wave impedance, which is used to balance the data fidelity term
Figure FDA0003159529690000013
and the size of the role played by the low-rank regular term in the inversion, the larger the value, the greater the role played by the low-rank regular term; 步骤三、目标函数的求解Step 3, the solution of the objective function 对于式(2)的目标函数，分两步进行求解，首先求解最优低秩近似，即求解(1)的最优值，式(1)的最优低秩近似
Figure FDA0003159529690000014
写成：For the objective function of equation (2), it is solved in two steps. First, the optimal low-rank approximation is solved, that is, the optimal value of (1) is solved, and the optimal low-rank approximation of equation (1) is solved.
Figure FDA0003159529690000014
written as:
Figure FDA0003159529690000015
Figure FDA0003159529690000015
 使用奇异值分解和硬阈值收缩求解式(3)，令Φim的奇异值分解为：Solving equation (3) using singular value decomposition and hard threshold shrinkage, let the singular value decomposition of Φ i m be: Φim＝Αdiag(ω)ΒH (4)Φ i m=Αdiag(ω)Β H (4) 其中，A和B分别表示奇异值分解后产生的两个酉矩阵；ω是奇异值分解得到的奇异值向量；diag(ω)表示由奇异值分解得到的奇异值向量ω张成的对角矩阵；H表示共轭算子；最优低秩近似
Figure FDA0003159529690000021
的解写成：Among them, A and B respectively represent two unitary matrices generated by singular value decomposition; ω is the singular value vector obtained by singular value decomposition; diag(ω) represents the diagonal matrix formed by the singular value vector ω obtained by singular value decomposition ; H stands for conjugate operator; optimal low-rank approximation
Figure FDA0003159529690000021
The solution is written as:
Figure FDA0003159529690000022
Figure FDA0003159529690000022
 其中，HTλ()表示硬阈值函数，定义式为：Among them, HT λ () represents the hard threshold function, which is defined as:
Figure FDA0003159529690000023
Figure FDA0003159529690000023
 其中，Γ表示输入HTλ()函数中的数据；Among them, Γ represents the data in the input HT λ () function; 在求得Oi的最优解后，在式(2)目标函数中固定Oi，得到波阻抗反演结果的更新公式为：After obtaining the optimal solution of O i , fix O i in the objective function of equation (2), and the updated formula of the wave impedance inversion result is obtained as:
Figure FDA0003159529690000024
Figure FDA0003159529690000024
 式(7)是一个最小二乘问题，将式(7)整理为关于
Figure FDA0003159529690000025
的方程：Equation (7) is a least squares problem, and formula (7) is organized as
Figure FDA0003159529690000025
The equation for :
Figure FDA0003159529690000026
Figure FDA0003159529690000026
 其中，G*和Φi *分别为G和Φi的伴随算子，使用共轭梯度法快速求解得到反演结果
Figure FDA0003159529690000027
Among them, G * and Φ i * are the adjoint operators of G and Φ i , respectively, and the inversion results are obtained by using the conjugate gradient method to quickly solve
Figure FDA0003159529690000027
 步骤四将反演结果
Figure FDA0003159529690000028
按照与构成地震波阻抗m相反的过程，以初始波阻抗模型的尺寸，恢复成2维剖面或3维数据体，即完成了整个反演过程。Step 4 will invert the result
Figure FDA0003159529690000028
According to the reverse process of constituting the seismic wave impedance m, with the size of the initial wave impedance model, it is restored to a 2-dimensional section or a 3-dimensional data volume, that is, the entire inversion process is completed. 2.如权利要求1所述的基于波阻抗低秩正则化的地震波阻抗反演方法，其特征在于，η通过交叉验证的方法确定，即：根据反演结果与真实值之间的差异而调整，选择出使得反演误差最小的值即η的值。2. The seismic wave impedance inversion method based on low-rank regularization of wave impedance as claimed in claim 1, wherein η is determined by a cross-validation method, that is: adjusted according to the difference between the inversion result and the true value , and select the value that minimizes the inversion error, that is, the value of η. 3.如权利要求1所述的基于波阻抗低秩正则化的地震波阻抗反演方法，其特征在于，所述步骤一中，构建波阻抗的初始低秩近似矩阵步骤如下：先将初始的地震波阻抗m按列排列成一个长的列向量；然后确定一个作为参考的子数据块di，假设子数据块di长度为P；再依次从m中选取与di的欧氏距离最小的Q个子数据块；将Q个子数据块排列成一个矩阵，则完成初始低秩近似矩阵的构建。3. The seismic wave impedance inversion method based on low-rank regularization of wave impedance as claimed in claim 1, wherein in the step 1, the steps of constructing an initial low-rank approximate matrix of wave impedance are as follows: The impedance m is arranged in columns into a long column vector; then a sub-data block d i as a reference is determined, assuming that the length of the sub-data block d i is P; then select the Q with the smallest Euclidean distance from m in turn sub-data blocks; the Q sub-data blocks are arranged into a matrix to complete the construction of the initial low-rank approximate matrix.
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