A tensor higher-order singular value decomposition for prestack seismic data noise reduction and interpolation

Geophysics ◽  
2012 ◽  
Vol 77 (3) ◽  
pp. V113-V122 ◽  
Author(s):  
Nadia Kreimer ◽  
Mauricio D. Sacchi
Keyword(s):  
Fourth Order ◽  
Seismic Data ◽  
Temporal Frequency ◽  
Reduction Process ◽  
Higher Order ◽  
Singular Value ◽  
Low Rank ◽  
High Signal ◽  
Order Tensor ◽  
Fourth Order Tensor

A patch of prestack data depends on four spatial dimensions ([Formula: see text], [Formula: see text] midpoints and [Formula: see text], [Formula: see text] offsets) and frequency. The spatial data at one temporal frequency can be represented by a fourth-order tensor. In ideal conditions of high signal-to-noise ratio and complete sampling, one can assume that the seismic data can be approximated via a low-rank fourth-order tensor. Missing samples were recovered by reinserting data obtained by approximating the original noisy and incomplete data volume with new observations obtained via the rank-reduction process. The higher-order singular value decompostion was used to reduce the rank of the prestack seismic tensor. Synthetic data demonstrated the ability of the proposed seismic data completion algorithm to reconstruct events with curvature. The synthetic example allowed to quantify the quality of the reconstruction for different levels of noise and survey sparsity. We also provided a real data example from the Western Canadian sedimentary basin.

Calculation of Fourth-Order Tensor Product Expansion by Power Method and Comparison of it with Higher-Order Singular Value Decomposition

2013 ◽  
Vol 444-445 ◽  
pp. 703-711
Author(s):  
Akio Ishida ◽  
Takumi Noda ◽  
Jun Murakami ◽  
Naoki Yamamoto ◽  
Chiharu Okuma
Keyword(s):  
Singular Value Decomposition ◽  
Tensor Product ◽  
Fourth Order ◽  
Computation Time ◽  
Higher Order ◽  
Singular Value ◽  
Multidimensional Data ◽  
Power Method ◽  
Order Tensor ◽  
Value Decomposition

Higher-order singular value decomposition (HOSVD) is known as an effective technique to reduce the dimension of multidimensional data. We have proposed a method to perform third-order tensor product expansion (3OTPE) by using the power method for the same purpose as HOSVD, and showed that our method had a better accuracy property than HOSVD, and furthermore, required fewer computation time than that. Since our method could not be applied to the tensors of fourth-order (or more) in spite of having those useful properties, we extend our algorithm of 3OTPE calculation to forth-order tensors in this paper. The results of newly developed method are compared to those obtained by HOSVD. We show that the results follow the same trend as the case of 3OTPE.

scholarly journals Integrity basis for a second-order and a fourth-order tensor

1982 ◽  
Vol 5 (1) ◽  
pp. 87-96 ◽  
Author(s):  
Josef Betten
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Tensor Function ◽  
Order Tensor ◽  
Isotropic Tensor ◽  
Anisotropic Properties ◽  
Isotropic Tensor Function ◽  
Fourth Order Tensor

In this paper a scalar-valued isotropic tensor function is considered, the variables of which are constitutive tensors of orders two and four, for instance, characterizing the anisotropic properties of a material. Therefore, the system of irreducible invariants of a fourth-order tensor is constructed. Furthermore, the joint or simultaneous invariants of a second-order and a fourth-order tensor are found. In a similar way one can construct an integrity basis for a tensor of order greater than four, as shown in the paper, for instance, for a tensor of order six.

Assessing the Use of Tensor Closure Methods With Orientation Distribution Reconstruction Functions

Materials ◽  
2003 ◽  
Author(s):  
David A. Jack ◽  
Douglas E. Smith
Keyword(s):  
Distribution Function ◽  
Fourth Order ◽  
Lower Order ◽  
Series Representation ◽  
Orientation Distribution ◽  
Higher Order ◽  
Interaction Coefficients ◽  
Order Tensor ◽  
Orientation Tensor ◽  
Orientation Tensors

Orientation tensors are widely used to describe fiber distri-butions in short fiber reinforced composite systems. Although these tensors capture the stochastic nature of concentrated fiber suspensions in a compact form, the evolution equation for each lower order tensor is a function of the next higher order tensor. Flow calculations typically employ a closure that approximates the fourth-order orientation tensor as a function of the second order orientation tensor. Recent work has been done with eigen-value based and invariant based closure approximations of the fourth-order tensor. The effect of using lower order tensors tensors in process simulations by reconstructing the distribution function from successively higher order orientation tensors in a Fourier series representation is considered. This analysis uses the property that orientation tensors are related to the series expansion coefficients of the distribution function. Errors for several closures are investigated and compared with errors developed when using a reconstruction from the exact 2nd, 4th, and 6th order orientation tensors over a range of interaction coefficients from 10−4 to 10−1 for several flow fields.

An Improved Seismic Data Completion Algorithm using Low-Rank Tensor Optimization: Cost Reduction and Optimal Data Orientation

Geophysics ◽  
2021 ◽  
pp. 1-88
Author(s):  
Jonathan Popa ◽  
Susan E. Minkoff ◽  
Yifei Lou
Keyword(s):  
Fourier Transform ◽  
Singular Value Decomposition ◽  
Seismic Data ◽  
Convex Sets ◽  
Singular Spectrum Analysis ◽  
Singular Value ◽  
Low Rank ◽  
Traditional Methods ◽  
Value Decomposition ◽  
Conjugate Symmetry

Seismic data are often incomplete due to equipment malfunction, limited source and receiver placement at near and far offsets, and missing cross-line data. Seismic data contain redundancies as they are repeatedly recorded over the same or adjacent subsurface regions, causing the data to have a low-rank structure. To recover missing data one can organize the data into a multidimensional array or tensor and apply a tensor completion method. We can increase the effectiveness and efficiency of low-rank data reconstruction based on the tensor singular value decomposition (tSVD) by analyzing the effect of tensor orientation and exploiting the conjugate symmetry of the multidimensional Fourier transform. In fact, these results can be generalized to any order tensor. Relating the singular values of the tSVD to those of a matrix leads to a simplified analysis, revealing that the most square orientation gives the best data structure for low-rank reconstruction. After the first step of the tSVD, a multidimensional Fourier transform, frontal slices of the tensor form conjugate pairs. For each pair a singular value decomposition can be replaced with a much cheaper conjugate calculation, allowing for faster computation of the tSVD. Using conjugate symmetry in our improved tSVD algorithm reduces the runtime of the inner loop by 35% to 50%. We consider synthetic and real seismic datasets from the Viking Graben Region and the Northwest Shelf of Australia arranged as high-dimensional tensors. We compare tSVD based reconstruction to traditional methods, projection onto convex sets and multichannel singular spectrum analysis, and see that the tSVD based method gives similar or better accuracy and is more efficient, converging with runtimes that are an order of magnitude faster than the traditional methods. Additionally, we verify the most square orientation improves recovery for these examples by 10-20% compared to the other orientations.

Effective diffraction separation using the improved optimal rank-reduction method

Geophysics ◽  
2022 ◽  
pp. 1-85
Author(s):  
Peng Lin ◽  
Suping Peng ◽  
Xiaoqin Cui ◽  
Wenfeng Du ◽  
Chuangjian Li
Keyword(s):  
Seismic Data ◽  
Reduction Method ◽  
Singular Values ◽  
Vital Role ◽  
Singular Value ◽  
Low Rank ◽  
Small Scale ◽  
Frobenius Norm ◽  
Rank Reduction ◽  
Reduction Methods

Seismic diffractions encoding subsurface small-scale geologic structures have great potential for high-resolution imaging of subwavelength information. Diffraction separation from the dominant reflected wavefields still plays a vital role because of the weak energy characteristics of the diffractions. Traditional rank-reduction methods based on the low-rank assumption of reflection events have been commonly used for diffraction separation. However, these methods using truncated singular-value decomposition (TSVD) suffer from the problem of reflection-rank selection by singular-value spectrum analysis, especially for complicated seismic data. In addition, the separation problem for the tangent wavefields of reflections and diffractions is challenging. To alleviate these limitations, we propose an effective diffraction separation strategy using an improved optimal rank-reduction method to remove the dependence on the reflection rank and improve the quality of separation results. The improved rank-reduction method adaptively determines the optimal singular values from the input signals by directly solving an optimization problem that minimizes the Frobenius-norm difference between the estimated and exact reflections instead of the TSVD operation. This improved method can effectively overcome the problem of reflection-rank estimation in the global and local rank-reduction methods and adjusts to the diversity and complexity of seismic data. The adaptive data-driven algorithms show good performance in terms of the trade-off between high-quality diffraction separation and reflection suppression for the optimal rank-reduction operation. Applications of the proposed strategy to synthetic and field examples demonstrate the superiority of diffraction separation in detecting and revealing subsurface small-scale geologic discontinuities and inhomogeneities.

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