scholarly journals A hybrid finite-difference/low-rank solution to anisotropy acoustic wave equations

Geophysics ◽  
2019 ◽  
Vol 84 (2) ◽  
pp. T83-T91 ◽  
Author(s):  
Zhen-Dong Zhang ◽  
Tariq Alkhalifah ◽  
Zedong Wu
Keyword(s):  
Finite Difference ◽  
Pseudodifferential Operator ◽  
Fourier Transforms ◽  
Pseudodifferential Operators ◽  
Correction Term ◽  
P Wave ◽  
Low Rank ◽  
Low Rank Approximation ◽  
Time Step ◽  
Wavenumber Domain

P-wave extrapolation in anisotropic media suffers from SV-wave artifacts and computational dependency on the complexity of anisotropy. The anisotropic pseudodifferential wave equation cannot be solved using an efficient time-domain finite-difference (FD) scheme directly. The wavenumber domain allows us to handle pseudodifferential operators accurately; however, it requires either smoothly varying media or more computational resources. In the limit of elliptical anisotropy, the pseudodifferential operator reduces to a conventional operator. Therefore, we have developed a hybrid-domain solution that includes a space-domain FD solver for the elliptical anisotropic part of the anisotropic operator and a wavenumber-domain low-rank scheme to solve the pseudodifferential part. Thus, we split the original pseudodifferential operator into a second-order differentiable background and a pseudodifferential correction term. The background equation is solved using the efficient FD scheme, and the correction term is approximated by the low-rank approximation. As a result, the correction wavefield is independent of the velocity model, and, thus, it has a reduced rank compared with the full operator. The total computation cost of our method includes the cost of solving a spatial FD time-step update plus several fast Fourier transforms related to the rank. The accuracy of our method is of the order of the FD scheme. Applications to a simple homogeneous tilted transverse isotropic (TTI) medium and modified BP TTI models demonstrate the effectiveness of the approach.

The optimized expansion based low-rank method for wavefield extrapolation

Geophysics ◽  
2014 ◽  
Vol 79 (2) ◽  
pp. T51-T60 ◽  
Author(s):  
Zedong Wu ◽  
Tariq Alkhalifah
Keyword(s):  
Fourier Transforms ◽  
Transverse Isotropy ◽  
Anisotropic Media ◽  
P Wave ◽  
Low Rank ◽  
Optimization Approach ◽  
Time Step ◽  
Decomposition Approach ◽  
Time Migration ◽  
Wavefield Extrapolation

Spectral methods are fast becoming an indispensable tool for wavefield extrapolation, especially in anisotropic media because it tends to be dispersion and artifact free as well as highly accurate when solving the wave equation. However, for inhomogeneous media, we face difficulties in dealing with the mixed space-wavenumber domain extrapolation operator efficiently. To solve this problem, we evaluated an optimized expansion method that can approximate this operator with a low-rank variable separation representation. The rank defines the number of inverse Fourier transforms for each time extrapolation step, and thus, the lower the rank, the faster the extrapolation. The method uses optimization instead of matrix decomposition to find the optimal wavenumbers and velocities needed to approximate the full operator with its explicit low-rank representation. As a result, we obtain lower rank representations compared with the standard low-rank method within reasonable accuracy and thus cheaper extrapolations. Additional bounds set on the range of propagated wavenumbers to adhere to the physical wave limits yield unconditionally stable extrapolations regardless of the time step. An application on the BP model provided superior results compared to those obtained using the decomposition approach. For transversely isotopic media, because we used the pure P-wave dispersion relation, we obtained solutions that were free of the shear wave artifacts, and the algorithm does not require that [Formula: see text]. In addition, the required rank for the optimization approach to obtain high accuracy in anisotropic media was lower than that obtained by the decomposition approach, and thus, it was more efficient. A reverse time migration result for the BP tilted transverse isotropy model using this method as a wave propagator demonstrated the ability of the algorithm.

scholarly journals Simulating propagation of decoupled elastic waves using low-rank approximate mixed-domain integral operators for anisotropic media

Geophysics ◽  
2016 ◽  
Vol 81 (2) ◽  
pp. T63-T77 ◽  
Author(s):  
Jiubing Cheng ◽  
Tariq Alkhalifah ◽  
Zedong Wu ◽  
Peng Zou ◽  
Chenlong Wang
Keyword(s):  
Elastic Waves ◽  
Vector Fields ◽  
Integral Operators ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Low Rank ◽  
Elastic Displacement ◽  
Low Rank Approximation ◽  
Time Step ◽  
Domain Integral

In elastic imaging, the extrapolated vector fields are decoupled into pure wave modes, such that the imaging condition produces interpretable images. Conventionally, mode decoupling in anisotropic media is costly because the operators involved are dependent on the velocity, and thus they are not stationary. We have developed an efficient pseudospectral approach to directly extrapolate the decoupled elastic waves using low-rank approximate mixed-domain integral operators on the basis of the elastic displacement wave equation. We have applied [Formula: see text]-space adjustment to the pseudospectral solution to allow for a relatively large extrapolation time step. The low-rank approximation was, thus, applied to the spectral operators that simultaneously extrapolate and decompose the elastic wavefields. Synthetic examples on transversely isotropic and orthorhombic models showed that our approach has the potential to efficiently and accurately simulate the propagations of the decoupled quasi-P and quasi-S modes as well as the total wavefields for elastic wave modeling, imaging, and inversion.

scholarly journals Modeling of pseudoacoustic P-waves in orthorhombic media with a low-rank approximation

Geophysics ◽  
2013 ◽  
Vol 78 (4) ◽  
pp. C33-C40 ◽  
Author(s):  
Xiaolei Song ◽  
Tariq Alkhalifah
Keyword(s):  
Wave Mode ◽  
P Wave ◽  
Low Rank ◽  
Parameter Range ◽  
Laplacian Operator ◽  
Low Rank Approximation ◽  
P Waves ◽  
Rank Approximation ◽  
Dispersion Solution ◽  
Wavefield Extrapolation

Wavefield extrapolation in pseudoacoustic orthorhombic anisotropic media suffers from wave-mode coupling and stability limitations in the parameter range. We use the dispersion relation for scalar wave propagation in pseudoacoustic orthorhombic media to model acoustic wavefields. The wavenumber-domain application of the Laplacian operator allows us to propagate the P-waves exclusively, without imposing any conditions on the parameter range of stability. It also allows us to avoid dispersion artifacts commonly associated with evaluating the Laplacian operator in space domain using practical finite-difference stencils. To handle the corresponding space-wavenumber mixed-domain operator, we apply the low-rank approximation approach. Considering the number of parameters necessary to describe orthorhombic anisotropy, the low-rank approach yields space-wavenumber decomposition of the extrapolator operator that is dependent on space location regardless of the parameters, a feature necessary for orthorhombic anisotropy. Numerical experiments that the proposed wavefield extrapolator is accurate and practically free of dispersion. Furthermore, there is no coupling of qSv and qP waves because we use the analytical dispersion solution corresponding to the P-wave.

P- and S-decomposition in anisotropic media with localized low-rank approximations

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. C13-C26 ◽  
Author(s):  
Wenlong Wang ◽  
Biaolong Hua ◽  
George A. McMechan ◽  
Bertrand Duquet
Keyword(s):  
Fourier Transforms ◽  
Synthetic Data ◽  
3D Models ◽  
Low Rank ◽  
S Wave ◽  
Low Rank Approximation ◽  
Low Rank Approximations ◽  
Rank Approximation ◽  
Wave Decomposition ◽  
2D And 3D

We have developed a P- and S-wave decomposition algorithm based on windowed Fourier transforms and a localized low-rank approximation with improved scalability and efficiency for anisotropic wavefields. The model and wavefield are divided into rectangular blocks that do not have to be geologically constrained; low-rank approximations and P- and S-decomposition are performed separately in each block. An overlap-add method reduces artifacts at block boundaries caused by Fourier transforms at wavefield truncations; limited communication is required between blocks. Localization allows a lower rank to be used than global low-rank approximations while maintaining the same quality of decomposition. The algorithm is scalable, making P- and S-decomposition possible in complicated 3D models. Tests with 2D and 3D synthetic data indicate good P- and S-decomposition results.

Improved MR image denoising via low- rank approximation and Laplacian-of-Gaussian edge detector

2020 ◽  
Vol 14 (12) ◽  
pp. 2791-2798
Author(s):  
Xiaoqun Qiu ◽  
Zhen Chen ◽  
Saifullah Adnan ◽  
Hongwei He
Keyword(s):  
Image Denoising ◽  
Low Rank ◽  
Edge Detector ◽  
Low Rank Approximation ◽  
Mr Image ◽  
Rank Approximation

scholarly journals Fast Joint Multiband Reconstruction From Wideband Images Based on Low-Rank Approximation

2020 ◽  
Vol 6 ◽  
pp. 922-933
Author(s):  
M. Amine Hadj-Youcef ◽  
Francois Orieux ◽  
Alain Abergel ◽  
Aurelia Fraysse
Keyword(s):  
Low Rank ◽  
Low Rank Approximation ◽  
Rank Approximation

scholarly journals Low-Rank Approximation of Difference between Correlation Matrices Using Inner Product

2021 ◽  
Vol 11 (10) ◽  
pp. 4582
Author(s):  
Kensuke Tanioka ◽  
Satoru Hiwa
Keyword(s):  
Dimensional Reduction ◽  
Low Rank ◽  
Inner Product ◽  
Low Rank Approximation ◽  
Correlation Matrices ◽  
Rank Matrix ◽  
Clustering Approach ◽  
Rank Approximation ◽  
The Difference ◽  
Low Rank Matrix

In the domain of functional magnetic resonance imaging (fMRI) data analysis, given two correlation matrices between regions of interest (ROIs) for the same subject, it is important to reveal relatively large differences to ensure accurate interpretation. However, clustering results based only on differences tend to be unsatisfactory and interpreting the features tends to be difficult because the differences likely suffer from noise. Therefore, to overcome these problems, we propose a new approach for dimensional reduction clustering. Methods: Our proposed dimensional reduction clustering approach consists of low-rank approximation and a clustering algorithm. The low-rank matrix, which reflects the difference, is estimated from the inner product of the difference matrix, not only from the difference. In addition, the low-rank matrix is calculated based on the majorize–minimization (MM) algorithm such that the difference is bounded within the range −1 to 1. For the clustering process, ordinal k-means is applied to the estimated low-rank matrix, which emphasizes the clustering structure. Results: Numerical simulations show that, compared with other approaches that are based only on differences, the proposed method provides superior performance in recovering the true clustering structure. Moreover, as demonstrated through a real-data example of brain activity measured via fMRI during the performance of a working memory task, the proposed method can visually provide interpretable community structures consisting of well-known brain functional networks, which can be associated with the human working memory system. Conclusions: The proposed dimensional reduction clustering approach is a very useful tool for revealing and interpreting the differences between correlation matrices, even when the true differences tend to be relatively small.

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