rank reduction
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Geophysics ◽  
2022 ◽  
pp. 1-85
Author(s):  
Peng Lin ◽  
Suping Peng ◽  
Xiaoqin Cui ◽  
Wenfeng Du ◽  
Chuangjian Li

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.


2021 ◽  
Author(s):  
Tucker Carrington ◽  
Sangeeth Kallullathil

Present day computers do not have enough memory to store the high-dimensional tensors required when using a direct product basis to compute vibrational energy levels of a polyatomic molecule with more than about 5 atoms. One way to deal with this problem is to represent tensors using a tensor format. In this paper, we use CP format. Energy levels are computed by building a basis from vectors obtained by solving linear equations. The method can be thought of as a CP realization of a block inverse iteration method with multiple shifts. The CP rank of the tensors is fixed and the linear equations are solved with an Alternating Least Squares method. There is no need for rank reduction, no need for orthogonalization, and tensors with rank larger than the fixed rank used to solve the linear equations are never generated. The ideas are tested by computing vibrational energy levels of a 64-D bilinearly coupled model Hamiltonian and of acetonitrile(12-D).


2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Bernardo Fraiman ◽  
Héctor Parra De Freitas

Abstract We use a moduli space exploration algorithm to produce a complete list of maximally enhanced gauge groups that are realized in the heterotic string in 7d, encompassing the usual Narain component, and five other components with rank reduction realized via nontrivial holonomy triples. Using lattice embedding techniques we find an explicit match with the mechanism of singularity freezing in M-theory on K3. The complete global data for each gauge group is explicitly given.


Author(s):  
Milija Zupanski

AbstractNew method for ensemble data assimilation that incorporates state space covariance localization, global numerical optimization, and implied Bayesian inference, is presented. The method is referred to as the MLEF with State Space Localization (MLEF-SSL) due to its similarity with the Maximum Likelihood Ensemble Filter (MLEF). One of the novelties introduced in MLEF-SSL is the calculation of a reduced-rank localized forecast error covariance using random projection. The Hessian preconditioning is accomplished via Cholesky decomposition of the Hessian matrix, accompanied with solving triangular system of equations instead of directly inverting matrices. For ensemble update the MLEF-SSL system employs resampling of posterior perturbations. The MLEF-SSL was applied to Lorenz model II and compared to Ensemble Kalman Filter with state space localization and to MLEF with observation space localization. The observations include linear and nonlinear observation operators, each applied to integrated and point observations. Results indicate improved performance of MLEF-SSL, particularly in assimilation of integrated nonlinear observations. Resampling of posterior perturbations for ensemble update also indicates a satisfactory performance. Additional experiments were conducted to examine the sensitivity of the method to the rank of random matrix and to compare it to truncated eigenvectors of the localization matrix. The two methods are comparable in application to low-dimensional Lorenz model, except that the new method outperforms the truncated eigenvector method in case of severe rank reduction. The random basis method is simple to implement and may be more promising for realistic high-dimensional applications.


Geophysics ◽  
2021 ◽  
pp. 1-92
Author(s):  
Yangkang Chen ◽  
Sergey Fomel ◽  
Hang Wang ◽  
shaohuan zu

The prediction error filter (PEF) assumes that the seismic data can be destructed to zero by applying a convolutional operation between the target data and prediction filter in either time-space or frequency-space domain. Here, we extend the commonly known PEF in 2D or 3D problems to its 5D version. To handle the non-stationary property of the seismic data, we formulate the PEF in a non-stationary way, which is called the non-stationary prediction error filter (NPEF). In the NPEF, the coefficients of a fixed-size PEF vary across the whole seismic data. In NPEF, we aim at solving a highly ill-posed inverse problem via the computationally efficient iterative shaping regularization. The NPEF can be used to denoise multi-dimensional seismic data, and more importantly, to restore the highly incomplete aliased 5D seismic data. We compare the proposed NPEF method with the state-of-the-art rank-reduction method for the 5D seismic data interpolation in cases of irregularly and regularly missing traces via several synthetic and real seismic data. Results show that although the proposed NPEF method is less effective than the rank-reduction method in interpolating irregularly missing traces especially in the case of low signal to noise ratio (S/N), it outperforms the rank-reduction method in interpolating aliased 5D dataset with regularly missing traces.


Author(s):  
Peng Lin ◽  
Jingtao Zhao ◽  
Suping Peng ◽  
Xiaoqin Cui ◽  
Chuangjian Li ◽  
...  

Geophysics ◽  
2021 ◽  
pp. 1-96
Author(s):  
Yapo Abolé Serge Innocent Oboué ◽  
Yangkang Chen

Noise and missing traces usually influence the quality of multidimensional seismic data. It is, therefore, necessary to e stimate the useful signal from its noisy observation. The damped rank-reduction (DRR) method has emerged as an effective method to reconstruct the useful signal matrix from its noisy observation. However, the higher the noise level and the ratio of missing traces, the weaker the DRR operator becomes. Consequently, the estimated low-rank signal matrix includes a unignorable amount of residual noise that influences the next processing steps. This paper focuses on the problem of estimating a low-rank signal matrix from its noisy observation. To elaborate on the novel algorithm, we formulate an improved proximity function by mixing the moving-average filter and the arctangent penalty function. We first apply the proximity function to the level-4 block Hankel matrix before the singular value decomposition (SVD), and then, to singular values, during the damped truncated SVD process. The relationship between the novel proximity function and the DRR framework leads to an optimization problem, which results in better recovery performance. The proposed algorithm aims at producing an enhanced rank-reduction operator to estimate the useful signal matrix with a higher quality. Experiments are conducted on synthetic and real 5-D seismic data to compare the effectiveness of our approach to the DRR approach. The proposed approach is shown to obtain better performance since the estimated low-rank signal matrix is cleaner and contains less amount of artifacts compared to the DRR algorithm.


Author(s):  
Wei Chen ◽  
Xingye Liu ◽  
Omar M. Saad ◽  
Yapo Abole Serge Innocent Oboue ◽  
Liuqing Yang ◽  
...  

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