Erratic Noise Research Articles

Robust dictionary learning for erratic noise-corrupted seismic data reconstruction

Dictionary learning methods adaptively train their bases from the given data in an iterative manner; hence, they can capture more detailed features and achieve sparser representation than a method that uses a fixed basis. However, there also exists a good chance of erratic noise corrupting the dictionary because of the insufficiency of the L1-norm regularization in distinguishing the signal and erratic noise, causing the inaccuracy of both dictionary learning and its application. A robust dictionary learning method is proposed to reconstruct the missing data even in the presence of strong erratic noise. Specifically, a projected operator, which is constructed with the robust Huber misfit, is used to damp erratic noise to a low-amplitude level. In the dictionary learning step, erratic noise is gradually reduced to an acceptable level with the help of this projected operator, from which the signal prototypes (or atoms) can be safely extracted in each iteration. Then, the learned dictionary is used to iteratively estimate the signal from the noisy and under-sampled data, performing noise attenuation as well as data interpolation at non-recording locations. We test the proposed dictionary learning method using 2D and 3D noisy seismic examples and compare it with other state-of-the-art methods. Numerical results demonstrate its superior effectiveness at recovering missing data and increasing signal-to-noise ratio in the presence of erratic noise.

Robust principal component analysis using facial reduction

We introduce a novel approach for robust principal component analysis (RPCA) for a partially observed data matrix. The aim is to recover the data matrix as a sum of a low-rank matrix and a sparse matrix so as to eliminate erratic noise (outliers). This problem is known to be NP-hard in general. A classical approach to solving RPCA is to consider convex relaxations. One such heuristic involves the minimization of the (weighted) sum of a nuclear norm part, that promotes a low-rank component, with an ell _1 norm part, to promote a sparse component. This results in a well-structured convex problem that can be efficiently solved by modern first-order methods. However, first-order methods often yield low accuracy solutions. Moreover, the heuristic of using a norm consisting of a weighted sum of norms may lose some of the advantages that each norm had when used separately. In this paper, we propose a novel nonconvex and nonsmooth reformulation of the original NP-hard RPCA model. The new model adds a redundant semidefinite cone constraint and solves small subproblems using a PALM algorithm. Each subproblem results in an exposing vector for a facial reduction technique that is able to reduce the size significantly. This makes the problem amenable to efficient algorithms in order to obtain high-level accuracy. We include numerical results that confirm the efficacy of our approach.

Iterative Double Laplacian-Scaled Low-Rank Optimization for Under-Sampled and Noisy Signal Recovery

Recovering signal from under-sampled and erratic noise-corrupted seismic data is indeed a challenging task because of its difficulty in simultaneous modeling of erratic noise and missing signal. Assuming that the recorded data are the superposition of low-rank and sparse components, many related works have been reported using a hybrid rank-sparsity constraint. Those published works typically detect the rank and erratic noise using empirical and global thresholds, which often fail to well characterize the varying sparsity and easily cause biased estimation in case of their nonstationary distribution. We propose an iterative double Laplacian-scaled low-rank optimization to adaptively select the sparsity and rank regularizer parameters for robust signal recovery. Comparing with the published approaches with global threshold, Laplacian-scaled mixture, which is obtained by multiplying Laplacian variable with a Gamma variable, is used to locally model the sparsity of erratic noise and the low-rank feature of signal. Then, the expectation–maximization (EM) algorithm is used to transform the Laplacian-scaled mixture problem into a localized reweighted $\ell _{1}$ minimization scheme. The weighted coefficient appearing in its EM solver provides a variable constraint to locally address the rank and erratic noise, and hence, their regularizer parameters can dynamically reflect the different importance of those coefficients. We tested the effectiveness of the proposed method using under-sampled synthetic and field data that are corrupted by erratic noise and used other state-of-the-art methods as comparisons. The results showed that more exact estimations of the signal and erratic noise can be obtained using the proposed method.

Normalized shaping regularization for robust separation of blended data

Simultaneous source acquisition has attracted more and more attention from geophysicists because of its cost savings, whereas it also brings some challenges that have never been addressed before. Deblending of simultaneous source data is usually considered as an underdetermined inverse problem, which can be effectively solved with a least-squares (LS) iterative procedure between data consistency ([Formula: see text]-norm) and regularization ([Formula: see text]-norm or [Formula: see text]-norm). However, when it comes to abnormal noise that follows non-Gaussian distribution and possesses high-amplitude features (e.g., erratic noise, swell noise, and power line noise), the [Formula: see text]-norm is a nonrobust statistic that can easily lead to suboptimal deblended results. Although abnormal noise can be attenuated in the common source domain at first, it is still challenging to apply a coherency-based filter due to the sparse receiver or crossline sampling, e.g., that commonly found in ocean bottom node (OBN) acquisition. To address this problem, we have developed a normalized shaping regularization to make the inversion-based deblending approach robust for the separation of blended data when abnormal noise exists. Its robustness comes from the normalized shaping operator defined by the confidence interval of normal distribution, which minimizes the abnormal risk to a normal level to satisfy the assumption of LS shaping regularization. In special cases, the proposed approach will revert to the classic LS shaping regularization once the normalized coefficient is large enough. Experimental results on synthetic and field data indicate that the proposed method can effectively restore the separated records from blended data at essentially the same convergence rate as the LS shaping regularization for the abnormal noise-free scenario, but it can obtain better deblending performance and less energy leakage when abnormal noise exists.

Robust Seismic Image Interpolation with Mathematical Morphological Constraint.

Seismic image interpolation is a currently popular research subject in modern reflection seismology. The interpolation problem is generally treated as a process of inversion. Under the compressed sensing framework, various sparse transformations and low-rank constraints based methods have great performances in recovering irregularly missing traces. However, in the case of regularly missing traces, their applications are limited because of the strong spatial aliasing energies. In addition, the erratic noise always poses a serious impact on the interpolation results obtained by the sparse transformations and low-rank constraints-based methods. This is because the erratic noise is far from satisfying the statistical assumption behind these methods. In this study, we propose a mathematical morphology-based interpolation technique, which constrains the morphological scale of the model in the inversion process. The inversion problem is solved by the shaping regularization approach. The mathematical morphological constraint (MMC)-based interpolation technique has a satisfactory robustness to the spatial aliasing and erratic energies. We provide a detailed algorithmic framework and discuss the extension from 2D to higher dimensional version and the back operator in the shaping inversion. A group of numerical examples demonstrates the successful performance of the proposed technique.

Robust tensor-completion algorithm for 5D seismic-data reconstruction

Multidimensional seismic data reconstruction has emerged as a primary topic of research in the field of seismic data processing. Although there exists a large number of algorithms for multidimensional seismic data reconstruction, they often adopt the [Formula: see text] norm to measure the discrepancy between observed and reconstructed data. Strictly speaking, these algorithms assume well-behaved noise that ideally follows a Gaussian distribution. When erratic noise contaminates the seismic traces, a 5D reconstruction must adopt a robust criterion to measure the difference between observed and reconstructed data. We develop a new formulation to the parallel matrix factorization tensor completion method and adapt it for coping with erratic noise. We use synthetic and field-data examples to examine our robust reconstruction technique.

Estimation of δ and C13 of organic-rich shale from laser ultrasonic technique measurement

The laser ultrasonic technique (LUT) has many advantages in dealing with the anisotropy of organic-rich shale over the transducer ultrasonic contacting measurement. We have developed a systematic procedure to estimate anisotropic parameter [Formula: see text] and elastic parameter [Formula: see text] in organic-rich shale from LUT measurement. A novel filtering method called the structural-oriented space-varying median filter (SOSVMF) is proposed for removing the erratic noise from the recorded data. We apply the proposed algorithm onto one synthetic example and real recorded data in artificial bakelite and organic-rich shale by using LUT to show the successful performance. Compared with the original recorded data, the denoised waveform by SOSVMF will also help to obtain more accurate parameter estimations of the anisotropic parameter [Formula: see text] and elastic parameter [Formula: see text]. More importantly, the presented systematic procedure may extend the application of LUT to the study of attenuation anisotropy in organic-rich shale and scale physical modeling. In addition, the surface-modification method may be an effective method to enhance the laser-ultrasonic conversion efficiency.

Signal-Preserving Erratic Noise Attenuation via Iterative Robust Sparsity-Promoting Filter

Sparse domain thresholding filters operating in a sparse domain are highly effective in removing Gaussian random noise under Gaussian distribution assumption. Erratic noise, which designates non-Gaussian noise that consists of large isolated events with known or unknown distribution, also needs to be explicitly taken into account. However, conventional sparse domain thresholding filters based on the least-squares (LS) criterion are severely sensitive to data with high-amplitude and non-Gaussian noise, i.e., the erratic noise, which makes the suppression of this type of noise extremely challenging. In this paper, we present a robust sparsity-promoting denoising model, in which the LS criterion is replaced by the Huber criterion to weaken the effects of erratic noise. The random and erratic noise is distinguished by using a data-adaptive parameter in the presented method, where random noise is described by mean square, while the erratic noise is downweighted through a damped weight. Different from conventional sparse domain thresholding filters, definition of the misfit between noisy data and recovered signal via the Huber criterion results in a nonlinear optimization problem. With the help of theoretical pseudoseismic data, an iterative robust sparsity-promoting filter is proposed to transform the nonlinear optimization problem into a linear LS problem through an iterative procedure. The main advantage of this transformation is that the nonlinear denoising filter can be solved by conventional LS solvers. Tests with several data sets demonstrate that the proposed denoising filter can successfully attenuate the erratic noise without damaging useful signal when compared with conventional denoising approaches based on the LS criterion.

Iteratively re‐weighted and refined least squares algorithm for robust inversion of geophysical data

ABSTRACTA robust metric of data misfit such as the ℓ1‐norm is required for geophysical parameter estimation when the data are contaminated by erratic noise. Recently, the iteratively re‐weighted and refined least‐squares algorithm was introduced for efficient solution of geophysical inverse problems in the presence of additive Gaussian noise in the data. We extend the algorithm in two practically important directions to make it applicable to data with non‐Gaussian noise and to make its regularisation parameter tuning more efficient and automatic. The regularisation parameter in iteratively reweighted and refined least‐squares algorithm varies with iteration, allowing the efficient solution of constrained problems. A technique is proposed based on the secant method for root finding to concentrate on finding a solution that satisfies the constraint, either fitting to a target misfit (if a bound on the noise is available) or having a target size (if a bound on the solution is available). This technique leads to an automatic update of the regularisation parameter at each and every iteration. We further propose a simple and efficient scheme that tunes the regularisation parameter without requiring target bounds. This is of great importance for the field data inversion where there is no information about the size of the noise and the solution. Numerical examples from non‐stationary seismic deconvolution and velocity‐stack inversion show that the proposed algorithm is efficient, stable, and robust and outperforms the conventional and state‐of‐the‐art methods.

Inversion of seismic arrival times with erratic noise using robust Tikhonov–TV regularization

Inversion of seismic arrival times with erratic noise using robust Tikhonov–TV regularization

Advanced Deblending Scheme for Independent Simultaneous Source Data

Independent Simultaneous Source (ISS®) technology is an attractive way to provide high acquisition production rates and affordable higher density seismic imaging. In this paper, we propose an advanced deblending scheme to address the source separation requirement of ISS® data. The deblending scheme includes a) 3D low-rank decomposition based iterative noise modelling, and subtraction, b) signal-to-noise ratio map guided residual denoise and c) erratic noise attenuation. The application is tested on actual shallow water OBC dataset, which was pseudo-blended using a realistic dual vessel acquisition plan. The result shows that this advanced deblending scheme can successfully recover individual source responses. The deblended data have a minimal difference compared with the actual unblended data.

Robustf‐xprojection filtering for simultaneous random and erratic seismic noise attenuation

ABSTRACTLinear prediction filters are an effective tool for reducing random noise from seismic records. Unfortunately, the ability of prediction filters to enhance seismic records deteriorates when the data are contaminated by erratic noise. Erratic noise in this article designates non‐Gaussian noise that consists of large isolated events with known or unknown distribution. We propose a robustf‐xprojection filtering scheme for simultaneous erratic noise and Gaussian random noise attenuation. Instead of adopting the ℓ2‐norm, as commonly used in the conventional design off‐xfilters, we utilize the hybrid‐norm to penalize the energy of the additive noise. The estimation of the prediction error filter and the additive noise sequence are performed in an alternating fashion. First, the additive noise sequence is fixed, and the prediction error filter is estimated via the least‐squares solution of a system of linear equations. Then, the prediction error filter is fixed, and the additive noise sequence is estimated through a cost function containing a hybrid‐norm that prevents erratic noise to influence the final solution. In other words, we proposed and designed a robust M‐estimate of a special autoregressive moving‐average model in thef‐xdomain. Synthetic and field data examples are used to evaluate the performance of the proposed algorithm.

Multidimensional simultaneous random plus erratic noise attenuation and interpolation for seismic data by joint low-rank and sparse inversion

We have developed an efficient convex optimization strategy enabling the simultaneous attenuation of random and erratic noise with interpolation in prestack seismic data. For a particular analysis window, frequency slice spatial data were reorganized into a block Toeplitz matrix with Toeplitz blocks as in Cadzow/singular spectrum analysis methods. The signal and erratic noise were, respectively, modeled as low-rank and sparse components of this matrix, and then a joint low-rank and sparse inversion (JLRSI) enabled us to recover the low-rank signal component from noisy and incomplete data thanks to joint minimization of a nuclear norm term and an [Formula: see text]-norm term. The convex optimization framework, related to recent developments in the field of compressed sensing, enabled the formulation of a well-posed problem as well as the use of state-of-the-art algorithms. We proposed an alternating directions method of multipliers scheme associated with an efficient singular value thresholding kernel. Numerical results on field data illustrated the effectiveness of the JLRSI approach at recovering missing data and increasing the signal-to-noise ratio.

Robust reduced-rank filtering for erratic seismic noise attenuation

Singular spectrum analysis (SSA) or Cadzow reduced-rank filtering is an efficient method for random noise attenuation. SSA starts by embedding the seismic data into a Hankel matrix. Rank reduction of this Hankel matrix followed by antidiagonal averaging is utilized to estimate an enhanced seismic signal. Rank reduction is often implemented via the singular value decomposition (SVD). The SVD is a nonrobust matrix factorization technique that leads to suboptimal results when the seismic data are contaminated by erratic noise. The term erratic noise designates non-Gaussian noise that consists of large isolated events with known or unknown distribution. We adopted a robust low-rank factorization that permitted use of the SSA filter in situations in which the data were contaminated by erratic noise. In our robust SSA method, we replaced the quadratic error criterion function that yielded the truncated SVD solution by a bisquare function. The Hankel matrix was then approximated by the product of two lower dimensional factor matrices. The iteratively reweighed least-squares method was used to approximately solve for the optimal robust factorization. Our algorithm was tested with synthetic and real data. In our synthetic examples, the data were contaminated with band-limited Gaussian noise and erratic noise. Then, denoising was carried out by means of [Formula: see text] deconvolution, the classical SSA method, and the proposed robust SSA method. The [Formula: see text] deconvolution and the classical SSA method failed to properly eliminate the noise and to preserve the desired signal. On the other hand, the robust SSA method was found to be immune to erratic noise and was able to preserve the desired signal. We also tested the robust SSA method with a data set from the Western Canadian Sedimentary Basin. The results with this data set revealed improved denoising performance in portions of data contaminated with erratic noise.

Simultaneous source separation using a robust Radon transform

We adopted the robust Radon transform to eliminate erratic incoherent noise that arises in common receiver gathers when simultaneous source data are acquired. The proposed robust Radon transform was posed as an inverse problem using an [Formula: see text] misfit that is not sensitive to erratic noise. The latter permitted us to design Radon algorithms that are capable of eliminating incoherent noise in common receiver gathers. We also compared nonrobust and robust Radon transforms that are implemented via a quadratic ([Formula: see text]) or a sparse ([Formula: see text]) penalty term in the cost function. The results demonstrated the importance of incorporating a robust misfit functional in the Radon transform to cope with simultaneous source interferences. Synthetic and real data examples proved that the robust Radon transform produces more accurate data estimates than least-squares and sparse Radon transforms.

When “keep it simple” turns into dangerous misconceptions: A Brazilian case study.

Communicating acoustics to naive subjects by means of the news media requires a good balance between “keep it simple” and correctness. Unfortunately in Brazil, the balance is usually not maintained and “keep it simple” is often even substituted by erratic use of terms and concepts, not only in news media material [Paul, S., Proceedings of the XXIII Encontro da Soc. Brasileira de Acstica (2010)]. The analysis of Brazilian news media material, but also folders, brochures, etc., issued by equipment suppliers, shows terrific errors regarding even simples concepts such as the definition of sound. Unfortunately it can be even shown that erratic use of terms and concepts in news media, brochures, etc., negatively affects the Brazilian legislative, causing erratic noise pollution laws. To change this specific situation in Brazil, a joint effort of the Brazilian acoustics community is required, e.g., on (1) issuing a guide on “communicating acoustics correctly,” (2) introducing a Brazilian standard that defines terms and concepts in acoustics (first proposals made by Souza and Paul: [Ferramentas para uma padronização dos termos utilizados em acxfa;stica e vibrações, Acstica&Vibrações, 42, (2010)], and (3) thorough revision of Brazilian print/online material.