Truncated Singular Value Decomposition Research Articles

An improved pulse sequence and inversion algorithm of [formula omitted] spectrum

The nuclear magnetic resonance transversal relaxation time is widely applied in geological prospecting, both in laboratory and downhole environments. However, current methods used for data acquisition and inversion should be reformed to characterize geological samples with complicated relaxation components and pore size distributions, such as samples of tight oil, gas shale, and carbonate. We present an improved pulse sequence to collect transversal relaxation signals based on the CPMG (Carr, Purcell, Meiboom, and Gill) pulse sequence. The echo spacing is not constant but varies in different windows, depending on prior knowledge or customer requirements. We use the entropy based truncated singular value decomposition (TSVD) to compress the ill-posed matrix and discard small singular values which cause the inversion instability. A hybrid algorithm combining the iterative TSVD and a simultaneous iterative reconstruction technique is implemented to reach the global convergence and stability of the inversion. Numerical simulations indicate that the improved pulse sequence leads to the same result as CPMG, but with lower echo numbers and computational time. The proposed method is a promising technique for geophysical prospecting and other related fields in future.

Reconstruction of the Radar Image From Actual DDMs Collected by TechDemoSat-1 GNSS-R Mission

In this study, a new high-level remote sensing image product is generated by exploiting Global Navigation Satellite System-Reflectometry (GNSS-R) measurements collected by the TechDemoSat-1 (TDS-1) mission. This product consists of normalized radar cross-section (NRCS) measurements arranged in a gridded format. The product is obtained reconstructing NRCSs from actual TDS-1 delay Doppler maps (DDMs) using a deconvolution method based on the two-dimensional (2-D) truncated singular-value decomposition (TSVD). The proposed method is here tested, for the first time, on actual DDMs that include nonhomogeneous areas related to land/sea transition. Experimental results demonstrate that: 1) the gridded product can be successfully reconstructed from actual DDMs; 2) this product is suitable for imaging remote sensing purposes since it exhibits a uniform spatial resolution of $8 \times 30$ km within an area of around $40 \times 40$ km; 3) land/sea transitions, which can be hardly recognized in the DDM, can be well located in the gridded product.

Retracted: Multi-step damped multichannel singular spectrum analysis for simultaneous reconstruction and denoising of 3D seismic data

Multichannel singular spectrum analysis (MSSA) is an effective approach for simultaneous seismic data reconstruction and denoising. MSSA utilizes truncated singular value decomposition (TSVD) to decompose the noisy signal into a signal subspace and a noise subspace and weighted projection onto convex sets (POCS)-like method to reconstruct the missing data in the appropriately constructed block Hankel matrix at each frequency slice. However, there still exists some residual noise in signal space due to two major factors: the deficiency of traditional TSVD and the iteratively inserted observed noisy data during the process of weighted POCS like iterations. In this paper, we first further extend the recently proposed damped MSSA (DMSSA) for random noise attenuation, which is more powerful in distinguishing between signal and noise, to simultaneous reconstruction and denoising. Then combined with DMSSA, we propose a multi-step strategy, named multi-step damped MSSA (MS-DMSSA), to efficiently reduce the inserted noise during the POCS like iterations, thus can improve the final performance of simultaneous reconstruction and denoising. Application of the MS-DMSSA approach on 3D synthetic and field seismic data demonstrates a better performance compared with the conventional MSSA approach.

Crustal-scale reflection imaging and interpretation by passive seismic interferometry using local earthquakes

We have developed an application of passive seismic interferometry (SI) using P-wave coda of local earthquakes for the purpose of crustal-scale reflection imaging. We processed the reflection gathers retrieved from SI following a standard seismic processing in exploration seismology. We applied SI to the P-wave coda using crosscorrelation, crosscoherence, and multidimensional deconvolution (MDD) approaches for data recorded in the Malargüe region, Argentina. Comparing the results from the three approaches, we found that MDD based on the truncated singular-value decomposition scheme gave us substantially better structural imaging. Although our results provided higher resolution images of the subsurface, they showed less clear images for the Moho in comparison with previous seismic images in the region obtained by the receiver function and global-phase SI. Above the Moho, we interpreted a deep thrust fault and the possible melting zones, which were previously indicated by active-seismic and magnetotelluric methods in this region, respectively. The method we developed could be an alternative option not only for crustal-scale imaging, e.g., in enhanced geothermal systems, but also for lithospheric-scale as well as basin-scale imaging, depending on the availability of local earthquakes and the frequency bandwidth of their P-wave coda.

Damped multichannel singular spectrum analysis for 3D random noise attenuation

Multichannel singular spectrum analysis (MSSA) is an effective algorithm for random noise attenuation in seismic data, which decomposes the vector space of the Hankel matrix of the noisy signal into a signal subspace and a noise subspace by truncated singular value decomposition (TSVD). However, this signal subspace actually still contains residual noise. We have derived a new formula of low-rank reduction, which is more powerful in distinguishing between signal and noise compared with the traditional TSVD. By introducing a damping factor into traditional MSSA to dampen the singular values, we have developed a new algorithm for random noise attenuation. We have named our modified MSSA as damped MSSA. The denoising performance is controlled by the damping factor, and our approach reverts to the traditional MSSA approach when the damping factor is sufficiently large. Application of the damped MSSA algorithm on synthetic and field seismic data demonstrates superior performance compared with the conventional MSSA algorithm.

An open-source Matlab code package for improved rank-reduction 3D seismic data denoising and reconstruction

Simultaneous seismic data denoising and reconstruction is a currently popular research subject in modern reflection seismology. Traditional rank-reduction based 3D seismic data denoising and reconstruction algorithm will cause strong residual noise in the reconstructed data and thus affect the following processing and interpretation tasks. In this paper, we propose an improved rank-reduction method by modifying the truncated singular value decomposition (TSVD) formula used in the traditional method. The proposed approach can help us obtain nearly perfect reconstruction performance even in the case of low signal-to-noise ratio (SNR). The proposed algorithm is tested via one synthetic and field data examples. Considering that seismic data interpolation and denoising source packages are seldom in the public domain, we also provide a program template for the rank-reduction based simultaneous denoising and reconstruction algorithm by providing an open-source Matlab package.

Noise Estimation for Images using Eigen Values and Frobenius norm

Performance of image denoising algorithm is critically dependent on the accuracy of noise level estimation. In the estimation process, restricting the influence of image content on the estimation dataset is a major challenge. In this paper, we propose a novel method which involves the application of Frobenius norm as the basis of content energy measurement. It involves the truncation of SVD (Singular Value Decomposition) components thereby effectively restricting the influence of image details. Application of linear regression for determining content parameter enhances the application scope of the proposed method. The experimental results demonstrates the effectiveness of the proposed approach. Keywords—Frobenius norm, noise estimation, singular value decomposition, linear regression, additive white gaussian noise I. INTRODUCTION Application of Image Denoising or Filtering algorithms is desirable for images corrupted with additive white gaussian noise. This step is applied before many processing algorithms such as image segmentation, restoration etc. Many of these filtering or denoising algorithms rely on the information about the incurred level of noise which is assumed to be known apriori. In real-time applications, this information is obtained by estimation techniques. Accuracy and reliability of estimation algorithms thus affects the filtering or denoising processes concomitantly. More often the noise is assumed to be zero mean additive white gaussian in nature (3), whose distribution is characterized by following equation:

Simultaneous geologic scenario identification and flow model calibration with group-sparsity formulations

Adopting representative geologic connectivity scenarios is critical for reliable modeling and prediction of flow and transport processes in subsurface environments. Geologic scenarios are often developed by integrating several sources of information, including knowledge of the depositional environment, qualitative and quantitative data such as outcrop and well logs, and process-based geologic modeling. In general, flow and transport response data are usually not included in constructing geologic scenarios for a basin. Instead, these data are typically matched using a given prior geologic scenario as constraint. Since data limitations, modeling assumptions and subjective interpretations can lead to significant uncertainty in the adopted geologic scenarios, flow and transport data may also be useful for constraining the uncertainty in proposed geologic scenarios. Constraining geologic scenarios with flow-related data opens an interesting and challenging research area, which goes beyond the traditional model calibration formulations where the geologic scenario is assumed given. In this paper, a novel concept, known as group-sparsity regularization, is proposed as an effective formulation to constrain the uncertainty in the prior geologic scenario during subsurface flow model calibration. Given a collection of model realizations from several plausible geologic scenarios, the proposed method first applies the truncated singular value decomposition (TSVD) to compactly represent the models from each geologic scenario. The TSVD basis for representing each scenario forms a distinct group. The proposed approach searches over these groups (i.e., geologic scenarios) to eliminate inconsistent groups that are not supported by the observed flow/pressure data. The group-sparsity regularization minimizes a l1/l2mixed norm, where the l2-norm quantifies the contribution of each group and operates on the coefficients within the groups while the l1-norm, having a selection property, is implemented across the groups to identify the significant groups. Numerical results demonstrate the effectiveness of the proposed method and its important properties.

A stopping criterion for iterative regularization methods

We present a discrepancy-like stopping criterium for iterative regularization methods for the solution of linear discrete ill-posed problems. The presented criterium terminates the iterations of the iterative method when the residual norm of the computed solution becomes less or equal to the residual norm of a regularized Truncated Singular Value Decomposition (TSVD) solution. We present two algorithms for the automatic computation of the TSVD residual norm using the Discrete Picard Condition. The first algorithm uses the SVD coefficients while the second one uses the Fourier coefficients. In this work, we mainly focus on the Conjugate Gradient Least Squares method, but the proposed criterium can be used for terminating the iterations of any iterative regularization method. Many numerical tests on some selected one dimensional and image deblurring problems are presented and the results are compared with those obtained by state-of-the-art parameter selection rules. The numerical results show the efficiency and robustness of the proposed criterium.

An improved TSVD-based Levenberg–Marquardt algorithm for history matching and comparison with Gauss–Newton

Abstract Parameterization based on truncated singular value decomposition (TSVD) of the dimensionless sensitivity matrix has been shown to be an efficient approach for history matching. With TSVD parameterization, the search direction is computed as a linear combination of a few principal right singular vectors. As the sensitivity matrix is not explicitly computed, this parameterization is appropriate for large-scale history-matching problems. Moreover, previous work presented theoretical evidence that TSVD of the dimensionless sensitivity matrix provides the optimal parameterization in terms of uncertainty reduction. TSVD has been used in the randomized maximum likelihood (RML) framework to generate multiple conditional realizations of reservoir models. In this work, we investigate the effect of TSVD in the search direction obtained by the application of the Gauss–Newton and the Levenberg–Marquardt (LM) methods. In particular, we show that the TSVD-based LM algorithm converges to appropriate estimates because it gradually resolves the important features of the true model. We also introduce an improved implementation of a TSVD-based LM algorithm for generating multiple realizations of reservoir models conditioned to production data. Our experiments indicate that the computational cost of the new implementation is on the order of 2/3 of the cost of the previous implementation.

High-Resolution Cortical Dipole Imaging Using Spatial Inverse Filter Based on Filtering Property.

Cortical dipole imaging has been developed to visualize brain electrical activity in high spatial resolution. It is necessary to solve an inverse problem to estimate the cortical dipole distribution from the scalp potentials. In the present study, the accuracy of cortical dipole imaging was improved by focusing on filtering property of the spatial inverse filter. We proposed an inverse filter that optimizes filtering property using a sigmoid function. The ability of the proposed method was compared with the traditional inverse techniques, such as Tikhonov regularization, truncated singular value decomposition (TSVD), and truncated total least squares (TTLS), in a computer simulation. The proposed method was applied to human experimental data of visual evoked potentials. As a result, the estimation accuracy was improved and the localized dipole distribution was obtained with less noise.

A Novel Method for Force Identification Based on the Discrete Cosine Transform

Force identification is a classical inverse problem, in which the measured data and the mathematical models of mechanical structures are used to determine the applied force. However, the identified force may seriously diverge from the true solution due to the unknown noise contaminating the measured data and the inverse of the ill-posed transfer matrix characterizing the mechanical structure. In this paper, a novel method based on the discrete cosine transform (DCT) in the time domain is proposed for force identification, which overcomes the deficiency of the ill-posedness of the transfer matrix. The unknown force is expanded by a set of cosine basis functions and then the original governing equation is reformulated to find the coefficient of each cosine basis function. Furthermore, a modified generalized cross-validation (GCV) criterion for determining the regularization parameter is developed for the truncated singular value decomposition (TSVD), Chebyshev polynomial, and DCT solutions. Numerical simulation reveals that compared with the L-curve criterion, the modified GCV criterion is quite robust in the presence of noise. Finally, a clamped-free shell structure that is excited by an impact hammer is selected as an example to validate the performance of the proposed method. Experimental results demonstrate that compared with the TSVD-based and Chebyshev-based methods, the DCT-based method combined with the modified GCV criterion can reconstruct the force time history and identify the peak of impact force with high accuracy. Additionally, the identification of force location using the DCT-based method is also discussed.

Joint inversion of [formula omitted] spectrum combining the iterative truncated singular value decomposition and the parallel particle swarm optimization algorithms

With more information than the conventional one dimensional (1D) longitudinal relaxation time (T1) and transversal relaxation time (T2) spectrums, a two dimensional (2D) T1–T2 spectrum in a low field nuclear magnetic resonance (NMR) is developed to discriminate the relaxation components of fluids such as water, oil and gas in porous rock. However, the accuracy and efficiency of the T1–T2 spectrum are limited by the existing inversion algorithms and data acquisition schemes. We introduce a joint method to inverse the T1–T2 spectrum, which combines iterative truncated singular value decomposition (TSVD) and a parallel particle swarm optimization (PSO) algorithm to get fast computational speed and stable solutions. We reorganize the first kind Fredholm integral equation of two kernels to a nonlinear optimization problem with non-negative constraints, and then solve the ill-conditioned problem by the iterative TSVD. Truncating positions of the two diagonal matrices are obtained by the Akaike information criterion (AIC). With the initial values obtained by TSVD, we use a PSO with parallel structure to get the global optimal solutions with a high computational speed. We use the synthetic data with different signal to noise ratio (SNR) to test the performance of the proposed method. The result shows that the new inversion algorithm can achieve favorable solutions for signals with SNR larger than 10, and the inversion precision increases with the decrease of the components of the porous rock.

Probing the influential factors of NMR T1–T2 spectra in the characterization of the kerogen by numerical simulation

The low field nuclear magnetic resonance (NMR) spectroscopy has been widely used to characterize the longitudinal and transversal relaxation (T1–T2) spectrum of unconventional resources such as shale gas and tight oil containing significant proportions of kerogen and bitumen. However, it requires exquisite design of the acquisition model and the inversion algorithm due to the fast relaxation nature of the kerogen and bitumen. A new direct two dimensional (2D) inversion algorithm combined the iterative truncated singular value decomposition (TSVD) and the Akaiake Information Criterion (AIC) is presented to perform the data inversion efficiently. The fluid component decomposition (FCD) is applied to construct the forward T1–T2 model of the kerogen, and numerical simulations are conducted to investigate factors which may influence inversion results including echo spacing, recovery time series, signal to noise ratio (SNR), and the maximal iteration time.Results show that the T2 component is heavily impaired by the echo spacing, whereas the T1 component is influenced by the recovery time series but with limited effects. The inversion precision is greatly affected by the quality of the data. The inversed spectrum deviates from the model seriously when the SNR of the artificial noise is lower than 50, and the T2 component is more sensitive to the noise than the T1 component. What’s more, the maximal iteration time can also affect the inversion result, especially when the maximal iteration time is smaller than 500. Proper acquisition and inversion parameters for the characterization of the kerogen are obtained considering the precision and the computational cost.

TSVD as a Statistical Estimator in the Latent Semantic Analysis Paradigm

The aim of this paper is to present a new point of view that makes it possible to give a statistical interpretation of the traditional latent semantic analysis (LSA) paradigm based on the truncated singular value decomposition (TSVD) technique. We show how the TSVD can be interpreted as a statistical estimator derived from the LSA co-occurrence relationship matrix by mapping probability distributions on Riemanian manifolds. Besides, the quality of the estimator model can be expressed by introducing a figure of merit arising from the Solomonoff approach. This figure of merit takes into account both the adherence to the sample data and the simplicity of the model. In our model, the simplicity parameter of the proposed figure of merit depends on the number of the singular values retained after the truncation process, while the TSVD estimator, according to the Hellinger distance, guarantees the minimal distance between the sample probability distribution and the inferred probabilistic model.

The application of cubic B-spline collocation method in impact force identification

The accurate real-time characterization of impact event is vital during the life-time of a mechanical product. However, the identified impact force may seriously diverge from the real one due to the unknown noise contaminating the measured data, as well as the ill-conditioned system matrix. In this paper, a regularized cubic B-spline collocation (CBSC) method is developed for identifying the impact force time history, which overcomes the deficiency of the ill-posed problem. The cubic B-spline function by controlling the mesh size of the collocation point has the profile of a typical impact event. The unknown impact force is approximated by a set of translated cubic B-spline functions and then the original governing equation of force identification is reduced to find the coefficient of the basis function at each collocation point. Moreover, a modified regularization parameter selection criterion derived from the generalized cross validation (GCV) criterion for the truncated singular value decomposition (TSVD) is introduced for the CBSC method to determine the optimum regularization number of cubic B-spline functions. In the numerical simulation of a two degrees-of-freedom (DOF) system, the regularized CBSC method is validated under different noise levels and frequency bands of exciting forces. Finally, an impact experiment is performed on a clamped-free shell structure to confirm the performance of the regularized CBSC method. Experimental results demonstrate that the peak relative errors of impact forces based on the regularized CBSC method are below 8%, while those based on the TSVD method are approximately 30%.

Compressive Sensing in High-resolution 3D SAR Tomography of Urban Scenarios

In modern high resolution SAR data, due to the intrinsic side-looking geometry of SAR sensors, layover and foreshortening issues inevitably arise, especially in dense urban areas. SAR tomography provides a new way of overcoming these problems by exploiting the back-scattering property for each pixel. However, traditional non-parametric spectral estimators, e.g. Truncated Singular Value Decomposition (TSVD), are limited by their poor elevation resolution, which is not comparable to the azimuth and slant-range resolution. In this paper, the Compressive Sensing (CS) approach using Basis Pursuit (BP) and TWo-step Iterative Shrinkage/Thresholding (TWIST) are introduced. Experimental studies with real spotlight-mode TerraSAR-X dataset are carried out using both BP and TWIST, to demonstrate the merits of compressive sensing approaches in terms of robustness, computational efficiency, and super-resolution capability.

A modified Tikhonov regularization method

Tikhonov regularization and truncated singular value decomposition (TSVD) are two elementary techniques for solving a least squares problem from a linear discrete ill-posed problem. Based on these two techniques, a modified regularization method is proposed, which is applied to an Arnoldi-based hybrid method. Theoretical analysis and numerical examples are presented to illustrate the effectiveness of the method.

Some matrix nearness problems suggested by Tikhonov regularization

The numerical solution of linear discrete ill-posed problems typically requires regularization, i.e., replacement of the available ill-conditioned problem by a nearby better conditioned one. The most popular regularization methods for problems of small to moderate size are Tikhonov regularization and truncated singular value decomposition (TSVD). By considering matrix nearness problems related to Tikhonov regularization, several novel regularization methods are derived. These methods share properties with both Tikhonov regularization and TSVD, and can give approximate solutions of higher quality than either one of these methods.

A comparison of regularization operators for noisy gamma-ray tomographic reconstruction

The severe effects of several types of noise present in tomographic image reconstruction are well known. Some of these are a consequence of the ill-posedness of this inverse problem. In this work, we investigate the impact of a Tikhonov regularization on the solution of a gamma-ray tomography reconstruction by means of a least squares numerical method. The theoretical methodology is considered in a broad sense as a Tikhonov regularization, but also includes the Morozov concept used specifically for the delta parameter control. The reconstruction quality shows effective improvement when this technique is applied to simple gamma-ray tomography algorithms. Furthermore, the impact of these regularization techniques on the solutions of linear systems of equations is significant. An ART (Algebraic Reconstruction Technique)-type algorithm was used for the reconstruction of simulated data utilizing built-in Matlab functions. These were compared with data obtained through a regularization implemented with TSVD (Truncated Singular Value Decomposition), as well as data obtained through hybrid algorithms such as TSVD plus Toepelitz, tridiagonal and identity operators. The quality of the resulting reconstruction is evaluated through RMSE (Root Mean Square Error). Direct comparisons suggest that for a high noise level and high delta parameter the TSVD plus tridiagonal operator is the best choice.