Wavelet Estimation Research Articles

Wavelets Based Simulation and Visualization Approach for Unmixing of Hyperspectral Data

This paper discusses the use of wavelet based simulation and visualization methods to unmix multi-component, intimate, and particulate mixtures. We describe a new approach to simulate the hyperspectral data based on an interpolating wavelet estimation. We choose a very constructive family of a wavelet basis, Duabechies wavelets, of order one and two to analyze the data. We used the discrete wavelet transform (DWT), which the basic tool needed for studying time series via wavelets and plays a role analogue in the theory of spectral analysis. The present paper present parameters and algorithms that successfully simulate the binary system of Na2SO4-MgSO4. This system was selected because of it is importance to us on analogue for Mars salt and salt regolith.

Wavelet regression estimations with strong mixing data

Using a wavelet basis, we establish in this paper upper bounds of wavelet estimation on $$ L^{p}({\mathbb {R}}^{d}) $$ risk of regression functions with strong mixing data for $$ 1\le p<\infty $$ . In contrast to the independent case, these upper bounds have different analytic formulae for $$p\in [1, 2]$$ and $$p\in (2, +\infty )$$ . For $$p=2$$ , it turns out that our result reduces to a theorem of Chaubey et al. (J Nonparametr Stat 25:53–71, 2013); and for $$d=1$$ and $$p=2$$ , it becomes the corresponding theorem of Chaubey and Shirazi (Commun Stat Theory Methods 44:885–899, 2015).

To: “A simple and efficient inversion approach for estimation of primaries, surface-related multiples, and wavelets,” Xintao Chai, GEOPHYSICS, doi: 10.1190/geo2017-0578.1.

To: “A simple and efficient inversion approach for estimation of primaries, surface-related multiples, and wavelets,” Xintao Chai, G<scp>EOPHYSICS</scp>, doi: 10.1190/geo2017-0578.1.

To: “A simple and efficient inversion approach for estimation of primaries, surface-related multiples, and wavelets,” Xintao Chai, Geophysics, doi: 10.1190/geo2017-0578.1.

<b>To: “A simple and efficient inversion approach for estimation of primaries, surface-related multiples, and wavelets,”</b> Xintao Chai, G<scp>eophysics</scp>, doi: 10.1190/geo2017-0578.1.

Asymptotic normality of a wavelet estimator for asymptotically negatively associated errors

In this paper, we consider the nonparametric regression model Yni=g(ti)+εni,1≤i≤n and n≥1, where {ti} are non-random design points, and g(⋅) is an unknown Borel measurable function defined on [0, 1]. Under some general conditions, we study the asymptotic normality of the wavelet estimator of g(⋅), where the random errors {εni} are asymptotically negatively associated (ANA, for short) random variables. In addition, a simulation study is provided to evaluate the finite sample performance of the wavelet estimator.

The mean consistency of wavelet estimators for convolutions of the density functions

In practical applications, people sometimes do not know whether the estimated function is smooth, and it is reasonable to consider the consistency of an estimator. Furthermore, the acquired data are usually contaminated by various random noises. In this paper, we develop the wavelet estimators for m -fold convolutions of the unknown density functions and consider their L p ( 1 ≤ p < ∞ ) consistency under noiseless and additive noise situations, respectively. Finally, simulation studies illustrate the good performances of our nonparametric wavelet estimators.

Wavelet estimation of semiparametric errors in variables model

Most of the work on wavelet estimation when the variables are measured with errors have centered around nonparametric approaches which cause curse of dimensionality. In this paper it is aimed to avoid this complexity using wavelet semiparametric errors in variables regression model. Using theoretical arguments for nonparametric wavelet estimation a wavelet approach is represented to estimate partially linear errors in variables model which is a semiparametric model when explanatory variable of nonparametric part has measurement error. Assuming that the measurement error has a known distribution we derive an estimator of the linear parts' parameter. In simulation study derived method is compared with no measurement error case.

Accuracy of reconstruction of the multidimensional probability density by wavelet estimates of one-dimensional projections

Accuracy of reconstruction of the multidimensional probability density by wavelet estimates of one-dimensional projections

Nonparametric density estimation for spatial data with wavelets

Nonparametric density estimators are studied for d-dimensional, strongly spatial mixing data which are defined on a general N-dimensional lattice structure. We consider linear and nonlinear hard thresholded wavelet estimators derived from a d-dimensional multi-resolution analysis. We give sufficient criteria for the consistency of these estimators and derive rates of convergence in Lp′ for p′∈[1,∞). For this reason, we study density functions which are elements of a d-dimensional Besov space Bp,qs(Rd). We also verify the analytic correctness of our results in numerical simulations.

Diving-wave FWI methodology applied to a Colombian Caribbean data set

Building accurate velocity models is of major interest for oil and gas companies because it improves the prospect evaluation and reduces the risk of geohazards. Full-waveform inversion (FWI) has become a popular method for finding high-resolution and accurate velocity and has been successfully applied to several offshore case studies. We develop a methodology that allows us to apply diving-wave FWI to a case study from the Colombian Caribbean area. The proposed diving-wave FWI methodology includes the wavelet estimation, velocity updates, and quality control (QC) processes. The QC is performed using the cost function, the crosscorrelation of the observed and synthetic gathers, and the analytical trace information. The first QC tool indicates the difference between observed and synthetic gathers; the other two verify that the observed and synthetic data are not cycle skipped. The velocity model obtained with the proposed methodology is the first successful case study performed in the Colombian Caribbean region. From the obtained model, we conclude that FWI is able to build a velocity model with better-resolved shallow depth areas that assist the subsequent pore pressure prediction and imaging processes.

A simple and efficient inversion approach for estimation of primaries, surface-related multiples, and wavelets

Multiples can strongly interfere with primaries if they are not properly handled. An estimation of primaries by sparse inversion (EPSI) method was previously proposed to overcome the inaccuracies in prediction-subtraction techniques. However, the application of the iterative inversion method for the estimation of primaries is limited by the convergence rate and algorithmic complexity, given the expensive computational cost per iteration. A robust EPSI (REPSI) method was recently proposed to eliminate the free parameters involved in the original EPSI method at the expense of employing a complicated spectral-projected gradient l1 (SPGl1) algorithm. In addition, wavelet variation between shots is seldom considered in multiple estimation. Using the feedback primary-multiple model, we propose a simple and efficient inversion approach to estimate primaries, surface-related multiples, and shot-constant and shot-variant wavelets, without any requirements on the subsurface model and assumptions on the wavelet. We ...

Regression estimation under strong mixing data

This paper studies multivariate wavelet regression estimators with errors-in-variables under strong mixing data. We firstly prove the strong consistency for non-oscillating and Fourier-oscillating noises. Then, a convergence rate is provided for non-oscillating noises, when an estimated function has some smoothness. Finally, the consistency and convergence rate are discussed for a practical wavelet estimator.

Sensitivity Improvement of Acoustic Partial Discharge Detection Measurements through Wavelet Analysis

Condition monitoring of power equipment is a vital stepin extending the lifetime of existing equipment andreducing costs for utilities while minimizing the risk ofunscheduled outages. Partial discharge (PD) monitoringhas evolved as a reliable mean of determiningdeterioration in insulation systems. Acoustic emissiondetection techniques are usually utilized for PDdetection mainly in oil-filled transformers offering theadvantage of being immune to electrical noise and amethod to localize PDs. In this work it was attempted toimprove the sensitivity of acoustic measurementsthrough wavelet analysis and estimation of the thresholdvalue from actual measurements of the noise, whichproved to be more effective compared to otherestimation values. The analysis was performed onlaboratory measurements from a 36 kV condenserbushing known to exhibit PD activity acquired with alow cost PD acoustic sensor developed at KTH. As anext step the results have to be verified by onlinemeasurements, which can result in the addition of anCondition monitoring of power equipment is a vital step in extending the lifetime of existing equipment and reducing costs for utilities while minimizing the risk of unscheduled outages. Partial discharge (PD) monitoring has evolved as a reliable mean of determining deterioration in insulation systems. Acoustic emission detection techniques are usually utilized for PD detection mainly in oil-filled transformers offering the advantage of being immune to electrical noise and a method to localize PDs. In this work it was attempted to improve the sensitivity of acoustic measurements through wavelet analysis and estimation of the threshold value from actual measurements of the noise, which proved to be more effective compared to other estimation values. The analysis was performed on laboratory measurements from a 36 kV condenser bushing known to exhibit PD activity acquired with a low cost PD acoustic sensor developed at KTH. As a next step the results have to be verified by online measurements, which can result in the addition of an onboard signal-processing box for improved sensitivity.

Strong Lp convergence of wavelet deconvolution density estimators

Using compactly supported wavelets, Giné and Nickl [Uniform limit theorems for wavelet density estimators, Ann. Probab. 37(4) (2009) 1605–1646] obtain the optimal strong [Formula: see text] convergence rates of wavelet estimators for a fixed noise-free density function. They also study the same problem by spline wavelets [Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections, Bernoulli 16(4) (2010) 1137–1163]. This paper considers the strong [Formula: see text] convergence of wavelet deconvolution density estimators. We first show the strong [Formula: see text] consistency of our wavelet estimator, when the Fourier transform of the noise density has no zeros. Then strong [Formula: see text] convergence rates are provided, when the noises are severely and moderately ill-posed. In particular, for moderately ill-posed noises and [Formula: see text], our convergence rate is close to Giné and Nickl’s.

Adaptive wavelet estimation of a function from an m-dependent process with possibly unbounded m

ABSTRACTThe estimation of a multivariate function from a stationary m-dependent process is investigated, with a special focus on the case where m is large or unbounded. We develop an adaptive estimator based on wavelet methods. Under flexible assumptions on the nonparametric model, we prove the good performances of our estimator by determining sharp rates of convergence under two kinds of errors: the pointwise mean squared error and the mean integrated squared error. We illustrate our theoretical result by considering the multivariate density estimation problem, the derivatives density estimation problem, the density estimation problem in a GARCH-type model and the multivariate regression function estimation problem. The performance of proposed estimator has been shown by a numerical study for a simulated and real data sets.

Wavelet estimation of functional coefficient regression models

The area of nonlinear time series models has experienced a great development since the 1980s. Although there is a wide range of parametric nonlinear time series models, in general, we do not know if the postulated model is the most appropriated one for a specific data set. This situation highlights the importance of nonparametric models. An interesting nonparametric model to fit nonlinear time series is the well-known functional coefficient regression model. Nonparametric estimations by, e.g., local linear regression and splines, are developed in the literature. In this work, we study the estimation of such a model using wavelets. It is a proposal that takes into account both, classical and warped wavelets. We present the rates of convergence of the proposed estimators and carry out simulation studies to evaluate automatic procedures (among AIC, AICc and BIC) for selecting the coarsest and finest levels to be used during the estimation process. Moreover, we illustrate the methodology with an application to a real data set, where we also calculate multi-step-ahead forecasts and compare the results with other methods known in the literature.

Asymptotic properties of wavelet estimators in partially linear errors-in-variables models with long-memory errors

While the random errors are a function of Gaussian random variables that are stationary and long dependent, we investigate a partially linear errors-in-variables (EV) model by the wavelet method. Under general conditions, we obtain asymptotic representation of the parametric estimator, and asymptotic distributions and weak convergence rates of the parametric and nonparametric estimators. At last, the validity of the wavelet method is illuminated by a simulation example and a real example.

Wawelet transform application for classification of solutions of reaction-diffusion systems

Girer–Meynxardt type system of reaction-diffusion with classical derivatives and Bryusselyator system with fractional time derivatives are investigated. On the basis of computer simulations it is shown that qualitatively different types of oscillatory solutions may arise due to instability in these systems. Wavelet transformations are applied to analyze and classify the solutions of such systems, A comparative estimation of wavelet transforms of oscillatory and chaotic solutions is given and it is shown that such method of classification of solutions is effective.

Integration of broadband seismic data into reservoir characterization workflows: A case study from the Campos Basin, Brazil

Towed-streamer marine broadband data have been key contributors to recent petroleum exploration history, in new frontiers and in mature basins around the world. They have improved the characterization of reservoirs by reducing the uncertainty in structural and stratigraphic interpretation and by providing more quantitative estimates of reservoir properties. Dedicated acquisition, processing, and quality control (QC) methods have been developed to capitalize on the broad bandwidth of the data and allow their rapid integration into reservoir models. Using a variable-depth steamer data set acquired in the Campos Basin, Brazil, we determine that particular care that should be taken when processing and inverting broadband data to realize their full potential for reservoir interpretation and uncertainty management in the reservoir model. In particular, we determine the QC implemented and interpretative processing approach used to monitor data improvements during processing and preconditioning for elastic inversion. In addition, we evaluate the importance of properly modeling the low frequencies during wavelet estimation. We find the benefits of carefully processed broadband data for structural interpretation and describe the application of acoustic and elastic inversions cascaded with Bayesian lithofacies classification, to provide clear interpretative products with which we were able to demonstrate a reduction in the uncertainty of the prediction and characterization of Santonian oil sandstones in the Campos Basin.

The influence of errors in the source wavelet on inversion‐based surface‐related multiple attenuation

ABSTRACTWe investigate the influence of source wavelet errors on inversion‐based, surface‐related multiple attenuation, in order to address how the inverted primary impulse response, estimated primaries, and predicted multiples are affected by the estimated wavelet. In theory, errors in estimated wavelet can lead to errors in the upgoing waves. Because of smoothness and the band‐limitedness characteristics of the estimated wavelet, errors in the upgoing waves are usually not white and random. Theoretical analysis and two synthetic examples demonstrate that (i) when the overall amplitude scalar of the estimated wavelet is underestimated, the inversion of the primary impulse response suffers from instability, which will distort the estimation of primaries, and (ii) when the wavelet is overestimated, the estimated primaries will simply mimic the recorded upgoing waves. Nevertheless, the quality of the estimated primaries in the region above the first‐order, water‐bottom multiples is independent of the estimated wavelet. Synthetic results illustrate that inversion‐based, surface‐related multiple attenuation with a known wavelet is stable, since slight inaccuracy in amplitude spectrum and/or phase spectrum of the given wavelet or the corresponding upgoing waves will not lead to considerable deviation in the waveforms of the inverted results from those of the references. Furthermore, shot‐to‐shot wavelet variations, with maximum amplitude difference of 5% and maximum phase difference of 10°, create just slight artefacts in both the inverted primary impulse response and the estimated primaries. Moreover, the sensitivity test of estimation of primaries by sparse inversion method involving wavelet estimation shows that this method can stably and alternately update the wavelet and the primary impulse response; however, different choices of the initial wavelet can lead to different final inverted results.