Wavelet Estimation Research Articles

Minimax wavelet estimation for multisample heteroscedastic nonparametric regression

ABSTRACTThe problem of estimating the baseline signal from multisample noisy curves is investigated. We consider the functional mixed-effects model, and we suppose that the functional fixed effect belongs to the Besov class. This framework allows us to model curves that can exhibit strong irregularities, such as peaks or jumps for instance. The lower bound for the minimax risk is provided, as well as the upper bound of the minimax rate, that is derived by constructing a wavelet estimator for the functional fixed effect. Our work constitutes the first theoretical functional results in multisample nonparametric regression. Our approach is illustrated on realistic simulated datasets as well as on experimental data.

Parameters effective on estimating a nonstationary mixed-phase wavelet using cumulant matching approach

Seismic inversion is a process performed to remove the effects of propagated wavelets in order to recover the acoustic impedance. To obtain valid velocity and density values related to subsurface layers through the inversion process, it is highly essential to perform reliable wavelet estimation such as cumulant matching approach. For this purpose, the seismic data were windowed in this work in such a way that two consecutive windows were only one sample apart. Also, we did not consider any fixed wavelet for any window and let the phase of each wavelet rotate in each sample in the window. Comparing the fourth order cumulant of the whitened trace and fourth-order moment of the all-pass operator in each window generated a cost function that should be minimized with a non-linear optimization method. In this regard, parameters effective on the estimation of the nonstationary mixed-phase wavelets were tested over the created nonstationary seismic trace at 0.82s and 1.6s. Besides, we compared the consequences of each parameter on estimated wavelets at two mentioned times. The parameters studied in this work are window length, taper type, the number of iteration, signal-to-noise ratio, bandwidth to central frequency ratio, and Q factor. The results show that applying the optimum values of the effective parameters, the average correlation of the estimated mixed-phase wavelets with the original ones is about 87%. Moreover, the effectiveness of the proposed approach was examined on a synthetic nonstationary seismic section with variable Q factor values alongside the time and offset axis. Eventually, the cumulant matching method was applied on a cross line of the migrated data from a 3D data set of an oilfield in the Persian Gulf. Also, the effect of the wrong Q estimation on the estimated mixed-phase wavelet was considered on the real data set. It is concluded that the accuracy of the estimated wavelet relied on the estimated Q and more than 10% error in the estimated value of Q is acceptable. Eventually, an 88% correlation was found between the estimated mixed-phase wavelets and the original ones for three horizons. The estimated wavelets applied to the data and the result of deconvolution processes was presented.

Wavelet density deconvolution estimations with heteroscedastic measurement errors

This paper discusses the mean consistency of both theoretical and practical wavelet estimators under deconvolution model with heteroscedastic measurement errors. When the model degenerates to the classical deconvolution problem, our results coincide with Geng & Wang’s theorem (2015).

Wavelet estimation in time-varying coefficient time series models with measurement errors

ABSTRACTThe article studies a time-varying coefficient time series model in which some of the covariates are measured with additive errors. In order to overcome the bias of estimator of the coefficient functions when measurement errors are ignored, we propose a modified least squares estimator based on wavelet procedures. The advantage of the wavelet method is to avoid the restrictive smoothness requirement for varying-coefficient functions of the traditional smoothing approaches, such as kernel and local polynomial methods. The asymptotic properties of the proposed wavelet estimators are established under the α-mixing conditions and without specifying the error distribution. These results can be used to make asymptotically valid statistical inference.

Convergence Rates of Multivariate Regression Estimators with Errors-In-Variables

ABSTRACTThis paper studies the regression estimation with errors-in-variables. We first extend Meister’s theorems (Meister, 2009. Deconvolution Problems in Nonparametric Statistics. Springer, Berlin) from one to multi-dimensional setting, when a noise density has no zeros in the Fourier domain. Then motivated by the work of Delaigle and Meister (Delaigle, Meister, 2011. Nonparametric function estimation under Fourier-oscillating noise. Statistica Sinica 21, 1065–1092), we show a desired convergence rate of a kernel estimator for Fourier-oscillating noises. Finally, two technical conditions are removed, when a wavelet estimator is used.

Wavelet estimation of density for censored data with censoring indicator missing at random

ABSTRACTIn this paper, we define the nonlinear wavelet estimator of density for the right censoring model with the censoring indicator missing at random (MAR), and develop its asymptotic expression for mean integrated squared error (MISE). Unlike for kernel estimator, the MISE expression of the estimator is not affected by the presence of discontinuities in the curve. Meanwhile, asymptotic normality of the estimator is established. The proposed estimator can reduce to the estimator defined by Li [Non-linear wavelet-based density estimators under random censorship. J Statist Plann Inference. 2003;117(1):35–58] when the censoring indicator MAR does not occur and a bandwidth in non-parametric estimation is close to zero. Also, we define another two nonlinear wavelet estimators of the density. A simulation is done to show the performance of the three proposed estimators.

Supersmooth density estimations over L p risk by wavelets

This paper studies wavelet estimations for supersmooth density functions with additive noises. We first show lower bounds of L p risk (1 ≤ p < ∞) with both moderately and severely ill-posed noises. Then a Shannon wavelet estimator provides optimal or nearly-optimal estimations over L p risks for p ≥ 2, and a nearly-optimal result for 1 < p < 2 under both noises. In the nearly-optimal cases, the ratios of upper and lower bounds are determined. When p = 1, we give a nearly-optimal estimation with moderately ill-posed noise by using the Meyer wavelet. Finally, the practical estimators are considered. Our results are motivated by the work of Pensky and Vidakovic (1999), Butucea and Tsybakov (2008), Comte et al. (2006), Lacour (2006) and Lounici and Nickl (2011).

Cross-correlation least-squares reverse time migration in the pseudo-time domain

The least-squares reverse time migration (LSRTM) method with higher image resolution and amplitude is becoming increasingly popular. However, the LSRTM is not widely used in field land data processing because of its sensitivity to the initial migration velocity model, large computational cost and mismatch of amplitudes between the synthetic and observed data. To overcome the shortcomings of the conventional LSRTM, we propose a cross-correlation least-squares reverse time migration algorithm in pseudo-time domain (PTCLSRTM). Our algorithm not only reduces the depth/velocity ambiguities, but also reduces the effect of velocity error on the imaging results. It relieves the accuracy requirements on the migration velocity model of least-squares migration (LSM). The pseudo-time domain algorithm eliminates the irregular wavelength sampling in the vertical direction, thus it can reduce the vertical grid points and memory requirements used during computation, which makes our method more computationally efficient than the standard implementation. Besides, for field data applications, matching the recorded amplitudes is a very difficult task because of the viscoelastic nature of the Earth and inaccuracies in the estimation of the source wavelet. To relax the requirement for strong amplitude matching of LSM, we extend the normalized cross-correlation objective function to the pseudo-time domain. Our method is only sensitive to the similarity between the predicted and the observed data. Numerical tests on synthetic and land field data confirm the effectiveness of our method and its adaptability for complex models.

A fully automated algorithm of baseline correction based on wavelet feature points and segment interpolation

Baseline correction is a very important part of pre-processing. Baseline in the spectrum signal can induce uneven amplitude shifts across different wavenumbers and lead to bad results. Therefore, these amplitude shifts should be compensated before further analysis. Many algorithms are used to remove baseline, however fully automated baseline correction is convenient in practical application. A fully automated algorithm based on wavelet feature points and segment interpolation (AWFPSI) is proposed. This algorithm finds feature points through continuous wavelet transformation and estimates baseline through segment interpolation. AWFPSI is compared with three commonly introduced fully automated and semi-automated algorithms, using simulated spectrum signal, visible spectrum signal and Raman spectrum signal. The results show that AWFPSI gives better accuracy and has the advantage of easy use.

Toward a robust workflow for deep crustal imaging by FWI of OBS data: The eastern Nankai Trough revisited

AbstractCrustal‐scale imaging by the full‐waveform inversion (FWI) of long‐offset seismic data is inherently difficult because the large number of wavelengths propagating through the crust makes the inversion prone to cycle skipping. Therefore, efficient crustal‐scale FWI requires an accurate starting model and a stable workflow minimizing the nonlinearity of the inversion. Here we attempt to reprocess a challenging 2‐D ocean‐bottom seismometer (OBS) data set from the eastern Nankai Trough. The starting model is built by first‐arrival traveltime tomography (FAT), which is FWI assisted for tracking cycle skipping. We iteratively refine the picked traveltimes and then reiterate the FAT until the traveltime residuals remain below the cycle‐skipping limit. Subsequently, we apply Laplace‐Fourier FWI, in which progressive relaxation of time damping is nested within frequency continuation to hierarchically inject more data into the inversion. These two multiscale levels are complemented by a layer‐stripping approach implemented through offset continuation. The reliability of the FWI velocity model is assessed by means of source wavelet estimation, synthetic seismogram modeling, ray tracing modeling, dynamic warping, and checkerboard tests. Although the viscoacoustic approximation is used for wave modeling, the synthetic seismograms reproduce most of the complexity of the data with a high traveltime accuracy. The revised FWI scheme produces a high‐resolution velocity model of the entire crust that can be jointly interpreted with migrated images derived from multichannel seismic data. This study opens a new perspective on the design of OBS crustal‐scale experiments amenable to FWI; however, a further assessment of the optimal OBS spacing is required for reliable FWI.

Application of the algorithm for time of arrival estimation of the shock waves produced by projectiles of different calibers

The topic of this paper is review of application of the algorithm for time of arrival estimation of shock waves produced by projectiles of different calibers. Time of arrival estimation of the shock waves is based on algorithm, which consists of wavelet decomposition, identification of the narrow data segment, which holds shock wave by wavelet coefficients and estimation the time of arrival of the shock wave by conventional statistical methods. Sources of the shock waves were projectiles which caliber varied from characteristic rifles, 7.62 mm, through large gun’s calibers up to 120 mm. Quality of the time of arrival estimate is verified on significant number of experiments. Estimated time of arrival is used in solving of the position of weapon when fire off happened. Localization of the shock wave sources was done by networked and in space distributed acoustic sensors. Discrete probability method, based on time differences of the time of arrivals obtained from multiple synchronized acoustic sensors, was used for localization of the weapon and these results are compared with their locations obtained by DGPS method.

A practical workflow for model-driven seismic inversion

Seismic inversion transforms reflectivity data into layer property which adds valuable interpretive benefits for reservoir characterization. Although seismic inversion is routinely conducted in reservoir interpretation projects, there are still pitfalls in using this technology. If not careful, risks exist which may lead to incorrect estimation of reservoir properties and in the worst case to wrong drilling. In this paper, a model-driven inversion is used as an example to devise a practical workflow for typical post-stack inversion projects. The devised workflow consists of (1) log calibration, (2) selection of a seismic trace near a well, (3) well tie analysis, (4) wavelet estimation, (5) seismic amplitude calibration, (6) low frequency model building, (7) inversion job parameterization, (8) volume inversion and (9) interpretation. The technical background of all steps is reviewed and methods of problem solving presented. Emphasis is placed on how to quality control each stage of a multi-step process so that uncertainty is reduced to a minimum. With careful data preparation and job parameterization, model-driven inversion produces both absolute and relative impedances, useful for more accurate stratigraphic interpretation and reservoir property determination.

Pre-stack full waveform inversion of ultra-high-frequency marine seismic reflection data

The full waveform inversion (FWI) of seismic reflection data aims to reconstruct a detailed physical properties model of the subsurface, fitting both the amplitude and the traveltime of the reflections generated at physical discontinuities in the propagation medium. Unlike reservoir-scale seismic exploration, where seismic inversion is a widely adopted remote characterization tool, ultrahigh-frequency (UHF, 0.2–4.0 kHz) multichannel marine reflection seismology is still most often limited to a qualitative interpretation of the reflections’ architecture. Here we propose an elastic FWI methodology, custom-tailored for pre-stack UHF marine data in vertically heterogeneous media to obtain a decimetric-scale distribution of P-impedance, density and Poisson’s ratio within the shallow subseabed sediments. We address the deterministic multiparameter inversion in a sequential fashion. The complex trace instantaneous phase is first inverted for the P-wave velocity to make up for the lack of low frequency in the data and reduce the nonlinearity of the problem. This is followed by a short-offset P-impedance optimization and a further step of full offset range Poisson’s ratio inversion. Provided that the seismogram contains wide reflection angles (>40°), we show that it is possible to invert for density and decompose a posteriori the relative contribution of P-wave velocity and density to the P-impedance. A broad range of synthetic tests is used to prove the potential of the methodology and highlights sensitivity issues specific to UHF seismic. An example application to real data is also presented. In the real case, trace normalization is applied to minimize the systematic error deriving from an inaccurate source wavelet estimation. The inverted model for the top 15 m of the subseabed agrees with the local lithological information and core-log data. Thus, we can obtain a detailed remote characterization of the shallow sediments using a multichannel sub-bottom profiler within a reasonable computing cost and with minimal pre-processing. This has the potential to reduce the need of extensive geotechnical coring campaigns.

Minimax robust designs for wavelet estimation of nonparametric regression models with autocorrelated errors

We discuss the construction of designs for estimation of nonparametric regression models with autocorrelated errors when the mean response is to be approximated by a finite order linear combination of dilated and translated versions of the Daubechies wavelet bases with four vanishing moments. We assume that the parameters of the resulting model will be estimated by weighted least squares (WLS) or by generalized least squares (GLS). The bias induced by the unused components of the wavelet bases, in the linear approximation, then inflates the natural variation of the WLS and GLS estimates. We therefore construct our designs by minimizing the maximum value of the average mean squared error (AMSE). Such designs are said to be robust in the minimax sense. Our illustrative examples are constructed by using the simulated annealing algorithm to select an optimal [Formula: see text]-point design, which are integers, from a grid of possible values of the explanatory or design variable [Formula: see text]. We found that the integer-valued designs we constructed based on GLS estimation, have smaller minimum loss when compared to the designs for WLS or ordinary least squares (OLS) estimation, except when the correlation parameter [Formula: see text] approaches 1.

TIME‐VARYING SEISMIC WAVELET ESTIMATION FROM NONSTATIONARY SEISMIC DATA

AbstractSeismic wavelet estimation is an important part of seismic data processing and interpretation, whose reliability is directly related to the results of deconvolution and inversion. The methods for seismic wavelet estimation can be classified into two basic types: deterministic and statistical. By combining the deterministic spectral coherence method and the statistical skewness attribute method, the amplitude and phase of the time‐varying wavelet are estimated separately. There is no assumption on the wavelet's time‐independent nature or the phase characteristic. The advantage of this method is the ability to estimate time‐varying phase. Phase‐only corrections can then be applied by means of a time‐varying phase rotation. Alternatively, amplitude and phase deconvolution can be achieved to enhance the resolution and support the fine reservoir prediction and description. We illustrate the method with both synthetic and real data examples. Synthetic examples certify its feasibility while real data example demonstrates the ability to estimate the time‐varying characteristic of wavelets.

Wavelet reconstruction ofEandBmodes for CMB polarization and cosmic shear analyses

We present new methods for mapping the curl-free (E-mode) and divergence-free (B-mode) components of spin 2 signals using spin directional wavelets. Our methods are equally applicable to measurements of the polarisation of the cosmic microwave background (CMB) and the shear of galaxy shapes due to weak gravitational lensing. We derive pseudo and pure wavelet estimators, where E-B mixing arising due to incomplete sky coverage is suppressed in wavelet space using scale- and orientation-dependent masking and weighting schemes. In the case of the pure estimator, ambiguous modes (which have vanishing curl and divergence simultaneously on the incomplete sky) are also cancelled. On simulations, we demonstrate the improvement (i.e., reduction in leakage) provided by our wavelet space estimators over standard harmonic space approaches. Our new methods can be directly interfaced in a coherent and computationally-efficient manner with component separation or feature extraction techniques that also exploit wavelets.

Adaptive multifocus image fusion using block compressed sensing with smoothed projected Landweber integration in the wavelet domain.

The need for image fusion in current image processing systems is increasing mainly due to the increased number and variety of image acquisition techniques. Image fusion is the process of combining substantial information from several sensors using mathematical techniques in order to create a single composite image that will be more comprehensive and thus more useful for a human operator or other computer vision tasks. This paper presents a new approach to multifocus image fusion based on sparse signal representation. Block-based compressive sensing integrated with a projection-driven compressive sensing (CS) recovery that encourages sparsity in the wavelet domain is used as a method to get the focused image from a set of out-of-focus images. Compression is achieved during the image acquisition process using a block compressive sensing method. An adaptive thresholding technique within the smoothed projected Landweber recovery process reconstructs high-resolution focused images from low-dimensional CS measurements of out-of-focus images. Discrete wavelet transform and dual-tree complex wavelet transform are used as the sparsifying basis for the proposed fusion. The main finding lies in the fact that sparsification enables a better selection of the fusion coefficients and hence better fusion. A Laplacian mixture model fit is done in the wavelet domain and estimation of the probability density function (pdf) parameters by expectation maximization leads us to the proper selection of the coefficients of the fused image. Using the proposed method compared with the fusion scheme without employing the projected Landweber (PL) scheme and the other existing CS-based fusion approaches, it is observed that with fewer samples itself, the proposed method outperforms other approaches.

Wavelet estimation in varying coefficient models for censored dependent data

In this paper, we discuss the estimation of varying coefficient models based on censored data by wavelet technique when the survival and the censoring times are from a stationary α-mixing sequence. For the wavelet estimator of varying coefficient functions, the strong uniform convergence rate is derived and the asymptotic normality is established under the mild conditions. The strong uniform convergence rate we obtained is comparable with the optimal convergence rate of the nonparametric estimation in nonparametric models.

Comparison between deterministic and statistical wavelet estimation methods through predictive deconvolution: Seismic to well tie example from the North Sea

Wavelet estimation as well as seismic-to-well tie procedures are at the core of every seismic interpretation workflow. In this paper we perform a comparative study of wavelet estimation methods for seismic-to-well tie. Two approaches to wavelet estimation are discussed: a deterministic estimation, based on both seismic and well log data, and a statistical estimation, based on predictive deconvolution and the classical assumptions of the convolutional model, which provides a minimum-phase wavelet. Our algorithms, for both wavelet estimation methods introduce a semi-automatic approach to determine the optimum parameters of deterministic wavelet estimation and statistical wavelet estimation and, further, to estimate the optimum seismic wavelets by searching for the highest correlation coefficient between the recorded trace and the synthetic trace, when the time–depth relationship is accurate. Tests with numerical data show some qualitative conclusions, which are probably useful for seismic inversion and interpretation of field data, by comparing deterministic wavelet estimation and statistical wavelet estimation in detail, especially for field data example. The feasibility of this approach is verified on real seismic and well data from Viking Graben field, North Sea, Norway. Our results also show the influence of the washout zones on well log data on the quality of the well to seismic tie.

Sparse Multichannel Blind Deconvolution of Seismic Data via Spectral Projected-Gradient

In this work, an efficient numerical scheme is presented for seismic blind deconvolution in a multichannel scenario. The proposed method iterate with wo steps: first, wavelet estimation across all channels and second, refinement of the reflectivity estimate simultaneously in all channels using sparse deconvolution. The reflectivity update step is formulated as a basis pursuit denoising problem and a sparse solution is obtained with the spectral projected-gradient algorithm - faithfulness to the recorded traces is constrained by the measured noise level. Wavelet re-estimation has a closed form solution when performed in the frequency domain by finding the minimum energy wavelet common to all channels. Nothing is assumed known about the wavelet apart from its time duration. In tests with both synthetic and real data, the method yields sparse reflectivity series and stable wavelet estimates results compared to existing methods with significantly less computational effort.