Wavelet Estimation Research Articles

A time-varying wavelet extraction using local similarity

Seismic wavelet extraction is an integral part of high-resolution seismic analysis. However, most extraction methods ignore the time-varying characteristic of wavelets introduced by attenuation, scattering, and other physical processes during propagation. We have developed a time-varying wavelet extraction method based on local similarity. This method estimates the amplitude spectra by spectral modeling in the time-frequency domain. We estimated the phase of each spectrum in two steps: First, the phase range was estimated by the bispectrum of the high-order cumulants, and then the phase spectrum at every point was extracted with additional local similarity optimization. The extracted nonstationary wavelet improved the resolution of the wavelet estimation in the adjacent layers. We have determined the practicability and reliability of the proposed method using a numerical simulation, and we have compared the results of this method with those of the adaptive segmentation method.

Wavelet-Based Density Estimation in Presence of Additive Noise under Various Dependence Structures

We study the following model: . The aim is to estimate the distribution of X when only are observed. In the classical model, the distribution of is assumed to be known, and this is often considered as an important drawback of this simple model. Indeed, in most practical applications, the distribution of the errors cannot be perfectly known. In this paper, the author will construct wavelet estimators and analyze their asymptotic mean integrated squared error for additive noise models under certain dependent conditions, the strong mixing case, the β-mixing case and the ρ-mixing case. Under mild conditions on the family of wavelets, the estimator is shown to be -consistent and fast rates of convergence have been established.

Quasi-Positive Delta Sequences and Their Applications in Wavelet Approximation

A sufficient literature is available for the wavelet error of approximation of certain functions in theL2-norm. There is no work in context of multiresolution approximation of a function in the sense of sup-error. In this paper, for the first time, wavelet estimator for the approximation of a functionfbelonging toLipα[a,b]class under supremum norm has been obtained. Working in this direction, four new theorems on the wavelet approximation of a functionfbelonging toLipα,0<α≤1class using the projectionPmfof its wavelet expansions have been estimated. The calculated estimator is best possible in wavelet analysis.

Improving homomorphic wavelet estimation by compensating for residual NMO stretching on stack section

Wavelet estimation is a common step in seismic data processing and inversion. Homomorphic wavelet estimation has long utilized as a method that uses a seismic stack section with no phase presumption. Forming a stack section, normal move-out (NMO) correction must be applied on common midpoint (CMP) gathers, although it introduces NMO stretching. After stacking, residual of the NMO stretching may affect the stack section even after muting the highly stretched zone of the NMO corrected CMP gather. Presence of significant residual NMO stretching changes the spectral characteristics of data in time direction, by different degrees. Considering that in homomorphic process the wavelet is estimated based on the spectral characteristics of data, compensating for the residual NMO stretching, can improve the accuracy of the process. Here, we introduce a fast method of calculating the amount of residual NMO stretching and compensating for its effect on wavelet estimation. The proposed method needs limited prestack information like offsets and velocity function and include no prestack processing. We apply the proposed method on synthetic and real datasets and demonstrate the improvement of the estimated wavelet.

Seismic Screening for Hydrocarbon Prospects using Rock-Physics Attributes

Rock-physics templates (RPT), in combination with seismic AVO inversion data, can be used to screen for hydrocarbon prospects during exploration. With the improved quality and increased use of elastic seismic inversion, there has recently been a paradigm change in prospect mapping in the oil industry, and quantitative interpretation has become a widely used jargon. Rock-physics models are essential in that they help in converting elastic parameters from inversion data to reservoir parameters. Rock physics models plays an important role in many of the stages of seismic AVO inversions; including the petrophysical log evaluations, well ties and wavelet estimation, the setting of parameter constraints during the inversion, and the interpretation of the inversion results in terms of reservoir properties. Rock-physics models can also be used to quality check and modify the low-frequency model used in seismic AVO inversion, and to assess the quality and uncertainties of inverted elastic parameters. In this presentation we will demonstrate the use of rock-physics models during the different stages of seismic inversion, and how these can improve our ability to reveal hydrocarbon-associated anomalies.

A Comparative Study of the Performance of Estimating Long-Memory Parameter Using Wavelet-Based Entropies

In this paper, we discuss the performance of four estimators in the wavelet domain in order to estimate the parameter of stationary long-memory models. The goal of our article is to construct a wavelet estimate for the fractional differencing parameter d where the selection of the optimal level of the multiresolution decomposition is given by three entropies-based approaches, as alternative to subjective determination of the multiscale wavelet decomposition methodology. We have shown by Monte Carlo experiments that the concentration in an $$l^{p}$$lp norm entropy-based procedure improves considerably the other suggested entropy-based determination of the optimal decomposition level considered. The simulation results also show that the concentration in an $$l^{p}$$lp norm entropy-related criterion and the maximum scale decomposition method performs better in most cases and provides evidence of the power of the wavelet methods. We then applied wavelet-entropy estimators to some daily stock market indices.

Nonparametric estimation of a quantile density function by wavelet methods

In this paper nonparametric wavelet estimators of the quantile density function are proposed. Consistency of the wavelet estimators is established under the Lp risk. A simulation study illustrates the good performance of our estimators.

Strong consistency of wavelet estimators for errors-in-variables regression model

This paper studies the strong consistency of some estimators for an errors-in-variables regression model. We first provide an extension of Meister’s theorem. Then, the same problem is dealt with under the Fourier-oscillating noises. Finally, we prove two strong consistency theorems for wavelet estimators corresponding to non-oscillating and Fourier-oscillating noises.

Efficient 3-D frequency-domain mono-parameter full-waveform inversion of ocean-bottom cable data: application to Valhall in the visco-acoustic vertical transverse isotropic approximation

Computationally efficient 3-D frequency-domain full waveform inversion (FWI) is applied to ocean-bottom cable data from the Valhall oil field in the visco-acoustic vertical transverse isotropic (VTI) approximation. Frequency-domain seismic modelling is performed with a parallel sparse direct solver on a limited number of computer nodes. A multiscale imaging is performed by successive inversions of single frequencies in the 3.5–10 Hz frequency band. The vertical wave speed is updated during FWI while density, quality factor QP and anisotropic Thomsen's parameters δ and ϵ are kept fixed to their initial values. The final FWI model shows the resolution improvement that was achieved compared to the initial model that was built by reflection traveltime tomography. This FWI model shows a glacial channel system at 175 m depth, the footprint of drifting icebergs on the palaeo-seafloor at 500 m depth, a detailed view of a gas cloud at 1 km depth and the base cretaceous reflector at 3.5 km depth. The relevance of the FWI model is assessed by frequency-domain and time-domain seismic modelling and source wavelet estimation. The agreement between the modelled and recorded data in the frequency domain is excellent up to 10 Hz although amplitudes of modelled wavefields propagating across the gas cloud are overestimated. This might highlight the footprint of attenuation, whose absorption effects are underestimated by the homogeneous background QP model (QP = 200). The match between recorded and modelled time-domain seismograms suggests that the inversion was not significantly hampered by cycle skipping. However, late arrivals in the synthetic seismograms, computed without attenuation and with a source wavelet estimated from short-offset early arrivals, arrive 40 ms earlier than the recorded seismograms. This might result from dispersion effects related to attenuation. The repeatability of the source wavelets inferred from data that are weighted by a linear gain with offset is dramatically improved when they are estimated in the FWI model rather than in the smooth initial model. The two source wavelets, estimated in the FWI model from data with and without offset gain, show a 40 ms time-shift, which is consistent with the previous analysis of the time-domain seismograms. The computational efficiency of our frequency-domain approach is assessed against a recent time-domain FWI case study performed in a similar geological environment. This analysis highlights the efficiency of the frequency-domain approach to process a large number of sources and receivers with limited computational resources, thanks to the efficiency of the substitution step performed by the direct solver. This efficiency can be further improved by using a block-low rank version of the multifrontal solver and by exploiting the sparsity of the source vectors during the substitution step. Future work will aim to update attenuation and density at the same time of the vertical wave speed.

L p $L^{p}$ convergence rate of wavelet estimators for the bias and multiplicative censoring model

Using wavelets methods, Abbaszadeh, Chesneau, Doosti studied the density estimation problem under bias and multiplicative censoring (Stat. Probab. Lett. 82:932-941, 2012), and obtains the convergence rate of wavelet estimators in $L^{2}$ norm for a density function in Besov space. This paper deals with $L^{p}$ risk estimation with $1\leq p<\infty$ based on wavelet bases. Motivated by the work of Youming and Junlian (2014), we construct new estimators: a linear one and a nonlinear adaptive one; an upper bound of wavelet estimators on $L^{p}$ risk for a density function in Besov space is provided, which generalizes Abbaszadeh et al.’s theorems. It turns out that the nonlinear adaptive estimator obtains faster rate of convergence than the linear one for $r< p$ .

Source wavelet estimation using common mid-point gathers

The source wavelet estimation is a crucial process for full waveform inversion because we can hardly know source information from observed data. In order to implement the full waveform inversion in the common mid-point (CMP) domain, source wavelet estimation may be performed using CMP gathers for efficiency and consistency. In this paper, we introduce the source wavelet estimation using CMP gather data. Since the CMP gather is the time–domain data, the source wavelet is transformed into the frequency domain to apply full Newton method. Finally, it is transformed back into the time domain. Through numerical examples, we show how sensitive the source wavelet inversion is to both the direct waves on CMP gathers and the uncertainty of initial velocity. We can succeed in accurately estimating source wavelet if we introduce time damping and the CMP data has direct waves in spite of an incorrect initial velocity. This is because direct waves, which are key elements for source estimation, propagate through water layer of which velocity information is precisely known. Therefore, we have to exploit direct wave for successful source wavelet estimation in full waveform inversion.

Compensating for source and receiver ghost effects in full waveform inversion and reverse time migration for marine streamer data

In conventional marine seismic surveys, due to the time-delayed reflections from the sea surface on both source and receiver sides, ghosts present in recorded seismograms and lead to both phase and spectrum distortions (especially near certain frequency notches). To achieve a high-quality broad-band image/velocity model with conventional reverse time migration (RTM)/full waveform inversion (FWI) that adopts a synthetic zero-phase source wavelet and absorbing surface condition during wavefield modelling, marine seismic data have to be pre-processed to remove ghost effects. However, seismic deghosting is not a trivial task. Instead of employing an external deghosting process, we propose a strategy to compensate for ghost effects during FWI and RTM, which consists of two parts: first, to address phase distortions due to ghost effects by means of obtaining an accurate source wavelet estimation and adopting an appropriate surface boundary condition in both forward and backward wave propagation to appropriately generate ghosts; secondly, to build a compensation operator in the adjoint state computation to mitigate spectrum distortions caused by dominant ghost effects. To demonstrate the success and robustness of the proposed strategy, we present both synthetic experiments and field examples, which suggest that this strategy can lead to successful applications of FWI/RTM directly on marine seismic data without an extra deghosting process.

Nonparametric regression on random fields with random design using wavelet method

We consider non-linear wavelet-based estimators of spatial regression functions with (known) random design on strictly stationary random fields, which are indexed by the integer lattice points in the \(N\)-dimensional Euclidean space and are assumed to satisfy some mixing conditions. We investigate their asymptotic rates of convergence based on thresholding of empirical wavelet coefficients and show that these estimators achieve nearly optimal convergence rates within a logarithmic term over a large range of Besov function classes \(B^{s}_{p,q}\). Therefore, wavelet estimators still achieve nearly optimal convergence rates for random fields and provide explicitly the extraordinary local adaptability.

Linear Multifractional Stable Motion: Wavelet Estimation of H(·) and α Parameters*

Stoev and Taqqu introduced a linear multifractional stable motion (LMSM), an extension of a linear fractional stable motion (LFSM) such that the Hurst parameter H becomes a function H(t). The stability parameter α determines tail heaviness of marginal distributions of LMSM. Under some conditions, Stoev and Taqqu showed that H(t 0) is its self-similarity exponent at a t 0 ≠ 0; also, recently, Ayache and Hamonier established that H(t 0)−1/α and min t∈I H(t)−1/α are its local Hölder exponent at t 0 and uniform Hölder exponent on a compact interval I. We construct, strongly consistent wavelet estimators of min t∈I H(t), H(t 0), and α when α ∈ (1, 2) and H(·) is smooth with values in $$ \left[\underset{\bar{\mkern6mu}}{H},\overline{H}\right]\subset \left(1/\alpha, 1\right) $$ .

On Wavelet Estimation of the Derivatives of a Density Based on Biased Data

Here, we consider wavelet based estimation of the derivatives of a probability density function under random sampling from a weighted distribution and extend the results regarding the asymptotic convergence rates under the i.i.d. setup studied in Prakasa Rao (1996) to the biased-data setup. We compare the performance of the wavelet based estimator with that of the kernel based estimator obtained by differentiating the Efromovich (2004) kernel density estimator through a simulation study.

The mean consistency of wavelet density estimators

The wavelet estimations have made great progress when an unknown density function belongs to a certain Besov space. However, in many practical applications, one does not know whether the density function is smooth or not. It makes sense to consider the mean $L_{p}$ -consistency of the wavelet estimators for $f\in L_{p}$ ( $1\leq p\leq\infty$ ). In this paper, the authors will construct wavelet estimators and analyze their $L_{p}(\mathbb{R})$ performance. They prove that, under mild conditions on the family of wavelets, the estimators are shown to be $L_{p} $ ( $1\leq p\leq\infty$ )-consistent for both noiseless and additive noise models.

Nonlinear wavelet density estimation with data missing at random when covariates are present

In this paper, we construct the nonlinear wavelet estimator of a density with data missing at random when covariables are present, and provide an asymptotic expression for the mean integrated squared error (MISE) of the estimator. Unlike for kernel estimators, the MISE expression of the wavelet-based estimator still holds when the density function is piecewise smooth. Also, the asymptotic normality of the estimator is established.

Wavelet and multifractal estimation of the intermittent photic stimulation response in the electroencephalogram of patients with dyscirculatory encephalopathy

The task is to elucidate quantitative characteristics in electroencephalographic (EEG) patterns giving the possibility to estimate the disruptions of the functional state of the central nervous system caused by dyscirculatory encephalopathy of different severity. For solving the task the background and reactive EEG patterns are analyzed by the continuous wavelet transform and the wavelet-transform modulus maxima methods. The EEG responses to intermittent photic stimulation are used as reactive patterns. There are no statistical differences between the width of the singularity spectra of background and reactive patterns for all the subjects. Therefore, the degree of multifractality determined by this parameter does not change considerably during the photic stimulation. By contrast, the coefficients of photic driving and holding and the energy increase times gained in EEG patterns of patients with dyscirculatory encephalopathy differ significantly from the parameters determined for the healthy subjects. The reactive patterns have been demonstrated to have the different photic driving of beta, theta and alpha ranges for the patients of various groups. Frequencies of the theta range are reproduced mainly in the EEG patterns of patients with hypertension disease and with vertebrobasilar insufficiency. The maximal photic driving reaction of the beta range is noticed in the group with the vegetovascular dystonia. Frequencies of the alpha range are predominantly reproduced by the group with hypertension disease. The study demonstrates the opportunity to estimate quantitatively the dynamics of changes in energy characteristics of EEG patterns for various groups of patients having cerebrovascular disturbances. The results can be applied for an appropriate choice of treatment for such patients with the regard for their photic driving reactions.

Joint Bayesian wavelet and well-path estimation in the impedance domain

We addressed the problem of the well-to-seismic tie as a Bayesian inversion for the wavelet and well path in the impedance domain. The result of the joint inversion is a set of wavelets for multiple angle stacks, and a corresponding well path. The wavelet optimally links the impedance data along the well to the seismic data along the optimized well path in the seismic time domain. Starting with prior distribution for wavelet and well path, the method calculates the posterior distribution of conditioning the prior distributions with the seismic and well-log data. This is done by iteratively inverting the seismic data with the current wavelet, to obtain an impedance cube around the well. In a second step, the seismic impedances are projected onto the well path. By minimizing the misfit between the inverted seismic impedances and the impedances derived from the well log, the wavelet and well path are optimized. Comparing the well and seismic data in the impedance domain enables the method to work on short and noisy well logs. Another advantage of this method is its ability to derive wavelets for multiple angle stacks and multiple well locations simultaneously. We tested the method on synthetic and real data examples. The algorithm performed well in the synthetic examples, in which we had control over the modeling wavelet, and the wavelets derived for a real data example showed consistently good seismic-to-well ties for six angle stacks and seven wells. The main algorithm we developed was aimed to linearize the problem. We compared the posterior distribution of the linearized result with a sampling-based result in a real data example and found good agreement.

Accelerating patch-based directional wavelets with multicore parallel computing in compressed sensing MRI

Compressed sensing MRI (CS-MRI) is a promising technology to accelerate magnetic resonance imaging. Both improving the image quality and reducing the computation time are important for this technology. Recently, a patch-based directional wavelet (PBDW) has been applied in CS-MRI to improve edge reconstruction. However, this method is time consuming since it involves extensive computations, including geometric direction estimation and numerous iterations of wavelet transform. To accelerate computations of PBDW, we propose a general parallelization of patch-based processing by taking the advantage of multicore processors. Additionally, two pertinent optimizations, excluding smooth patches and pre-arranged insertion sort, that make use of sparsity in MR images are also proposed. Simulation results demonstrate that the acceleration factor with the parallel architecture of PBDW approaches the number of central processing unit cores, and that pertinent optimizations are also effective to make further accelerations. The proposed approaches allow compressed sensing MRI reconstruction to be accomplished within several seconds.