Wavelet Estimation Methods Research Articles

Estimation of the depth-variant seismic wavelet based on the modified unscaled S-transform

Seismic inversion is one of the key techniques used for reservoir characterization. Depth-domain seismic inversion eliminates the cumulative errors associated with depth-to-time and time-to-depth conversions, thus providing geologists and reservoir engineers with an intuitive basis for geological interpretation. The method has received increasing attention in the field of reservoir characterization. Extracting accurate depth-domain seismic wavelets is a prerequisite for successful depth-domain seismic inversion. However, the depth-domain wavelet is velocity-dependent and exhibits significant non-stationarity, which leads to the failure of seismic wavelet estimation methods based on the stationary convolutional model. To this end, we propose a modified wavenumber-domain unscaled S-transform (MWUST) method to accomplish accurate estimation of depth-domain seismic wavelets. The proposed method enhances the accuracy of wavenumber components by removing the linear wavenumber-dependent term from the S-transform. Furthermore, it introduces slope and intercept parameters to improve the depth resolution at low wavenumbers, thereby yielding a more reliable depth–wavenumber spectrum. Subsequently, the relationship between the non-stationary depth-domain seismic wavelet and the depth–wavenumber spectrum is established, allowing for the accurate extraction of non-stationary wavelets under the assumption that the depth-domain reflectivity is a random sequence. Synthetic and real data applications have been used to verify the effectiveness of the proposed method.

Nonparametric statistical inference for stochastic optimal control problems and its applications for financial investment

Against a backdrop of financial decision-making, this study examines nonparametric statistical inference problems in stochastic optimal control problems. First, we construct a nonparametric estimation model using the wavelet estimation method for drift and diffusion coefficients. In this study, we focus on stochastic linear–quadratic (SLQ) problems, solved using Riccati equations, and apply this to a cash management problem within the linear–quadratic framework. The results show the effectiveness of the wavelet method using a numerical case.

Source-Independent Waveform Inversion Method for Ground Penetrating Radar Based on Envelope Objective Function

For the full waveform inversion, it is necessary to provide an accurate source wavelet for forwarding modeling in the iteration. The source wavelet estimation method based on deconvolution technology can solve this problem to some extent, but we find that the estimated source wavelet is not accurate and needs to be manually corrected repeatedly in the iteration. This process is highly operator-intensive, and the update process is time-consuming and increases the potential for errors. We propose a source-independent waveform inversion (SIEWI) scheme for cross-hole GPR data, and use the envelope objective function combined with this method to effectively reduce the nonlinearity of inversion. The residual field used by SIEWI to construct the gradient inherits the characteristics of the envelope wavefield. Compared with full waveform inversion (FWI), SIEWI is more robust and less sensitive to frequency components and inaccurate source wavelet. To avoid cycle jumping, the multi-scale strategy effectively utilizes the properties of convolutional wavefields. In one iteration, the wavefield is decomposed into multiple frequency bands through multiple convolutions in the time domain to construct a multi-scale inversion strategy that preferentially inverts low-frequency information.

Nonstationary deconvolutive Radon transform

Different versions of the Radon transform (RT) are widely used in seismic data processing to focus the recorded seismic events. Multiple separation, data interpolation, and noise attenuation are some of the RT applications in seismic processing workflows. Unfortunately, conventional RT methods cannot focus events perfectly in the RT domain. This problem arises due to the blurring effects of the source wavelet and the nonstationary nature of the seismic data. Sometimes, the distortion results in a big difference between the original data and its inverse transform. We have developed a nonstationary deconvolutive RT (DecRT) to handle these two issues. Our algorithm takes advantage of a nonstationary convolution technique that builds on the concept of block convolution and the overlap method, in which the convolution operation is defined separately for overlapping blocks. Therefore, it allows the Radon basis function to take arbitrary shapes in the time and space directions. In addition, we introduce a nonstationary wavelet estimation method to determine time-space-varying wavelets. The wavelets and the Radon panel are estimated simultaneously and in an alternating way. Numerical examples demonstrate that our nonstationary DecRT method can significantly improve the sparsity of Radon panels. Hence, inverse RT does not suffer from the distortion caused by unfocused seismic events.

A Note on the Nonparametric Estimation of the Conditional Mode by Wavelet Methods

The purpose of this note is to introduce and investigate the nonparametric estimation of the conditional mode using wavelet methods. We propose a new linear wavelet estimator for this problem. The estimator is constructed by combining a specific ratio technique and an established wavelet estimation method. We obtain rates of almost sure convergence over compact subsets of Rd. A general estimator beyond the wavelet methodology is also proposed, discussing adaptivity within this statistical framework.

Time-varying wavelet estimation and deconvolution for nonstationary data based on a FWE function

• We present a nonstationary convolution model constituted of the reflectivity series and a time-varying wavelet matrix. • We propose a novel wavelet estimation method toextract time-varying wavelet at every instant for nonstationary seismic data . • We achieve blind deconvolution without assumptions about factor Q value and the source wavelet.

On the seismic wavelet estimative and reflectivity recovering based on linear inversion: Well-to-seismic tie on a real data set from Viking Graben, North Sea

Tying seismic data to well data is critical in reservoir characterization. In general, the main factors controlling a successful seismic well tie are an accurate time-depth relationship and a coherent wavelet estimate. Wavelet estimation methods are divided into two major groups: statistical and deterministic. Deterministic methods are based on using the seismic trace and the well data to estimate the wavelet. Statistical methods use only the seismic trace and generally require assumptions about the wavelet’s phase or a random process reflectivity series. We have compared the estimation of the wavelet for seismic well tie purposes through least-squares minimization and zero-order quadratic regularization with the results obtained from homomorphic deconvolution. Both methods make no assumption regarding the wavelet’s phase or the reflectivity. The best-estimated wavelet is used as the input to sparse-spike deconvolution to recover the reflectivity near the well location. The results show that the wavelets estimated from both deconvolutions are similar, which builds our confidence in their accuracy. The reflectivity of the seismic section is recovered according to known stratigraphic markers (from gamma-ray logs) present in the real data set from the Viking Graben field, Norway.

Wavelet Estimation of Semi-parametric Regression Model

The semi-parametric regression model combines parametric and nonparametric regression. However, non-parametric estimation may provide flexible solutions to the problems suffers by the regression model, but the problem of dimensionality that this estimator suffers, which occurs due to the increasing number of explanatory variables, still remain, this, in turn, may reduce the accuracy of the estimation process. Estimate the non-parametric part of the semi-parametric models that can be studied using conventional non-parametric methods such as the Spline Smoothing and Kernel Smoothing. However, there are other non-parametric methods that can be used, therefore, in this paper, the semi-parametric regression model was estimated by employing the wavelet estimate for the soft threshold, according to the "Speckman" method, and then comparing it with the two methods, Nadaraya-Watson and Local Linear, through the implementation of simulation experiments that included different sample sizes and threshold values. The parametric part estimation of the partially linear model according to the least-squares method was not identical to those estimates using the Speckman method, that is because the least-squares method was not appropriate for the uneven nature of the number of weekly work hours. Simulation experiments have demonstrated the efficiency of the wavelet estimation method and its superiority over other methods. The above estimation methods were applied to real data related to the study of the production value for the public industrial sector in Iraq, and some factors affect it, such as the value of industrial supplies, the total wages of workers, and the number of workers.

Intrinsic Wavelet Regression for Curves of Hermitian Positive Definite Matrices

Intrinsic wavelet transforms and wavelet estimation methods are introduced for curves in the non-Euclidean space of Hermitian positive definite matrices, with in mind the application to Fourier spectral estimation of multivariate stationary time series. The main focus is on intrinsic average-interpolation wavelet transforms in the space of positive definite matrices equipped with an affine-invariant Riemannian metric, and convergence rates of linear wavelet thresholding are derived for intrinsically smooth curves of Hermitian positive definite matrices. In the context of multivariate Fourier spectral estimation, intrinsic wavelet thresholding is equivariant under a change of basis of the time series, and nonlinear wavelet thresholding is able to capture localized features in the spectral density matrix across frequency, always guaranteeing positive definite estimates. The finite-sample performance of intrinsic wavelet thresholding is assessed by means of simulated data and compared to several benchmark estimators in the Riemannian manifold. Further illustrations are provided by examining the multivariate spectra of trial-replicated brain signal time series recorded during a learning experiment. Supplementary materials for this article are available online.

Time-varying wavelet estimation and its applications in deconvolution and seismic inversion

Wavelet holds an essential role in seismic data processing and characterization, for examples deconvolution and seismic inversion. Unfortunately, wavelet is an unknown data. Several existing methods attempt to estimate and extract the wavelet from seismic data. However, the methods give only a single wavelet from one seismic trace. When seismic data are non-stationer, single wavelet usage will cause a problem, that is raising the error. This paper proposes a time-varying wavelet estimation method to accommodate this problem. It uses matrix diagonalization to estimate a set of wavelets. Next, the time-varying wavelet is applied to deconvolution and seismic inversion. The experiment shows that time-varying wavelet improves the results in both deconvolution and seismic inversion. The errors decreased and spectrum bandwidth broadened.

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.

Wavelets and estimation of long memory in nonstationary models: Does anything beat the exact local whittle estimator?

ABSTRACTIn this article, we analyze the performance of five estimation methods for the long memory parameter d. The goal of our article is to construct a wavelet estimate for the fractional differencing parameter in nonstationary long memory processes that dominate the well-known estimate of Shimotsu and Phillips (2005). The simulation results show that the wavelet estimation method of Lee (2005) with several tapering techniques performs better under most cases in nonstationary long memory. The comparison is based on the empirical root mean squared error of each estimate.

Comparison of wavelet estimation methods

Wavelet estimation is a very important task in seismic data processing and analysis such as deterministic deconvolution, seismic-to-well tie, and seismic inversion, among others. We investigated the wavelets estimated from four different methods: (1) the wavelet estimated from the seafloor signal; (2) the wavelet estimated fully from well-log data; (3) the wavelet estimated using seismic and well-log data; and (4) the wavelet estimated from sparse-spike deconvolution. The wavelets estimated from 2-D seismic data using the four methods are quite comparable to one another. The results of the deconvolution and inversion of the 2-D seismic data using the four wavelets show that the wavelet estimated from the seafloor signal can be as effective as those estimated from the more rigorous methods.

A novel seismic wavelet estimation method

Abstract A novel seismic wavelet estimation scheme is proposed based on S transform and Least Squares (LS) method. The scheme can reconstruct the coherent components after estimating major wavelet even when the signal-to-noise ratio (SNR) is low. The main frequency and delay along the event should be estimated. The proposed scheme can estimate seismic source wavelet with minor distortion. It can be realized as follows: firstly uses an S transform to estimate the delay and coarse frequency information, secondly applies a filter designed in S transform domain, thirdly gets the data in time domain by applying an inverse S transform to the filtered data, fourthly uses a windowed FFT unit to get the accuracy frequency estimation, and fifthly reconstructs the seismic source wavelet after estimating the energy based on LS method in time domain. Furthermore, the space–time-variant characteristic of the source wavelet is taken into account. An iteration scheme is designed to estimate all components of wavelet. The simulation results show that the proposed method can estimate the source wavelet effectively when the SNR is higher than 0 dB.

Wavelet Estimation Method for Short-Term Prediction of Deep Pit Settlement

A wavelet estimation method is presented herein to estimate deep pit settlement. In this method, the pit settlement is decomposed into the trend settlement and the stochastic settlement by using Wavelet Analysis based on the characteristic of influencing factor. The model identifier is established by using artificial neural network (ANN), and trained to approximate the trend settlement. Then, the prediction controller developed could be applied for estimating the actual settlement. Finally, the verification examples show that the WIAN is an effective tool for predicting the pit settlement dynamically , high precision could be expected and achieved.

Full Bayesian wavelet inference with a nonparametric prior

In this paper, we introduce a new Bayesian nonparametric model for estimating an unknown function in the presence of Gaussian noise. The proposed model involves a mixture of a point mass and an arbitrary (nonparametric) symmetric and unimodal distribution for modeling wavelet coefficients. Posterior simulation uses slice sampling ideas and the consistency under the proposed model is discussed. In particular, the method is shown to be computationally competitive with some of best Empirical wavelet estimation methods.

Wavelets in functional data analysis: Estimation of multidimensional curves and their derivatives

A wavelet-based method is proposed to obtain accurate estimates of curves in more than one dimension and of their derivatives. By means of simulation studies, this novel method is compared to another locally-adaptive estimation technique for multidimensional functional data, based on free-knot regression splines. This comparison shows that the proposed method is particularly attractive when the curves to be estimated present strongly localized features. The multidimensional wavelet estimation method is thus applied to multi-lead electrocardiogram records, where strongly localized features are indeed expected.

How reliable is statistical wavelet estimation?

Well logs often are used for the estimation of seismic wavelets. The phase is obtained by forcing a well-derived synthetic seismogram to match the seismic, thus assuming the well log provides ground truth. However, well logs are not always available and can predict different phase corrections at nearby locations. Thus, a wavelet-estimation method that reliably can predict phase from the seismic alone is required. Three statistical wavelet-estimation techniques were tested against the deterministic method of seismic-to-well ties. How the choice of method influences the estimated wavelet phase was explored, with the aim of finding a statistical method which consistently predicts a phase in agreement with well logs. It was shown that the statistical method of kurtosis maximization by constant phase rotation consistently is able to extract a phase in agreement with seismic-to-well ties. A statistical method based on a modified mutual-information-rate criterion was demonstrated to provide frequency-dependent phase wavelets where the deterministic method could not. Time-varying statistical wavelets also were estimated with good results — a challenge for deterministic approaches because of the short logging sequence. It was concluded that statistical techniques can be used as quality control tools for the deterministic methods, as a way of extrapolating phase away from wells, or to act as standalone tools in the absence of wells.

Frequency-domain reverse-time migration with source estimation

Although artificially generated seismic sources such as dynamite, vibroseis, and air guns are used in seismic exploration, it is not easy to exactly recover the source wavelet in field recording or in data processing. For this reason, seismic data processing often assumes that an explosive-source wavelet can be described by a well-known function (e.g., a Ricker wavelet), a near-offset trace, or a deconvolved wavelet. In frequency-domain waveform inversion, it has been proven that a source wavelet can be estimated by an optimization method, and incorporating the source wavelet estimation into an inversion algorithm yields better inversion results. We have developed source wavelet estimation into 2D two-way frequency-domain reverse-time migration. The source wavelet is first estimated independently of reverse-time migration by an optimization method such as the full Newton method. It is then used in reverse-time migration. This source-wavelet-incorporated reverse-time migration algorithm is applied to three model data sets: a simple anticline model, the Institut Français du Petrole (IFP) Marmousi model, and the BP 2004 EAGE model. Numerical examples were used to show that better migration images can be obtained by using an estimated source wavelet than those obtained by using a designatured wavelet without source estimation. To enhance the migrated images, the Laplacian filter is applied and migrated images are scaled using the diagonal of the pseudo-Hessian matrix. It is expected that the source wavelet estimation method can be directly applied to 2D or 3D time-domain two-way reverse-time migration.

Wavelet estimation by matching well-log, VSP, and surface-seismic data

In this paper, we present a method of wavelet estimation by matching well-log, VSP, and surface-seismic data. It’s based on a statistical model in which both input and output are contaminated with additive random noise. A coherency matching technique is used to estimate the wavelet. Measurements of goodness-of-fit and accuracy provide tools for quality control. A practical example suggests that our method is robust and stable. The matching and estimation of the wavelet is reliable within the seismic bandwidth. This method needs no assumption on the wavelet amplitude and phase and the main advantage of the method is its ability to determine phase.