Wavelet Estimation Methods Research Articles

BOOTSTRAP WAVELET IN THE NONPARAMETRIC REGRESSION MODEL WITH WEAKLY DEPENDENT PROCESSES

This paper introduces a method of bootstrap wavelet estimation in a nonparametric regression model with weakly dependent processes for both fixed and random designs. The asymptotic bounds for the bias and variance of the bootstrap wavelet estimators are given in the fixed design model. The conditional normality for a modified version of the bootstrap wavelet estimators is obtained in the fixed model. The consistency for the bootstrap wavelet estimator is also proved in the random design model. These results show that the bootstrap wavelet method is valid for the model with weakly dependent processes.

Thresholding algorithms, maxisets and well-concentrated bases

The aim of this paper is to synthetically analyse the performances of thresholding and wavelet estimation methods. In this connection, it is useful to describe the maximal sets where these methods attain a special rate of convergence. We relate these “maxisets” to other problems naturally arising in the context of non parametric estimation, as approximation theory or information reduction. A second part of the paper is devoted to isolate two very special properties especially shared by wavelet bases, which allow them to behave almost as in an Hilbertian context even for Lp risks.

Wavelets in statistics: A review

The field of nonparametric function estimation has broadened its appeal in recent years with an array of new tools for statistical analysis. In particular, theoretical and applied research on the field of wavelets has had noticeable influence on statistical topics such as nonparametric regression, nonparametric density estimation, nonparametric discrimination and many other related topics. This is a survey article that attempts to synthetize a broad variety of work on wavelets in statistics and includes some recent developments in nonparametric curve estimation that have been omitted from review articles and books on the subject. After a short introduction to wavelet theory, wavelets are treated in the familiar context of estimation of «smooth» functions. Both «linear» and «nonlinear» wavelet estimation methods are discussed and cross-validation methods for choosing the smoothing parameters are addressed. Finally, some areas of related research are mentioned, such as hypothesis testing, model selection, hazard rate estimation for censored data, and nonparametric change-point problems. The closing section formulates some promising research directions relating to wavelets in statistics.

Zero‐phasing seismic data without wells in offshore West Africa: Reducing uncertainty and variability of the wavelet

Traditionally, many companies have zero‐phased seismic data using operators based on wavelets estimated from well matching. However, in new areas of exploration, other methods must be used. A data example from deep‐water West Africa is discussed in which the use of two independent methods of estimating the wavelet (physical modeling and statistical evaluation) allows for increased confidence in the resulting zero‐phasing. The two methods are complimentary in that they compensate for the inadequacies of each other's assumptions. The combination of the two to reduce uncertainty in wavelet estimation is the main objective. The acquisition and processing of the data are of great importance in ensuring wavelet stability in the section and allowing the wavelet estimation methods to be used reliably. Careful application of gapped deconvolutions in the processing sequence allows for wavelet compression and consistency over large areas of data that can then be zero‐phased using a single operator, even though the data covers diverse geology and water depths.

An algorithm for successive identification of reflections

A new algorithm for successive identification of seismic reflections is proposed. Generally, the algorithm can be viewed as a curve matching method for images with specific structure. However, in the paper, the algorithm works on seismic signals assembled to constitute an image in which the investigated reflections produce curves. In numerical examples, the authors work on signals assembled in CMP gathers. The key idea of the algorithm is to estimate the reflection curve parameters and the reflection coefficients along these curves by combining the multipulse technique and the generalized Radon transform. The multipulse technique is used for wavelet identification in each trace, and the generalized Radon transform is used to coordinate the wavelet identification between the individual traces. Furthermore, a stop criterion and a reflection validation procedure are presented. The stop criterion stops the reflection estimation when the actual estimated reflection is insignificant. The reflection validation procedure ensures that the estimated reflections follow the shape of the investigated reflection curves. The algorithm is successfully used in two numerical examples. One is based on a synthetic CMP gather, whereas the other is based on a real recorded CMP gather. Initially, the algorithm requires an estimate of the wavelet that can be performed by any wavelet estimation method.

Estimation of a point source wavelet in elastic media

A new method of wavelet estimation in elastic or acoustic media is presented. The method is based on the simple procedure of weighted summation of seismic traces for all the distances of source‐receivers, with a horizontal offset r as a weight. The model treated consists of a homogeneous elastic (or acoustic) layer with the free surface above and a half‐space below. The Lamé parameters and the density of the half‐space can be any function of the depth. A P‐wave point source operates in the layer, and the vertical displacement field or the vertical particle velocity field (or the pressure in the acoustic case) is measured by two horizontal lines of receivers located at two depth levels in the same layer. To obtain the wavelet‐estimation algorithm, the Fourier‐Hankel transform of the field is used. It is shown that there are two possibilities of data measuring: (1) when both the lines of the receivers are below the source and (2) when one of the lines is above the source. Numerical examples show that the proposed method gives a correct estimate of the source wavelet.

Determination of the signature of a dynamite source using source scaling, Part 2: Experiment

In April 1990 we performed an experiment in the Netherlands to test the theory of the determination of the signature of a dynamite source using the scaling law. The theory says that the source signature may be determined from the recorded seismic data using two shots of different charge size at the shotpoint; we used 125 g and 500 g charges. The theory was put at risk with a 250-g test charge at each shotpoint. According to the theory, the test record should be different from the other two and, apart from the noise, should be predictable from them. This experiment was repeated 95 times at approximately 50 m shotpoint intervals, using a 240-channel recording system. The results corroborate the theory within an acceptable error. The second‐derivative of the volume injection function was extracted as the source signature; it varied slightly from shot to shot and was minimum phase. This new method of seismic data acquisition allows the signature of the dynamite source to be obtained from the data, uncontaminated by the earth, and avoids the assumptions that must be made in statistical wavelet estimation methods. If there is good shot‐to‐shot repeatability, the second shot is only needed occasionally for calibration.

Wave theoretic seismic processing.

The goal of seismic processing is to determine the character of the subsurface of the earth from wave-field measurements on the surface. In this inversion problem, both the target and the propagating medium are unknown. Also, the character of the energy source is not typically known. In this paper, the evolution of methods which address various aspects of the problem is described. Approximate linear inverse scattering approaches often can produce useful results. Nonlinear inversion approaches have the potential of providing certain important practical benefits not provided by linear approaches, although the former are typically less robust than the latter. Examples of linear and nonlinear inversion methods illustrate the strengths and weaknesses of each. Real data quality issues such as aperture, spatial sampling, and noise will impact what algorithm is appropriate and what goal level can be achieved with confidence. A recent wavelet estimation method [A. B. Weglein and B. G. Secrest, Geophysics 55, No. 7 (1990)] provides an approach for identifying the source signature and radiation pattern.

Why don’t we measure seismic signatures?

There are three related problems with our approach to signature deconvolution. First, there is a confusion among geophysicists about the basis of the convolutional model itself, which leads to doubts about the value of measurements of the source signature. Secondly, it is not generally recognized that statistical methods of wavelet estimation are unreliable. Thirdly, many explorationists are unaware that it is practical in many cases to make meaningful measurements of the source signature. The convolutional model of the reflection seismogram applies only for a point source, and is the convolution of the source signature with the impulse response of the earth, of Green’s function, which contains all possible arrivals, including reflections, refractions, multiples and diffractions. Stabilized deconvolution of the data with a known band‐limited signature is straightforward. The signature can be obtained by independent measurements, as described in the literature. The recovery of the elastic layer parameters from the band‐limited impulse response of the earth, after removal of the source signature by deconvolution, is the problem of inversion, and is not discussed in this paper. The theory of wave propagation does not support the commonly held view that a reflection seismogram can be regarded as a convolution of a wavelet with the series of normal‐incidence primary reflection coefficients. This is true of both prestack and poststack data. Poststack seismic inversion schemes, based on this model, that use well logs to extract the wavelet for predicting lateral variations in lithology away from the wells, rely on the wavelet to be laterally invariant. Even if there is perfect shot‐to‐shot repeatability, this model must yield a different wavelet at every well, and therefore the extracted wavelet does vary laterally. These schemes are therefore self‐contradictory and, in the worst cases, their results are likely to be worthless. Published methods for determining the source signature from measurements for the land vibrator, marine seismic source arrays, and dynamite on land are summarized. None of these methods appears to be in use. A Vibroseis example is included to show that the signal transmitted into the ground by the vibrators does not closely resemble the predetermined sweep, as is normally assumed. The transmitted signal could be determined in processing from measurements of the vibrator behaviour that are made in production for vibrator control, if only these measurements were recorded. Normally they are not. Instead of using measurements to determine the signature, the exploration industry relies on wavelet estimation methods that depend on both a model and statistical assumptions that have no theoretical justification.

Wavelet estimation for a multidimensional acoustic or elastic earth

A new and general wave theoretical wavelet estimation method is derived. Knowing the seismic wavelet is important both for processing seismic data and for modeling the seismic response. To obtain the wavelet, both statistical (e.g., Wiener‐Levinson) and deterministic (matching surface seismic to well‐log data) methods are generally used. In the marine case, a far‐field signature is often obtained with a deep‐towed hydrophone. The statistical methods do not allow obtaining the phase of the wavelet, whereas the deterministic method obviously requires data from a well. The deep‐towed hydrophone requires that the water be deep enough for the hydrophone to be in the far field and in addition that the reflections from the water bottom and structure do not corrupt the measured wavelet. None of the methods address the source array pattern, which is important for amplitude‐versus‐offset (AVO) studies. This paper presents a method of calculating the total wavelet, including the phase and source‐array pattern. When the source locations are specified, the method predicts the source spectrum. When the source is completely unknown (discrete and/or continuously distributed) the method predicts the wavefield due to this source. The method is in principle exact and yet no information about the properties of the earth is required. In addition, the theory allows either an acoustic wavelet (marine) or an elastic wavelet (land), so the wavelet is consistent with the earth model to be used in processing the data. To accomplish this, the method requires a new data collection procedure. It requires that the field and its normal derivative be measured on a surface. The procedure allows the multidimensional earth properties to be arbitrary and acts like a filter to eliminate the scattered energy from the wavelet calculation. The elastic wavelet estimation theory applied in this method may allow a true land wavelet to be obtained. Along with the derivation of the procedure, we present analytic and synthetic examples.

AN INVESTIGATION OF THE SPECTRAL PROPERTIES OF PRIMARY REFLECTION COEFFICIENTS*

ABSTRACTThis paper investigates the form of the nonwhiteness found in the reflection coefficients from a wide variety of rock sequences around the world. In all but one case densities are taken as constant due to the paucity of suitable density data. The reflection sequences are pseudo‐white only above a corner frequency, below which their power spectrum falls away according to a power law fβ, where β is between 0.5 and 1.5. This spectrum can be adequately modelled in practice very simply with an ARMA (1, 1) process which acts on a white innovation sequence. The corollary of this is that before wavelet estimation methods are applied (all of which‐except those based on synthetic seismograms—presuppose white reflection sequences) or deconvolution filters estimated, seismic traces should be filtered with the inverse of this process.Interestingly, the estimated ARMA processes group themselves into two clearly differentiated categories, having very different indices of predictability (or, strictly, indices of linear determinism). The two categories apparently correlate precisely with two kinds of sedimentation: one which consists largely of sequences of rocks with repeating properties, called “repetitive” in this paper but perhaps loosely describable as “cyclic”, and the other which is randomly bedded with no apparent pattern of components. The former has indices of predictability which are two to four times as great as those of the latter. Another, probably related, property is that β for the repetitive sequences tends to be greater than that for non‐repetitive rock columns.The observed power spectra are shown to be consistent with a simple model for the logarithm of acoustic impedance consisting of a mixture of processes where the distribution of (time) scale parameters is reciprocal.Detailed effects of block‐averaging and sampling the logs are shown to depend on the type of sequence under examination.

An examination of the exponential decay method of mixed‐phase wavelet estimation

Signal processing theory states that an isolated wavelet which is causal and mixed phase may be converted to minimum phase by applying an exponential decay of amplitude with time. The exponential decay might therefore be a useful preprocessing step for seismic wavelet estimation since many estimation methods require that the wavelet in the data be minimum phase. This is the basis of a method proposed by Taner and Coburn (1980). The wavelets in a seismic trace, however, are generally not isolated, but instead are convolved with a densely populated reflection coefficient series causing severe wavelet overlap. Wavelet estimation is generally done using a window of data from the seismic trace which excludes refractions, surface waves, and data with poor signal‐to‐noise ratios. Due to the wavelet overlap, the window generally truncates wavelets at the window edges. When exponential decay is applied to the window, these truncated wavelets dominate the wavelet estimation methods. When no wavelet truncation occurs, the exponential decay converts each wavelet to minimum phase, and complete wavelets dominate the data window. If the reflection series is uncorrelated, then the autocorrelations of these data windows, when averaged over many traces, give an average autocorrelation which equals that of the decayed wavelet. This autocorrelation gives the correct minimum‐phase estimated wavelet. When truncation of wavelets does occur, the autocorrelation of the decayed data window does not equal that of the decayed wavelet, and an erroneous wavelet is estimated. Therefore, the exponential decay method is only useful for seismic wavelet estimation when data windows may be chosen such that no wavelets are truncated at the window onset.

Ghosting and marine signature deconvolution: A prerequisite for detailed seismic interpretation

Detailed lithologic interpretation of seismic sections and/or pseudo‐sonic logs generated from seismic data requires that the seismic trace can be modeled as a reflection series convolved with a zero‐phase broadband wavelet. Ghosting and marine signature deconvolution processing is a prerequisite for assuring that the seismic wavelet on a marine CDP section will be zero phase. A deterministic approach to deconvolution is centered around the concept of abandoning the purely statistical method of wavelet estimation and actually measuring the seismic wavelet. A proper signature recording for marine data is, therefore, a crucial component of deterministic deconvolution. Another important element in the deterministic deconvolution sequence is the application of a deghosting filter to remove near‐surface reflections. Proper application of a deghosting filter significantly improves the correlation between log synthetics and the seismic trace. It has been found that statistical deconvolution schemes, because of the number of statistical hypotheses required to produce a deconvolution filter, produce residual wavelets that are highly variable in character and whose average phases cover the entire phase spectrum, modulo 2π. Examples of a Gulf Coast marine line which was shot with Aquapulse™, air gun, and Maxipulse™ sources by the RV Hollis Hedberg are presented to demonstrate the differences between statistical and deterministic deconvolution processing sequences. It will be shown, using sonic logs from wells adjacent to the seismic line, that the deterministic deconvolution sections for all three sources are close to zero phase while the statistical deconvolution sections have residual average phase errors between 180 and 270 degrees. The deterministic deconvolution sections have a high degree of correlation among themselves and to the wells adjacent to the line, while the statistical deconvolution sections correlate poorly to each other and to the wells. Synthetic seismograms and their impedance logs, and the seismic sections and their corresponding pseudo‐sonic logs, are used to demonstrate how deconvolution influences lithologic interpretation. ™Western Geophysics Co.

THE OLD AND THE NEW IN SEISMIC DECONVOLUTION AND WAVELET ESTIMATION*

AbstractVarious seismic deconvolution operators can be determined by estimating a seismic wavelet and subsequently designing an appropriate inverse filter which converts the wavelet to a spike. Seismic wavelets and deconvolution operators must be estimated in a time adaptive sense due to the nonstationarity of the seismic trace. The wavelet estimation methods considered in this paper either use the assumption of a minimum phase wavelet and a random impulse response, or the assumption that the wavelet cepstrum is readily separable from the cepstrum of the seismic trace. The former assumption is required in using the Hilbert transform and Wiener‐Levinson wavelet estimations, while the latter assumption is used in homomorphic deconvolution. These wavelet estimates can be used in the design of multichannel Wiener and Kalman deconvolution operators. Multichannel usage of homomorphic deconvolution can also be implemented through various types of cepstral stacking.The discussion of deconvolution filter design focuses on the problems of filter length, degree of prewhitening, and nonstationarity. In designing time adaptive deconvolution filters, the autocorrelation function can be used to monitor the nonstationarity of the seismic trace. The autocorrelation function, which is used in the computation of least squares inverse filters, can be estimated in an optimum fashion by using the maximum entropy method. Differences between minimum phase Wiener deconvolution and maximum entropy deconvolution become more pronounced for shorter data gates. As a result, the maximum entropy approach is preferred for time adaptive deconvolution.In this paper, the performance of various wavelet estimators and inverse filters is discussed using real and synthetic seismic data. Discussions of homomorphic deconvolution and maximum entropy prediction error filtering are merged with descriptions of conventional approaches to deconvolution.