Showing papers on "Wavelet published in 1984"

Decomposition of Hardy functions into square integrable wavelets of constant shape

Abstract: An arbitrary square integrable real-valued function (or, equivalently, the associated Hardy function) can be conveniently analyzed into a suitable family of square integrable wavelets of constant shape, (i.e. obtained by shifts and dilations from any one of them.) The resulting integral transform is isometric and self-reciprocal if the wavelets satisfy an “admissibility condition” given here. Explicit expressions are obtained in the case of a particular analyzing family that plays a role analogous to that of coherent states (Gabor wavelets) in the usual $L_2 $ -theory. They are written in terms of a modified $\Gamma $-function that is introduced and studied. From the point of view of group theory, this paper is concerned with square integrable coefficients of an irreducible representation of the nonunimodular $ax + b$-group.

Phase estimation using the bispectrum

Abstract: The information which is of importance in the reflection seismic method resides in the reflectivity series. In order to extract this information about the subsurface, the blurring effect of the seismic wavelet must first be removed. Since this signature is generally unknown, various wavelet estimation schemes have been developed. The one most currently used in the seismic industry is based on the assumption that the seismic wavelet has the minimum-phase property. This restrictive assumption is often incorrect. The purpose of this paper is to explore the use of the bispectrum in order to obviate the minimum-phase requirement. Specifically, using synthetic examples, we develop and compare three different algorithms which determine the wavelet phase from the bispectrum of the reflection seismogram. An important aspect of the problem not treated before is the application of the bispectral technique to band-limited data.

Predictive deconvolution and the zero‐phase source

Abstract: Predictive deconvolution is commonly applied to seismic data generated with a Vibroseisr® source. Unfortunately, when this process invokes a minimum‐phase assumption, the phase of the resulting trace will not be correct. Nonetheless, spiking deconvolution is an attractive process because it restores attenuated higher frequencies, thus increasing resolution. For detailed stratigraphic analyses, however, it is desirable that the phase of the data be treated properly as well. The most common solution is to apply a phase‐shifting filter that corrects for errors attributable to a zero‐phase source. The phase correction is given by the minimum‐phase spectrum of the correlated Vibroseis wavelet. Because no minimum‐phase spectrum truly exists for this bandlimited wavelet, white noise is added to its amplitude spectrum in order to design the phase‐correction filter. Different levels of white noise, however, produce markedly different results when field data sections are filtered. A simple argument suggests that th...

A computationally fast approach to maximum-likelihood deconvolution

Abstract: In this paper we derive and implement a maximum-likelyhood deconvolution (MLD) algorithm, based on the same channel and statistical models used by Kormylo and Medel (1983a), that leads to many fewer computations than their MLD algorithm. Both algorithms can simultaneously estimate a nonminimum phase wavelet and statistical parameters, detect locations of significant reflectors, and deconvolve the data. Our MLD algorithm is implemented by a two-phase block component method (BCM). The phase-1 block functions like a coarse adjustment of unkown quantities and provides a set of initial conditions for the phase-2 block, which functions like a fine adjustment of unknown quantities. We demonstrate good performance of our algorithm for both synthetic and real data.--Modified journal abstract.

On errors of fit and accuracy in matching synthetic seismograms and seismic traces

Abstract: A synthetic seismogram that closely resembles a seismic trace recorded at a well may not be at all reliable for, say, stratigraphic interpretation around the well. The most accurate synthetic seismogram is, in general, not the one that displays the smallest errors of fit to the trace but the one that best estimates the noise on the trace. If the match is confined to a short interval of interest or if the seismic reflection wavelet is allowed to be unduly long, there is considerable danger of forcing a spurious fit that treats the noise on the trace as part of the seismic reflection signal instead of making a genuine match with the signal itself. This paper outlines tests that allow an objective and quantitative evaluation of the accuracy of any match and illustrates their application with practical examples. The accuracy of estimation is summarized by the normalized mean square error (NMSE) in the estimated reflection signal, which is shown to be (/n)(PN/PS) where PS/PN is the signal-to-noise power ratio and n is the spectral smoothing factor. That is, the accuracy varies directly with the ratio of the power in the signal (taken to be the synthetic) to that in the noise on the seismic trace, and the smoothing acts to improve the accuracy of the predicted signal. The construction of confidence intervals for the NMSE is discussed. Guidelines for the choice of the spectral smoothing factor n are given. The variation of wavelet shape due to different realizations of the noise component is illustrated, and the use of confidence intervals on wavelet phase is recommended. Tests are described for examining the normality and stationarity of the errors of fit and their independence of the estimated reflection signal.

Wavelet estimation

Abstract: An important problem in seismic exploration is the estimation of and correction for the seismic wavelet. A seismic signal may be modeled as a convolutional model with the wavelet as one component. The wavelet propagated by the seismic energy source is complicated by transmission and recording filters. Some filters in the system can be deterministically defined while others are more conjectural. The estimation of the wavelet is useful in two major ways. Borehole measurements are used to model the surface seismograms. The wavelet used in the model needs to match that of the seismogram to correlate the two measurements. Conversely, the estimated wavelet can be used to design inverse filters which make the seismogram approach the borehole measures. Some well-known methods for estimation of the wavelet are based on assumptions about the wavelet or the earth reflectivity. Examples of the methods indicate success on some data even though each makes different assumptions. The methods serve to point out basic problems in reliably estimating the wavelet from the seismogram. Basic problems include noise, band-limiting, nonstationarity, uncertain theoretical models, assumption failure, and widely diverse geological sequences of the earth. Quality control or evaluation of the performance of an estimation algorithm is a nontrivial problem. The estimation of the wavelet from a seismic recording remains an area of challenging research and importance in exploration for hydrocarbons.

Performance of minimum-variance deconvolution filter

Abstract: Recently, we observed zero phase and undershoot patterns in data processed by a minimum-variance deconvolution (MVD) filter. These observations motivated a careful analysis of the MVD filter, which, as we demonstrate in this paper, explains both the zero phase and undershoot patterns. This analysis also connects the MVD filter with the well-known prediction-error filter [6], and Berkhout's two-sided least-squares inverse filter [7]. We show that the performance of the MVD filter depends heavily on the bandwidth of the source wavelet and signal-to-noise ratio, and only slightly on data length.

A critique of seismic deconvolution methods

Abstract: Different seismic pulse compression methods are evaluated. These include several algorithms for computing prediction error filters: Wiener filtering, Burg's method, the l 1 norm criterion, Kalman filtering, and two time-adaptive methods. Algorithms which do not assume a minimum-phase condition for the seismic wavelet include minimum entropy, homomorphic, and zero-phase deconvolution. The sensitivity of these algorithms is examined for various earth reflectivity functions, source waveforms, and signal distortions. The results indicate that standard Wiener predictive deconvolution is robust under a wide variety of input conditions. However, a substantial improvement in pulse compression can be obtained by the Burg algorithm under conditions of short data segments and by minimum entropy deconvolution for seismograms consisting of mixed-phase wavelets combined with sparse reflectivity series.

Statistical pulse compression

Abstract: The seismic method in petroleum exploration is an echo-location technique to detect interfaces between the subsurface sedimentary layers of the earth. The received seismic reflection record (field trace), in general, may be modeled as a linear time-varying (LTV) system. However, in order to make the problem tractable, we do not deal with the entire field trace as a single unit, but instead subdivide it into time gates. For any time gate on the trace, there is a corresponding vertical section of rock layers within the earth, such that the primary (direct) reflections from these layers all arrive within the gate. Each interface between layers is characterized by a local (or Fresnel) reflection coefficient, which physically must be less than unity in magnitude. Under the hypothesis that the vertical earth section has small reflection coefficients, then within the corresponding time gate the LTV model of the seismic field trace reduces to a linear time-invariant (LTI) system. This LTI system, known as the convolutional model of the seismic trace, says that the field trace is the convolution of a seismic wavelet with the reflection coefficient series. If, in addition, the reflection coefficient series is white, then all the spectral shape of the trace within the gate can be attributed to the seismic wavelet. Thus the inverse wavelet can be computed as the prediction error operator (for unit prediction distance) by the method of least squares. The convolution of this inverse wavelet with the field trace yields the desired reflection coefficients. This statistical pulse compression method, known as predictive deconvolution with unit prediction distance, is also called spike deconvolution. Alternatively, predictive deconvolution with greater prediction distance can be used, and it is known as gapped deconvolution. Other pulse compression methods used in seismic processing are signature deconvolution, wavelet processing, and minimum entropy deconvolution.

Timing circuit for acoustic flow meters

Abstract: A timing means for determining the time of travel of ultrasonic pulses between a transmitter and a receiver immersed in a fluid. The first time instant is determined when the transmitter transmits an ultrasonic pulse in the fluid. A second time instant is determined when the ultrasonic wavelet has been detected, converted to electrical signal, and the zero crossing time of the electrical wavelet after a selected positive peak is determined. Timing means are provided to determine the time interval between the first and second time instants. This is done by having a coarse time clock and counter. The number of full clock periods before the second instant is determined, and a ramp of voltage is generated having a known rate of rise. The ramp is negatively biased and the bias voltage is varied until the ramp voltage reaches zero at the second instant. The transmitter upstream is fired, and the wavelet is detected at the downstream transducer. Simultaneously, or sequentially the downstream transducer is powered to generate a wavelet and the second wavelet is detected at the upstream transducer. The rate of flow of the fluid is a function of the difference between these two time intervals.

Estimation of bubble pulse wavelets for deconvolution of marine seismograms

Abstract: Summary. We describe a method for estimating bubble pulse wavelets directly from marine seismograms, and illustrate its use with our l1 (least absolute values) deconvolution algorithm. The method builds wavelet estimates in three stages, the first two of which use information derived from individual seismograms. The usual minimum phase assumption is not employed in our method, since direct measurements of source wavelets demonstrate that this assumption can be inappropriate in practice.

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

Abstract: 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 occur...

Measurement of sound-speed gradients in deep-ocean sediments using l_{1} deconvolution techniques

Abstract: A method is described for measuring the sound speed and the sound-speed gradient of surficial sea floor sediment from bottom-reflected signals recorded in marine seismic experiments. The technique makes use of the ocean-bottom impulse responses that are deconvolved from the data by means of a novel curve-fitting algorithm based on the l_{1} norm (least absolute value) criterion. The algorithm constructs the impulse response by extracting spikes one at a time in a manner that causes the l_{1} error to decrease by the maximum amount possible as each spike is chosen. The l_{1} curve-fitting approach is a completely general strategy for deconvolution, and our algorithm can be used with data obtained from any type of marine seismic source. Since our experiments have been carried out with small explosive charges, we have also developed a method for estimating the bubble-pulse wavelet directly from the recorded bottom-reflected signal. In this paper, the l_{1} algorithm is used to deconvolve impulse responses for data obtained in an experiment in the Alaskan Abyssal Plain. The sediment-sound-speed gradient determined from these results is typical of other values reported for turbidite abyssal plains where the surficial sediments are composed of unconsolidated silty deposits.