Showing papers on "Wavelet published in 1982"

Wave propagation and sampling theory—Part I: Complex signal and scattering in multilayered media

Abstract: From experimental studies in digital processing of seismic reflection data, geophysicists know that a seismic signal does vary in amplitude, shape, frequency and phase, versus propagation time To enhance the resolution of the seismic reflection method, we must investigate these variations in more detail. We present quantitative results of theoretical studies on propagation of plane waves for normal incidence, through perfectly elastic multilayered media. As wavelet shapes, we use zero-phase cosine wavelets modulated by a Gaussian envelope and the corresponding complex wavelets. A finite set of such wavelets, for an appropriate sampling of the frequency domain, may be taken as the basic wavelets for a Gabor expansion of any signal or trace in a two-dimensional (2-D) domain (time and frequency). We can then compute the wave propagation using complex functions and thereby obtain quantitative results including energy and phase of the propagating signals. These results appear as complex 2-D functions of time and frequency, i.e., as “instantaneous frequency spectra. ’ ’ Choosing a constant sampling rate on the logarithmic scale in the frequency domain leads to an appropriate sampling method for phase preservation of the complex signals or traces. For this purpose, we developed a Gabor expansion involving basic wavelets with a constant time duration/mean period ratio. For layered media, as found in sedimentary basins,

Wave propagation and sampling theory—Part II: Sampling theory and complex waves

Abstract: Morlet et al (1982, this issue) showed the advantages of using complex values for both waves and characteristics of the media. We simulated the theoretical tools we present here, using the Goupillaud-Kunetz algorithm. Now we present sampling methods for complex signals or traces corresponding to received waves, and sampling methods for complex characterization of multilayered or heterogeneous media. Regarding the complex signals, we present a twodimecsional(2-D) method of sampling in the time-frequency domain using a special or “extended” Gabor expansion on a set of basic wavelets adapted to phase preservation. Such a 2-D expansion permits us to handle in a proper manner instantaneous frequency spectra. We show the differences between “wavelet resolution” and “sampling grid resolution.” We also show the importance of phase preservation in high-resolution seismic. Regarding the media, we show how analytical studies of wave propagation in periodic structured layers could help when trying to characterize the physical properties of the layers and their large scale granularity as a result of complex deconvolution. Analytical studies of wave propagation in periodic structures are well known in solid state physics, and lead to the so-called “Bloch waves.” The introduction of complex waves leads to replacing the classical wave equation by a Schriidinger equation. Finally, we show that complex wave equations, Gabor expansion, and Bloch waves are three different ways of ‘introducing the tools of quantum mechanics in highresolution seismic (Gabor, 1946; Kittel, 1976, Morlet, 1975). And conversely, the Goupillaud-Kunetz algorithm and an extended Gabor expansion may be of some use in solid state physics.

The limits of resolution of zero-phase wavelets

Abstract: This investigation deals with resolving reflections from thin beds rather than the detection of events that may or may not be resolved. Resolution is approached by considering a thinning bed and how accurately measured times on a seismic trace represent actual, vertical two-way traveltimes through the bed. Theoretical developments are in terms of frequency and time rather than wavelength and thickness because the latter two variables require knowledge of interval velocities. These results are compared with similar studies by Rayleigh, Ricker (19.53), and Widess (1973, 1980). We show that the temporal resolution of a broadband wavelet with a white spectrum is controlled by its highest terminal frequency f,,, and the resolution limit approximates I/ ( I .5 fU), provided the wavelet’s band ratio exceeds two octaves. The practical limit of resolution, however, occurs at a one-quarter wavelength condition and approximates I /( 1.4 fJ. The resolving power of zero-phase wavelets can be compared quantitatively once a wavelet is known in the time domain.

Linear inverse theory and deconvolution

Abstract: Seismic source wavelet deconvolution can be treated within the framework of the Backus‐Gilbert (BG) inverse theory. A time shift‐invariant version of this theory leads to the Wiener shaping filter, which has enjoyed widespread use for source wavelet deconvolution in exploration seismology. The model of the BG theory is the ground impulse response, the BG mapping kernel is the source wavelet, and the BG resolving kernel is the convolution between the source wavelet and the Wiener shaping filter. BG inversion involves the minimization of an optimality criterion under a set of constraints. The application of the BG “filter energy” or “noise output power” constraint to Wiener filter design leads to the familiar prewhitening parameter that stabilizes the filter on the one hand, but degrades resolution on the other. The BG “unimodular” constraint produces an unbiased estimate of the model, or ground impulse response. These constraints provide novel insights into the performance of deconvolution filters.

Quantifying resolving power of seismic systems

Abstract: A quantitative formulation of vertical resolving power of seismic exploration systems is presented and is offered as a proposed characteristic, or standard, resolving power identified with individual systems. The formulation broadens the classical concept of resolution by taking into account the reflection waveform and the noise, in addition to the classical time variable. The principal feature in the formulation is the stipulation that the intratrace distribution of reflections and of noise be treated as random (Gaussian) distribution, which is regarded as the most general representation for seismic sections as a whole.Through this quantification of vertical resolving power and therefore of intratrace reflection quality, a number of elemental reflection properties that have been described only qualitatively in the past are expressed by simple formulas. The quantification is consistent with the concept that the resolving power of a noise-free zero-phase system with a flat spectral band response is proportional to the bandwidth. The derived basic formula for the proposed characteristic resolving power is a 2 m /E, where a m is the maximum (absolute) amplitude of the signal wavelet of a seismogram interval, E is the energy of the signal wavelet, and noise is neglected. The quantification of the reflection properties, including taking the noise into account, stems from this formula.The classical concept of resolution, which considers only the time variable, such as the dominant period of signal wavelets, is applicable essentially only in cases of two noise-free equal-strength reflections. In contrast, the proposed formulation of resolving power accommodates a wide scope of applications and might be considered basic to seismic systems. I present theoretical material for evaluating the merits of the proposal. Suitable comparisons by seismic modeling would be useful in the overall evaluation.

Analytic minimum entropy deconvolution

Abstract: The standard model of the reflection seismogram assumes an impulsive, scalar, reflectivity series convolved with a seismic wavelet. Under certain, not unlikely, conditions a reflection coefficient can become complex and the resulting seismogram may be shown to contain phase‐shifted wavelets. When this is the case, scalar deconvolution approaches fail. To overcome this problem, Levy and Oldenburg (1982, this issue) have proposed a technique based on the analytic signal formulation. We couple the Levy and Oldenburg approach with a complex version of the minimum entropy algorithm and apply this new technique to synthetic and real examples.

The deconvolution of phase‐shifted wavelets

Abstract: The assumption that a seismogram can be represented as a convolution of a source wavelet with a set of real impulses breaks down when the wavelet is phase shifted upon reflection from a boundary. For plane waves and plane layers, this effect occurs only for wide‐angle supercritical reflections, but it may also occur in normal incidence seismograms when either the impinging wavefront or the reflective boundary is curved. We show that seismograms containing time‐displaced, phase‐shifted replications of the source wavelet can be deconvolved to recover both the amplitude and phase of the reflectivity coefficients. The method begins by writing the analytic seismogram as the convolution of a complex reflectivity function with an analytic source wavelet; linear inverse theory is then used to carry out the deconvolution.

The effects of noise on minimum-phase Vibroseis deconvolution

Abstract: Ristow and Jurczyk (1975) proposed a mixed‐phase Vibroseis® inverse filter which is the usual minimum phase spiking deconvolution filter convolved with a Weiner‐Levinson minimum phase wavelet having the same amplitude spectrum as the Vibroseis wavelet. A problem exists since (for large time‐bandwidth products) the Vibroseis signal approximates a band‐limited signal and noise may have to be added to ensure convergence of the Wiener‐Levinson algorithm. This processing noise level can alter the resulting minimum phase wavelet. Since the deconvolution filter is influenced by the ambient or environmental noise as well as by the processing noise, the proposed correction to spiking deconvolution may not always yield meaningful results. It is shown that although the Vibroseis wavelet may span several octaves, it is not only band‐limited but can be approximated by a narrow‐band signal representation. In this formulation, the center frequency for the wavelet is considered to be the average of the high and low frequ...

Estimation of parameters in lossless layered media systems

Abstract: In this paper we develop a procedure for estimating the parameters (reflection coefficients and travel times) of lossless layered media systems. The estimation criterion is maximum likelihood. It is shown that maximum-likelihood estimates of the parameters of a layer may be obtained by first determining the maximum-likelihood estimate of the upgoing waveform (one of the states) in that layer, and then estimating the parameters from the estimated upgoing waveform as if the estimated waveform is the output of the layer under consideration. A maximum-likelihood state estimator is developed. It is shown that the state waveforms are composed of overlapping wavelets, and estimation of the parameters of a layer is essentially equivalent to estimation of the amplitude and time-delay of the first wavelet in the upgoing waveform of the layer under consideration. A suboptimal maximum-likelihood parameter estimator is developed. It consists of two filters connected in series: a matched filter to treat noise; and a transversal equalizing-filter to correct for wavelet overlapping effects. Numerical results which illustrate the performance of this maximum-likelihood procedure for different noise levels are presented.

Windowing and estimation variance in deconvolution

Abstract: Spiking deconvolution operators are computed using the Levinson and orthonormal lattice filter algorithms. By comparing the amplitude and phase spectra of the operators, the differences between the two methods are shown to be strictly tied to the differences in how the algorithms handle the windowing problem. Also, the two methods are equivalent if the windowing problem can be overcome through the use of multipass deconvolution prior to the application of an interpretive wavelet. The most significant effect of windowing is to introduce errors in the estimate of the phase spectrum. When a filter is computed from a single trace, estimation variance in the filter is high. Spatial averaging in the filter design process can overcome a large part of this problem and produce sections with better lateral continuity. The parameters averaged spatially are the autocorrelations and the inner products for the Levinson and orthonormal lattice filter algorithms, respectively. This suggests that short window, time, and s...

Many-body effects and the diffusion of high-density electrons and holes in semiconductors

Abstract: Exchange and correlation effects at high electron and hole densities can cause a reduction of the energy band gap in semiconductors. The authors demonstrate theoretically that the ambipolar diffusion coefficient is therefore less than that derived from the conventional Boltzmann transport equation even for densities as low as 1015 cm-3. The diffusion coefficient density dependence is also significantly different from that derived by Wavelet's manybody theory, primarily because of differences in the statistical mechanical description of the carrier distributions.

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...

Effect of a nonwhite reflection series spectrum on pulse estimation

Abstract: where S,(f) is the magnitude squared of the seismic pulse’s Fourier transform, and S,(f) is the reflection series power spectrum. A common approach to wavelet extraction is to measure the smoothed periodogram of seismic data averaged over a number of adjacent traces, and to equate the resulting spectral estimate with the spectrum of the pulse. The validity of this approach depends upon satisfaction of two conditions: (1) a good quality estimate of the seismic spectrum can be obtained; and (2) the reflection series spectrum is white. The purpose of the work reported here was to determine whether these conditions are in fact satisfied in typical geologic contexts. From the theory of the smoothed periodogram, it is shown that the first of the above assumptions is severely strained in many situations where this spectral estimator is used. Using a data base of 21 wells, it is also shown that the second assumption is violated quite often. We have found that the combined effect of estimator variance and spectrum nonwhiteness degrades the pulse estimate enough to produce distortions in one-dimensional inverse models computed with the pulse.

Thin Bed Stratigraphy from Complex Trace Attributes: ABSTRACT

Abstract: Both model seismic data and broad-band field data acquired to delineate complicated stratigraphy have been converted to displays of the instantaneous attributes of the complex seismic trace. Attribute sections enhance the interpretation of conventional sections not only by qualitatively highlighting specific properties of conventional displays, but also by quantitatively defining wavelet characteristics like dominant frequency, and stratigraphic variables like formation thickness. An example of the quantitative use of complex attributes in wavelet definition is the phenomenon that the maximum instantaneous frequency of a zero-phase Ricker wavelet is synchronous with the central peak of the wavelet and exactly equal to the frequency corresponding to the center of gravity o the wavelet's amplitude spectrum. Peak instantaneous frequency thus is a physical meaningful measure of the spectral content of a zero-phase Ricker wavelet. If the signal in a seismic section can be approximated by zero-phase Ricker wavelets, and if the geophysicist can identify occasional wavelet peaks in the sections which are uncontaminated by noise or interference, instantaneous frequencies at these samples are direct estimates of a significant and absolute spectral characteristic of the signal. An example of the quantitative use of attribute sections in seismic stratigraphy is their application to estimation of the thickness of thin, porous sands. Pods of porous sand which are encased in high-velocity material and whose thicknesses are of the order of half the peak-to-peak period of the dominant seismic energy show up as anomalously high amplitude zones on instantaneous amplitude displays. These anomalies result from the well-known amplitude tuning effect which occurs when reflection coefficients of opposite polarity a half-period apart are convolved with a seismic wavelet. As sand members thin to a quarter-period of the dominant seismic energy, the thinning is revealed by an anomalous increase in instantaneous frequency. This behavior results from the less well-known but eq ally important phenomenon of frequency tuning by thinning beds. Frequency tuning reaches a maximum when sand thickness is about a quarter-period and remains evident as the sand continues to thin. The instantaneous frequency section thus can be a sensitive analytical tool for investigating stratigraphic sequences composed of very thin layers. Frequency and amplitude tunings are accompanied by changes in the character of the complex of interfering reflections from various impedance boundaries in a formation of thin beds, and these changes are highlighted by the instantaneous phase display. End_of_Article - Last_Page 624------------

On Quantitative Analysis of Periodicity in Neonatal Respiratory Signals

Abstract: A quantitative wavelet periodicity measure was defined. This measure allows the quantitative description of wavelet periodicity in the neonatal respiratory signal. An iterative algorithm for on-line estimation of the periodicity measure may be useful as one of the features for automatic evaluation of the state of consciousness in premature infants.