Showing papers on "Wavelet published in 1980"
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TL;DR: In this paper, the best fit wavelet is defined as the ratio of the cross spectrum between the synthetic spike sequence and the seismic trace to the power spectrum of the synthetic wavelet, and the statistics of the match are related to the ordinary coherence function.
Abstract: A crucial step in the use of synthetic seismograms is the estimation of the filtering needed to convert the synthetic reflection spike sequence into a clearly recognizable approximation of a given seismic trace. In the past the filtering has been effected by a single wavelet, usually found by trial and error, and evaluated by eye. Matching can be made more precise than this by using spectral estimation procedures to determine the contribution of primaries and other reflection components to the seismic trace. The wavelet or wavelets that give the least squares best fit to the trace can be found, the errors of fit estimated, and statistics developed for testing whether a valid match can be made. If the composition of the seismogram is assumed to be known (e.g. that it consists solely of primaries and internal multiples) the frequency response of the best fit wavelet is simply the ratio of the cross spectrum between the synthetic spike sequence and the seismic trace to the power spectrum of the synthetic spike sequence, and the statistics of the match are related to the ordinary coherence function. Usually the composition cannot be assumed to be known (e.g. multiples of unknown relative amplitude may be present), and the synthetic sequence has to be split into components that contribute in different ways to the seismic trace. The matching problem is then to determine what filters should be applied to these components, regarded as inputs to a multichannel filter, in order to best fit the seismic trace, regarded as a noisy output. Partial coherence analysis is intended for just this problem. It provides fundamental statistics for the match, and it cannot be properly applied without interpreting these statistics. A useful and concise statistic is the ratio of the power in the total filtered synthetic trace to the power in the errors of fit. This measures the overall goodness-of-fit of the least squares match. It corresponds to a coherent (signal) to incoherent (noise) power ratio. Two limits can be set on it: an upper one equal to the signal-to-noise ratio estimated from the seismic data themselves, and a lower one defined from the distribution of the goodness-of-fit ratios yielded by matching with random noise of the same bandwidth and duration as the seismic trace segment. A match can be considered completely successful if its goodness-of-fit reaches the upper limit; it is rejected if the goodness-of-fit falls below the lower one.
128 citations
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TL;DR: In this paper, the authors considered a quasi-linear relation between the thickness of a coal seam and its reflection response to a seismic signal and investigated the range within which this quasilinearity exists, for a continuous sine wave and for a seismic wavelet.
Abstract: Thin layers are considered from the point of view of the quasi-linear relation that exists between their thickness and their reflection response to a seismic signal. The range within which this quasi-linearity exists is investigated; for a continuous sine wave, this is done by means of the equation for the response given in Rayleigh (1945), and for a seismic wavelet by means of a synthetic seismogram program. For a wavelet, the limiting value of the dominant frequency is found to be smaller than that for a continuous sine wave, the difference being in the order of magnitude of 15 percent.Within the linearity range, a thin layer may be replaced by an equivalent layer which gives the same reflection response but differs in thickness and in acoustic impedance. In the construction of synthetic seismograms over coal seams, this equivalent replacement may be utilized to replace the seams by layers, for which the two-way traveltime is equal to an integer number of sampling intervals; by this procedure the usual rounding-off errors are avoided. The method of equivalent replacement is also applicable when the host rock above and below the seam have different velocities.
41 citations
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TL;DR: In this article, a scaling law for point sources in an homogeneous isotropic medium is derived and the scaling law provides the relationship between the two source signatures and permits the earth impulse response to be extracted from the seismograms without any usual assumptions about phase.
Abstract: We present a new method for the extraction and removal of the source wavelet from the reflection seismogram. In contrast to all other methods currently in use, this one does not demand that there be any mathematically convenient relationship between the phase spectrum of the source wavelet and the phase spectrum of the earth impulse response. Instead, it requires a fundamental change in the field technique such that two different seismograms are now generated from each source-receiver pair: the source and receiver locations stay the same, but the source used to generate one seismogram is a scaled version of the source used to generate the other. A scaling law provides the relationship between the two source signatures and permits the earth impulse response to be extracted from the seismograms without any of the usual assumptions about phase. We derive the scaling law for point sources in an homogeneous isotropic medium. Next, we describe a method for the solution of the set of three simultaneous equations and test it rigorously using a variety of synthetic data and two types of synthetic source waveform: damped sine waves and non-minimum-phase air gun waveforms. Finally we demonstrate that this method is stable in the presence of noise.
33 citations
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TL;DR: In this article, a theoretical design method for overcoming this problem is presented using two scaled arrays, where two independent seismograms are generated by the two scaled array for each pair of source-receiver locations, the source wavelets being related by the scaling law.
Abstract: A seismic source array is normally composed of elements spaced at distances less than a wavelength while the overall dimensions of the array are normally of the order of a wavelength. Consequently, unpredictable interaction effects occur between element and the shape of the far field wavelet, which is azimuth-dependent, can only be determined by measurements in the far field. Since such measurements are very often impossible to make, the shape of the wavelet—particularly its phase spectrum—is unknown. A theoretical design method for overcoming this problem is presented using two scaled arrays. The far field source wavelets from the source arrays have the same azimuth dependence at scaled frequencies, and the far field wavelets along any azimuth are related by a simple scaling law. Two independent seismograms are generated by the two scaled arrays for each pair of source-receiver locations, the source wavelets being related by the scaling law. The technique thus permits the far field waveform of an array to be determined in situations where it is impossible to measure it. Furthermore it permits the array design criteria to be changed: instead of sacrificing useful signal energy for the sake of the phase spectrum, the array may be designed to produce a wavelet with desired amplitude characteristics, without much regard for phase.
22 citations
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TL;DR: In this article, a generalized linear inverse (GLI) method was proposed to estimate the earth's impulse response and the wavelet of a source wavelet in a debubbled, deconvolved seismogram.
Abstract: On seismograms recorded at sea bubble pulse oscillations can present a serious problem to an interpreter. We propose a new approach, based on generalized linear inverse theory, to the solution of the debubbling problem. Under the usual assumption that a seismogram can be modelled as the convolution of the earth's impulse response and a source wavelet we show that estimation of either the wavelet or the impulse response can be formulated as a generalized linear inverse problem. This parametric approach involves solution of a system of equations by minimizing the error vector (ΔX = X obs – X cal ) in a least squares sense. One of the most significant results is that the method enables us to control the accuracy of the solution so that it is consistent with the observational errors and/or known noise levels.
The complete debubbling procedure can be described in four steps: (1) apply minimum entropy deconvolution to the observed data to obtain a deconvolved spike trace, a first approximation to the earth's response function; (2) use this trace and the observed data as input for the generalized linear inverse procedure to compute an estimated basic bubble pulse wavelet; (3) use the results of steps 1 and 2 to construct the compound source signature consisting of the primary pulse plus appropriate bubble oscillations; and (4) use the compound source signature and the observed data as input for the generalized linear inverse method to determine the estimated earth impulse response—a debubbled, deconvolved seismogram. We illustrate the applicability of the new approach with a set of synthetic seismic traces and with a set of field seismograms. A disadvantage of the procedure is that it is computationally expensive. Thus it may be more appropriate to apply the technique in cases where standard analysis techniques do not give acceptable results. In such cases the inherent advantages of the method may be exploited to provide better quality seismograms.
6 citations
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TL;DR: This discussion describes the effects of noise on the application of a new method of Vibroseis deconvolution and shows that zero phase cepstral filtering is a robust wavelet estimation approach for noisy data.
Abstract: A new method of Vibroseis deconvolution has been recently proposed by the authors. This discussion describes the effects of noise on the application of this method. The initial deconvolution step involves estimating the spectrum of the Vibroseis wavelet by homomorphic filtering. It is shown that noise causes problems with phase estimation. Hence, the Vibroseis wavelet is assumed to be zero phase. Examples demonstrate that zero phase cepstral filtering is a robust wavelet estimation approach for noisy data. The second step of the deconvolution method forms an impulse response model by a spectral extension method. Although this step can improve the resolution of seismic arrivals, it must be applied with caution in view of the deleterious effects of noise.
3 citations
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01 Apr 1980TL;DR: This work presents a formulation for the ideal reflection process, when the source and receiver are in different layers, and gives the response as a sum of all replica pulse wavelets due to a source pulse signal, or equivalently as the sum of impulse wavelets in the reflection sequence model.
Abstract: The fundamental tool for simulation and interpretive work on acoustical test data towards extraction of spatial and physical parameters of multilayered media, such as marine sediments or geological strata, is the ideal layered media reflection model. We present here a formulation for the ideal reflection process, when the source and receiver are in different layers. Layer transition times are not restricted to being equal. The wavelet index notation used aids in giving very tractable and compact results. The model gives the response as a sum of all replica pulse wavelets due to a source pulse signal, or equivalently as the sum of impulse wavelets in the reflection sequence model. The new features of this model format are (1) complete details of reflected wavelets to arbitrary reflection multiplicity, (2) systematic count and aggregation of all multipath wavelets with identical arrival times, (3) compact expressions for wavelet amplitudes and delay times. The model is pertinent, for example, for simulation and analysis of seismic responses to source pulses triggered in exploration boreholes.
1 citations