Showing papers on "Wavelet published in 1972"

Signal detection and extraction by cepstrum techniques

Abstract: A technique for decomposing a composite signal of unknown multiple wavelets overlapping in time is described. The computation algorithm incorporates the power cepstrum and complex cepstrum techniques. It has been found that the power cepstrum is most efficient in recognizing wavelet arrival times and amplitudes while the complex cepstrum is invaluable in estimating the form of the basic wavelet and its echoes, even if the latter are distorted. Digital data-processing problems such as the detection of multiple echoes, various methods of linear filtering the complex cepstrum, the picket-fence phenomenon, minimum-maximum phase situations, and amplitude- versus phase-smoothing for the additive-noise case are examined empirically and where possible theoretically, and are discussed. A similar investigation is performed for some of the preceding problems when the echo or echoes are distorted versions of the wavelet, thereby giving some insight into the complex problem of separating a composite signal composed of several additive stochastic processes. The threshold results are still empirical and the results should be extended to multi-dimensional data. Applications are the decomposition or resolution of signals (e.g., echoes) in radar and sonar, seismology, speech, brain waves, and neuroelectric spike data. Examples of results are presented for decomposition in the absence and presence of noise for specified signals. Results are tendered for the decomposition of pulse-type data appropriate to many systems and for the decomposition of brain waves evoked by visual stimulation.

Homomorphic deconvolution of some teleseismic events

Abstract: Homomorphic filtering (Oppenheim et al. , 1968), recently applied to seismology by Ulrych (1971), has been used to deconvolve three earthquake and one underground nuclear explosion seismograms recorded at leduc, Alberta. Comparison of actual recordings with seismograms synthesized using deconvolved wavelets together with a crustal response computed from a well-defined crustal model shows striking correlations. This improved wavelet extraction procedure is expected to be an aid in Q , source mechanism, and depth studies as well as crustal deconvolution investigations.

Complex demodulation for transient wavelet detection and extraction

Abstract: Research in fields such as communication, speech, oceanography, seismic exploration, economics, and biomedical data processing is often directed toward the analysis of nonstationary or transient data. Complex demodulation is shown to be a valuable method that can be used to decompose a composite signal composed of differing transient wavelets and to estimate spectra. For the latter application it is shown that several other techniques recently advanced in the literature are special cases of complex demodulation. The algorithm discussed can achieve the decomposition of a certain class of noisy composite signals composed of nonidentical unknown multiple wavelets overlapping in time, namely those signals with reasonably well-defined independent resonances in the spectrum. The decomposition estimates the arrival time, peak, envelope, and frequency of the damped oscillatory transient wavelet. The procedure has been tested extensively and several selected experimental results are tendered. It has been found that for wavelets of the type t^{k}e^{-at} \sin (\omegat) , k =0, 1 , that an uncertainty relationship for the product of the 3-dB bandwidth and the time duration of the wavelet must be satisfied. An error analysis has established a relationship between envelope-and phase-estimation errors to wavelet and filter parameters. The results obtained via complex demodulation are discussed relative to those obtained via inverse filtering, the complex cepstrum, and the chirp z transform.

Adaptive Decomposition of a Composite Signal of Identical Unknown Wavelets in Noise

Abstract: An algorithm is discussed which decomposes a noisy composite signal of identical but unknown multiple wavelets overlapping in time. The decomposition determines the number of wavelets present, their epochs, amplitudes, and an estimate of the basic wavelet shape. The algorithm is an adaptive decomposition filter which is a combination tion of three separate filters. One is an adaptive cross-correlation filter which resolves the composite signal from noise by an iteration procedure; this is followed by a wavelet extraction filter which ferrets out the basic wavelet form, and last there appears an inverse filter which achieves decomposition of the composite signal in the time domain. The decomposition algorithm can be applied to echoes and overlapping wavelets which might arise in radar, sonar, seismology, or electro-physiology. The proposed theory has been thoroughly simulated and selected experimental results are presented to demonstrate the technique. These include decomposition of brain waves evoked by visual stimulation.

Continuous velocity estimation and seismic wavelet processing

Abstract: An economic computer program can stack the data from several adjoining common depth points over a wide range of both dip and normal moveout. We can extract from this a set of seismic wavelets, each possessing a determined dip and normal moveout, which represent the original seismic data in an approximate and compressed form. The seismic wavelets resulting from the processing of a complete seismic line are stored for a variety of subsequent uses, such as the following: 1) Superimpose the wavelets, or a subset of them, to form a record section analogous to a conventional common-depth-point stacked section. This facilitates the construction of record sections consisting dominantly of either multiple or primary reflections. Other benefits can arise from improved signal-to-random-noise ratio, the concurrent display of overlapping primary wavelets with widely different normal moveouts, and the elimination of the waveform stretching that occurs on the long offset traces with conventional normal moveout removal. 2) By displaying each picked wavelet as a short dip-bar located at the correct time and spatial position and annotated with the estimated rms velocity, we can exhibit essentially continuous rms-velocity data along each reflection. This information can be utilized for the estimation of interval and average velocities. For comparative purposes this velocity-annotated dip-bar display is normally formed on the same scale as the conventional common-depth-point stack section.

Computer-designed Wiener Filters For Seismic Data

Abstract: This paper is concerned with differences in the frequency content of signal and noise on seismic traces. In order to develop a filter which has applicability over some considerable spatial range, special consideration is given to basic differences in the shape and the frequency content of individual signal and noise wavelets (events) on these traces. Therefore, a so-called “wavelet” Wiener filter is introduced which suppresses “noise” wavelets and enhances “signal” wavelets; this filter can be contrasted with an ordinary Wiener filter which discriminates between signal and noise on the additional basis of the statistics of the repetition of wavelets along the seismic trace. A technique for the automatic derivation of (wavelet) Wiener filters for seismic data by a digital computer is developed. The filters are time dependent to the extent that independent filters are derived at a sequence of data windows which are specified by the operator; criteria for selecting the window positions are given. The overall technique for the computer derivation of Wiener filters is demonstrated with synthetic and actual seismic data. A discussion of the wavelet Wiener filter and its relation to the ordinary Wiener filter is appended to this paper with a discussion of the effects of finite data windows on the eduction of the wavelet Wiener filter.

Deconvolution of refraction seismograms from large underwater explosions

Abstract: The P-wave spectra of seismograms from large underwater explosions are frequently dominated by reverberations. When this is so, a simple reverberation model similar to that of Backus (1959) gives a good approximation to the source spectrum. The basic wavelet determined by this method is not necessarily minimum‐delay. A few promising deconvolutions have been carried out, revealing a sequence of arrivals of comparable amplitudes separated by short time intervals. Synthetic seismograms which have been constructed from these spike sequences differ very little from the field records. However, the technique often yields output seismograms which are not easily interpreted. A study using synthetic seismograms suggests five reasons for this: 1) low signal‐to‐noise ratios, which result in too narrow a frequency band in which signal is predominant; 2) pulses arriving at the recorder having undergone phase changes during transmission, especially in combination with 3) very close spacing of the sequence of arrivals, c...

Estimation Of A Time-varying Seismic Autocorrelation Function

Abstract: The autocorrelation function of a seismic trace provides information about the generating wavelet needed for optimal processing. The classical time‐averaging technique used to estimate the autocorrelation, however, fails for the commonly encountered time‐varying autocorrelation resulting from progressive wavelet distortion. To estimate a time‐varying autocorrelation, an iterative computational procedure is proposed from which a sequence of progressively improved estimates of the autocorrelation is obtained. In the time‐invariant case, the algorithm reduces to the usual time‐average procedure applied to several different “windows” along the trace, and convergence occurs on the first step. For a “slowly” varying autocorrelation, however, the procedure tends to iteratively correct the bias error resulting from time averaging. It is proven that if the time variation for each “lag” value can be modeled by a polynomial of degree N, the sequence of estimates converges (in expected value) to the solution in no mo...