Showing papers on "Wavelet published in 1974"

Resolution comparison of minimum-phase and zero-phase signals

Abstract: Despite their intuitive appeal, minimum‐phase wavelets are not the shortest wavelets achievable on a seismic section. For several amplitude spectra typical of processed seismic sections, both minimum‐phase and zero‐phase wavelets are presented. In each case, several measures of length reveal that the zero‐phase wavelet is shorter than the minimum‐phase wavelet corresponding to the same amplitude spectrum. Furthermore, the zero‐phase wavelet has smaller side lobes than the corresponding minimum‐phase wavelet. Synthetic seismograms were generated using both the zero‐phase and minimum‐phase signals as inputs. In each case, the seismogram generated with the zero‐phase input signal had better resolution. This relation is demonstrated quantitatively and is also visually obvious on the seismograms. In addition to comparing the wavelets’ resolution capabilities, the accuracies permitted in estimating reflection times were compared. The zero‐phase wavelets resulted in more accurate estimates of both reflection tim...

Estimation of the primary seismic pulse

Abstract: A seismic trace after application of suitable amplitude recovery may be treated as a stationary time-series. Such a trace, or a portion of it, is modelled by the expression where j represents trace number on the record, t is time, αj is a time delay, α (t) is the seismic wavelet, s(t) is the reflection impulse response of the ground and nj is uncorrelated noise. With the common assumption that s(t) is white, random, and stationary, estimates of the energy spectrum (or auto-correlation function) of the pulse α(t) are obtained by statistical analysis of the multitrace record. The time-domain pulse itself is then reconstituted under the assumption of minimum-phase. Three techniques for obtaining the phase spectrum have been evaluated: (A) use of the Hilbert transform, (B) Use of the z-transform, (C) a fast method based on inverting the least-squares inverse of the wavelets, i.e. inverting the normal time-domain deconvolution operator. Problems associated with these three methods are most acute when the z-transform of α(t) has zeroes on or near the unit circle. Such zeroes result from oversampling or from highly resonant wavelets. The behaviour of the three methods when the energy spectra are perturbed by measurement errors is studied. It is concluded that method (A) is the best of the three. Examples of reconstituted pulses are given which illustrate the variability from trace-to-trace, from shot-to-shot, and from one shot-point medium to another. There is reasonable agreement between the minimum-phase pulses obtained by this statistical analysis of operational records and those estimated from measurements close to the source. However, this comparison incorporates a “fudge-factor” since an allowance for absorption has to be made in order to attenuate the high frequencies present in the pulse measured close to the shot.

Tunnel effect in the motion of deformable wavelets

Abstract: In a previous paper it was shown that the tunnel effect does not occur in the motion of an undeformable wavelet which we obtained by solving the Schrodinger equation and interpreted as representing a single particle. In the present supplementary paper we solve the same Schrodinger equation while modifying one of the restrictive conditions assumed previously, and obtain wavelets which deform in force fields. Tunnel effects are seen in the motion of these deformable wavelets. An apparent inaccuracy pointed out mathematically in these solutions is shown to be insignificant under conditions which are feasible in the physical sense.

A KALMAN FILTER APPROACH TO OF SEISMIC SlGNALSt

Abstract: It is common practice to model a reflection seismogram as a convolution of the reflectivity function of the earth and an energy waveform referred to as the seismic wavelet. The objective of the deconvolution technique described here is to extract the reflectivity function from the reflection seismogram. The most common approach to deconvolution has been the design of inverse filters based on Wiener filter theory. Some of the disadvantages of the inverse filter approach may be overcome by using a state variable representation of the earth’s reflectivity function and the seismic signal THE DECONVOLUTION generating process. die problem is formulated in discrete state variable form to facilitate digital computer processing of digitized seismic signals. The discrete form of the Kalman filter is then used to generate an estimate of the reflectivity function. The principal advantages of this technique are its capability for handling continually timevarying models, its adaptability to a large class of models, its suitability for either single or multichannel processing, and its potentially high-resolution capabilities. Examples based on both synthetic and field seismic data illustrate the feasibility of the method.