Spiking Deconvolution Research Articles

An autoregressive filter model for constant Q attenuation

The Earth’s filter response to seismic wave propagation can be approximated by constant Q attenuation and is dispersive or minimum phase. A finite‐length autoregressive (AR) filter model is a good representation for constant Q attenuation with minimum phase. Coefficients of the AR filter are the Wiener‐Levinson inverse filter coefficients for the sampled constant Q auto‐correlation. Conventional spike deconvolution approximates fairly well the inverse filter on this ground if the minimum‐phase attenuation law holds true. Wavelet response to attenuation was analyzed using the AR filter model. The amplitude of the filtered impulse response decreases at nearly 1/t, where t is traveltime, and is sensitive to the loss of low frequencies. The wavelets’ peak amplitude is fractionally time delayed with .92/Q. Velocity dispersion of seismic waves should not contribute to the mis‐tie observed between the conventional check shot and the borehole sonic log when the first arrivals are picked the same way.

Predictive deconvolution and the zero‐phase source

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 the amount of white noise used should match that added in designing the (minimum‐phase) spiking deconvolution operator. This choice, however, also produces inconsistent results; field data again show that the phase treatment is sensitive to the amount of added white noise. Synthetic data tests show that the standard phase‐correction procedure breaks down when earth attenuation is severe. Deterministically reducing the earth‐filter effects before deconvolution improved the resulting phase treatment for the synthetic data. After application of the inverse attenuation filter to the field data, however, phase differences again remain for different levels of added white noise. These inconsistencies are attributable to the phase action of spiking deconvolution. This action is dependent upon the shape of the signal spectrum as well as the spectral shape and level of contaminating noise. Thus, in practice the proper treatment of phase in data-dependent processing requires extensive knowledge of the spectral characteristics of both signal and noise. With such knowledge, one could apply deterministic techniques that either eliminate the need for statistical deconvolution or condition the data so as to satisfy better the statistical model assumed in data‐dependent processing.

Statistical pulse compression

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.

The effects of noise on minimum‐phase Vibroseis deconvolution

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 frequencies. The phase associated with this center frequency is independent of time but depends upon both the signal bandwidth and the deconvolution noise platform. Finally, this paper examines distortions in the deconvolved wavelet arising from both processing and environmental noise‐induced variations in the phase and envelope delay. The reflection sequence is assumed to be white, and a minimum phase, nearly constant Q, earth model is assumed. Curves are presented which show the residual phase error as a function of attenuation, processing noise, and the environmental signal‐to‐noise (S/N) ratio. It was found that although both types of noise will cause some residual phase error, phase compensation can correct for many of the phase distortions and polarity reversals that may be present in the deconvolved data only if the environmental noise is smaller than the processing noise.

Windowing and estimation variance in deconvolution

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 space varying deconvolution is undesirable and that the geology is better defined by using operators which result from averaging over several independent estimates. Analysis of the phase spectra of the deconvolution operators computed for synthetic and real data indicates that selection of the interpretive wavelet based on the phase spectrum of the deconvolution operator is the best approach to reject the frequencies which will have a large phase error in the deconvolved section.

OPTIMUM MIXED DELAY SPIKING FILTERS

Recent increased knowledge of mixed delay seismic source input wavelets requires extension of industry understanding of high‐resolution digital spiking filters. A close analysis of widely existing minimum delay deconvolution processes can be used to gain insights and estimates in mixed delay cases. This paper presents a time‐domain method wherein a double application of standard (zero delay spiking) deconvolution processes yields understanding of the mixed delay nature of a wavelet given for spiking. It is shown that such deconvolution of both a given mixed delay wavelet and the reversal of this given wavelet can be combined to estimate the complete variable delay suite of spiking filter performances for the given mixed delay wavelet. The intuitive basis of the method lies in the observation that the reversal of a wavelet is equivalent to inversion of the zeros of the wavelet Z‐transform in the complex plane. Thus, (zero delay spiking) deconvolution of the wavelet emphasizes the minimum‐phase portion of the wavelet while such deconvolution of the reversed wavelet emphasizes the maximum‐phase portion of the original wavelet.

VIBROSEIS DECONVOLUTION*

AbstractVibroseis deconvolution can be performed either before or after correlation. As regards the deconvolution before correlation, the Vibroseis deconvolution operator can be described as convolution of a spike deconvolution operator with a minimum‐phase filter operator with bandpass properties. As regards the deconvolution after correlation, the deconvolution operator can be shown to be the convolution of three operators: spike deconvolution operator and two‐fold convolution with a minimum phase operator. Time‐varying Vibroseis deconvolution can particularly well be described and performed after correlation.