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Seismic Deconvolution Based on Fractionally Integrated Noiseby Submitted to the Department of Earth, Atmospheric, and Planetary Sciences on April 1995 in partial fulfillment of the requirements for the degree of Master of Science ABSTRACT
Seismic deconvolution is nowadays, and has been for some time, an integral
part of geophysical data processing. The objective of seismic deconvolution
is to recover the earth's reflectivity from the seismic trace by removing
the effects of source reverberations. The most widely-used deconvolution method
is by far that based on Weiner filtering. The conventional implementation
of this method assumes that the earth's reflectivity series is modeled by
a white noise process, in order to make the problem of calculating the deconvolution
filter more tractable. However, the earth's reflectivity is observed to have
power spectra that are actually proportional to frequency; in other words,
to have a richer content of high frequency, or to exhibit blueness. In this thesis we propose to model reflection coefficients by a process that
mimics the behavior observed of the earth more closely than white noise. This
process is called fractionally integrated noise, and is defined as the process
whose fractional differencing gives rise to white noise. The stochastic properties
of fractionally integrated noise approximate those observed of data derived
from typical well logs much better than random white noise. For instance, the
power spectrum is proportional to frequency, and the auto correlation function
falls off less rapidly than a unit pulse. We develop an efficient method for modifying the conventional Weiner deconvolution
scheme to use fractionally integrated noise and do away with the assumption
of white noise. The method is implemented in such a way that the computational
overhead is minimal and that it is a generalization of the conventional method,
so that it reduces to the conventional scheme when the underlying fractionally
integrated noise process reduces to white noise. Also, the proposed implementation
can be thought of as a preliminary filter that corrects for blueness; and in
this case deconvolution methods other than Weiner filtering can be used in its
second stage. We analyze the computational requirements for the proposed implementation
and also suggest ways to estimate the parameter of the underlying process. We study the effectiveness of the generalized deconvolution based on fractionally
integrated noise by applying it to synthetic traces derived from real well log
data and comparing its output to the exact reflection coefficients used to produce
the synthetic traces. We also compare it to the outcome obtained from applying
the conventional Weiner deconvolution method. The results are quite favorable
and show the generalized method to have a clear edge over the conventional method.
The generalized method also appear to be quite robust in the sense that it outperforms
the conventional method over a wide range of the estimate of the underlying
process parameter. Return to Theses Return to ERL Home Updated: June, 1999
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