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Seismic Deconvolution Based on Fractionally Integrated Noise
Muhammed M. Saggaf
Submitted
to the Department of Earth, Atmospheric, and Planetary Sciences in 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.
