Stratigraphic filtering, Part I, Derivation of the O'Doherty-Anstey formula

Abstract

Abstract We present a new and more detailed derivation of the formula due to O'Doherty and Anstey (1971) for filtering a transmitted wavelet by short-period multiples. We use a continuous rather than discrete formulation and regard the impedance as a random variable. The mean pressure represents the downgoing wavelet as progressively modified by short-period multiples, while the deviations from the mean field are essentially the upcoming reflections. Standard procedures and approximations lead to the dispersion relation of the mean pressure field. To describe the stratigraphic filtering, we introduce a dimensionless complex quantity F such that a wavelet which has traveled a time Delta T is modified by the filter A(omega ) = exp iomega (FDelta T). From the Kramers-Kronig relation appropriate for a causal earth, F has real and imaginary partsEquationandEquationwhere Q and delta t define the apparent attenuation and time delay, both of which may depend upon frequency, R is the spectrum of reflection coefficients, and M is the spectrum of the impedance fluctuations. The first equation means that the apparent attenuation depends only on the impedance fluctuations with spatial period half the seismic wavelength; the second means that the stratigraphic filter is minimum-phase. We also show that changes of impedance on a spatial scale much larger than a seismic wavelength modify the amplitude so as to conserve energy, but they do not filter the waveform.