Stratigraphic filtering, Part II: Model spectra

Abstract

We describe filtering by short‐period multiples in one dimension using a combination of the O’Doherty‐Anstey (1971) formula with a stochastic model in which the autocorrelation of acoustic impedance decreases exponentially with lag [Formula: see text] while the spectrum of reflection coefficients rises with frequency up to a corner, then is approximately constant, [Formula: see text] An impedance log with these statistics is a generalization of the classic random telegraph wave. The parameter [Formula: see text], the mean‐square fractional fluctuation of impedance, is typically less than 1 percent, although we show an example where it is as high as 13 percent. The corner frequency [Formula: see text] is inversely related to average bed thickness. Typical values are 50–100 Hz, at the upper end of the normal seismic band. The cyclic stratification discussed by O’Doherty and Anstey (1971) and others corresponds to [Formula: see text] above the seismic band. Some logs have more reflection power at low frequencies than predicted from high frequencies by this model. We describe in both frequency and time domains the filtering of a seismic wavelet by multiples in such a stratigraphic sequence. The impulse response has a direct arrival, followed by a long tail of multiply scattered energy. The greater the impedance fluctuations or the longer the traveltime, the more amplitude is transferred from the primary to the multiples. We discuss in less detail several other models, including periodic and nearly periodic bedding. We also include two numerical applications of the O’Doherty‐Anstey formula; we show on a specific logged interval the formula’s remarkable accuracy for both attenuation and time delay; and we describe the reduction in computed time delay due to coarser sampling of the log.