Similarity and the inverse Q filter: The Pareto‐Levy stretch

Abstract

The seismic impulse response of a constant Q attenuating material has the same mathematical form as the density function for the Pareto‐Levy probability law. Since this law is stable it follows that the seismic pulse is closed under convolution. This means that if the distance from the source x is the sum of two distances [Formula: see text] then the pulse at x is the convolution of the pulse at [Formula: see text] with the one at [Formula: see text]. Its amplitude is proportional to [Formula: see text] where the parameter [Formula: see text] depends only on Q with α = 1 corresponding to a loss‐free material (Q = ∞). As the pulse loses amplitude it stretches along the time axis so that its area is preserved. This scaling between time and distance is illustrated by the stationary phase approximation for the constant Q pulse. The approximation becomes increasingly more accurate for small values of Q until [Formula: see text] where it is exact; it becomes the solution to the heat equation. Log‐stretch inverse Q filtering of seismic data is performed by logarithmically stretching the time axis of the data about a reference time so that all broad‐band pulses have the same shape. A time‐stationary inverse filter is then applied in the stretch domain. However the stretching operation is nonlinear and so band‐limited signals will not have the same shape in the stretch domain—the bandwidths of the stretched wavelets will depend upon wavelet position. To avoid energy loss the bandwidth of the inverse filter must be broader than that of the reference wavelet. For events arriving ahead of the reference wavelet the low‐frequency limit must be lowered, while the high‐frequency limit must be increased for later arrivals. Furthermore, Pareto‐Levy scaling suggests [Formula: see text] stretching corrects the residual traveltime errors that accompany the log‐stretch method of inverse Q filtering.