Application of the tau-p transform (slant-stack) in processing seismic reflection data

Abstract

The tau-p transform, or slant-stack, converts seismic records (i.e. time versus offset records) into a space in which many seismic events are well separated. In particular, ground-roll transforms to a point at time zero, refractions transform to points at their zero offset intercept times and reflection hyperbolae transform to ellipses. Significantly, the ellipses do not cross one another, even if the reflection hyperbolae do cross each other in record space. The tau-p space has different dimensions than conventional record space. Record space can be thought of as traces with units of time and with each trace representing a given offset x. In the tau-p space, tau (in dimensions of time), actually represents zero offset reflection time. Each trace corresponds to a particular ray parameter, p. This parameter p can be thought of in many different ways: it is a ray parameter sin i/v; it is the dt/dx, or the slope of the arriving event; and it is the horizontal slowness of the event across the recording array. Further, each trace represents a single angle of incidence at the surface; thus, events can be separated according to angle of incidence. Since each trace represents a particular angle of incidence, multiples are exactly periodic in tau-p space. In addition, velocity analysis can be readily performed in the tau-p space. In fact, the Wiechert-Herglotz-Bateman inversion of refraction data is essentially determining velocities in p space. Possible applications of the tau-p transform in a seismic processing sequence include: ground-roll isolation, isolation of refractions, limitation of angles of incidence (beam-steering), separation of P and mode-converted SV waves, combining multi-component recording, interpolation and resampling of the data, multiple attenuation, and velocity analysis. Some of these applications can be readily performed in the normal processing sequence, while others require special considerations in the tau-p space. Many of these applications are possible because an inverse transform, that is a transform from tau-p space back to X-t record space, is readily accomplished. Multiple attenuation in tau-p space is possible in two different modes: (1) multiples are exactly periodic and can thus be addressed precisely using conventional time series analysis, and (2) some multiples are very dependent upon angle of incidence. Those that are dependent upon angle of incidence can be separated by simply eliminating angles of incidence associated with the strong multiples. In summary, the use of the tau-p transform, or slant-stack, presents an opportunity to examine problems that are difficult in conventional seismic record space.