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 recordi...