INVERSION METHODS FOR τ‐ p MAPS OF NEAR OFFSET DATA—LINEAR INVERSION *

A bstract Conventional velocity analysis, based on the ideas of rms velocity and hyperbolic reflection events in the x‐t domain, is restricted in validity to near vertical incidence. Thus analysis of near‐offset datasets usually requires the muting of wide‐angle reflections from shallow interfaces before the rms velocities are determined. The ray‐theoretical integral for the delay time τ, which depends on the slowness p and the velocity function, is valid for all angles. The wide‐angle reflections can be used to improve the accuracy of the derived velocity function in the near surface region, if the recorded x‐t data are mapped into the τ‐p domain. By representing the velocity function between reflectors as a series of gradient zones, i.e. regions with a uniform increase in velocity with depth, the recovery of the velocities may be posed as a matrix linear inverse problem for the slopes of the gradient zones. In order to convert the problem to a linear one, the velocity discontinuities at the reflecting interfaces must be fixed in advance. Their positions are based on the behaviour of the τ‐ p map of the data. Finding a stable velocity model may require several iterations with the reflecting interfaces at different positions. An understanding of the workings of the inversion algorithm allied with an analysis of the causes of instability aids the search for a stable model.