Several elastic inversion methods have been developed in the past decade to estimate the elastic parameters and density of the subsurface media. Among these methods, it is possible to define a subgroup, loosely called Amplitude versus Offset (AVO), to refer to a specific class of methods which are based on the Zoeppritz or Knot's equations (or an approximation to them). Such equations relate the reflection and transmission coefficients, of a plane wave reflected by the interface, with some of the elastic properties of the two media involved, as a function of the angle of incidence. However, the Reflectivity versus Angle (RvA) relation obtained by these equations is not directly available from the recorded data, which contains basically Amplitude versus Offset (AVO) information. The usual way to overcome this restriction is to replace the angular dependence of the equations by a horizontal slowness (p) dependence, since the latter is a constant of propagation in a plane layered half-space. Therefore, the transformation of the original AVO data into Reflectivity versus p (Rvp) data is an important first step for the implementation of some of these methods.
Since this paper is mainly focused on the retrieval of amplitude information, I will, for all purposes, consider the subsurface as a horizontally layered medium, within the spatial range spanned by one record (i.e. within a CMP range), which is the same basic assumption of the standard CMP stacking process (Kolb and Chapel,1989). Under this assumption the horizontal slowness (or Snell parameter) is a constant of propagation. Although the conventional slant-stack transform is a commonly used method to obtain the desired plane-wave decomposition of the arriving wavefield, there are some limitations that must be considered: Conventional slant-stacks assume a line source geometry, and the proper way to deal with the actual point source data is the use of Cylindrical slant-stacks (Brysk and McCowan, 1986); a differential moveout is still required to align the response of a reflector point (with the inevitable stretch); the finite aperture of the data requires the use of weighting on the integration and some undesirable events can be amplified in this process; summing over a large interval results in an decrease of the signal/noise ratio. Local slant-stacks can be used to solve the last two problems, but then, the geometrical spreading correction (implicit in the plane-wave decomposition process) is lost.
I propose a transform, similar to the slant-stack, that maps the data into the vertical-traveltime--horizontal-slowness domain (t0-p), instead of into the usual intercept-time--horizontal-slowness domain (). This transform properly handles the above mentioned limitations of the standard slant-stack, performing thus some of the required steps to convert the recorded data into the reflectivity series.