A slant‐stack procedure for point‐source data

Abstract

The proper implementation of the τ-p method for surface data excited by a point source requires a cylindrical slant stack. Usually the common (Cartesian) slant stack is computed instead as an approximation to the geometrically correct procedure. Here we describe a formulation of the cylindrical slant stack as a weighted sum of Cartesian slant stacks; our cylindrical slant stack is computationally efficient to perform. We show how, although the usefulness of the slant stack is most easily seen with Cartesian coordinates, it can also be used with Fourier‐Bessel transforms. An example of the method shows results computed from data recorded on the West Florida Shelf. Severe edge‐effect noise which overwhelms the Cartesian slant stack is attenuated by the cylindrical slant‐stacking. Applications of the cylindrical slant stack to other seismological calculations, such as Lamb’s problem, are also discussed. In particular, we prove that the plane‐wave reflection coefficients apply exactly in the τ-p domain; hence an amplitude‐versus‐offset analysis is unambiguous in that domain.

Point‐source τ‐p transform: A review and comparison of computational methods

“…In the TD2D formulation, we follow Brysk and McCowan (1986) in representing the cylindrical slant stack as a weighted integral of the 2-D slant stacks. In the TDTS method, devised here, we represent the point-source τ -p transform as a 2-D slant stack (weighted by offset x) plus a correction term.…”

“…Our method is similar to that of Brysk and McCowan (1986) except that we use a cubic spline interpolation of the function Y , whereas they used a linear interpolation. The integral over q is divided into segments of length p,…”

“…The two signal contributions in equation (14), S(τ − qx) and S(τ + qx), were interpreted in Brysk and McCowan (1986) as representing, respectively, negative and positive range data. From the above derivation (and in comparison with Figure 3) we see the alternative interpretation that the first term is associated with the incoming energy (Figure 4b), whereas the second term is associated with the outgoing energy ( Figure 4c).…”

“…Analyses such as full wave inversion, amplitude variation with offset (AVO) analysis, and dispersive analysis, for surface data excited by a point source require the cylindrical implementation of the τ -p transform (e.g., Chapman, 1981;Henry et al, 1981;Phinney et al, 1981;Treitel et al, 1982;Harding, 1985;Brysk and McCowan, 1986;Wilson, 1986;Benoliel et al, 1987;Kappus et al, 1990;Dietrich, 1990;Fokkema et al, 1992;Zhao et al, 1994). The cylindrical τ -p transform is derived from the solution of the wave equation for a point source in a 3-D vertically stratified medium.…”

“…We compare the Bessel-Fourier transform method with several variants of the equivalent time-domain convolution, discussed previously in Brysk and McCowan (1986). They showed that the timedomain cylindrical slant stack may be expressed as a weighted integral of the conventional 2-D slant stacks for a line-source.…”