The theoretical basis for prestack migration by equivalent offset

Abstract

The method of prestack by offset (EOM) forms common scatter point (CSP) gathers for each migrated trace and then images those gathers with a algorithm. The major benefits are that the CSP gathers are formed by trace mappings at constant and that trace binning can be conveniently done as the gathers are formed. Furthermore, the CSP gathers are very sensitive velocity analysis instruments. To provide a foundation in scalar wave theory, the Fourier dual algorithm to EOM, called wavenumber or EWM, is derived from Fourier theory. Both EWM and EOM are based on the algebraic combination of a double square root equation into a single square root. This result defines wavenumber or offset. EWM is found to be an exact reformulation of prestack f-k migration. The CSP gathers are shown to be formed by a Fourier mapping, at constant frequency, of the unmigrated spectrum followed by an inverse Fourier transform. The imaging expression (for each CSP gather) which results from this analysis is formally identical to post with the result retained only at zero offset. Through a numerical simulation, the impulse responses of EOM and EWM are shown to be kinematically identical. Amplitude scale factors, which are exact in the constant velocity EWM theory, are implemented approximately in variable velocity EOM. INTRODUCTION The modern theory of seismic wavefield imaging (migration) is generally acknowledged to rest on theoretical developments from the late 1970's and early 1980's such as Stolt (1978) Schneider (1978) and Gazdag (1978). Conventional seismic data processing is usually separated into prestack and poststack processes where stack refers to the common midpoint (CMP) stacking technique. Though seismic data is manifestly a wavefield, wavefield imaging techniques are usually confined to the poststack realm for economic and other practical reasons in spite of the general recognition that prestack is theoretically preferable. This has lead to the development of DMO (dip moveout) theory which enhances the conventional image by improving the input to poststack migration. Hale (1983) put DMO theory on a firm theoretical basis by deriving its relationship to the prestack theory formulated in Stolt (1978). Hale proved that, for constant velocity, prestack is fully achieved by the cascade of three imaging steps: NMO removal, DMO correction, (stack) and poststack migration. Extension of the DMO theory to non-constant velocity has proven possible for v(z) (Hale and Artley, 1993) but problematic for v(x,z). Thus, the theory is well formulated as a time migration method and has been of great practical benefit to seismic exploration. Bancroft and Geiger (1994) and Bancroft et al. (1995) introduced an alternative technique initially called common scatterpoint (CSP) and now called offset (EOM). In a companion paper (Bancroft et al. 1996), we detail the domain implementation and illustrate the method with real data examples. The essence of the EOM technique is to bypass CMP stacking completely by forming a new kind of gather which assumes a common subsurface scatterpoint rather than a common source-receiver midpoint. In a 2-D medium with constant velocity, v, the Margrave, Bancroft, and Geiger 23-2 CREWES Research Report — Volume 8 (1996) expression for traveltime, t, from source to receiver via a scatterpoint at depth z and x=0 (figure 1) is called the double square root (DSR) equation and is written in terms of midpoint, x, and half-offset, h, as: vt = z + x+h 2 + z + x–h 2 = 2 z +he 2 (1) This equation also defines equivalent offset, he, by asserting that the DSR can be written as a single square root. In appendix A it is shown that he is given exactly by: he 2 = x + h – 4xh vt (2) x z 2h Source Receiver