MIGRATION TO ZERO OFFSET (DMO) FOR A CONSTANT VELOCITY GRADIENT: AN ANALYTICAL FORMULATION¹

Abstract

DIETRICH, M. and COHEN, J.K. 1993. Migration to zero offset (DMO) for a constant velocity gradient: an analytical formulation. Geophysical Prospecting 41, 621-643.Migration to zero offset (MZO) is a prestack partial migration process that transforms finite-offset seismic data into a close approximation to zero-offset data, regardless of the reflector dips that are present in the data. MZO is an important step in the standard processing sequence of seismic data, but is usually restricted to constant velocity media. Thus, most MZO algorithms are unable to correct for the reflection point dispersal caused by ray bending in inhomogeneous media.We present an analytical formulation of the MZO operator for the simple possible variation of velocity within the earth, i.e. a constant gradient in the vertical direction. The derivation of the MZO operator is carried out in two steps. We first derive the equation of the constant traveltime surface for linear V(z) velocity functions and show that the isochron can be represented by a fourth-degree polynomial in x, y and z. This surface reduces to the well-known ellipsoid in the constant-velocity case, and to the spherical wavefront obtained by Slotnick in the coincident source-receiver case.We then derive the kinematic and dynamic zero-offset corrections in parametric form by using the equation of the isochron. The weighting factors are obtained in the high-frequency limit by means of a simple geometric spreading correction. Our analytical results show that the MZO operator is a multivalued, saddle-shaped operator with marked dip moveout effects in the cross-line direction. However, the amplitude analysis and the distribution of dips along the MZO impulse response show that the most important contributions of the MZO operator are concentrated in a narrow zone along the in-line direction. In practice, MZO processing requires approximately the same trace spacing in the in-line and cross-line directions to avoid spatial aliasing effects.

Prestack migration error in transversely isotropic media

“…The inverse problem consists of finding the coordinates (x, z) given the traveltime t, velocity v, and slope p = dt/dx at midpoint relative to the diffractor location. Dietrich and Cohen (1993) reduced the above equation to…”

2.5-D true‐amplitude Kirchhoff migration to zero offset in laterally inhomogeneous media

“…The problem of a depth-dependent velocity has been addressed more recently. Dietrich and Cohen (1993) derived the analytic expression for the stacking curve in a medium with a constant vertical velocity gradient. They also heuristically suggested a DMO weight function which, however, does not correctly take into account the amplitude effect of the stacking process itself.…”

The common reflecting element (CRE) method revisited

“…highly depends on the accuracy of the velocity model. Different imaging techniques, including the common midpoint (CMP) stack (Yilmaz, 1987), migration to zero offset (MZO) (Dietrich and Cohen, 1993;Tygel et al, 1998), poststack migration (Stolt, 1978;Schneider, 1978), and prestack true-amplitude migration (Bleistein, 1987;Schleicher et al, 1993) require different degrees of accuracy of the macrovelocity model to construct the respective image in either the time or depth domains. Therefore, two key issues to be addressed in seismic reflection imaging are to identify the best imaging technique for an insufficiently known macrovelocity model and to determine how the original estimate of the macrovelocity model can be refined as part of the imaging procedure.…”