








Migration to Zero Offset (DMO) for a Constant Velocity Gradient: An Analytical Formulation






Organization:  Center for Wave Phenomena, Colorado School of Mines 












Migration to zero offset (MZO) is a prestack partial migration process that transforms finiteoffset seismic data into a close approximation to zerooffset 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 with 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 fourthdegree polynomial in x, y and z. This surface reduces to the wellknown ellipsoid in the constantvelocity case, and to the spherical wavefront obtained by Slotnick in the coincident sourcereceiver case. We then derive the kinematic and dynamic zerooffset corrections in parametric form by using the equation of the isochron. The weighting factors are obtained in the highfrequency limit by means of a simple geometric spreading correction. Our analytical results show that the MZO operator is a multivalued, saddleshaped operator with marked dip moveout effects in the crossline 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 inline direction. In practice, MZO processing requires approximately the same trace spacing in the inline and crossline directions to avoid spatial aliasing effects.













   
 
