The conventional approach to the seismic migration theory Berkhout (1985); Claerbout (1985) employs the downward continuation concept. According to this concept, migration extrapolates upgoing reflected waves, recorded on the surface, to the place of their reflection to form an image of subsurface structures. When post-stack migration is performed in the time domain, it possesses peculiar properties, which can lead to a different viewpoint on migration. One of the most interesting properties is an ability to decompose the time migration procedure into a cascade of two or more migrations with smaller migration velocities. This remarkable property is described by Rothman, Levin, and Rocca 1985 as residual migration. Larner and Beasley 1987 have generalized the method of residual migration to one of cascaded migration. Cascading finite-difference migrations overcomes the dip limitations of conventional finite-difference algorithms Larner and Beasley (1987); cascading Stolt-type f-k migrations expands their range of validity to the case of a vertically varying velocity Beasley et al. (1988). Further theoretical generalization sets the number of migrations in a cascade to infinity, making each step in the velocity space infinitely small. This leads to the partial differential equation in the time-midpoint-velocity space, discovered by Claerbout 1986. Claerbout's equation describes the process of velocity continuation, which fills the velocity space in the same manner as a set of constant-velocity migrations. Slicing in the migration velocity space can serve as a method of velocity analysis for migration with nonconstant velocity Fowler (1988).
In this paper, I generalize the velocity continuation concept to the case of prestack migration, connecting it with the theory of prestack residual migration Etgen (1990). I provide a simplified kinematic derivation of the velocity continuation equation, which is alternative though closely related to the previously published derivations Claerbout (1986); Fomel (1994); Levin (1986a). Though this derivation is purely kinematic, it leads to differential equations with reasonable dynamic properties. In practice, one can accomplish dynamic velocity continuation by integral, finite-difference, or Fourier-domain methods.
Besides the idea of reviving SEP's results from the epoch of ``nonreproducible research,'' this paper is motivated by the challenge of velocity analysis for prestack 3-D depth migration. Though prestack velocity continuation cannot provide a complete solution to this problem in the areas of lateral velocity variation, it can serve as a useful transformation for simplifying the kinematic features of prestack data and for preconditioning the velocity inversion.