Velocity continuation and the anatomy of residual prestack time migration |

The conventional approach to 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.
Post-stack time migration 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 et al. (1985) as *residual migration*.
Larner and Beasley (1987) 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 infinitesimally small. This leads to a 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, 1988,1984; Mikulich and Hale, 1992; Shurtleff, 1984).

The concept of velocity continuation was introduced in the earlier
publications (Fomel, 1994,1997). Hubral et al. (1996) and
Schleicher et al. (1997)
use the term *image waves* to describe a
similar idea. Adler (2002) generalizes it to the case of variable
background velocities under the name *Kirchhoff image propagation*. The
importance of this concept lies in its ability to predict changes in the
geometry and intensity of reflection events on seismic images with the change
of migration velocity. While conventional approaches to migration velocity
analysis methods take into account only vertical movement of reflectors
(Deregowski, 1990; Liu and Bleistein, 1995),
velocity continuation attempts to describe both
vertical and lateral movements, thus providing for optimal focusing in
velocity analysis applications (Fomel, 2001,2003b).

In this paper, I describe the velocity continuation theory for the case of prestack time migration, connecting it with the theory of prestack residual migration (Etgen, 1990; Stolt, 1996; Al-Yahya and Fowler, 1986). By exploiting the mathematical theory of characteristics, a simplified kinematic derivation of the velocity continuation equation leads to a differential equation with correct dynamic properties. In practice, one can accomplish dynamic velocity continuation by integral, finite-difference, or spectral methods. The accompanying paper (Fomel, 2003b) introduces one of the possible numerical implementations and demonstrates its application on a field data example.

The paper is organized into two main sections. First, I derive the kinematics of velocity continuation from the first geometric principles. I identify three distinctive terms, corresponding to zero-offset residual migration, residual normal moveout, and residual dip moveout. Each term is analyzed separately to derive an analytical prediction for the changes in the geometry of traveltime curves (reflection events on migrated images) with the change of migration velocity. Second, the dynamic behavior of seismic images is described with the help of partial differential equations and their solutions. Reconstruction of the dynamical counterparts for kinematic equations is not unique. However, I show that, with an appropriate selection of additional terms, the image waves corresponding to the velocity continuation process have the correct dynamic behavior. In particular, a special boundary value problem with the zero-offset velocity continuation equation produces the solution identical to the conventional Kirchoff time migration.

Velocity continuation and the anatomy of residual prestack time migration |

2014-04-01