A Kirchhoff migration analog to least-squares slant
stacking is summarized in Figure . Since we
are using time-invariant transforms, a separate least-squares
problem is set up for each frequency. In the slant stack
case, this problem involved transformation to a one-dimensional
space (ray parameter *p*), since only one parameter is needed
to describe a linear moveout trajectory, its slope.

In the migration case, two parameters are needed to
describe a migration hyperbola: the position
beneath which the apex of the hyperbola lies (*x*), and
another parameter that describes the curvature of the
hyperbola. If we fix velocity, this second parameter can
be called *t _{0}*. For a given constant velocity, the hyperbola
will have greater curvature for an early

This means that the least-squares problem is likely to be larger than in the slant-stack case, and is typically underdetermined. But in principle there is nothing wrong with the transform space having an extra dimension.

Figure 2

11/17/1997