It is interesting to note that many integral operators routinely used
in seismic data processing have the form of operator (25)
with the Green's function
The impulse response (30) is typical for different forms of Kirchhoff migration and datuming as well as for velocity transform, integral offset continuation, DMO, and AMO. Integral operators of that class rarely satisfy the unitarity condition, with the Radon transform (slant stack) being a notable exception. In an earlier paper (Fomel, 1996), I have shown that it is possible to define the amplitude function for each kinematic path so that the operator becomes asymptotically pseudo-unitary. This means that the adjoint operator coincides with the inverse in the high-frequency (stationary-phase) approximation. Consequently, equation (28) is satisfied to the same asymptotic order.
Using asymptotically pseudo-unitary operators, we can apply formula
(29) to find an explicit analytic form of the interpolation
function , as follows:
For a simple example, let us consider the case of zero-offset time
migration with a constant velocity . The summation path
in this case is an ellipse
While opening a curious theoretical possibility, seismic imaging interpolants have an undesirable computational complexity. Following the general regularization framework of Chapter , I shift the computational emphasis towards appropriately chosen regularization operators discussed in Chapter . For the forward interpolation method, all data examples in this dissertation use either the simplest nearest neighbor and linear interpolation or a more accurate B-spline method, described in the next section.