Integral (stacking) operators play a very important role in seismic data processing. The most common applications are common midpoint stacking, Kirchhoff-type migration, and dip moveout. Other examples include (listed in random order) Kirchhoff-type datuming, back-projection tomography, slant stack, velocity transform, offset continuation, and azimuth moveout (AMO). The role of the integral methods increases with the development of prestack three-dimensional processing because they appear flexible toward irregularities in the data geometry.
Often an integral operator represents the forward modeling problem, and we need to invert it to solve for the model. In this paper, I consider two different approaches to inversion. The first is least-square inversion, which requires constructing the adjoint counterpart of the modeling operator. The second approach is asymptotic inversion, which aims to reconstruct the high-frequency (discontinuous) parts of the model. I compare the two approaches and introduce the notion of what I call the asymptotic pseudo-unitary operator to tie them together.
The first part of this paper contains a formal definition of a stacking operator and reviews the theory of asymptotic inversion, following the fundamental results of Beylkin 1985 and Goldin . According to this theory, the high-frequency asymptotic inverse of a stacking operator is also a stacking operator with a different summation path and weighting. To connect this theory with the theory of adjoint operators, I prove that the adjoint of a stacking operator can also be included in the class of stacking operators. The stacking (``pull'') adjoint has the same summation path as the asymptotic inverse but a different weighting function. These two results combine together to form the theory of asymptotic pseudo-unitary integral operators. I apply this theory to define a general preconditioning operator for least-square inversion.
Finally, I consider such examples of commonly used stacking operators as wave-equation datuming, migration, velocity transform, and offset continuation.