Asymptotic pseudounitary stacking operators |

Mathematical analysis of the inverse problem for operator
(1) shows that only in rare cases can we obtain an
analytically exact inversion. A well-known example is the Radon
transform, which has acquired a lot of different aliases in
geophysical literature: slant stack, tau-p transform, plane wave
decomposition, and controlled directional reception (CDR) transform
(Gardner and Lu, 1991). In this case,

Radon obtained a result similar to the theoretical inversion of operator (1) with the summation path (2) and the weighting function (3) in 1917, but his result was not widely known until the development of computer tomography. According to Radon (1917), the inverse operator has the form

where

(5) | |||

(6) |

is a one-dimensional convolution operator with the spectrum :

and is the dimensionality of and (usually 1 or 2). In Russian geophysical literature, a similar result for the inversion of the CDR transform was published by Nakhamkin (1969).

Extension of Radon's result to the general form of integral operator
(1) (*generalized Radon transform*) is possible
via asymptotic analysis of the inverse problem. In the general case,
Beylkin (1985) and Goldin (1988) have shown that asymptotic
inversion can
reconstruct discontinuous parts of the model. These are the parts
responsible for the asymptotic behavior of the model at high
frequencies. Since the discontinuities are associated with wavefronts
and reflection events at seismic sections, there is a certain
correspondence between asymptotic inversion and such standard goals of
seismic data processing as kinematic equivalence and amplitude
preservation.

The main theorem of asymptotic inversion can be formulated as follows (Goldin, 1988). The leading-order discontinuities in are reconstructed by an integral operator of the form

where the summation path is obtained simply by solving the equation

for (if such an explicit solution is possible). The correctly chosen summation path reconstructs the geometry of the discontinuities. To recover the amplitude, we must choose the correct weighting function, which is constrained by the equation (Goldin, 1988; Beylkin, 1985)

where

(11) | |||

(12) |

The solution assumes that differential forms and exist and are bounded and non-vanishing

Asymptotic pseudounitary stacking operators |

2013-03-03