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It is interesting to note that many integral operators routinely used
in seismic data processing have the form of operator ()
with the Green's function
 
(55) 
where we have split the variable x into the onedimensional
component z (typically depth or time) and the mdimensional
component (typically a lateral coordinate with m equal
1 or 2). Similarly, the variable y is split into t and
. The function represents the summation
path , which captures the kinematic properties of the operator, and
A is the amplitude function. In the case of m=1, the fractional
derivative is defined
as the operator with the frequency response , where
is the temporal frequency Samko et al. (1993).
The impulse response () 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 (1996b), I have shown that it is possible to define
the amplitude function A for each kinematic path so that
the operator becomes asymptotically pseudounitary . This means
that the adjoint operator coincides with the inverse in the
highfrequency (stationaryphase) approximation. Consequently,
equation () is satisfied to the same asymptotic order.
Using asymptotically pseudounitary operators, we can apply formula
() to find an explicit analytic form of the interpolation
function W, as follows:
 

 (56) 
Here the amplitude function A is defined according to the general
theory of asymptotically pseudoinverse operators as
 
(57) 
where
 
(58) 
 (59) 
and is the dual
summation path, obtained by solving equation for t
(assuming that an explicit solution is possible).
For a simple example, let us consider the case of zerooffset time
migration with a constant velocity v. The summation path in this case is an ellipse
 
(60) 
and the dual summation path is a hyperbola
 
(61) 
The corresponding pseudounitary amplitude function is found from
formula () to be Fomel (1996b)
 
(62) 
Substituting formula () into (), we derive
the corresponding interpolation function
 
(63) 
where , and . For m=1 (the twodimensional
case), we can apply the known properties of the delta function to
simplify formula () further to the form
 
(64) 
The result is an interpolant for zerooffset seismic sections. Like
the sinc interpolant in equation (), which is based on
decomposing the signal into sinusoids, equation () is
based on decomposing the zerooffset section into hyperbolas.
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 Bspline method, described in the
next section.
Next: Interpolation with convolutional bases
Up: Continuous case and seismic
Previous: Continuous case and seismic
Stanford Exploration Project
12/28/2000