Next: Discussion
Up: Continuous case and seismic
Previous: Continuous case and seismic
It is interesting to note that a wide class of integral operators,
routinely used in seismic data processing, have the form of operator
(22) with the ``Green'' function
 
(27) 
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.
Impulse response (27) 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 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 A for each kinematic path so that
the operator becomes asymptotic pseudounitary . This means that
the adjoint operator coincides with the inverse in the highfrequency
(stationaryphase) approximation. Consequently, equation
(25) is satisfied to the same asymptotic order.
Using asymptotic pseudounitary operators, we can apply formula
(26) to find an explicit analytic form of the interpolation
function W, as follows:
 

 (28) 
Here the amplitude function A is defined according to the general
theory of asymptotic pseudoinverse operators as
 
(29) 
where
 
(30) 
 (31) 
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
 
(32) 
and the dual summation path is a hyperbola
 
(33) 
The corresponding pseudounitary amplitude function is found from
formula (29) to be Fomel (1996)
 
(34) 
Substituting formula (34) into (28), we derive
the corresponding interpolation function
 
(35) 
where , and . For m=1 (the twodimensional
case), we can apply the known properties of the delta function to
simplify formula (35) further to the form
 
(36) 
The result is an interpolator for zerooffset seismic sections. Like
the sinc interpolator in formula (19) that is based on
decomposing the signal into sinusoids, interpolation (36)
is based on decomposing the zerooffset section into hyperbolas.
Next: Discussion
Up: Continuous case and seismic
Previous: Continuous case and seismic
Stanford Exploration Project
11/11/1997