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# THE INTEGRAL OPERATOR FOR OFFSET CONTINUATION

The integral solution of problem (1)-(3) is obtained in Appendix A with the help of the classic methods of mathematical physics. It takes the explicit form

 (4)
where the Green's functions'' G0 and G1 are expressed as
 (5) (6)
and the parameter is
 (7)
H stands for the Heavyside step-function.

From formulas (5) and (6) one can see that the impulse response of the offset continuation operator is discontinuous in the time-offset-midpoint space on a surface defined by the equality

 (8)
that describes the wavefronts'' of the offset continuation process. In terms of the theory of characteristics Courant (1962), the surface corresponds to the characteristic conoid formed by bi-characteristics of equation (1) - time rays'' Fomel (1995) emerging from the point (Figure 1).

offcon
Figure 1
Constant-offset sections of the characteristic conoid - offset continuation fronts'' (left), and branches of the conoid used in the integral OC operator (right). The upper part of the plots (small times) corresponds to continuation to smaller offsets; the lower part (large times) corresponds to larger offsets.

As a second-order differential equation of the hyperbolic type, equation (1) describes two different processes. The first process is forward'' continuation from smaller to larger offsets; the second one is reverse'' continuation in the opposite direction. These two processes are clearly separated in the high-frequency asymptotics of operator (4). To obtain the asymptotic representation, it is sufficient to note that is the impulse response of the causal half-order integration operator, and that is asymptotically equivalent to (t, a >0). Thus, the asymptotic form of the integral offset continuation operator becomes

 (9)
Here the signs +'' and -'' correspond to the type of continuation (the sign of h-h1); and stand for the operators of causal and anticausal half-order differentiation and integration applied with respect to the time variable tn; the summation paths correspond to the two non-negative sections of the characteristic conoid (8) (Figure 1):

 (10)
where , and ; is the midpoint separation (the integration parameter); and and are the following weighting functions:

 (11) (12)
Expression (10) for the summation path of the OC operator was obtained previously by Stovas and Fomel 1993 and Biondi and Chemingui . A somewhat different form of it is proposed by Bagaini et al. 1994. I describe the kinematic interpretation of formula (10) in Appendix B.

The limit of expression (10) for the output offset h approaching zero can be evaluated by L'Hospitale's rule. As one would expect, it coincides with the well-known expression for the summation path of the integral DMO operator Deregowski and Rocca (1981)

 (13)

Next: OFFSET CONTINUATION AND DMO Up: Fomel: Offset continuation Previous: THE CAUCHY PROBLEM
Stanford Exploration Project
11/12/1997