Next: OFFSET CONTINUATION AND DMO
Up: Fomel: Offset continuation
Previous: THE CAUCHY PROBLEM
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'' G_{0} and G_{1} are expressed as
 
(5) 
 (6) 
and the parameter is
 
(7) 
H stands for the Heavyside stepfunction.
From formulas (5) and (6) one can see that the
impulse response of the offset continuation operator is discontinuous
in the timeoffsetmidpoint 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
bicharacteristics of equation (1)  ``time rays''
Fomel (1995) emerging from the point (Figure 1).
offcon
Figure 1
Constantoffset 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 secondorder 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 highfrequency
asymptotics of operator (4). To obtain the asymptotic
representation, it is sufficient to note that is the impulse response of the causal halforder
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 hh_{1}); and
stand for the operators of causal and
anticausal halforder differentiation and integration applied with
respect to the time variable t_{n}; the summation paths
correspond to the two nonnegative
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 wellknown 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