Theory of differential offset continuation |

*Given a post-NMO constant-offset section at half-offset *

Equation (1) belongs to the hyperbolic type, with the offset coordinate being a ``time-like'' variable and the midpoint coordinate and the time being ``space-like'' variables. The last condition (63) is required for the initial value problem to be well-posed (Courant, 1962). From a physical point of view, its role is to separate the two different wave-like processes embedded in equation (1), which are analogous to inward and outward wave propagation. We will associate the first process with continuation to a larger offset and the second one with continuation to a smaller offset. Though the offset derivatives of data are not measured in practice, they can be estimated from the data at neighboring offsets by a finite-difference approximation. Selecting a propagation branch explicitly, for example by considering the high-frequency asymptotics of the continuation operators, can allow us to eliminate the need for condition (63). In this section, I discuss the exact integral solution of the OC equation and analyze its asymptotics.

The integral solution of problem (62-63)
for equation (1) is obtained in with the help of
the classic methods of mathematical physics
(Fomel, 2001,1994). It takes the explicit form

where the Green's functions and are expressed as

and the parameter is

stands for the Heaviside step-function.

From equations (65) and (66) 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

cont
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.
Figure 5. |
---|

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 (64). 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
. Thus, the asymptotical form of
the integral offset-continuation operator becomes

Here the signs ``'' and ``'' correspond to the type of continuation (the sign of ), and stand for the operators of causal and anticausal half-order differentiation and integration applied with respect to the time variable , the summation paths correspond to the two non-negative sections of the characteristic conoid (68) (Figure 5):

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

Expression (70) for the summation path of the OC operator was obtained previously by Stovas and Fomel (1996) and Biondi and Chemingui (1994). A somewhat different form of it is proposed by Bagaini and Spagnolini (1996). I describe the kinematic interpretation of formula (70) in Appendix B.

In the high-frequency asymptotics, it is possible to replace the two
terms in equation (69) with a single term
(Fomel, 2003a). The single-term expression is

(73) |

A more general approach to true-amplitude asymptotic offset continuation is developed by ().

The limit of expression (70) for the output offset
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)

Theory of differential offset continuation |

2014-03-26