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Equation (1) describes a continuous process of
reflected wavefield continuation in the time-offset-midpoint domain.
In order to find an integral-type operator that performs the one-step
offset continuation, I consider the following initial-value (Cauchy)
problem for equation (1):
Given a post-NMO constant-offset section at half-offset h1
| |
(80) |
and its first-order derivative with respect to offset
| |
(81) |
find the corresponding section P(0)(tn,y) at offset h.
Equation (1) belongs to the hyperbolic type, with
the offset coordinate h being a ``time-like'' variable and the
midpoint coordinate y and the time tn being ``space-like''
variables. The last condition (81) is required for the
initial value problem to be well-posed (). 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 (81). In this section, I discuss the exact
integral solution of the OC equation and analyze its asymptotics.
The integral solution of problem (80-81)
for equation (1) is obtained in
Appendix with the help of the classic methods of
mathematical physics. It takes the explicit form
| |
|
| (82) |
where the Green's functions G0 and G1 are expressed as
| |
(83) |
| (84) |
and the parameter is
| |
(85) |
H stands for the Heavyside step-function.
From equations (83) and (84) 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
| |
(86) |
which describes the ``wavefronts'' of the offset continuation process.
In terms of the theory of characteristics (), the surface
corresponds to the characteristic conoid formed by the
bi-characteristics of equation (1) - time rays
emerging from the point . The
common-offset slices of the characteristic conoid are shown in the
left plot of Figure 7.
con
Figure 7
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 (82). To obtain
the asymptotical 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 asymptotical form of
the integral offset-continuation operator becomes
| |
|
| (87) |
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 (86) (Figure
7):
| |
(88) |
where , and ; is
the midpoint separation (the integration parameter), and and are the following weighting functions:
| |
(89) |
| (90) |
Expression (88) for the summation path of the OC
operator was obtained previously by (, ) and
(, ). A somewhat different form of it
is proposed by (). I describe the kinematic
interpretation of formula (88) in
Appendix .
In the high-frequency asymptotics, it is possible to replace the two
terms in equation (87) with a single term
(). The single-term expression is
| |
(91) |
where
| |
(92) |
| (93) |
A more general approach to true-amplitude asymptotic offset
continuation is developed by ().
The limit of expression (88) 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
()
| |
(94) |
I discuss the connection between offset continuation and DMO in the
next section.
Next: Offset continuation and DMO
Up: Offset continuation for reflection
Previous: Kirchhoff model and the
Stanford Exploration Project
12/30/2000