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I have implemented second-order finite differencing to solve the
wave equation for all the computations done in this paper.
As for migration, there are various possible methods for
wavefield extrapolation of datuming. The finite-difference two-way
wave equation method I describe in this paper is just one of them. The
other recursive wavefield extrapolation methods,
like the one-way paraxial wave equation (Claerbout, 1976, 1985),
the Kirchhoff-integral method (Berryhill, 1979, 1984),
and the f-k phase shift method (Gazdag, 1978) are also workable.
But an accurate method of datuming
wavefield extrapolation must be based on the wave equation.
Even for solving the two-way wave equation,
one can use various numerical methods,
such as the high-order finite-difference method (Dablain, 1986; Etgen, 1986)
and the accurate pseudospectral method (Kosloff and Baysal, 1982).
One simply chooses a sufficiently accurate and efficient method
for the practical situation.
The finite-difference method can process irregular
topography efficiently and can honor arbitrary velocity structures. Two
preferable methods of datuming wavefield extrapolation are the two-way wave
equation implemented in the space-time domain by the finite-difference method
(McMechan, 1983) and the one-way 45- and 65-degree
paraxial wave equation implemented in
the space-frequency domain by the finite-difference method (Claerbout, 1985).
High-order finite differencing reduces the number of spatial samples per
wavelength for satisfactory numerical precision. And it can handle
an irregular topography efficiently.
In using a high-order finite-difference scheme to solve the wave equation,
one should gradually drop the order of central finite differencing
to the second-order near the recording boundary
for the derivative perpendicular to
the recording boundary, to avoid edge oscillation and
ensure that the numerical solution conforms to the physical wave phenomena.
This idea has been implemented in Mo (1992) for migration of
crosswell seismic data.
Another option suggested to me by John Etgen (1992) is to pad
with zero traces outside the recording datum, and let the high-order
finite-differencing operators grab zero values outside the recording datum.

The 2-D descriptions here can be applied straightforwardly to the 3-D case.
I am not so sure about application
to datuming for multi-component data.
In that case, one needs to solve the coupled elastic wave equations.
But I do not see any barriers to doing so.
I believe the surface reflection multi-component
data can also be regarded as recorded at an elastic
absorbing boundary, if one thinks the data contain only primary reflections.

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** Up:** Mo: Datuming
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Stanford Exploration Project

11/17/1997