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The NMO correction time for the small offset-spread approximation is given
by the well-known hyperbolic equation Yilmaz (1987):
| |
(8) |

where *x* is the trace offset, *t*_{x} is the two-way travel time at offset *x*,
*t*_{0} is the two-way travel time at zero offset (normal incidence trace) and
*V*_{s} is the stacking velocity. Clearly, for a given trace different samples
will have different NMO correction times even if the velocity is constant.
Shallow events on the farthest trace with the slowest velocity have the maximum
NMO-correction time whereas deep events on the near traces with the fastest
velocity will have the minimum NMO-correction time. It is also important to
note that in general some fractional sample interpolation will be required
since we cannot expect the values of to be integer multiples
of the sampling interval.
In order to apply the non-stationary filtering algorithm we need to
recast the NMO equation as an all-pass non-stationary filter
that will simply shift each sample by the given value of . This
can easily be achieved in the frequency domain by a linear phase shift with
slope proportional to the value of . In principle, any value
of can be handled, so no fractional interpolation is
required. For the sake of efficiency, however, it is convenient to precompute
a given number of values. The accuracy of the implicit
fractional interpolation is determined by the number of precomputed
values and so can be controlled as an input parameter.
Clearly, this parameter controls the trade-off between accuracy and speed
of computation.

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** Up:** Theory Overview
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Stanford Exploration Project

6/8/2002