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The NMO correction time for the small offset-spread approximation is given
by the well-known hyperbolic equation ():
where x is the trace offset, tx is the two-way travel time at offset x,
t0 is the two-way travel time at zero offset (normal incidence trace) and
Vs 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 668#668 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 668#668. This
can easily be achieved in the frequency domain by a linear phase shift with
slope proportional to the value of 668#668. In principle, any value
of 668#668 can be handled, so no fractional interpolation is
required. For the sake of efficiency, however, it is convenient to precompute
a given number of 668#668 values. The accuracy of the implicit
fractional interpolation is determined by the number of precomputed
668#668 values and so can be controlled as an input parameter.
Clearly, this parameter controls the trade-off between accuracy and speed
of computation.
Next: Description of the Algorithm
Up: Theory Overview
Previous: Time-variant Filtering
Stanford Exploration Project
6/7/2002