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## Forward and Inverse NMO

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, 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 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.

Next: Description of the Algorithm Up: Theory Overview Previous: Time-variant Filtering
Stanford Exploration Project
6/8/2002