Decomposing a gather by velocity

A process will be defined that can partition a CMP gather, both reflections and head waves, into one part with RMS velocity greater than that of some given model $\bar v ( z)$ and another part with velocity less than $\bar v ( z)$.

After such a partitioning, the low-velocity noise could be abandoned. Or the earth velocity could be found through iteration, by making the usual assumption that the velocity spectrum has a peak at earth velocity. As will be seen later, various data interpolation, lateral extrapolation, and other statistical procedures are also made possible by the linearity and invertibility of the partitioning of the data by velocity.

The procedure is simple. Begin with a common-midpoint gather, zero the negative offsets, and then downward continue according to the velocity model $\bar v ( z)$.The components of the data with velocity less than $\bar v ( z)$ will overmigrate through zero offset to negative offsets. The components of the data with velocity greater than $\bar v ( z)$ will undermigrate. They will move toward zero offset but they will not go through. So the low-velocity part is at negative offset and the high-velocity part is at positive offset. If you wish, the process can then be reversed to bring the two parts back to the space of the original data.

Obviously, the process of multiplying data by a step function may create some undesirable diffractions, but then, you wouldn't expect to find an infinitely sharp velocity cutoff filter. Clearly, the false diffractions could be reduced by using a ramp instead of a step. An alternative to zeroing negative h would be to go into $( k_h , \omega )$-space and zero the two quadrants of sign disagreement between k_h and $\omega$.

This partitioning method unfortunately does not, by itself, provide a velocity spectrum. Energy away from h = 0 is unfocused and not obviously related to velocity. The need for a velocity spectrum motivates the development of other processes.