Anti-alias characteristic of dip moveout

You might think that if (y,h,t)-space is sampled along the y-axis at a sample interval $\Delta y$,then any final migrated section P(y,z) would have a spatial resolution no better than $\Delta y$.This is not the case.

The basic principle at work here has been known since the time of Shannon. If a time function and its derivative are sampled at a time interval $2 \Delta t$,they can both be fully reconstructed provided that the original bandwidth of the signal is lower than $1/(2 \Delta t)$.More generally, if a signal is filtered with m independent filters, and these m signals are sampled at an interval $m \Delta t$,then the signal can be recovered.

Here is how this concept applies to seismic data. The basic signal is the earth model. The various filtered versions of it are the constant-offset sections. Recall that the CDP reflection point moves up dip as the offset is increased. Further details can be found in a paper by Bolondi, Loinger, and Rocca [1982], who first pointed out the anti-alias properties of dip moveout. At a time of increasing interest in 3-D seismic data, special attention should be paid to the anti-alias character of dip moveout.

EXERCISES:

1. Describe the effect of the Jacobian in Hale's dip moveout process.