Considering a data set with *n*_{x} receivers with *n*_{t} time samples,
Table lists the
number of operations required to construct shot-gathers by
cross-correlating the traces of the transmission wavefield and
migration of either the raw volume or the data volume produced by
correlation. Under the conditions presented above, the *n*_{ts} time
samples needed for the shot-gathers is the same as the number of
frequencies required for direct migration. This will be orders of
magnitude less than the *n*_{t} time samples collected during a passive
seismic experiment in most cases. Finally, it is possible that not all
receiver stations need be correlated when processing a large, high
frequency experiment. will be a subsampling factor, less than 1,
that controls the number of traces in, or aperture of, the correlated
shot-gathers.

Migration costs scale according to the size of the input data set and
the size of the image domain through which the data is extrapolated.
*X* will represent a scalar multiplier due to the computer overhead
costs of the migration strategy used. This will vary from a factor of
5 to *n*_{x} depending on the algorithm and accuracy required, but will
be common to either direct migration or the migration of the
correlated shot-gathers.

The size of the image space is assumed to be
*n*_{x} samples areally by *n*_{z} samples in depth. No inverse
Fourier transforms are required to prepare for migration, as the
shot-gathers are needed to be functions of frequency for many
migration algorithms. Also, I assume that the field passive seismic
data fulfills the model of short source functions unevenly dispersed
along the time axis of the duration of data collection.

operations | |

Multiply | |

Gather Mig. | |

Raw Mig. |

With these costs in mind, the ratio of the sum of the first two rows
to the last must be balanced to decide which choice requires the least
amount of computation operations. When , the costs of
producing an image by either method is the same. The meaning of this
situation is shot-gathers of one trace, *i.e.* a constant-offset
(post-stack sized) migration. Thus,
the passive seismic experiment acts similarly to a natural
phase-encoding of active seismic shot-gathers that
Romero et al. (2000) explains as a method to reduce the cost of shot-profile
migration schemes. This can be thought of as performing a full
prestack migration for the cost of a zero-offset migration.

Therefore, if it is appropriate to assume that potential subsurface seismic sources are reasonably short in duration, and that the length of the passive experiment is dictated by the requirement to collect a sufficient number of them to illuminate our image space, we can save substantial computation cost, orders of magnitude, by migrating the raw data directly. Even if one is concerned that such a severe decimation of the frequency axis might be detrimental, many safety multiples can be carried without affecting the speed-up of direct migration of the data.

5/3/2005