Conclusion

Processing windowed subsets of a passive survey may be advantageous. If time-localized events are present, such as teleseismic arrivals, one can process smaller time windows when sure of significant contribution to the image. Without knowing of if or how many sources are active within the bulk of the data, long correlations of the raw data are an almost inevitable approximation (equation 3) to the rigorous definition for synthesizing shot-gathers from transmission wavefields (equation 1). Accepting this reality, first aliasing the time data reduces the computation cost for a DFT by $1/n_\tau$ (where $n_\tau$ is the number of samples in the long input trace) without any further approximation. Rather, it simply capitalizes on the original approximation without having to assume uncorrelable white source functions.

The inherent aliasing within the approximation sums the source functions within the output. This superposition of sources does not produce $R({\bf x}_r,{\bf x}_s, \omega)$ under realistic situations. Instead the result is, $\sum_{{\bf x}_s} R({\bf x}_r,{\bf x}_s,\varpi)$. This data volume can only be migrated with an algorithm that can accept generalized source functions (parameterized by space and time), and uses a correlation imaging condition. Both of these conditions are enjoyed by shot-profile migration.

Migrating all sources at the same time removes the redundant information from a reflector as a function of incidence angle. This makes velocity updating after migration impossible. At this early stage, I contend that passive surveys will only be conducted in actively studied regions where very good velocity models are already available. If this becomes a severe limit, the incorporation of planewave migration strategies can fill the offset dimension of the image.

In practice, the length of the aliased windows should probably be several times longer than the minimum time to the deepest reflector. Multiple sources within this time are handled perfectly by direct migration, and the risk of adding the end of the reflection series to the beginning of the record will be minimized. The decision can be determined by whatever compute resources are available for the size of the data set collected. However, if the time support of the wavefield migrated is many times longer than appropriate for the deepest reflector of interest, aphysical correlation lags will introduce coherent noise into the image that will look exactly like reflections.