If this model holds true, then estimating the source function reduces
to estimating a minimum-phase function with the same (*w*,*k*_{x},*k*_{y})
spectrum as the original data: multi-dimensional spectral
factorization.

Helical boundary conditions Claerbout (1998) provide a framework for converting a multi-dimensional problem into an equivalent problem in only one dimension, and allow us to solve the three-dimensional spectral factorization problem efficiently.

We perform the spectral factorization rapidly in the frequency domain in three steps. Firstly, we transform the multi-dimensional signal to an equivalent one-dimensional signal using helical boundary conditions. Secondly, we perform one-dimensional spectral factorization with Kolmogoroff's 1939 algorithm. Finally, we remap the impulse response back to three-dimensional space. We reduce wrap-around effects by padding the spatial axes.

Figure 3 shows the impulse response derived from
Kolmogoroff spectral factorization as a function of radial distance
from the impulse. It looks very similar to the cross-correlation
time-distance seismogram shown in Figure 4, and those
displayed by Kosovichev (1999).
However, for the dataset described above, this operation was
approximately twenty times faster than cross-correlating every
trace in either () or ().
The speed-up becomes apparent when you consider that cross-correlating
every trace with every other trace requires operations, whereas one-dimensional spectral factorization requires
only operations where *N*=*N*_{x} *N*_{y} *N*_{t}.

kolstack
Impulse response derived by
Kolmogoroff spectral factorization binned as a function of radial
distance from the impulse.
Figure 3 |

xcorr
Impulse response derived by
cross-correlation binned as a function of radial
distance from the impulse. More noise is present in this Figure
compared to Figure 3 because less data was used in
the calculation.
Figure 4 |

Figure 5

4/20/1999