If this model holds true, then estimating the source function reduces to estimating a minimum-phase function with the same (w,kx,ky) 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=Nx Ny Nt.
Figure 3 Impulse response derived by Kolmogoroff spectral factorization binned as a function of radial distance from the impulse.
Figure 4 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.