Multi-dimensional factorization

Kolmogorov spectral factorization, as described above, is a purely one-dimensional theory. By applying helical boundary conditions Claerbout (1998b), a three-dimensional stochastic dopplergram can be mapped into an equivalent one-dimensional dataset, and the entire cube can be factored with Kolmogorov.

The concept of helical boundary conditions is reviewed in Chapter  and demonstrated in Figure , which shows the mapping of small five-point two-dimensional filter into one dimension. For this application, however, rather than map a two-dimensional function, I map the entire three-dimensional MDI dataset into one dimension, and apply Kolmogorov spectral factorization on the entire super-trace.

The spatial axes need to be padded to reduce wrap-around effects. This spatial wrap-around is not an artifact of the Fourier transform, but rather it is an artifact of the helical boundary conditions. In this respect, there would be little advantage to choosing a time-domain spectral factorization algorithm Wilson (1969) over Kolmogorov.

To summarize, I perform the spectral factorization in three steps. Firstly, I transform the cube of data to an equivalent one-dimensional super-trace via helical boundary conditions. Secondly, I perform one-dimensional spectral factorization with Kolmogorov's frequency domain method. Finally, I remap the impulse response back to three-dimensional space.