A time-space implementation of AMO as a time-variant operator is still not practical for 3D data because of its computing costs. Several authors Bale and Jakubowicz (1987); Biondi and Ronen (1986); Bolondi et al. (1984); Notfors and Godfrey (1987) have described a logarithmic stretching of the time axis that can convert a non-Fourier transform implementation to a Fourier transform combined with a phase shift.
The log-stretch transform makes AMO a time-invariant operator, which means it only depends on the difference between the input and output time. A transformation of the log-stretched data to the Fourier domain is then a convenient way to process the data in the space. Furthermore, since each frequency inversion is completely independent, one needs to solve many small systems instead of solving one huge system of equations. The inversions can be done in parallel by as many processing units as number of frequencies. In practice, several frequency bands from a useful bandwidth of the data are carried in parallel.