Jeff P. Grossman, Gary F. Margrave, Michael P. Lamoureux
The assumption of a minimum-phase seismic wavelet is central to the Wiener and Gabor deconvolution algorithms. The concept of minimum phase is well understood by the geophysics community for the case of digital signals. The extension of minimum phase to analog signals, however, has not received sufficient attention in the literature. Certain well-known properties about digital minimum phase signals, such as convolutional invertibility, stability, and Hilbert transform relationships between Fourier amplitude and phase spectra, do not carry over so easily to the continuous setting.
We have found it necessary to develop the analog theory of minimum phase within the rather general framework of tempered distributions. Subsequently, the theory allows us to extend the concept of a minimum-phase filter (linear operator) to the general setting of nonstationary analog systems. Such a filter is used explicitly in theoretical development of Gabor deconvolution.