Minimum phase and attenuation models in continuous time

Michael P. Lamoureux, Peter C. Gibson, Gary F. Margrave

We extend earlier work on minimum phase attenuation models in discrete-time signal processing to the continuous time setting, where real physical processes occur. This includes the propagation of seismic energy through the earth and allows for the modelling of attenuation processes.

Minimum phase signals are characterized by an energy condition, equivalent to an outer function identification in the complex half-plane. Certain physical processes preserve the minimum phase property, and as such, the operators must arise mathematically as productconvolution operators of a very restrictive form. The basic mathematical model shows Q-attenuation arises as a simple consequence of the minimum phase preservation property for seismic signal propagation. In contrast to stationary filter processes, in Q-attenuation, not one but two data measurements are necessary for a complete determination of the attenuation characteristics. But only two.

This work is a summary of a sequence of papers on minimum phase properties.