Generalized minimum-phase relations for memory functions associated with wave phenomena

Memory functions occur as temporal convolution operators in governing equations of wave propagation and generally account for the instantaneous and non-instantaneous responses of a medium. The specific memory function that is causal and stable, and the inverse of which is causal and stable as well, is conventionally referred to as a minimum-phase (MP) function. Its amplitude and phase spectra are not independent, but related through MP-relations; that is, Kramers-Kronig relations between the amplitude and phase spectra. In this paper, we derive generalized MP-relations for a memory function that does not necessarily meet the stability requirements; such functions are often encountered in the geophysical context. We still address the function as MP because its phase spectrum exhibits minimum group delay, like that of a conventional MP function. We successfully tested the derived relations for the well-known Maxwell, Kelvin-Voigt and Zener compressibility models used in acoustics/elastodynamics, the dynamic permeability used in poroelasticity and the electrokinetic coefficient used in coupled acoustics and electromagnetics. In these fields, the derived relations can be applied for the determining the involved memory function using numerical or laboratory experiments; only the amplitude or the phase spectrum needs to be measured and the other can be calculated. The relations also have applications in effective-medium theory and for any other wave phenomenon that employs memory functions.