Test of the relation between temporal and spatial Q by Knopoff et al.

The quality factor is a dimensionless measure of the energy loss per cycle of wave modes in an attenuation medium. Accurate measurement is important in various fields, from seismological studies to detect zones of partial melting to the geophysics of reservoirs to study rock properties such as porosity, fluid properties and saturation and permeability. In seismology, the quality factors measured for normal (standing) modes and propagating waves differ, as well those of equivalent experiments based on resonant rods and ultrasonic pulses performed in the laboratory. These measurements result in temporal and spatial quality factors respectively. A relationship between these two different quality factors and between the corresponding attenuation factors was proposed by Knopoff et al. sixty years ago. The conversion factor is basically the ratio between the phase velocity and the group velocity, while for the attenuation factor is the group velocity. We test these relations, which hold for low-loss solids, for body waves, using a Kelvin–Voigt rheology and a constant $Q$ model, which provide explicit expressions of the temporal and spatial quality factors and velocities involved in these relations. The proposed theory provides the basis for a complete characterization of temporal and spatial quality factors and velocity dispersion based on arbitrary stress–strain relationships.