Is a moving average field a proper macro scale measure?

Abstract

A number of derivations of the Biot equations governing the acoustics of fluid-filled, porous solids, which are based on a more complete formulation that applies to all length scales, accept moving averages of the response fields that enter the more complete formulation as the response fields that enter the Biot equations. This raises a question: Does a moving average field incorporate only macro scale variation? A moving average field is presented as one extreme of a class of fields that is formed from sets of discrete local spatial averages. The set of local spatial averages for a moving average field is accomplished for locations that are separated by a vanishing distance. The opposite extreme of a field of local averages is formed from a set of local averages accomplished for locations that are separated by a distance equal to a linear measure of the region of the spatial average. Explicitly demonstrated is that the moving average of a field that contains both macro and micro scale variation will itself contain both macro and micro scale variation. The relative suppression of the micro scale variation compared to the macro scale variation, which obtains in one representation of the moving average field, is only apparent; the micro scale variation can be recovered by an appropriate signal processing. This is in contrast to a wavelet-defined field of local averages, an example of the other extreme, for which the suppression of the micro scale variation is absolute. The issue is significant for a derivation of a prediction model that purports to output macro scale response fields.