Effective macroscale measures of microscale heterogeneity a multiresolution approach

Since the articulated operator describes the effects of the microstructure on the macroscale response, it is interpreted as an effective properties description of the complex scatterer, or effective material. Our purpose is two-fold. As a first objective, we report the self consistent derivation of the formulation smooth. That is, a formulation of the scattering problem, that is written on a macroscale grid, and governs the macroscale component of the response field for a complex scatterer. The second objective is the effective properties study. That is, an investigation as to the degree to which the effective properties operator or some of its components are invariant under classes of variations of the macro- or micro-structures of a complex scatterer, and the precise characterization of these classes. In the context of the second objective, two fundamental results are reported.