Spatial Variability of Electric Field Implied by Common Dielectric Effective Medium Models

Remote sensing measurements of Earth materials are always made at scales much larger than individual grains and cavities, yielding only upscaled effective properties. An “effective medium” is an idealized uniform material that has the same measured properties as the real mixture. A uniform electric field applied to the ideal effective medium remains uniform within the sample; however, the same electric field applied to the composite results in fine-scale spatial variations of field strength within the sample, which depend on the properties of the constituents, their volume fractions, and their microgeometries. We derived analytic expressions for the electric field strength heterogeneity implicit in commonly used dielectric effective medium models. Only two-phase, statistically isotropic, low-loss materials, e.g., ice, snow, minerals, and freshwater in the microwave UHF band are considered. The method applies to singly or biconnected phases. The results confirm the uniform field in the isolated phase of material lying on the Hashin–Shtrikman (HS) bounds; the continuous phase field variance increases with a decreasing volume fraction, approaching a well-defined limit as the fraction becomes vanishingly small. Expressions are found for field variance in higher-order composites of coated spheres, providing realizations of composites lying between the HS bounds, and illustrating field nonuniqueness when microstructure is unknown. The mean and variance of the field strength in popular effective medium models are also examined. Not only do the effective properties predicted by these models differ so do the electric field strength spatial variability, especially when the volume fraction of inclusions increases.