Inverse bounds for microstructural parameters of composite media derived from complex permittivity measurements

Bounds on the volume fraction of the constituents in a two-component mixture are derived from measurements of the effective complex permittivity of the mixture, using the analyticity of the effective property. First-order inverse bounds for general anisotropic materials, as well as second-order bounds for isotropic mixtures, are obtained. By exploiting an analytic representation of the effective complex permittivity, the problem of estimating the structural parameters is reduced to a problem of evaluating the moments and support of a measure containing information about the geometrical structure of the material. Rigorous bounds on the volume fraction are found by inverting firstand second-order (Hashin–Shtrikman) forward bounds on the complex permittivity. The inverse bounds are applied to measurements of the effective complex permittivity of sea ice, which is a three-component mixture of ice, brine and air. The sea ice is treated via the two-component theory applied to a mixture of brine and an ice/air composite. The bounds on the brine volume of sea ice derived from the effective permittivity measurements are in excellent agreement with data from experiments. The inversion of forward bounds on the complex permittivity of composite media provides a basis for a theory of inverse homogenization for recovering microstructural parameters from bulk property measurements. Such results are applicable to problems in remote sensing, medical imaging and non-destructive testing of materials.