RESUMEN
An analytic continuation method for obtaining rigorous bounds on the effective complex permittivity ε * of polycrystalline composite materials is developed. It is assumed that the composite consists of many identical anisotropic crystals, each with a unique orientation. The key step in obtaining the bounds involves deriving an integral representation for ε *, which separates parameter information from geometrical information. Forward bounds are then found using knowledge of the single crystal permittivity tensor and mean crystal orientation. Inverse bounds are also developed, which recover information about the mean crystal orientation from ε *. We apply the polycrystalline bounds to sea ice, a critical component of the climate system. Different ice types, which result from different growth conditions, have different crystal orientation and size statistics. These characteristics significantly influence the fluid transport properties of sea ice, which control many geophysical and biogeochemical processes important to the climate and polar ecosystems. Using a two-scale homogenization scheme, where the single crystal tensor is numerically computed, forward bounds for sea ice are obtained and are in excellent agreement with columnar sea ice data. Furthermore, the inverse bounds are also applied to sea ice, helping to lay the groundwork for determining ice type using remote sensing techniques.
RESUMEN
This paper deals with the recovery of porosity of bone from measurements of its effective electrical properties. The microstructural information is contained in the spectral measure in the Stieltjes representation of the bone effective complex permittivity or complex conductivity and can be recovered from the measurements over a range of frequencies. The problem of reconstruction of the spectral measure is very ill-posed and requires the use of regularization techniques. We apply the method to the effective electrical properties of cancellous bone numerically calculated using micro-CT images of human vertebrae. The presented method is based on an analytical approach and does not rely on correlation analysis nor on any a priori model of the bone micro-architecture. However the method requires a priori knowledge of the properties of the bone constituents (trabecular tissue and bone marrow). These properties vary from patient to patient. To address this issue, a sensitivity analysis of the technique was performed. Normally distributed random noise was added to the data to simulate uncertainty in the properties of the constituents and possible experimental errors in measurements of the effective properties. The values of porosity calculated from effective complex conductivity are in good agreement with the true values of bone porosity even assuming high level errors in the estimation of the bone components. These results prove the future potential of electrical impedance spectroscopy for in vivo monitoring of level and treatment of osteoporosis.