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.
On the Stability of Symmetric Periodic Orbits of the Elliptic Sitnikov Problem