Multiscale Electromagnetic Imaging of a Random Medium
Tuesday, April 23, 2002 - 11:00am - 12:00pm
Elena Cherkaev (The University of Utah)
In many heterogeneous materials such as geophysical porous media, the scale of the microstructure is much smaller than the wavelength of the electromagnetic signal. In this situation, the wave cannot resolve all the details of the structure, and the response of the medium is homogenized. The talk discusses an approach to electromagnetic imaging which ties together properties of the medium on two different scales: Complex permittivity found on the coarse scale imaging step provide data for microscale inversion which recovers information about the fine structure of the random medium. Information about the microgeometry is contained in the spectral measure in the Stieltjes integral representation of the effective complex permittivity of the medium. The problem of reconstruction of the spectral measure has unique solution, however it is extremely ill-posed. Several stabilization techniques are discussed such as quadratically constrained minimization, regularization using nonnegativity constraint, and reconstruction in the class of functions of bounded variation. The reconstructed spectral function can be used to estimate geometric parameters of the structure as well as to evaluate other effective properties of the same medium, such as thermal or hydraulic conductivity.