The retrieval of a buried cylindrical obstacle by a constrained modified gradient method in the H-polarization case and for Maxwellian materials

The retrieval of an unknown, possibly inhomogeneous, penetrable cylindrical obstacle buried entirely in a known homogeneous half-space-the constitutive material parameters of the obstacle and of its embedding obey a Maxwell model-is considered from single- or multiple-frequency aspect-limited data collected by ideal sensors located in air above the embedding half-space, when a small number of time-harmonic transverse electric (TE)-polarized line sources-the magnetic field H is directed along the axis of the cylinder-is also placed in air. The wavefield is modelled from a rigorous H-field domain integral-differential formulation which involves the dot product of the gradients of the single component of H and of the Green function of the stratified environment times a scalar-valued contrast function which contains the obstacle parameters (the frequency-independent, position-dependent relative permittivity and conductivity). A modified gradient method is developed in order to reconstruct the maps of such parameters within a prescribed search domain from the iterative minimization of a cost functional which incorporates both the error in reproducing the data and the error on the field built inside this domain. Non-physical values are excluded and convergence reached by incorporating in the solution algorithm, from a proper choice of unknowns, the condition that the relative permittivity be larger than or equal to 1, and the conductivity be non-negative. The efficiency of the constrained method is illustrated from noiseless and noisy synthetic data acquired independently. The importance of the choice of the initial values of the sought quantities, the need for a periodic refreshment of the constitutive parameters to avoid the algorithm providing inconsistent results, and the interest of a frequency-hopping strategy to obtain finer and finer features of the obstacle when the frequency is raised, are underlined. It is also shown that though either the permittivity map or the conductivity map can be obtained for a fair variety of cases, retrieving both of them may be difficult unless further information is made available.