The paper deals with a microwave‐tomography‐based solution algorithm tailored for use with GPR data‐processing applications. The algorithm tackles an inverse scattering problem in the frequency domain through the use of a linear model of the electromagnetic scattering, based on the Born approximation. In particular, we evaluate the reconstruction capabilities of the linear inversion algorithm in terms of the retrievable spatial variations of the unknown contrast function, whilst considering the problem of choosing an optimal frequency measurement step, theoretically, using diffraction tomography arguments. A numerical analysis of the technique is performed by means of the singular‐value decomposition tool, which allows us to extend the theoretical results to more realistic cases involving lossy soils. Finally, we present a series of reconstructions, obtained using synthetic and experimental data, which show the performance of the method under realistic conditions.
The problem of determining radiated electromagnetic fields from phaseless distributions on one or more surfaces surrounding the source is considered. We first examine the theoretical aspects and basic points of an appropriate formulation and show the advantage of tackling the problem as the inversion of the quadratic operator, which, by acting on the real and imaginary parts of the field, provides square amplitude distributions. Next, useful properties and representations of both fields and square amplitude distributions are introduced, thus making it possible to come to a convenient finite-dimensional model of the problem, to recognize its ill-posed nature and, finally, to define an appropriate generalized solution. Novel uniqueness conditions for the solution of the problem and questions regarding the attainment of the generalized solution are discussed. The geometrical properties of the functional set corresponding to the range of the quadratic operator relating the unknowns to the data are examined. The question of avoiding local minima problems in the search for the generalized solution is carefully discussed and the crucial role of the ratio between the dimension of the data representation space and that of the unknowns is emphasized.
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