Reconstruction of Underground Dielectric Objects Using the Conjugate Gradient Method

An iterative inversion algorithm of imaging two-dimensional underground dielectric objects in cross-borehole geometry is presented. We define a cost functional by the norm of the difference between the measured scattered field and the calculated one for an estimated object function. Then the inverse scattering problem is cast into an optimization problem where the object function is determined by minimizing the functional. The Fréchet differential of the functional is employed to obtain its gradient. Applying the conjugate gradient method to the optimization problem, one can derive an iterative algorithm for getting the object function. Computer simulations are performed for lossy and homogeneous dielectric cylinders. The numerical results demonstrarte that the reconstructions of complex indices of refraction based on the proposed algorithm show good agreement with the true profiles even if the Born or the Rytov approximation fails and the scattered field data contains a measurement error. The results also confirm that the iterative procedure has the property of fast convergence.