Constrained least-squares reconstruction of multidimensional objects buried in inhomogeneous elastic media

The ultrasonic probing by a time-harmonic line (point) source of a multidimensional object buried in an inhomogeneous elastic background structure is considered. The goal is the imaging and quantitative characterization of the object from the scattered field measured along a receiver array. This leads to a multidimensional continuous inverse scattering problem which is nonlinear and ill-posed. The authors linearize the problem within the Born approximation for inhomogeneous background, and seek a minimum-norm least-square solution to the (discretized) integral equations. This solution is based on a singular value decomposition of the forward operator matrix. A priori information can be incorporated into the algorithm to enhance the accuracy and improve the resolution of the recovered object characteristics. The method is illustrated on a 2-D problem where constrained least-squares inversion of the object characteristics is performed from synthetic data. A Tikhonov regularization scheme is also examined and compared to the minimum-norm least-square estimation.