In this thesis, an efficient wavefield imaging technique of Forward/Backward Time-Stepping (FBTS) in the Time Domain (TD) is developed, employing the Discontinuous Galerkin Method (DGM) as the spatial-discretization technique. The FBTS method is based on calculating gradients of a TD cost functional with respect to the constitutive parameters of the target, and updating the modeled target using Conjugate Gradient (CG) method. The DGM and Runge-Kutta method are used to solve the mentioned problem, respectively, in space and time. We call this TD - Microwave Imaging (MWI) technique DGM-FBTS and present it, for the first time, for dispersive media.

The electromagnetic (EM) two-dimensional non-dispersive Transverse Magnetic (TM) DGM-FBTS is compared to Frequency Domain (FD) imaging algorithms for synthetic and experimental imaging targets in terms of computational cost, the quality of results, and robustness. For the first time, a direct comparison of TD and single-frequency FD MWI, DGM-Contrast Source Inversion (CSI) and DGM-Gauss Newton Inversion (GNI) schemes are used as the FD counterparts, all implemented in Matlab. For experimental data, the DGM-FBTS algorithm shows a robust noise performance, generating higher-resolution results than the two FD methods.

The DGM-FBTS formulation is also modified for quantitative ultrasound imaging with preliminary results presented, as a foundation for future experimental work in this area. This ultrasound imaging technique is validated briefly in this work, and it shows to be promising in quantitative wavefield imaging.

Finally, a novel development and investigation of quantitative hybrid time- and frequency-domain techniques is presented, focusing on enhancing the performance of both FD and TD schemes by improving the inversion speed and image resolution, respectively. These hybrid schemes are tested/validated on experimental data to study their accuracy in the presence of measurement noise and system modeling error. These results show improvement of image resolution compared to stand-alone FD algorithms (especially for complicated targets) and improvement in computational time by an average of 44% compared to stand-alone TD algorithm.