The thesis is devoted to study inverse problems related to acoustic and elastic source localization in attenuating media from boundary measurements and their applications in biomedical imaging. We present efficient and stable algorithms to compensate for the effects of attenuation on image resolution. We develop Radon transform based algorithms to recover initial pressure distribution in attenuating media with and without imposed boundary conditions. We apply stationary phase theorem on an ill-conditioned attenuation operator to rectify attenuation effect and use TV-Tikhonov regularization methods to handle partial measurement problems. We revisit time reversal methods for loss-less media and extend them to attenuating media. As the attenuated waves are not time reversible, we use the strategy of back-propagating the regular approximations of adjoint attenuated waves to reconstruct sources stably with first order attenuation correction. For acoustic media, we present an alternative strategy based on data pre-processing for higher order corrections. As the data in elastic media consists of coupled shear and pressure waves, we propose an original approach based on weighted Helmholtz decomposition. Further, we introduce efficient weighted imaging algorithms for locating acoustic noise sources by cross correlation techniques and using regularized back-propagators for attenuation correction. We also localize spatially correlated noise sources and estimate correlation matrix between them. In order to extend elastic anomaly detection algorithms to visco-elastic media, we derive a closed form expression for an isotropic visco-elastic Green function. Then, we propose an attenuation correction technique for a quasi-incompressible medium and prove that one can access, approximately, the ideal (inviscid) Green function from the visco-elastic one by inverting an ordinary diff erential operator. Finally, we provide some anisotropic visco-elastic Green functions, with an aim to extend our results to anisotropic media.
