Anomalous Diffusion of Acoustic Fields in Periodic and Random Scattering Media and Its Application to Remote Sensing

In recent years, several experimental studies have shown that field transport processes in non-homogeneous media can occur according to anomalous and hybrid mechanisms that are not accounted for in traditional methods of analysis. These unconventional transport mechanisms have been observed in many diverse physical systems, ranging from porous and heterogeneous materials to biological tissues and granular materials. Current modeling techniques, based on integer order differential models, cannot effectively capture this hybrid behavior, while direct numerical simulations represent often an impractical route, due to both the size of the computational problem and the coexistence of multiple spatial and temporal scales. On the other hand, fractional order differ-integral models have shown the capability to predict accurately the complex nature of these transport processes. In part, this capability is a direct consequence of the many intrinsic properties of these operators such as memory, non-locality, and multiscale behavior that all combined allow describing hybrid transport processes within a unified mathematical formulation. The main objective of this thesis is twofold and consists in investigating the leading causes of the occurrence of anomalous diffusion of acoustic wave fields propagating through inhomogeneous and highly scattering media and in developing computational tools capable of modeling accurately its effects. Acoustic wave fields provide the foundation of many disciplines of practical interest ranging from imaging to remote sensing and non-destructive evaluation. The ability to accurately model the acoustic field transport in complex media is paramount to enhance the quality of the reconstructed images and to reduce the occurrence of artifacts. While these are important driving factors for the development of new imaging technologies and very challenging issues in any inverse problem, addressing them in the context of complex scattering media still poses substantial and unsolved technical problems. The most direct effect of these limitations is the lack of reliable methodologies for the acoustic based sensing of complex media. In order to provide a comprehensive description of acoustic transport mechanisms in complex scattering media, we develop a generalized fractional-order acoustic theory capable of describing the propagation of wave fields in periodic and disordered media. Particular attention is devoted to the analysis and characterization of anomalous acoustic transport and to uncovering the link between both geometric and material parameters and the fractional order of the governing continuum model. The unique properties of fractional-order models and their ability to capture and simulate complex physical acoustic processes suggest they offer great potential for applications to inverse acoustic problems, and beyond. In an effort to illustrate these unique capabilities and the many opportunities provided by these models, we also propose and numerically test a novel acoustic tomographic technique that builds on both the physics of hybrid propagation and on the characteristics of fractional order models to obtain a promising imaging technique capable, at least on the basis of available numerical results, to distinctly over perform more traditional acoustic imaging methods.