The focus of this research is the development of a comprehensive theory to analyze (inverse) scattering problems with waves in random media generated by (random) ambient noise sources. In particular, the investigators are interested in the analysis of measurements of such waves if the (multiple) scattering takes place in the interior of a domain and the measurements are taken on its boundary. Milestones include: (1) an analysis framework for the scattering of jointly coherent and incoherent waves in a mixed model consisting of a macro multi-scale component, a random microscale component and coherent conormal singularities (the word ``incoherent'' is used advisedly; many would prefer the term ``diffuse'' if the phase relation between source of ambient seisms and the field that is generated is deterministic); and (2) the design of functionals of cross correlations and beyond, and associated sensitivity maps statistically stably resolving (changes in) the various (random) model parameters. This project develops methods for extracting information from ``noise''. The idea is to look at higher order statistical patterns in noise-generated data. As an example, imagine estimating the distance to a wall by analyzing noise reflected off the wall rather than by analyzing the delayed echo of a coherent signal. In the project much more complicated situations are considered in which noise generated waves have propagated through a complex (random) medium before they are measured. The investigators will develop new techniques for describing waves that are affected by such a medium, in addition to algorithms that effectively process the mentioned data numerically to extract relevant statistical information and subsequent imaging. These will have applications in remote sensing, medical imaging and seismic imaging using ambient noise sources; the investigators will focus on the latter.
With improved theory the investigators can develop inverse scattering tools that have the potential to revolutionize the characterization of structures and processes (e.g., flow, tremors, seismogenesis) in the subsurface, on spatial scales relevant for regional studies (e.g., structure and microseismic activity of crust and upper mantle lithosphere; hazards) and hydrocarbon production (energy). The results can furthermore be applied to monitoring CO_2 sequestration. Large and dense sensor arrays that have become operative recently provide unprecedented opportunities for exploiting new theoretical insights to unravel, hitherto unexplored, complexities in the subsurface.