During this presentation an overview will be given on how recent developments in mathematics -- such as multiscale and multi-directional signal representations, "compressive sampling" and percolation theory -- can be used to recover and explain images of the Earth subsurface. First, the recovery of wavefronts from incomplete data is discussed, followed by a brief tour of the seismic imaging problem. The output of this procedure leads to images of the reflectors (=singularities) that are related to the type of transitions that occur in the Earth subsurface. We conclude by making the argument that these transitions can be modeled as critical points in the elastic properties of bi-compositional mixtures.
This is joint work with Yves Bernabe (MIT) and my students Gilles Hennenfent, Mohammad Maysami and Peyman Moghaddam.