In this chapterwe give a briefsurvey of basic results on the dispersive decay due to P. D’Ancona, M. Beals, M. Beceanu, M. B. Erdo?an, L. Fanelli, M. Goldberg,W. R. Green, V. A. Marchenko, B. Marshall, W. Strauss, S. Wainger, I. Rodnianski, W. Schlag, D. Tataru, K. Yajima, and others. Moreover, we present a new short and simplified proof of the fundamental results on the L^1-L^\infty dispersive decay forthe Schrödingerequation established by J.-L. Journé, A. Soffer, and C. D. Sogge.

The theoretical foundations of the description of heterogeneity are established for both body waves and modal fields. We first consider perturbations of the wavefield using Born series and show how such concepts can be combined with the use of reflection and transmission operators to provide a flexible treatment of structures with varying heterogeneity in different zones of the model. Although the various modes of surface waves propagate independently in simple structure, the presence of heterogeneity induces cross-coupling that modifies the wavefield. Because the nature of heterogeneity differs in the various parts of the Earth, the interaction of different classes of heterogeneity has an important role in shaping the nature of the full seismic wavefield.