Examples of asymptotic analysis for understanding and building seismic wave-equation imaging tools

Migration velocity analysis aims at determining an optimal velocity macro-model to focus energy in the subsurface. The replacement of the adjoint operator by the inverse operator in the extended subsurface offset domain offers a more robust velocity analysis approach. In order to better understand the non-linear relationship between the data and the velocity macro-model, we review here the behaviour of different wave-equation based operators through an asymptotic analysis. It is a very useful tool for interpreting and even building some imaging algorithms. In particular, we establish a mapping between the data and the subsurface image domains. This mapping offers new possibilities to better understand how velocity analysis behaves.