Tomographic migration velocity analysis by differential semblance optimization with respect to scattering-angle/azimuth in anisotropic elastic media

Linearized asymptotic inversion of seismic data maps, given the smooth variations in medium properties such as phase velocities (the background medium), known information on the wavefront set of the data into the unknown singular component of medium variations (the medium contrast). Such inversion was formulated in terms of a Maslov extension of Generalized Radon Transforms, and allows the formation of caustics in the smoothly varying background medium. In the present paper, we analyze the functional dependencies of the linearized asymptotic inversion operator on the background medium properties. We do this by introducing the Fréchet derivatives of this operator, and by analyzing their microlocal properties. Geometrical reduncies in the data, on the wavefront set, leads to the existence of annihilators – which when applied to the data with the true background medium would yield a vanishing outcome. Hence, such annihilators can be exploited to find the background medium properties. Combined with the Fréchet derivatives of the linearized inversion operator, we propose an iterative procedure to determine the smoothly varying background medium. The ideas presented in this paper are a microlocal reformulation of the procedure of ‘migrationvelocity-analysis’.