Microlocal analysis of imaging operators for effective common offset seismic reconstruction

The elliptic Radon transform (eRT) integrates functions over ellipses
in 2D and ellipsoids of revolution in 3D. It thus serves as a model for linearized
seismic imaging under the common offset scanning geometry where sources and
receivers are offset by a constant vector. As an inversion formula of eRT is unknown
we propose certain imaging operators (generalized backprojection operators)
which allow to reconstruct some singularities of the searched-for function
from seismic measurements. We calculate and analyze the top order symbols of
these imaging operators as pseudo-differential operators to understand how they
map, emphasize or de-emphasize singularities. We use this information to develop
local reconstruction operators that reconstruct relatively independently of depth
and offset. Numerical examples illustrate the theoretical findings.