Seismic Imaging with the Elliptic Radon Transform in 3D: Analytical and Numerical Aspects

A well-established method to investigate subsurface material parameters is to generate pressure waves on the surface and measure their reflections returning there at different points.

In this thesis, we consider a scanning geometry with constant distance from source to receiver in three space dimensions.
After linearisation this situation is modelled by the elliptic Radon transform which integrates over ellipsoids.

As an inversion formula of this transform is unknown, we propose a certain imaging operator appropriate to apply the method of the approximate inverse and develop a migration scheme to reconstruct singularities in the speed of sound.
Further, we calculate the top order symbol of the imaging operator as a pseudodifferential operator and analyse its behaviour in different situations.
We use the obtained results to achieve reconstructions of the subsurface, which are relatively independent of the distance to the surface and the offset.
Last, we present numerical experiments and test our implementation with data generated by solving the wave equation numerically.