Discrete almost symmetric wave packets and multi - scale geometrical representation of ( seismic ) waves . Anton

We discuss a multi-scale geometrical representation of (seismic) waves via a decomposition into wave packets. Wave packets can be thought of as certain localized “fat” plane waves. As a starting point we take the frame of continuous “curvelets” and associated transform. Different discretizations, and approximations, of this transform lead to different discrete wave packets. One such discretization leads to the introduction of digital curvelets. Here, we follow a different discretization while constructing discrete almost symmetric wave packets and making use of the USFFT. The resulting discrete transform is unitary. In our discretization, we directly approximate the underlying dyadic parabolic decomposition, its (rotational) symmetry, and approximate all the necessary decay estimates for continuous “curvelets” in phase space with arbitrary accuracy. Another relevant aspect of our discretization is the appearance of parameters that control the tiling of phase space corresponding with the dyadic parabolic decomposition, preserving the relative parabolic scaling, while adapting to the physical problem at hand. We consider applications in exploration seismology and global seismology. For these, we need transforms in higher dimensions, that is, in dimension n = 3, 4, 5.