Discrete Almost-Symmetric Wave Packets and Multiscale Geometrical Representation of (Seismic) Waves

We discuss a multiscale geometrical representation of (seismic) waves through a decomposition into wave packets. Wave packets can be thought of as certain localized “fat” plane waves. Here, we construct discrete almost-symmetric 3-D wave packets by using the unequally spaced fast Fourier transform. The resulting discrete transform is unitary, implying that the reconstruction operator is simply the adjoint of the decomposition operator. Another relevant aspect of the discretization scheme is the appearance of parameters that control the tiling of the phase space that corresponds with the dyadic parabolic decomposition, preserving the relative parabolic scaling while adapting to the physical problem at hand. We consider applications in exploration and global seismology, in particular for higher dimensional data regularization, seismic map migration, denoising, directional regularity analysis, and feature extraction.