February Fourier Talks 2008

Gary Margrave


Seismic Deconvolution and Imaging in the Gabor Domain


The Gabor transform, or windowed Fourier transform, is an effective technique to extend Fourier spectral theory to inherently nonstationary problems. A particularly simple formulation, based on a localizing window set constrained to form a partition of unity (POU), has proven very adaptable to seismic imaging applications. I will outline the Gabor theory and illustrate its connection to pseudodifferential operator theory. Then I will describe the application to two problems in seismic image construction: deconvolution and migration. In the first case, we develop a complex-valued Gabor multiplier that effectively corrects seismic data for both attenuation effects and source signature. The magnitude of the Gabor symbol of this operator is estimated from the data itself while the phase is constructed under the minimum phase assumption. In the second case, the problem of wavefield extrapolation in depth through laterally variable velocity is addressed through the construction of a non-uniform POU. This partition is constrained by an error criterion bounding lateral position error. The result is an effective pre-stack depth migration that generalizes directly to 3D. Both of these applications will be illustrated by data examples.

Collaborators: Michael P. Lamoureux (Professor of Mathematics), Carlos Montana (PhD candidate in geophysics), Yongwang Ma (PhD candidate in geophysics)