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)