Pre-2004

In about midyear of 2000, we became excited about the connection between Gabor analysis and pseudodifferential operators. Since then we have focused strongly on Gabor techniques as a practical means of implementation of pseudodifferential operators and developed a number of exciting results. The most mature is a new seismic deconvolution technique, called Gabor deconvolution, which enables deconvolution to deal with nonstationary signals. Our technique is able to achieve much higher resolution than standard methods, including Weiner deconvolution. In collaboration with CREWES, we have implemented it within the environment of a commercial software package and distributed it to both POTSI and CREWES sponsors.

We have also extended our work with Gabor theory to the subject of wavefield extrapolation. This technology is central to seismic imaging and our initial Gabor extrapolator is quite interesting. In the course of this work we have developed a novel approach to the construction of a nonuniform, adaptive Gabor frame. We have also come to realize that our approach to Gabor frame construction based on partitions of unity is not well known and are developing its mathematical properties.

Previous work by Margrave and others under CREWES funding has resulted in two wavefield extrapolators using pseudodifferential (actually Fourier integral) operators: generalized phase shift plus interpolation (PSPI) and non-stationary phase shift (NSPS). Numerical implementations yield images typically superior than standard seismic imaging methods. Recently we have also formulated what we call the Weyl extrapolator based on the Weyl pseudodifferential calculus.

Building on this work with Fourier integral wavefield extrapolators, we have recast the analytic expressions as singular integral operators and implemented these numerically. The singular integral extrapolators have some advantageous features including a computation cost that is independent of the velocity complexity and relative insensitivity to irregular spatial sampling. We call these operators by their geophysical name: recursive Kirchhoff wavefield extrapolators.

We have completed the initial development of a new seismic imaging engine that runs on a parallel Linux cluster. To accomplish this we hired two highly-skilled programmers from industry for two months and put them under the direction of Hugh Geiger (PDF). We asked them to implement our recursive wavefield extrapolators in C using MPI (message passing interface). At the same time, we built a parallel computation facility using Matlab with a public toolbox. The Matlab facility was completed first but we have now verified that the C/MPI facility is working and is providing the same answers as the Matlab facility but with a three-fold increase in computation speed. We are hopeful for greater speed increases as we optimize the code.

In the fall of 2003 we made a significant mathematical advance in that we have developed an exact solution to the wavefield extrapolation problem in a particularly interesting case: two dimensions, discrete sampling, but arbitrary lateral velocity variations. This allows us to compare our Fourier integral operators, that are all approximate, to the exact result. The exact result is computationally too demanding to be useful in an imaging scheme but it very valuable theoretically. We are now trying to extend this result to the continuous case.

Simply put, our improved methods for seismic imaging will lead to better economic success in the search for new oil and gas reserves. Better images, and better characterizations of the lithologies, lead to better decisions on where to drill. Our deconvolution code based on the Gabor transform has been released and is in commercial use; the feedback we have received from the industrial users is that it indeed gives better images than the current standard methods. We anticipate that the parallel implementation, and other new developments in our algorithms, will lead to further successes.