Highlights

Project highlights

2010:

• Summer students working on numerical implementations of nonstationary techniques applied to imaging and acoustics, including butterfly algorithms for Fourier Integral Operators used in wave propagation, and creation of synthetic musical notes.
• Internship with CGG/Veritas on an implementation of reverse time migration based on Gabor windows, with comparisons to high order pseudospectral methods, by PhD candidate Ben Wards.
• Several of our graduate students have completed, or are completing their theses in seimic imaging and moving on to research positions in industry.
• Presentations of POTSI work at several national and international conferences, including the annual meetings of the Canadian Society of Exploration Geophysicists (May, Calgary), the US Society for Exploration Geophysicists (October, Denver), the MITACS AGM (May, Edmonton), and the Canadian Mathematical Society (June, Fredericton).
• Generalized Chebeshev-type orthogonal polynomials arising from nonlinear transformations of the Lie group transform developed by Patera et al suggests a new way of understanding orthogonal systems.

2008 - 2009:

• Presentations of POTSI work at the annual meeting of the Society for Exploration Geophysicists in NOv, 2008, Las Vegas..
• Reverse time migration based on Gabor windows, with comparisons to high order pseudospectral methods and standard finite difference methods, by PhD candidate Ben Wards.
• Application of Gabor methods to mutliple suppression in seismic imaging.
• Development of Chebeshev-type approximations based on Patera's work on Lie group lattices.

2007 - 2008:

• A four month internship with Husky Energy is underway, with graduate student Chad Hogan, investigating a full waveform topography technique for 4D seismic surveys. Synthetic data models of the Pike Peak site is being used as a realistic test for the method.
• Reverse time migration based on Gabor windows is under development, with initial results to be presented at SEG in November. The technique offers faster wavefield extrpolations with few artifacts due to dispersion limitations.
• Masters student S. Ismail has completed his thesis on nonstationary filtering. A mathematical framework for analyzing Gabor filters is presented, one of the key technologies used in Gabor decon and Gabor wavefield extrapolation.

2006 - 2007:

• The FOCI method for 2D depth migration has been implemented in FORTRAN for speed, with provisions for accommodating topography in the surface data. This algorithmprovides a fast, stable method for wavefield extrapolation used in the key step of migrations in the  seismic imaging problem. The topography accommodations allows its use with real data from land seismic surveys.
• FOCI has been implemented in the data processing architecture of one of our sponsors through a four-month internship. This allows a complete test of the FOCI algorithm with real data from commercial seismic surveys.
• Results on a new 3D wavefield extrapolation method using annular sums have been achieved. This is a significant improvement over the well-known Hall-McClellan methods, offering improved stability, computational speed, and reduction in lattice artifacts.
• An alternate stabilization method for f-x migration has been developed, using locally WKBJ operators that arise in classical scattering theory. Results have appeared in the Journal of Seismic Exploration.
• An adaptive Gabor method of wavefield extrapolation has been developed and demonstrated it with a 2D prestack migration code using a nonuniform partition of unity to perform spatial localization only where there is sufficient velocity variation. These partitions are adaptively designed to the local velocity structure such that the spatial positioning error is bounded below any desired threshold.
• A four month internship with Talisman Energy on the seismic footprint problem was completed by graduate student Joanna Cooper. This project created a large data set of synthetic data which will be used to determine how the choices of geophone and source spacing and layout affects the final image of the earth that is created through seismic processing.
• Development and testing of a fully elastic version of the popular acoustic migration scheme knows as GPSPI (Generalized Phase Shift Plus Interpolation) has been done.  Goals of this work are to more accurately investigate the phenomenon known as shear-wave splitting in structurally complex settings and to more accurately image multicomponent seismic data.
• A U.S. patent application has been filed for the transform methods on Lie groups.
• The work on Lie-group based transforms has been generalized to the n-variable case
• A new application for the Lie group transform methods has been developed, for secure encryption of data.
• Publications on the S-transform identify the precise connection between continuous wavelet transforms and the S-transform, which opens up the full power of established mathematical theory in wavelets to this particular implementation.
• Promising new techniques for rapid numerical integration have been developed, providing better performance that such classic algorithms as  Gauss-Kronrod integration.
• Summer schools, workshops, conference minisymposium have been organized by POTSI members, with details in the Event section.

2004 - 2005:

• The FOCI method (Forward operator conjugate inverse) developed by the POTSI group provides a fast, stable method for wavefield extrapolation used in the key step of migrations in the  seismic imaging problem. Our presentation on FOCI was awarded the Best Paper distinction at the CSEG National Convention. Descriptions of FOCI have been published in Geophysics and in the CSEG Recorder.
• A 2D depth migration algorithm based on the FOCI method has been transferred from POTSI to our industrial partners.
• The FOCI and Gabor decon algorithms have been implementation on our 10 node linux cluster, to take advantage of parallel processing for rapid processing of seismic data. The cluster takes advantaage of MPI software architecture  developed at POTSI.
• Extensive testing of Gabor decon with real seismic data has been completed. Full seismic processing of land data compares the performance of Gabor decon with the industry standard of surface-consistent Wiener decon with spectral whitening. The results indicated Gabor decon is competitive with the standard, both in terms of accuracy and computational speed, with someindication of areas for further refinement.
• A fully elastic version of the popular acoustic migration scheme knows as GPSPI (Generalized Phase Shift Plus Interpolation) has been implementated and tested, using the "elastic Marmousi" model dataset.  This work models shear-wave splitting in structurally complex settings and provides imaging from multicomponent seismic data.
• A fully functional prestack depth migration code based on a Gabor approximation to the locally homogeneous wavefield extrapolator has been developed and implemeted. Testing on the Marmousi synthetic is giving very good results, though the runtimes are presently longer than FOCI.
• Patent protection is being sought for the novel transform methods based on lattices in Lie groups, and applications to signal processing.
• Herrmann and Dupuis have devised a detection-estimation formulation for the characterization of sesimic reflectors, using recent results on expansions with respect to redundant dictionaries with parameterized prototype waveforms. By finding the unique expansion, important geological, geophysical and mineralogical information is derived from the waveforms.
• Sacchi has developed an inversion algorithm to test the feasibility of least-squares wave equation imaging methods, including the theoretical framework and numerical implementation of wave equation least-squares amplitude versus ray parameter imaging. The resulting image of the subsurface can be used for the estimation of rock physical parameters  and for migration velocity analysis.
• Kuehl and Sacchi  have posed the AVP imaging as an inverse problem, and utilized regularization strategies to estimate accurate models of the subsurface.  Physical constraints for a well-defined imaging problem were used to control the stability of the solution of the inverse problem.  The lateral smoothness of AVP gathers is used as a desirable feature for well-focused images in the ray-parameter domain.
• Sacchi reports that 3-D field data tests of AVP regularized least-squares
  migration/inversion appear to be the first attempts to validate inverted
  subsurface amplitude responses with synthetics generated from borehole
  data. Results are published in the Journal of Geophysics.
• POTSI has begun an in-depth collaboration with Dr. Lou Fishman (MDF International), to apply his theory of a path integral approach to wavefield extrapoation in oceanography to the seismic problem. Fishman ran a special course in Fall 2004, and returned in Fall 2005 as a PIMS Distinguished Lecturer, to present his work
• POTSI is working with the PIMS Collaborative Research Group in Inverse theory, headed by Gunther Uhlman and Gary Margrave. POTSI researchers are directly involved in the ongoing series of summer schools and workshops planned by the CRG, including the PIMS-MITACS-VIGRE Summer Graduate School held at the University of Washington in the summer of 2005.
• POTSI organized a week-long BIRS workshop on Time-frequency analysis and Nonstationary filtering in September 2005, in conjunction with the special year in time-frequency analysis run by the Edwin Schrodinger institute in Vienna.
• POTSI has initiated two new nodes, at York University, and the University of Montreal, to focus on the applications of mathematical techniques to other image processing problems, including medical imaging and target enhancement. New sponsors for this work include Lockheed Martin, and Merck Forsst.

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.