Research

Research

This project brings together mathematicians and geophysicists in a collaboration to build seismic imaging algorithms based on pseudodifferential operator theory and other modern mathematical analysis techniques. Primary objectives are:

• Improve the theoretical foundation for existing wavefield extrapolation methods (invented at Consortium for Research in Elastic Wave Exploration Seismology [CREWES]). These are acoustic one-way wavefield extrapolators.
• Extend the theory to predict higher-order one-way extrapolators and two-way extrapolators.
• Extend the theory to fully anisotropic and inhomogeneous elastic media.
• Develop stationary phase or other approximations to the oscillatory integrals from the theory.
• Develop efficient numerical approximations to the pseudodifferential operators.
• Analyze stability and dispersion of the numerical schemes.
• Develop time-adaptive deconvolution techniques.
• Investigate the theoretical linkages between Gabor theory and pseudodifferential operators.
• Create 3D computer codes for forward wavefield propagation (modelling) and inverse propagation (migration). It is expected that the forward code will use two-way operators (i.e. the forward and backward scattered wavefields will be coupled) while the inverse code will use one-way operators.

• Test these codes in a variety of settings. For example, physical-modelling data could be compared to the forward-modeled numerical data and could be migrated with the inverse code. The driving force for this research is the need for precise estimates of lithologies and pore fluids from seismic images and that calls for sophisticated imaging techniques. Currently, most relevant research is based on extending Green's function techniques to complex media. While certainly promising, this ignores the major theoretical developments in pseudodifferential operator theory and transform methods developed by mathematicians over the past 50 years. There are many technical reasons for anticipating that imaging methods based on pseudodifferential operators and related machinery will have advantages over existing methods.

Theoretical analysis in wavefield extrapolation:

The work already done has resulted in analytical expressions for four different wavefield extrapolators (of varying degrees of approximation) and points to clear paths to other (higher) forms. All four are Fourier integral operators and, though all four are approximate in some sense, any one is easily shown to more accurate than the best possible finite-difference or pseudo-spectral method. We have implemented these operators numerically in a variety of ways including: direct computation of the Fourier integral, calculation of the equivalent singular integral, and conversion to a Gabor filter. Recently we have developed an exact solution for the 2D discrete case using an eigenvalue decomposition in the Fourier domain.  This is useful to compare with the approximate forms. However, much remains to be done including:

• Establishing a firm theoretical basis for these extrapolators.Defining the relationship between the new extrapolators and established techniques such as the finite difference, pseudo spectral, and phase screen methods.
• Development of higher order approximations.
• Extension to two-way extrapolators (i.e. develop coupling between the forward and back scattered wavefields.)
• Extension to elastic media.

Ultimately, it is hoped that this work will lead to a methodology for generating approximate solutions to variable-coefficient partial differential equations that can be easily communicated to the applied science community. Something much like the established Fourier methods (i.e. separation of variables) that appear in mathematical physics texts is envisioned.

Creation of controlled test datasets:

This refers to using both numerical and physical modelling techniques to create high-fidelity seismic recordings from complex media. The numerical methods could include high-order pseudo-differential operator techniques or an established method such as finite differences. (However, it is quite difficult to maintain the necessary fidelity at high frequencies with finite differences.) Physical modelling, for which CREWES operates a facility, refers to the process of using transducers to acquire high frequency seismic data over scale models of complex media. Both numerical and physical modelling techniques are strongly challenged to produce data with sufficient fidelity to test the Fourier integral operator techniques. Thus both must be improved.

Numerical implementation:

We have completed the initial development of a seismic imagine engine that runs in parallel on a Linux cluster and are currently testing.  Our implementation program used simultaneous code development in Parallel Matlab and C/MPI and this greatly accelerated the process.  We hired two very skilled programmers from industry and they estimate that we accomplished several years worth of typical development in only two months.  We now have a world-class 3D imaging facility that uses recursive wavefield extrapolation as its fundamental technique.  It will serve as a foundation for further research and will allow us to quickly test new ideas of real datasets.

Testing of methods:

This stage will involve the use of sophisticates synthetics created within the project and real datasets from industry. It will involve working closely with industry scientists to ensure that realistic problems are addressed and meaningful results are achieved. It will be important to statistically assess the correlation between computed results and control data. In the case of the controlled test datasets, this means that computed seismic images will be statistically compared to known models. For real data, this involves comparison to (typically sparse) well control.