Many physical processes can be modelled by a linear partial differential operator. Classically, this corresponds to a convolution operation, or equivalently, the application of a Fourier filter. The scope of such operators is limited to the stationary stetting, but physical processes are never stationary. Part of our ob jective is to consider two ways to extend the Fourier theory: (1) via Gabor analysis, and (2) via pseudodifferential operator theory. The main emphasis is on the former, which we deem general enough to handle most, if not all, linear physical processes.
Two important seismic inverse problems, each modelled by a pseudodifferential operator, can be solved successfully via a Gabor filtering approach: (1) seismic deconvolution for a dissipative earth; and (2) wavefield extrapolation in an inhomogeneous earth.
The first solution relies on the existence of a well-defined concept of nonstationary minimum phase in the analog setting; the second can be overly computationally intensive, owing to excessive redundancies in the Gabor frame. Both of these issues can be addressed, respectively, by: (1) extending the idea of minimum phase to the class of tempered distributions; and (2) developing an adaptive, nonuniform sampling scheme to minimize the redundancy of the Gabor transform, while simultaneously respecting the inherent nonstationarity of the problem.
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