Generalized frames for Gabor operators in imaging

Michael P. Lamoureux, Gary F. Margrave and Peter C. Gibsony

ABSTRACT

In numerical wavefield propagation, it is useful to decompose a complex geological region into small local regions of nearly constant velocity, and propagate pieces of the wavefield through each region separately. The total wavefield is then obtained by reassembling all the pieces.

We show here how this decomposition/reassembling is captured mathematically using a windowing procedure which is accurately described by so-called generalized frames. By applying frame theory, we show that a collection of local wavefield propagators combined via a suitable partition of unity, remains a stable propagator, which is a highly desirable property in numerical simulations. These results apply more generally to combinations of linear operators that are useful for many nonstationary filtering operations.

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