A Preliminary Report on Some Recent Results in Born Inversion.
COLORADO SCHOOL OF MINES GOLDEN CENTER FOR WAVE PHENOMENA
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A research group at the Center for Wave Phenomena, Department of Mathematics, Colorado School of Mines, has been developing inversion algorithms for progressively more complex background velocities and source-receiver configurations. In each extension, the crucial issue has been the determination of certain properties of a matrix involving derivatives of the travel times. This report discusses and extends a recent result along these lines which appeared in a recent paper by Gregory Beylkin reduces the problem to consideration of a single canonical determinant, h, and then assumes that this determinant does not vanish. With this assumption, he establishes a very general inversion result. Consequently, future theoretical research will focus on the evaluation of h, on establishing conditions for its non-vanishing and on dealing with the phenomena arising when it does vanish. Beylkins paper uses powerful mathematical tools, such as the notions of pseudo-differential operators, generalized Radon transforms, and generalized back projections. Moreover, Beylkin frames his work in an N-dimensional space. Here, we dispense with much of this mathematical machinery and for convenience confine ourselves to the 3D case and its 2.5D specialization. We are able to espound Beylkins results by an approach similar to that presented earlier by Cohen and Hagin. However, we do not attempt to rigorously prove our results, but instead content ourselves with an intuitive derivation and the citing of Beylkins main theorem.
- Theoretical Mathematics