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Numerical Comparison of SVD and Propagator/Reflectivity Decomposition for the Acoustic Wave Equation

Abstract : In acoustic seismic-reflection experiments, because of the lack of low frequencies in the usual seismic sources, the reflection of the energy back to the surface is associated to the short wavelengths of the slowness (the «reflectivity» parameters), while the long wavelengths of the slowness (the «propagator» parameters) are associated to the kinematics of the arrival times of this energy. Such a decomposition between the propagator and reflectivity parameters is automatic when a linearized model is used, but can only approximately be satisfied in the case of the full (non linearized) acoustic model. In the 2D case, the embedded sequence of subspaces built by bilinear interpolation at the nodes of subgrids provides the slowness space with a hierarchical basis which allows a simple multiscale analysis and leads to the choice of two orthogonal subspaces for the propagator and reflectivity unknowns. This paper is devoted to the mathematical justification of the previous propagator/reflectivity decomposition by means of singular value decomposition (SVD) analysis of the Frechet derivative of the modeling operator mapping the slowness to the surface data in the case of the 2D acoustic wave equation. It is shown numerically that the propagator/reflectivity decomposition is associated to a truncated SVD.
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Valery Khajdukov, Victor Kostin, Vladimir Tcheverda, François Clément, Guy Chavent. Numerical Comparison of SVD and Propagator/Reflectivity Decomposition for the Acoustic Wave Equation. [Research Report] RR-2888, INRIA. 1996. ⟨inria-00073802⟩

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