Numerical Comparison of SVD and Propagator/Reflectivity Decomposition for the Acoustic Wave Equation
نویسندگان
چکیده
In acoustic seismic-reeection experiments, because of the lack of low frequencies in the usual seismic sources, the reeection of the energy back to the surface is associated to the short wavelengths of the slowness (the reeectivityy parameters), while the long wavelengths of the slowness (the propagatorr parameters) are associated to the kinematics of the arrival times of this energy. Such a decomposition between the propagator and reeectivity parameters is automatic when a linearized model is used, but can only approximately be satissed 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 reeectivity unknowns. This paper is devoted to the mathematical justiication of the previous propagator/reeectivity 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/reeectivity decomposition is associated to a truncated SVD. Comparaison nummrique de la ddcomposition en valeurs singuliires et de la ddcomposition propagateur/rrrectivitt pour l''quation des ondes acoustiques RRsumm : Lors d'expriences de sismique-rrrexion acoustique, en raison du faible contenu basse-frrquence des sources sismiques courantes, la rrrexion de l''nergie vers la surface est associie aux courtes longueurs d'ondes de la lenteur (les parammtres rrrectivittt), alors que les grandes longueurs d'ondes de la lenteur (les parammtres propagateurr) sont associies la cinnmatique des temps d'arrives de cette nergie. Une telle ddcomposition entre les parammtres propagateur et rrrectivitt est automatique quand un moddle linnariss est utiliss, mais peut seulement tre satisfaite de faaon approchhe dans le cas du moddle acoustique complet (non linnariss). Dans le cas 2D, la suite embootte des sous-espaces construits par interpolation bilinnaire aux nnuds de sous-grilles pourvoit l'espace des lenteurs d'une base hiirarchique qui permet une analyse multiichelle simple et mmne au choix de deux sous-espaces orthogonaux pour les inconnues propagateur et rrrectivitt. Cet article est ddvolu la justiication mathhmatique de la ddcomposition propagateur/rrrectivitt prrccdente au moyen d'une analyse par ddcomposition en valeurs singuliires de la ddrive de Frechet de l'oprateur de moddlisation associant les donnnes de surface la lenteur dans le cas de l''quation des ondes acoustiques. Il est montrr …
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