Fourier integral operators. II

Duistermaat, J. J.; Hörmander, Lars

doi:10.1007/bf02392165

Cited by 571 publications

(470 citation statements)

References 15 publications

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“…and using ∂ α S(α; x, η(α; x, ∂ x S(α; x, η))) = ∂ α S(α; x, η), we recover (39) and thus get (41). Since S is homogeneous of degree 1 in η, p 1 is a symbol of order 1.…”

Section: Methods 2 Evolution Operator From 'Migration' Operatormentioning

confidence: 79%

Evolution-equation approach to seismic image, and data, continuation

2008

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In reflection seismology one places point sources and point receivers on the earth's surface, forming an acquisition geometry. Each source generates acoustic waves in the subsurface, that are reflected where the medium properties vary discontinuously. The reflections that can be observed at the receivers are used to image these discontinuities or reflectors assuming a background medium. We analyze methods to circumvent the repeated imaging of reflectors under varying background media, or the repeated modelling of reflections under varying acquisition geometries. These methods involve the introduction of the notion of seismic continuation. Here, we develop the foundation of, and a comprehensive framework for seismic continuation while extending earlier approaches to allow for the formation of caustics. Traditionally, seismic continuation has been viewed from a geometrical (ray asymptotic) point of view; here, we introduce the notion of wave-equation continuation through the appearance of evolution equations.

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“…and using ∂ α S(α; x, η(α; x, ∂ x S(α; x, η))) = ∂ α S(α; x, η), we recover (39) and thus get (41). Since S is homogeneous of degree 1 in η, p 1 is a symbol of order 1.…”

Section: Methods 2 Evolution Operator From 'Migration' Operatormentioning

confidence: 79%

Evolution-equation approach to seismic image, and data, continuation

2008

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“…By Egorov's theorem, one may conjugate the operator via an invertible Fourier integral operator to the operator D xn . The construction of microlocal parametrices for operators of real principal type and the fact that singularities of solutions of these equations propagate along bicharacteristics follow directly from this result ( [10]). …”

Section: Historical Background To the Problemmentioning

confidence: 91%

Parametrices for symmetric systems with multiplicity

Nolan

Uhlmann

2007

Wave Motion

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“…OPERATEURS DE TYPE PRINCIPAL § 2*1 Le cas où p est réel On sait [4] que, si H n'est pas collinéaire à l'axe du cône S ^4 K5l» l'opérateur P de symbole p est "équivalent" à -c-, et qu'on d^l J 'J/ ^n peut construire deux paramétrixes propageant l'une le long des bicaractéristiques positives, l'autre le long des bicaractéristiques négatives.…”

Section: Rappels Sur La Methode De Melin-sjostrandunclassified

Quelques exemples d'operateurs pseudodifferentiels localement resolubles

Helffer

1978

Equations Aux Dérivées Partielles

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Fourier integral operators. II

Cited by 571 publications

References 15 publications

Evolution-equation approach to seismic image, and data, continuation

Evolution-equation approach to seismic image, and data, continuation

Parametrices for symmetric systems with multiplicity

Quelques exemples d'operateurs pseudodifferentiels localement resolubles

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