A microlocal analysis of migration

Kroode, A. P. E. ten; Smit, Dirk; Verdel, Arie

doi:10.1016/s0165-2125(98)00004-3

Cited by 74 publications

(69 citation statements)

References 6 publications

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“…We assume that F is a Fourier integral operator (FIO) -this assumption is commonly satisfied in seismic data applications [28][29][30][31][32]. Then F * is an FIO as well.…”

Section: Representations Of 'Migration' and 'Demigration' Operatorsmentioning

confidence: 99%

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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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“…We assume that F is a Fourier integral operator (FIO) -this assumption is commonly satisfied in seismic data applications [28][29][30][31][32]. Then F * is an FIO as well.…”

Section: Representations Of 'Migration' and 'Demigration' Operatorsmentioning

confidence: 99%

“…[24][25][26][27]. Modelling and imaging of seismic reflection data can be mathematically described in terms of Fourier integral operators (FIOs) [28][29][30][31][32]. The basic properties of FIOs are summarized in Section 2.…”

Section: Introductionmentioning

confidence: 99%

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

2008

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“…Hansen [56] analyzed the inversion in an acoustic medium with multipathing for both the Born approximation and the case of a smooth jump. Ten Kroode et al [99] extended the work of Hansen. Guillemin [51] discussed the Bolker condition in the context of generalized Radon transforms, that ensures invertibility of the modeling operator in the least-squares sense (see, e.g., De Hoop and Brandsberg-Dahl [33] and Stolk and De Hoop [91]).…”

Section: Synopsis Of Publicationsmentioning

confidence: 99%

“…Because Λ M N is a canonical relation that projects submersively (under Assumption 1) on the subsurface variables (x, ξ), the projection of (3-10) on T * Y \ 0 is immersive [62,Lemma 25.3.6 and (25.3.4)]. Therefore only the injectivity in the assumption need be verified [99]. In fact, it is precisely the injectivity condition that has been tacitly assumed in what seismologists call map migration.…”

Section: Linearized (Born) Inversionmentioning

confidence: 99%

“…In the context of the discussion in the previous subsection, we call these solutions the reciprocal-ray solutions: the ray originating at s and terminating at r coincides with the ray originating at r and terminating at s. These are the only solutions under the Bolker condition as discussed in Theorem 4.3. It was noted that then L and M intersect cleanly with excess e = n − 1 − c. In the more general situation when there are solutions with t M N = 0 (that correspond with the violation of travel time injectivity [99]) the normal operator will attain a contribution that is in general not microlocal (corresponding to nonreciprocal-ray solutions). To ensure that this contribution can be represented by a Fourier integral operator, we invoke the condition that L and M still intersect transversally (cleanly with excess 0).…”

Section: Phantom Images Artifacts Of Type Imentioning

confidence: 99%

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Microlocal analysis of seismic inverse scattering in anisotropic elastic media

Stolk¹,

Hoop²

2001

Comm Pure Appl Math

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Abstract. We review applications of microlocal analysis (MA) to reflection seismology. In this inverse method one attempts to estimate the index of refraction of waves in the earth from seismic data measured at the Earth's surface. Seismic imaging creates images of the Earth's upper crust using seismic waves generated by artificial sources and recorded into extensive arrays of sensors (geophones or hydrophones). The technology is based on a complex, and rapidly evolving, mathematical theory that employs advanced solutions to a wave equation as tools to solve approximately the general seismic inverse problem, with complications introduced by the heterogeneity and anisotropy of the Earth's crust. We describe several important developments using MA to generate these wave-solutions by manipulating the wavefields directly on their phase space. We also consider some recent applications of MA to global seismology.

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Inversion Formulas in Inverse Scattering

Nolan¹

2015

Encyclopedia of Applied and Computational Mathematics

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A microlocal analysis of migration

Cited by 74 publications

References 6 publications

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

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

Microlocal analysis of seismic inverse scattering in anisotropic elastic media

Inversion Formulas in Inverse Scattering

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