Geo Mathematical Imaging Group, Purdue University

Purdue University Mark
Maarten de Hoop

Maarten de Hoop
Director, GMIG
mdehoop@purdue.edu

Dept of Mathematics
Purdue University
150 N Univ St
West Lafayette, IN 47907
ph. (765) 496-7678
Fax (765) 496-1169


Barbara Doremire
Assistant to the Director
bjd@purdue.edu

Geometric Methods




In the category 'Geometric Methods', we exploit high frequency data, and are concerned with developing imaging and reconstruction procedures using techniques from Riemannian geometry and symplectic geometry. Results of our research program include:

  • We introduced extended isochron rays and an associated (single) evolution equation for extended imaging valid in the presence of caustics; this equation is tied to our program in wave-equation migration velocity analysis (MVA).

  • We obtained a curvilinear DSR condition which guarantees artifact-free extended imaging in the presence of caustics also for velocity inversion.

  • We developed the notion of velocity continuation in extended imaging in the presence of caustics in terms of a canonical transformation associated with an evolution equation.

  • We used surface-to-surface propagator matrices for anisotropic media to obtain expressions that relate reflector dip and curvature to first and second derivatives of the time-migration reflection time with respect to image point coordinates. The curvatures play a role in data reconstruction using wave packets, and velocity inversion using selected reflectors, but also provide structural geological information. We obtained a formulation of finite-offset time migration and time demigration in heterogeneous, anisotropic velocity models. We developed an approach to estimate the time-migration velocity matrix (or ellipse) which is directly related to the so-called shape operator in Riemannian geometry.

  • We generalized Dix and arrived at a direct (nonlinear) reconstruction of a Riemannian metric (velocity) using diffraction time expansions, closely related to the mentioned shape operator, as the data. The formulation is in terms of Riemannian curvature, makes use of Riemannian normal coordinates and the relationship between Jacobi fields and Riccati equations. The reconstruction involves higher-order time derivatives of the time-migration velocity matrix.

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    Geo-Mathematical Imaging Group, Purdue University
    150 N University Street, West Lafayette, IN 47907 USA     Phone: (765) 496-7678 - Fax: (765) 496-1169
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