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Journal of Mathematical Physics / Volume 26 / Issue 1

J. Math. Phys. 26, 99 (1985); http://dx.doi.org/10.1063/1.526755 (10 pages)

Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform

G. Beylkin

Schlumberger–Doll Research, Old Quarry Road, Ridgefield, Connecticut 06877‐4108

(Received 8 May 1984; accepted 20 July 1984)

This paper treats the linearized inverse scattering problem for the case of variable background velocity and for an arbitrary configuration of sources and receivers. The linearized inverse scattering problem is formulated in terms of an integral equation in a form which covers wave propagation in fluids with constant and variable densities and in elastic solids. This integral equation is connected with the causal generalized Radon transform (GRT), and an asymptotic expansion of the solution of the integral equation is obtained using an inversion procedure for the GRT. The first term of this asymptotic expansion is interpreted as a migration algorithm. As a result, this paper contains a rigorous derivation of migration as a technique for imaging discontinuities of parameters describing a medium. Also, a partial reconstruction operator is explicitly derived for a limited aperture. When specialized to a constant background velocity and specific source–receiver geometries our results are directly related to some known migration algorithms.

KEYWORDS and PACS

Keywords

inverse scattering problem, transformations, integral equations, analytical solution, mathematical operators, apertures, geometry, algorithms, phase shift, images, continuity equations, perturbation theory, born approximation, fluids, density, elasticity, solids, fourier transformation, weighting functions, wave propagation

PACS

  • 02.90.+p

    Other topics in mathematical methods in physics (restricted to new topics in section 02)

  • 11.80.-m

    Relativistic scattering theory

  • 41.20.Jb

    Electromagnetic wave propagation; radiowave propagation

  • 43.35.Pt

    Surface waves in solids and liquids

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.526755

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

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