Forward Scattering Series and Seismic Events: Far Field Approximations, Critical and Postcritical Events

Inverse scattering series is the only nonlinear, direct inversion method for the multidimensional, acoustic or elastic equation. Recently developed techniques for inverse problems based on the inverse scattering series [Weglein et al., Geophys., 62 (1997), pp. 1975-1989; Top. Rev. Inverse Problems,...

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Bibliographic Details
Published inSIAM journal on applied mathematics Vol. 64; no. 6; pp. 2167 - 2185
Main Authors Weglein, Arthur B., Nita, Bogdan G., Matson, Kenneth H.
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.01.2004
Subjects
Acoustics
Approximation
Coefficients
Experiments
Forward scattering
Integrands
Inverse problems
Inverse scattering
Propagation
Saddle points
Series convergence
Taylor series
Wave propagation
Wave reflection
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Summary:Inverse scattering series is the only nonlinear, direct inversion method for the multidimensional, acoustic or elastic equation. Recently developed techniques for inverse problems based on the inverse scattering series [Weglein et al., Geophys., 62 (1997), pp. 1975-1989; Top. Rev. Inverse Problems, 19 (2003), pp. R27-R83] were shown to require two mappings, one associating nonperturbative description of seismic events with their forward scattering series description and a second relating the construction of events in the forward to their treatment in the inverse scattering series. This paper extends and further analyzes the first of these two mappings, introduced, for 1D normal incidence, in Matson [J. Seismic Exploration, 5 (1996), pp. 63-78] and later extended to two dimensions in Matson [An Inverse Scattering Series for Attenuating Elastic Multiples from Multicomponent Land and Ocean Bottom Seismic Data, Ph.D. thesis, Department of Earth and Ocean Sciences, University of British Columbia, Vancouver, BC, Canada, 1997]. It brings a new and more rigorous understanding of the mathematics and physics underlying the calculation of terms in the forward scattering series and the events in the seismic model. The convergence of the series for ID acoustic models is examined, and the earlier precritical analysis is extended to critical and postcritical reflections. An explanation is proposed for the divergence of the series for postcritical incident planewaves.
ISSN:0036-1399
1095-712X
DOI:10.1137/S0036139903435619