Abstract
Recently, de Hoop and coworkers developed an asymptotic, seismic inversion formula for application in complex environments supporting multi-pathed and multi-mode wave propagation (DE HooP et al., 1999; DE HooP and BRANDSBERG-DAHL, 2000; STOLK and DE HooP, 2000). This inversion is based on the Born/Kirchhoff approximation, and employs the global, uniform asymptotic extension of the geometrical method of “tracing rays” to account for caustic phenomena. While this approach has successfully inverted the multicomponent, ocean-bottom data from the Valhall field in Norway, accounting for severe focusing effects (DE HooP and BRANDSBERG-DAHL, 2000), it is not able to account properly for wave phenomena neglected in the “high-frequency” limit (i.e., diffraction effects) and strong scattering effects. To proceed further and incorporate wave effects in a nonlinear inversion scheme, the theory of directional wavefield decomposition and the construction of the generalized Bremmer coupling series are combined with the application of modern phase space and path (functional) integral methods to, ultimately, suggest an inversion algorithm which can be interpreted as a method of “tracing waves.” This paper is intended to provide the seismic community with an introduction to these approaches to direct and inverse wave propagation and scattering, intertwining some of the most recent new results with the basic outline of the theory, and culminating in an outline of the extended, asymptotic, seismic inversion algorithm. Modeling at the level of the fixed-frequency (elliptic), scalar Helmholtz equation, exact and uniform asymptotic constructions of the well-known, and fundamentally important, square-root Helmholtz operator (symbol) provide the most important results.
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References
De Hoop, M. V. (1996), Generalization of the Bremmer Coupling Series, J. Math. Phys. 37, 3246–3282.
DE Hoop, M. V., Spencer, C., and Burridge, R. (1999), The Resolving Power of Seismic Amplitude Data: An Anisotropic Inversion’ Migration Approach, Geophys. 64, 852–873.
DE Hoop, M. V. and Brandsberg-Dahl, S. (2000), Maslov Asymptotic Extension of Generalized Radon
Transform Inversion in Anisotropie Elastic Media: A Least-squares Approach, Inverse Probs., 519–562.
DE Hoop, M. V. and Gautesen, A. K. (2000), Uniform Asymptotic Expansion of the Generalized Bremmer Series, SIAM J. Appl. Math. 60, 1302–1329.
DE Hoop, M. V., Fishman, L., and Jonsson, B. L. G. (2002), Acoustic Time-reversal Mirrors in the Framework of One-way Wave Theories, manuscript submitted for publication.
Fishman, L., Mccoy, J. J., and Wales, S. C. (1987), Factorization and Path Integration of the Helmholtz Equation: Numerical Algorithms, J. Acoust. Soc. Am. 81, 1355–1376.
Fishman, L. and Wales, S. C. (1987), Phase Space Methods and Path Integration: The Analysis and Computation of Scalar Wave Equations, J. Comp. Appl. Math. 20, 219–238.
Fishman, L. (1992), Exact and Operator Rational Approximate Solutions of the Helmholtz, Weyl Composition Equation in Underwater Acoustics-The Quadratic Profile, J. Math. Phys. 33, 1887–1914.
Fishman, L. (1993), One-way Wave Propagation Methods in Direct and Inverse Scalar Wave Propagation Modeling, Radio Sci. 28, 865–876.
Fishman, L., Gautesen, A. K., and Sun, Z. (1997), Uniform High frequency Approximations of the Square-root Helmholtz Operator Symbol, Wave Motion 26, 127–161.
Fishman, L., DE Hoop, M. V., and Van Stralen, M. J. N. (2000), Exact Constructions of Square-root Helmholtz Operator Symbols: The Focusing Quadratic Profile, J. Math. Phys. 41, 4881–4938.
Jordan, P. M. (2000), Constructions of Helmholtz Operator Symbols for Three-layer Composite Media Using the MZB Rational Square-root Approximation, unpublished.
Jordan, P. M. (2001), Comment on “Exact Constructions of Square-root Helmholtz Operator Symbols: The Focusing Quadratic Profile” [ J. Math. Phys. 41, 4881 (2000)], J. Math. Phys. 42, 4618–4623.
Lu, Y. Y. and Mclaughlin, J. R. (1996), The Riccati Method for the Helmholtz Equation, J. Acoust. Soc. Am. 100, 1432–1446.
Lu, Y. Y. (1999), One-way Large Range Step Methods for Helmholtz Waveguides, J. Comp. Phys. 152, 231–250.
Lu, Y. Y., Huang, J., and Mclaughlin, J. R. (2001), Local Orthogonal Transformation and One-way Methods for Acoustic Waveguides, Wave Motion, 34, 193–207.
Stolk, C. C. and De Hoop, M. V. (2000), Microlocal Analysis of Seismic Inverse Scattering in Anisotropic, Elastic Media, Center for Wave Phenomena, preprint.
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Fishman, L. (2002). Applications of Directional Wavefield Decomposition, Phase Space, and Path Integral Methods to Seismic Wave Propagation and Inversion. In: Pšenčík, I., Červený, V. (eds) Seismic Waves in Laterally Inhomogeneous Media. Pageoph Topical Volumes. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8146-3_13
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DOI: https://doi.org/10.1007/978-3-0348-8146-3_13
Publisher Name: Birkhäuser, Basel
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