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Forward Scattering and Volterra Renormalization for Acoustic Wavefield Propagation in Vertically Varying Media

Part of: Geophysics Integral equations, integral transforms Miscellaneous topics - Partial differential equations Inverse problems Geophysics (general) Scattering theory Volterra integral equations Nonlinear integral equations General mathematical topics and methods in quantum theory

Published online by Cambridge University Press:  21 July 2016

Jie Yao ,
Anne-Cécile Lesage ,
Fazle Hussain  and
Donald J. Kouri
Jie Yao*
Affiliation:
Department of Mechanical Engineering, Physics, Mathematics, University of Houston, Houston, Texas 77204, USA
Anne-Cécile Lesage*
Affiliation:
Department of Mechanical Engineering, Texas Tech University, Lubbock, Texas 79409, USA
Fazle Hussain*
Affiliation:
Department of Mechanical Engineering, Texas Tech University, Lubbock, Texas 79409, USA
Donald J. Kouri*
Affiliation:
Department of Mechanical Engineering, Physics, Mathematics, University of Houston, Houston, Texas 77204, USA
*
*Corresponding author. Email addresses:[email protected] (J. Yao), [email protected] (A.-C. Lesage), [email protected] (F. Hussain), [email protected] (D. Kouri)
*Corresponding author. Email addresses:[email protected] (J. Yao), [email protected] (A.-C. Lesage), [email protected] (F. Hussain), [email protected] (D. Kouri)
*Corresponding author. Email addresses:[email protected] (J. Yao), [email protected] (A.-C. Lesage), [email protected] (F. Hussain), [email protected] (D. Kouri)
*Corresponding author. Email addresses:[email protected] (J. Yao), [email protected] (A.-C. Lesage), [email protected] (F. Hussain), [email protected] (D. Kouri)
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Abstract

We extend the full wavefield modeling with forward scattering theory and Volterra Renormalization to a vertically varying two-parameter (velocity and density) acoustic medium. The forward scattering series, derived by applying Born-Neumann iterative procedure to the Lippmann-Schwinger equation (LSE), is a well known tool for modeling and imaging. However, it has limited convergence properties depending on the strength of contrast between the actual and reference medium or the angle of incidence of a plane wave component. Here, we introduce the Volterra renormalization technique to the LSE. The renormalized LSE and related Neumann series are absolutely convergent for any strength of perturbation and any incidence angle. The renormalized LSE can further be separated into two sub-Volterra type integral equations, which are then solved noniteratively. We apply the approach to velocity-only, density-only, and both velocity and density perturbations. We demonstrate that this Volterra Renormalization modeling is a promising and efficient method. In addition, it can also provide insight for developing a scattering theory-based direct inversion method.

Keywords

Acoustic modelingscattering theoryVolterra renormalizationvelocity and density variation

MSC classification

Secondary: 35R30: Inverse problems 45D05: Volterra integral equations 45G10: Other nonlinear integral equations 45Q05: Inverse problems 65R20: Integral equations 81Q15: Perturbation theories for operators and differential equations 81U40: Inverse scattering problems 86A15: Seismology
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 2 , August 2016 , pp. 353 - 373
Copyright
Copyright © Global-Science Press 2016 

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References

[1] Aki, K. and Richards, P.. Quantitative Seismology: Theory andMethods. Number v. 1 in A Series of books in geology. W. H. Freeman, 1980.Google Scholar
[2] De Wolf, D.. Electromagnetic reflection from an extended turbulent medium: Cumulative forward-scatter single-backscatter approximation. Antennas and Propagation, IEEE Transactions on, 19(2):254–262, Mar 1971.CrossRefGoogle Scholar
[3] De Wolf, D. A.. Renormalization of EM fields in application to large-angle scattering from randomly continuous media and sparse particle distributions. Antennas and Propagation, IEEE Transactions on, 33(6):608–615, Jun 1985.CrossRefGoogle Scholar
[4] Innanen, K. A.. A direct nonlinear inversion of primary wave data reflecting from extended, heterogeneous media. Inverse Problems, 24(3):035021, 2008.CrossRefGoogle Scholar
[5] Innanen, K. A.. Born series forward modelling of seismic primary and multiple reflections: an inverse scattering shortcut. Geophysical Journal International, 177(3):1197–1204, 2009.Google Scholar
[6] Jost, R. and Kohn, W.. Construction of a potential from a phase shift. Phys. Rev., 87:977–992, 1952.Google Scholar
[7] Keys, R. G.. Polarity reversals in reflections from layered media. GEOPHYSICS, 54(7):900–905, 1989.CrossRefGoogle Scholar
[8] Kouri, D. J. and Vijay, A.. Inverse scattering theory: Renormalization of the Lippmann-Schwinger equation for acoustic scattering in one dimension. Phys. Rev. E, 67:046614, Apr 2003.Google Scholar
[9] Lesage, A., Yao, J., Eftekhar, R., Hussain, F., and Kouri, D.. Inverse acoustic scattering series using the Volterra renormalization of the Lippmann-Schwinger equation. SEG Expanded Abstracts, 2013.Google Scholar
[10] Lesage, A.-C., Yao, J., Bodmann, B., Kouri, D. J., and Hussain, F.. Data partionning method for convergent Volterra Inverse Scattering Series, chapter 245, pages 1274–1279.Google Scholar
[11] Lesage, A.-C., Yao, J., Hussain, F., Wijesinghe, N., and Kouri, D. J.. Multi-dimensional Inverse acoustic scattering series using the Volterra renormalization of the Lippmann-Schwinger equation, chapter 597, pages 3118–3122. 2014.Google Scholar
[12] Matson, K. H.. An inverse scattering series method for attenuating elastic multiples from multicomponent land and ocean bottom seismic data. PhD thesis, University of British Columbia, 1997.Google Scholar
[13] Morse, P. and Feshbach, H.. Methods of theoretical physics. Number 2 in International series in pure and applied physics. McGraw-Hill, 1953.Google Scholar
[14] Moses, H. E.. Calculation of the scattering potential from reflection coefficients. Phys. Rev., 102:559–567, 1956.CrossRefGoogle Scholar
[15] Newton, R. G.. Scattering theory of waves and particles. Springer-Verlag, New York, 1982.Google Scholar
[16] Nita, B.G.. Forward scattering series and padè approximants for acoustic wavefield propagation in a vertically varying medium. Communications in Computational Physics, 3(1):180–202, 2008.Google Scholar
[17] Prosser, R. T.. Formal solutions of inverse scattering problems. Journal of Mathematical Physics, 10(10):1819–1822, 1969.CrossRefGoogle Scholar
[18] Prosser, R. T.. Formal solutions of inverse scattering problems. v. Journal of Mathematical Physics, 33(10):3493–3496, 1992.Google Scholar
[19] Ramirez, A. C. and Otnes, E.. Forward scattering series for 2-parameter acoustic media: Analysis and implications to the inverse scattering task specific subseries. Communications in Computational Physics, 3:136–159, 2008.Google Scholar
[20] Razavy, M.. Determination of the wave velocity in an inhomogeneous medium from the reflection coefficient. J. Acoust. Soc. Am., 58:956, 1975.Google Scholar
[21] Sams, W. N. and Kouri, D. J.. Noniterative solutions of integral equations for scattering. i. single channels. The Journal of Chemical Physics, 51(11):4809–4814, 1969.Google Scholar
[22] Shaw, S. A.. PhD thesis, University of Houston, 2005.Google Scholar
[23] Sifuentes, J.. Preconditioned Iterative Methods for Inhomogeneous Acoustic Scattering Applications. PhD thesis, Rice University, 2010.Google Scholar
[24] Weglein, A., Gasparotto, F., Carvalho, P., and Stolt, R.. An inverse scattering series method for attenuating multiples in seismic reflection data. Geophysics, 62(6):1975–1989, 1997.Google Scholar
[25] Weglein, A. B. and S. F., An inverse-scattering sub-series for predicting the spatial location of reflectors without the precise referencemedium and wave velocity. SEG Expanded Abstracts, 2001.Google Scholar
[26] Weglein, A. B., Matson, K. H., and Foster, D. J.. Imaging and inversion at depth without a velocity model: theory, concepts and initial evaluation. SEG Expanded Abstracts, 2000.CrossRefGoogle Scholar
[27] Weglein, A. B., Nita, B. G., and Matson, K. H.. Forward scattering series and seismic events: Far field approximations, critical and postcritical events. SIAM Journal on Applied Mathematics, 64(6):2167–2185, 2004.Google Scholar
[28] Weglein, A. B., Araújo, F. V., Carvalho, P. M., Stolt, R. H., Matson, K. H., Coates, R. T., C. D., , F. D. J., , S. S. A., , and Z. H., Inverse scattering series and seismic exploration. Inverse Problems, 19(6):R27, 2003.Google Scholar
[29] Wu, R. and Wang, B.. Inverse thin-slab propagator based on forward-scattering renormalization for wave-equation tomography. In 77th EAGE Conference and Exhibition 2015, 2015.Google Scholar
[30] Wu, R.-S.. Wave propagation, scattering and imaging using dual-domain one-way and one-return propagators. In Ben-Zion, Y., editor, Seismic Motion, Lithospheric Structures, Earthquake and Volcanic Sources: The Keiiti Aki Volume, Pageoph Topical Volumes, pages 509–539. Birkhäuser Basel, 2003.CrossRefGoogle Scholar
[31] Wu, R.-S. and Huang, L.. Reflected wave modeling in heterogeneous acoustic media using the de wolf approximation. Proc. SPIE, 2571:176–186, 1995.CrossRefGoogle Scholar
[32] R.-S.Wu, , Xie, X.-B., and Jin, S.. One-return propagators and the applications inmodeling and imaging. In Imaging,Modeling and Assimilation in Seismology, chapter 2, pages 65–105. 2012.Google Scholar
[33] Wu, R.-S., Xie, X.-B., and Wu, X.-Y.. One-way and one-return approximations (de wolf approximation) for fast elastic wave modeling in complex media. In Ru-Shan Wu, V. M. and Dmowska, R., editors, Advances in Wave Propagation in Heterogenous Earth, volume 48 of Advances in Geophysics, pages 265–322. Elsevier, 2007.Google Scholar
[34] Wu, R.-S. and Zheng, Y.. Non-linear partial derivative and its de wolf approximation for non-linear seismic inversion. Geophysical Journal International, 2014.Google Scholar
[35] Yao, J., Lesage, A.-C., Bodmann, B. G., Hussain, F., and Kouri, D. J.. Inverse scattering theory: Inverse scattering series method for one dimensional non-compact support potential. Journal of Mathematical Physics, 55(12):123512, 2014.CrossRefGoogle Scholar
[36] Yao, J., Lesage, A.-C., Bodmann, B. G., Hussain, F., and Kouri, D. J.. One dimensional acoustic direct nonlinear inversion using the volterra inverse scattering series. Inverse Problems, 30(7):075006, 2014.Google Scholar
[37] Zhang, H. and Weglein, A.. Direct nonlinear inversion of 1d acoustic media using inverse scattering subseries. Geophysics, 74(6):WCD29–WCD39, 2009.Google Scholar