Analytic continuation of perturbative solutions of acoustic 1D wave equation by means of Padé approximants

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

The forward scattering series is an important and useful tool in constructing perturbative solutions to wave equation and understanding their relationship to their non-perturbative counterparts. When it converges, the series describes the total wavefield everywhere in a given medium as propagations in a reference medium and interactions with point scatterers. The method can be viewed as constructing a mapping between non-perturbative solutions of wave events and their volume point scatterer description. This mapping was shown to be required by the recently developed techniques for inverse problems based on the inverse scattering series with applications to seismic exploration (Weglein, A.B., Gasparotto, F.A., Carvalho, P.M. and Stolt, R.H., 1997, An inverse scattering series method for attenuating multiples in seismic reflection data. Geophysics, 62, 1975--1989, Weglein, A.B., Araujo, F.V., Carvalho, P.M., Stolt, R.H., Matson, K.H., Coates, R., Corrigan, D., Foster, D.J., Shaw, S.A. and Zhang, H., 2003, Inverse scattering series and seismic exploration. Topical Review Inverse Problems, 19, R27--R83). The forward scattering series for a 1D acoustic medium and a normal incidence plane wave was shown in Matson, K.H., 1996, The relationship between scattering theory and the primaries and multiples of reflection seismic data. J. Seis. Expl., 5, 63--78 to converge for a ratio less than (Formula presented.) between the reference and the actual velocity. Same restricted convergence was obtained in Innanen, K.H., 2003, Methods for the treatment of acoustic and absorbtive/dispersive wavefield measurements, PhD Thesis, Department of Earth and Ocean Sciences, University of British Columbia, Vancouver, Canada for a visco-acoustic medium with or without dispersion. In this article, we propose an explanation for this divergence and an extension of the method able to construct the solution of the 1D wave equation for any velocity contrast between the actual and the reference medium for both acoustic and visco-acoustic cases. The method involves the analytic continuation of the forward scattering solution by computing a certain sequence of Padé approximants to the partial sums of the forward scattering series.

Original languageEnglish
Pages (from-to)41-58
Number of pages18
JournalApplicable Analysis
Volume86
Issue number1
DOIs
StatePublished - Jan 2007

Fingerprint

Analytic Continuation
Acoustic Waves
Forward scattering
Wave equations
Wave equation
Acoustics
Acoustic waves
Series
Scattering
Inverse Scattering
Inverse problems
Geophysics
Inverse Problem
Converge
Scattering Theory
Earth (planet)
Partial Sums
Plane Wave
Ocean
Incidence

Keywords

  • 2000 Mathematics Subject Classifications: 35J05
  • 34E10
  • 35P25
  • 35R12
  • Acoustic wavefields
  • Analytic continuation
  • Forward scattering series
  • Padé approximants

Cite this

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title = "Analytic continuation of perturbative solutions of acoustic 1D wave equation by means of Pad{\'e} approximants",
abstract = "The forward scattering series is an important and useful tool in constructing perturbative solutions to wave equation and understanding their relationship to their non-perturbative counterparts. When it converges, the series describes the total wavefield everywhere in a given medium as propagations in a reference medium and interactions with point scatterers. The method can be viewed as constructing a mapping between non-perturbative solutions of wave events and their volume point scatterer description. This mapping was shown to be required by the recently developed techniques for inverse problems based on the inverse scattering series with applications to seismic exploration (Weglein, A.B., Gasparotto, F.A., Carvalho, P.M. and Stolt, R.H., 1997, An inverse scattering series method for attenuating multiples in seismic reflection data. Geophysics, 62, 1975--1989, Weglein, A.B., Araujo, F.V., Carvalho, P.M., Stolt, R.H., Matson, K.H., Coates, R., Corrigan, D., Foster, D.J., Shaw, S.A. and Zhang, H., 2003, Inverse scattering series and seismic exploration. Topical Review Inverse Problems, 19, R27--R83). The forward scattering series for a 1D acoustic medium and a normal incidence plane wave was shown in Matson, K.H., 1996, The relationship between scattering theory and the primaries and multiples of reflection seismic data. J. Seis. Expl., 5, 63--78 to converge for a ratio less than (Formula presented.) between the reference and the actual velocity. Same restricted convergence was obtained in Innanen, K.H., 2003, Methods for the treatment of acoustic and absorbtive/dispersive wavefield measurements, PhD Thesis, Department of Earth and Ocean Sciences, University of British Columbia, Vancouver, Canada for a visco-acoustic medium with or without dispersion. In this article, we propose an explanation for this divergence and an extension of the method able to construct the solution of the 1D wave equation for any velocity contrast between the actual and the reference medium for both acoustic and visco-acoustic cases. The method involves the analytic continuation of the forward scattering solution by computing a certain sequence of Pad{\'e} approximants to the partial sums of the forward scattering series.",
keywords = "2000 Mathematics Subject Classifications: 35J05, 34E10, 35P25, 35R12, Acoustic wavefields, Analytic continuation, Forward scattering series, Pad{\'e} approximants",
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Analytic continuation of perturbative solutions of acoustic 1D wave equation by means of Padé approximants. / Nita, Bogdan G.

In: Applicable Analysis, Vol. 86, No. 1, 01.2007, p. 41-58.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Analytic continuation of perturbative solutions of acoustic 1D wave equation by means of Padé approximants

AU - Nita, Bogdan G.

PY - 2007/1

Y1 - 2007/1

N2 - The forward scattering series is an important and useful tool in constructing perturbative solutions to wave equation and understanding their relationship to their non-perturbative counterparts. When it converges, the series describes the total wavefield everywhere in a given medium as propagations in a reference medium and interactions with point scatterers. The method can be viewed as constructing a mapping between non-perturbative solutions of wave events and their volume point scatterer description. This mapping was shown to be required by the recently developed techniques for inverse problems based on the inverse scattering series with applications to seismic exploration (Weglein, A.B., Gasparotto, F.A., Carvalho, P.M. and Stolt, R.H., 1997, An inverse scattering series method for attenuating multiples in seismic reflection data. Geophysics, 62, 1975--1989, Weglein, A.B., Araujo, F.V., Carvalho, P.M., Stolt, R.H., Matson, K.H., Coates, R., Corrigan, D., Foster, D.J., Shaw, S.A. and Zhang, H., 2003, Inverse scattering series and seismic exploration. Topical Review Inverse Problems, 19, R27--R83). The forward scattering series for a 1D acoustic medium and a normal incidence plane wave was shown in Matson, K.H., 1996, The relationship between scattering theory and the primaries and multiples of reflection seismic data. J. Seis. Expl., 5, 63--78 to converge for a ratio less than (Formula presented.) between the reference and the actual velocity. Same restricted convergence was obtained in Innanen, K.H., 2003, Methods for the treatment of acoustic and absorbtive/dispersive wavefield measurements, PhD Thesis, Department of Earth and Ocean Sciences, University of British Columbia, Vancouver, Canada for a visco-acoustic medium with or without dispersion. In this article, we propose an explanation for this divergence and an extension of the method able to construct the solution of the 1D wave equation for any velocity contrast between the actual and the reference medium for both acoustic and visco-acoustic cases. The method involves the analytic continuation of the forward scattering solution by computing a certain sequence of Padé approximants to the partial sums of the forward scattering series.

AB - The forward scattering series is an important and useful tool in constructing perturbative solutions to wave equation and understanding their relationship to their non-perturbative counterparts. When it converges, the series describes the total wavefield everywhere in a given medium as propagations in a reference medium and interactions with point scatterers. The method can be viewed as constructing a mapping between non-perturbative solutions of wave events and their volume point scatterer description. This mapping was shown to be required by the recently developed techniques for inverse problems based on the inverse scattering series with applications to seismic exploration (Weglein, A.B., Gasparotto, F.A., Carvalho, P.M. and Stolt, R.H., 1997, An inverse scattering series method for attenuating multiples in seismic reflection data. Geophysics, 62, 1975--1989, Weglein, A.B., Araujo, F.V., Carvalho, P.M., Stolt, R.H., Matson, K.H., Coates, R., Corrigan, D., Foster, D.J., Shaw, S.A. and Zhang, H., 2003, Inverse scattering series and seismic exploration. Topical Review Inverse Problems, 19, R27--R83). The forward scattering series for a 1D acoustic medium and a normal incidence plane wave was shown in Matson, K.H., 1996, The relationship between scattering theory and the primaries and multiples of reflection seismic data. J. Seis. Expl., 5, 63--78 to converge for a ratio less than (Formula presented.) between the reference and the actual velocity. Same restricted convergence was obtained in Innanen, K.H., 2003, Methods for the treatment of acoustic and absorbtive/dispersive wavefield measurements, PhD Thesis, Department of Earth and Ocean Sciences, University of British Columbia, Vancouver, Canada for a visco-acoustic medium with or without dispersion. In this article, we propose an explanation for this divergence and an extension of the method able to construct the solution of the 1D wave equation for any velocity contrast between the actual and the reference medium for both acoustic and visco-acoustic cases. The method involves the analytic continuation of the forward scattering solution by computing a certain sequence of Padé approximants to the partial sums of the forward scattering series.

KW - 2000 Mathematics Subject Classifications: 35J05

KW - 34E10

KW - 35P25

KW - 35R12

KW - Acoustic wavefields

KW - Analytic continuation

KW - Forward scattering series

KW - Padé approximants

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SN - 0003-6811

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