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

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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
Issue number1
StatePublished - Jan 2007


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


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