Wave Splitting and the Reflection Operator for the Wave Equation

  • V. H. Weston
Conference paper
Part of the Inverse Problems and Theoretical Imaging book series (IPTI)


The problem of wave splitting in a non-homogeneous medium (with sufficiently smooth velocity) is R 3 is considered. The wave equation is factorized into an up and down-going wave system using certain integral and integral-differential operators. The equation for the reflection operator (which relates the up-going wave to a down-going wave) is obtained, and certain properties of the reflection operator are deduced, including the ideal set of measurements needed to determine the kernel of the reflection operator. The possible application to inverse problems is considered.


Inverse Problem Wave Equation Stratify Medium Reflection Operator Dirichlet Data 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Bellman R. AND Wing G. N., (1975), “An Introduction to Invariant Imbedding,” Wiley, New York.Google Scholar
  2. [2]
    Redheffer R., (1962), On the relation of transmission-line theory to scattering and. transfer, J. Math. Phys. 41, p. 1, Cambridge, MA.Google Scholar
  3. [3]
    Corones J. P., Davison M. E., and Krueger R. J., (1983), Wave splittings, invariant imbedding and inverse scattering, edited by A. J. Devaney, Proc. SPIE, in “Inverse Optics,” SPIE, Bellingham, WA, pp. 102–106.Google Scholar
  4. [4]
    Corones J. P., Krueger R. J., and Weston V. H., (1984), Some recent results in inverse scattering theory, edited by Santosa, Poa, Symes and Holland, in “Inverse Problems of Acoustic and Elastic Waves,” S.I.A.M., Philadelphia, pp. 65–81.Google Scholar
  5. [5]
    Corones J. P., Davison, M. E., and Krueger, R. J. (1983), Direct and inverse scattering in the time domain by invariant imbedding techniques, J. Acoust. Soc. Am. 74, p. 1535.CrossRefMATHADSMathSciNetGoogle Scholar
  6. [6]
    Corones J. P. and Krueger, R. J., (1983), Obtaining Scattering Kernels Using In-Variant Imbedding, J. Math. Anal. Appl. 95, P. 393.CrossRefMATHMathSciNetGoogle Scholar
  7. [7]
    Beezley E. A. AND Krueger R. J., (1985), An electromagnetic inverse problem for dispersive media, J. Math. Phys. 26, p. 317.Google Scholar
  8. [8]
    Kristensson G. and Krueger R. J., (1986), Direct and inverse scattering in the time domains for a dissipative wave operation equation I. Scattering Operators, J. Math. Phys. 27, p. 1667.MATHADSMathSciNetGoogle Scholar
  9. [9]
    Weston V. H., (1987), Factorization of the wave equation in higher dimensions, J. Math. Phys. 28, p. 1061.CrossRefMATHADSMathSciNetGoogle Scholar
  10. [10]
    Weston V. H., (1988), Factorization of the dissipative wave equations and inverse scattering, J. Math. Phys 29, p. 2205.CrossRefMATHADSMathSciNetGoogle Scholar
  11. [11]
    Weston V. H., (1989), Wave splitting and the reflection operator for the wave equation in R3, J. Math. Phys. 30, p. 2545.CrossRefMATHADSMathSciNetGoogle Scholar
  12. [12]
    Weston V. H., (1990), Square root of a second order hyperbolic differential operator and wave-splitting, to be published.Google Scholar
  13. [13]
    Weston V. H., (1988), Factorization of the wave equation in a nonplanar medium, J. Math. Phys. 29, p. 36.CrossRefMATHADSMathSciNetGoogle Scholar
  14. [14]
    Kreider K. L., (1989), A wave-splitting approach to the time dependent inverse scattering for the stratified cylinder, S.I.A.M. J. App. Math. 49, p. 932.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • V. H. Weston
    • 1
  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA

Personalised recommendations