Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bellman R. AND Wing G. N., (1975), “An Introduction to Invariant Imbedding,” Wiley, New York.
Redheffer R., (1962), On the relation of transmission-line theory to scattering and. transfer, J. Math. Phys. 41, p. 1, Cambridge, MA.
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.
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.
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.
Corones J. P. and Krueger, R. J., (1983), Obtaining Scattering Kernels Using In-Variant Imbedding, J. Math. Anal. Appl. 95, P. 393.
Beezley E. A. AND Krueger R. J., (1985), An electromagnetic inverse problem for dispersive media, J. Math. Phys. 26, p. 317.
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.
Weston V. H., (1987), Factorization of the wave equation in higher dimensions, J. Math. Phys. 28, p. 1061.
Weston V. H., (1988), Factorization of the dissipative wave equations and inverse scattering, J. Math. Phys 29, p. 2205.
Weston V. H., (1989), Wave splitting and the reflection operator for the wave equation in R3, J. Math. Phys. 30, p. 2545.
Weston V. H., (1990), Square root of a second order hyperbolic differential operator and wave-splitting, to be published.
Weston V. H., (1988), Factorization of the wave equation in a nonplanar medium, J. Math. Phys. 29, p. 36.
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.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Weston, V.H. (1990). Wave Splitting and the Reflection Operator for the Wave Equation. In: Sabatier, P.C. (eds) Inverse Methods in Action. Inverse Problems and Theoretical Imaging. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-75298-8_29
Download citation
DOI: https://doi.org/10.1007/978-3-642-75298-8_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-75300-8
Online ISBN: 978-3-642-75298-8
eBook Packages: Springer Book Archive