Abstract
We describe a rigorous layer stripping approach to inverse scattering for the Helmholtz equation in one dimension. In section 3, we show how the Ricatti ordinary differential equation, which comes from the invariant embedding approach to forward scattering, becomes an inverse scattering algorithm when combined with the principle of causality. In section 4 we discuss a method of stacking and splitting layers. We first discuss a formula for combining the reflection coefficients of two layers to produce the reflection coefficient for the thicker layer built by stacking the first layer upon the second. We then describe an algorithm for inverting this procedure; that is, for splitting a reflection coefficient into two thinner reflection coefficients. We produce a strictly convex variational problem whose solution accomplishes this splitting. Once we can split an arbitrary layer into two thinner layers, we proceed recursively until each reflection coefficients in the stack is so thin that the Born approximation holds (i.e. the reflection coefficient is approximately the Fourier transform of the derivative of the logarithm of the wave speed). We then invert the Born approximation in each thin layer.
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References
A. Bruckstein AND T. Kailath, Inverse scattering for discrete transmission-line models, SIAM Review, 29(3) (1987), 359–389.
Chen, Y. AND Rokhlin, V., On the Inverse Scattering Problem for the Helmholtz Equation in One Dimension, Inverse Problems 8, pp. 356–391 (1992)
Cheney, M., Isaacson, D., Somersalo, E., AND Isaacson, E., A layer stripping approach to impedance imaging, 7th Annual Review of Progress in Applied Computational Electromagnetics, Naval Postgraduate School, Monterrey, (1991).
J. P. Corones, R.J. Krueger, and Davison., Direct and inverse scattering in the time domain via invariant embedding equations, J. Acoust. Soc. Am. 74 (1983), 1535–1541.
H. Dym AND H. P. Mckean, “Fourier Series and Integrals”, Academic Press, New York (1972).
Somersalo, E., Layer stripping for time-harmonic Maxwell’s equations with high frequency. Inverse Problems 10 (1994), 449–466
Sylvester, J., A Convergent Layer Stripping Algorithm for the Radially Symmetric Impedance Tomography Problem, Communications in PDE 17, No.12, pp. 1955–1994 (1992)
Sylvester, J., Impedance tomography and layer stripping. Inverse problems: principles and applications in geophysics, technology, and medicine, 307–321, Math. Res., 74, Akademie-Verlag, Berlin, (Potsdam, 1993) 1993.
Sylvester, J., Winebrenner, D. AND Gylys-Colwell, F., Layer Stripping for the Helmholtz Equation, SIAM Journal of Applied Mathematics 56(3) pp. 736–754 June 1996
Sylvester, J., Winebrenner, D., Nonlinear and Linear Inverse Scattering, SIAM Journal of Applied Mathematics 59(2) pp. 669–699 April 1999
Symes, W. W., Impedance profile inversion via the first transport equation, J. Math. Anal. App., 94 (1983), 435–453.
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© 2000 Springer-Verlag/Wien
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Sylvester, J. (2000). Layer Stripping. In: Colton, D., Engl, H.W., Louis, A.K., McLaughlin, J.R., Rundell, W. (eds) Surveys on Solution Methods for Inverse Problems. Springer, Vienna. https://doi.org/10.1007/978-3-7091-6296-5_5
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DOI: https://doi.org/10.1007/978-3-7091-6296-5_5
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-83470-1
Online ISBN: 978-3-7091-6296-5
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