# Layer Stripping

## 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.

## Keywords

Reflection Coefficient Unit Disk Hardy Space Wave Speed Helmholtz Equation## Preview

Unable to display preview. Download preview PDF.

## References

- 1.A. Bruckstein AND T. Kailath,
*Inverse scattering for discrete transmission-line models*,*SIAM Review*,**29**(3) (1987), 359–389.MATHCrossRefMathSciNetGoogle Scholar - 2.Chen, Y. AND Rokhlin, V., On the Inverse Scattering Problem for the Helmholtz Equation in One Dimension,
*Inverse Problems***8**, pp. 356–391 (1992)CrossRefMathSciNetGoogle Scholar - 3.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).Google Scholar
- 4.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.MATHCrossRefMathSciNetGoogle Scholar - 5.H. Dym AND H. P. Mckean, “
*Fourier Series and Integrals*”, Academic Press, New York (1972).MATHGoogle Scholar - 6.Somersalo, E., Layer stripping for time-harmonic Maxwell’s equations with high frequency.
*Inverse Problems***10**(1994), 449–466CrossRefMathSciNetGoogle Scholar - 7.Sylvester, J., A Convergent Layer Stripping Algorithm for the Radially Symmetric Impedance Tomography Problem,
*Communications in PDE***17**, No.12, pp. 1955–1994 (1992)MATHCrossRefMathSciNetGoogle Scholar - 8.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.Google Scholar - 9.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 1996MATHCrossRefMathSciNetGoogle Scholar - 10.Sylvester, J., Winebrenner, D., Nonlinear and Linear Inverse Scattering,
*SIAM Journal of Applied Mathematics***59**(2) pp. 669–699 April 1999MATHMathSciNetGoogle Scholar - 11.Symes, W. W., Impedance profile inversion via the first transport equation,
*J. Math. Anal. App*., 94 (1983), 435–453.MATHCrossRefMathSciNetGoogle Scholar