### Abstract

A scheme is presented for the solution of inverse scattering problems for the one-dimensional Helmholtz equation. The scheme is based on a combination of the standard Riccati equation for the impedance function with a new trace formula for the derivative of the potential, and can be viewed as a frequency-domain version of the layer-stripping approach. The principal advantage of the procedure is that if the scatterer is be reconstructed has m>or=1 continuous derivatives, the accuracy of the reconstruction is proportional to 1/a^{m}, where a is the highest frequency for which scattering data are available. Thus a smooth scatterer is reconstructed very accurately from a limited amount of available data. The scheme has an asymptotic cost O(n^{2}), where n is the number of features to be recovered (as do classical layer-stripping algorithms), and is stable with respect to perturbations of the scattering data. The performance of the algorithm is illustrated by several examples.

Original language | English (US) |
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Article number | 002 |

Pages (from-to) | 365-391 |

Number of pages | 27 |

Journal | Inverse Problems |

Volume | 8 |

Issue number | 3 |

DOIs | |

State | Published - Dec 1 1992 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Signal Processing
- Mathematical Physics
- Computer Science Applications
- Applied Mathematics

### Cite this

*Inverse Problems*,

*8*(3), 365-391. [002]. https://doi.org/10.1088/0266-5611/8/3/002