### Abstract

Layer stripping based on the Riccati equation for the scattering matrix is a natural approach to the solution of the inverse scattering problem for the Helmholtz equation. But it has so far failed to work in two and three dimensions. This paper examines a cause of failure and establishes a Riccati equation for the scattering matrix on the level surfaces of the index of refraction. The new Riccati equation overcomes a common drawback shared by the traditional Riccati equations - the waves are unnaturally forced to propagate with a Green's function of an artificially fixed background wave number, and this man-made mismatch will mix and deform the propagating and evanescent modes, which skews the regularization of the backward solve of the Riccati equations for layer stripping.

Original language | English (US) |
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Pages (from-to) | 1745-1756 |

Number of pages | 12 |

Journal | Inverse Problems |

Volume | 21 |

Issue number | 5 |

DOIs | |

State | Published - Oct 1 2005 |

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### ASJC Scopus subject areas

- Applied Mathematics
- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

### Cite this

**Riccati equations for scattering matrices on level surfaces.** / Chen, Yu.

Research output: Contribution to journal › Article

*Inverse Problems*, vol. 21, no. 5, pp. 1745-1756. https://doi.org/10.1088/0266-5611/21/5/014

}

TY - JOUR

T1 - Riccati equations for scattering matrices on level surfaces

AU - Chen, Yu

PY - 2005/10/1

Y1 - 2005/10/1

N2 - Layer stripping based on the Riccati equation for the scattering matrix is a natural approach to the solution of the inverse scattering problem for the Helmholtz equation. But it has so far failed to work in two and three dimensions. This paper examines a cause of failure and establishes a Riccati equation for the scattering matrix on the level surfaces of the index of refraction. The new Riccati equation overcomes a common drawback shared by the traditional Riccati equations - the waves are unnaturally forced to propagate with a Green's function of an artificially fixed background wave number, and this man-made mismatch will mix and deform the propagating and evanescent modes, which skews the regularization of the backward solve of the Riccati equations for layer stripping.

AB - Layer stripping based on the Riccati equation for the scattering matrix is a natural approach to the solution of the inverse scattering problem for the Helmholtz equation. But it has so far failed to work in two and three dimensions. This paper examines a cause of failure and establishes a Riccati equation for the scattering matrix on the level surfaces of the index of refraction. The new Riccati equation overcomes a common drawback shared by the traditional Riccati equations - the waves are unnaturally forced to propagate with a Green's function of an artificially fixed background wave number, and this man-made mismatch will mix and deform the propagating and evanescent modes, which skews the regularization of the backward solve of the Riccati equations for layer stripping.

UR - http://www.scopus.com/inward/record.url?scp=25444473698&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=25444473698&partnerID=8YFLogxK

U2 - 10.1088/0266-5611/21/5/014

DO - 10.1088/0266-5611/21/5/014

M3 - Article

AN - SCOPUS:25444473698

VL - 21

SP - 1745

EP - 1756

JO - Inverse Problems

JF - Inverse Problems

SN - 0266-5611

IS - 5

ER -