### Abstract

This paper considers an inverse problem for a transversely isotropic three-dimensional acoustic medium, where there is one preferred direction called the fiber direction along which the wave propagates fastest and there is no preferred wave propagation direction in the isotropic plane, which is the plane orthogonal to the fiber direction. In this medium the parameters to be recovered are (1) the wave speed for a wave propagating in the direction along the fiber; (2) the wave speed for a wave propagating in any direction which is orthogonal to the fiber direction; and (3) the unit fiber direction itself. So four scalar functions are to be recovered. The data are the positions of four distinct wave fronts as the corresponding waves propagate through the medium. The mathematical relation, which is the Eikonal equation, between the wave front locations and the four unknown functions, is nonlinear. Here it is established, perhaps surprisingly, that corresponding to the given data set, there can be up to four possible solution quadruples. We present and implement an algorithm to compute each of the possible solutions and show our selection criteria to obtain the correct solution. The Eikonal equation, which relates the wave front positions to the unknown functions, is the same equation obtained for the horizontally polarized shear wave (SH wave) which propagates in a linear elastic system.

Original language | English |
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Pages (from-to) | 24-42 |

Number of pages | 19 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 68 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Dec 2007 |

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### Keywords

- Anisotropic wave equation
- Arrival time
- Elastography
- Fiber direction
- Inverse problem
- Transversely isotropic medium

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*SIAM Journal on Applied Mathematics*,

*68*(1), 24-42. https://doi.org/10.1137/060651252