### 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 |
---|---|

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 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*SIAM Journal on Applied Mathematics*,

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

**Anisotropy reconstruction from wave fronts in transversely isotropic acoustic media.** / Mclaughlin, Joyce R.; Renzi, Paul; Yoon, Jeong Rock.

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 68, no. 1, pp. 24-42. https://doi.org/10.1137/060651252

}

TY - JOUR

T1 - Anisotropy reconstruction from wave fronts in transversely isotropic acoustic media

AU - Mclaughlin, Joyce R.

AU - Renzi, Paul

AU - Yoon, Jeong Rock

PY - 2007/12/1

Y1 - 2007/12/1

N2 - 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.

AB - 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.

KW - Anisotropic wave equation

KW - Arrival time

KW - Elastography

KW - Fiber direction

KW - Inverse problem

KW - Transversely isotropic medium

UR - http://www.scopus.com/inward/record.url?scp=41849121497&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=41849121497&partnerID=8YFLogxK

U2 - 10.1137/060651252

DO - 10.1137/060651252

M3 - Article

AN - SCOPUS:41849121497

VL - 68

SP - 24

EP - 42

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 1

ER -