Anisotropy Reconstruction from Wave Fronts in Transversely Isotropic Acoustic Media

This paper considers an inverse problem for a transversely isotropic 3D 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, that 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 is the positions of four distinct wave fronts as the corresponding waves propagate through the medium. The mathematical relation, that is the Eikonal equation, between the wavefront 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, that relates the wave front positions to the unknown functions, is the same as one obtains for the SH wave which propagates in a linear elastic system.