Cartesian parametrization of anisotropic traveltime tomography

A new method for inverting P-wave traveltimes for seismic anisotropy on a l ocal scale is presented and tested. In this analysis, direction-dependent s eismic velocity is represented by a second- or fourth-order Cartesian tenso r, which is shown to be equivalent to decomposing a velocity surface using a basis set of Cartesian products of unit vectors. The new inversion method for P- and S-wave anisotropy from traveltime data is based on the tensor d ecomposition. The formulation is formally derived from a Taylor series expa nsion of a continuously extended, 3-D velocity function originally defined on the surface of the unit sphere. This approach allows us to solve a linea r inversion instead of the standard non-linear method. The resultant, linea rized, fourth-order traveltime equation is similar to a previous fourth-ord er result (Chapman & Pratt 1992), although our representation offers a natu ral second-order simplification. Conventional isotropic traveltime tomograp hy is a special case of our tensorial representation of velocities. P-wave velocity can be represented by a second-order tensor (matrix) as a first ap proximation, although S-wave traveltime tomography is intrinsically fourth order because of S-wave solution duality. Differences between isotropic and anisotropic parametrizations are investigated when velocity is represented by a matrix A. The trade-off between isotropy and anisotropy in practical tomography, which differs from the fundamental deficiency of anisotropic tr aveltime tomography (Mochizuki 1997), is shown to be similar to 1; that is, their effects are of the same order. We conclude that anisotropic consider ations may be important in velocity inversions where ray coverage is less t han optimal. On the other hand, when the ray directional coverage is comple te and balanced, effects of anisotropy sum to zero and the isotropic part g ives the result obtained from inverting for isotropic variations of velocit y alone. Synthetic test data sets are inverted, demonstrating the effective ness of the new inversion approach. When ray coverage is fairly complete, o riginal anisotropy is well recovered, even with random noise introduced, al though anisotropy ambiguities arise where ray coverage is limited. Random n oise was found to be less important than ray directional coverage in anisot ropic inversions.