Cartesian parametrization of anisotropic traveltime tomography

Abstract

A new method for inverting P-wave traveltimes for seismic anisotropy on a local scale is presented and tested. In this analysis, direction-dependent seismic velocity is represented by a second- or fourth-order Cartesian tensor, 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 decomposition. The formulation is formally derived from a Taylor series expansion of a continuously extended, 3-D velocity function originally defined on the surface of the unit sphere. This approach allows us to solve a linear inversion instead of the standard non-linear method. The resultant, linearized, fourth-order traveltime equation is similar to a previous fourth-order result (Chapman and Pratt 1992), although our representation offers a natural second-order simplification. Conventional isotropic traveltime tomography 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 approximation, although S-wave traveltime tomography is intrinsically fourth order because of S-wave solution duality. Differences between isotropic and anisotropic parameterizations 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 traveltime tomography (Mochizuki 1997), is shown to be ~ 1; that is, their effects are of the same order. We conclude that anisotropic considerations may be important in velocity inversions where ray coverage is less than optimal. On the other hand, when the ray directional coverage is complete and balanced, effects of anisotropy sum to zero and the isotropic part gives the result obtained from inverting for isotropic variations of velocity alone. Synthetic test data sets are inverted, demonstrating the effectiveness of the new inversion approach. When ray coverage is fairly complete, original anisotropy is well recovered, even with random noise introduced, although anisotropy ambiguities arise where ray coverage is limited. Random noise was found to be less important than ray directional coverage in anisotropic inversions

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