Inversion for Eigenvalues of Focal Region’s Elasticity Tensor from a Moment Tensor

In this talk, a new geometric inversion method is proposed for obtaining the elastic parameters of an anisotropic focal region. More precisely, the eigenvalues of a vertical transversely isotropic (VTI) elasticity tensor of a focal region are obtained, up to a constant, for a given only one moment tensor, with an accuracy depending on the strength of anisotropy. The reason of using only one moment tensor is that although there occurs a lot of earthquakes in the same focal region, the orientation of the sources are similar. Hence one do not obtain independent equations from each earthquake, in order to use it in the inversion of elastic parameters of the focal region. Moreover, this method can be applied for real-time inversion once the moment tensor of an earthquake is evaluated.

The inversion method relies on the geometric fact that a moment tensor can be written as a linear combination of the eigenvectors of the anisotropic focal region's elasticity tensor. Then, in the inversion, we use the fact that each coefficient of this unique decomposition is proportional to the eigenvalues of the focal region's elasticity tensor. Two approximations are used in this inversion method; in particular for the potency and for the source orientations.

The strength of anisotropy of the focal region determines how accurate these approximations are and hence it also determines the resolutions of the inverted eigenvalues. Because of the anisotropy of the focal region, the errors in the inversion do depend on the orientations of the dip and rakes angles, but not the strike angle since the focal region is VTI. The accuracy of the inversion for the five parameters of VTI are shown on the steographic projection. The results are very promising along some orientations as shown in the figures. The last section of the talk deals with the inversion of eigenvalues, up to a constant, for a given set of moment tensors; not only one moment tensor. It turns out that the best fit corresponds to the average of inversion results obtained for different orientations of the sources.