Inversion for eigenvalues of focal region's elasticity tensor from a moment tensor

In this paper, a new geometric inversion method is proposed for obtaining the elastic parameters of an anisotropic focal regions More precisely, the eigenvalues of a vertical transversely isotropic elasticity tensor of a focal region are inverted up to a constant for a given only one moment tensor with the accuracy depending on the strength of anisotropy. The reason for using only one moment tensor is that although there might occur several earthquakes in the same focal region, the orientation of the sources is similar to each other. Hence, one cannot obtain independent equations from each earthquake in order to use them 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 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 orientations of the dip and rake angles but not on the strike angle since the focal region is vertically transversely isotropic. The accuracy of the inversion for the five parameters of vertical transversely isotropic is shown for every 10°‐gridded orientation of the dip and rake angles on the stereographic projection. The results are very promising along some orientations as shown in the figures. The last section of the paper deals with the inversion of eigenvalues, up to a constant, for a given set of moment tensors; not by using only one moment tensor. It turns out that the best fit corresponds to the average of inversions obtained for different orientations of the sources.