Moment Tensor for Seismic Sources on a Bimaterial Interface: A Hyperfunction Approach

Abstract

Abstract In this article, we consider the problem of finding the effective moment tensor for shear slip along a bimaterial interface. When the slip occurs at the jump of the values of elastic parameters, it is not clear how the elasticity tensor of each side should be combined in a unique way to obtain the effective elasticity tensor that then can be used in the moment tensor expression. Moreover, the Green’s function of the medium is not differentiable at the bimaterial interface, which prevents writing down the moment tensor form of the representation theorem, therefore leading to an ambiguity. In this article, we first show that the derivative of the Green’s function can be obtained once the Green’s function is considered as a hyperfunction, which is an equivalent concept to generalized functions or distributions. The resulting Green’s function at the bimaterial interface is the average of the Green’s function derivatives evaluated on either side. Finally, to obtain the effective elastic parameters, we used the equivalence of the unambiguous potency form and the moment tensor form of the representation theorems constrained with the boundary conditions, which are the continuity of the traction acting on the fault and the tangential strain across the fault. A unique solution is obtained for the isotropic case as well as transversely isotropic and orthotropic media if the symmetry axes of the tensors are aligned with the fault orientation. However, for general anisotropy or arbitrarily oriented structures, the solution is nonunique, for which certain physical constraints such as positive definiteness of the elasticity tensor can be used to confine the solution.