The eigenvalue lune as a window on moment tensors

Abstract

A moment tensor is a symmetric matrix that expresses the source for a seismic event. The fundamental lune is a certain subset of the unit sphere whose points represent the source types for all moment tensors. Familiar source types such as double couple or pure isotropic have natural locations on the lune. Although the lune consists only of source types, it serves as a low-dimensional outline of moment tensor space as a whole. For each subset B of the lune we therefore consider the associated set [B]U of unit moment tensors that have their source types in B; we wish to get a sense of [B]U from looking at B. We succeed in calculating both the angular diameter and the volume of [B]U. We also calculate the angular diameter and the volume of the set [Λ]U of unit moment tensors that have source type Λ, and we plot the results as contours on the lune. We show that great circle arc lengths on the lune are closely related to angles between moment tensors, and that arc length on the lune gives a natural measure of difference in source type.We show how to calculate volume elements for a variety of moment tensor coordinates. Volumes are relevant in part because we equate fractional volume with the probability that expresses randomness.We thus can find the probability that a random moment tensor have its source type in a given subset of the lune. Our results have implications for estimating moment tensor uncertainties, but we do not pursue them in this paper.