Angle between principal axis triples

Abstract

The principal axis angle ξ0, or Kagan angle, is a measure of the difference between the orientations of two seismic moment tensors. It is the smallest angle needed to rotate the principal axes of one moment tensor to the corresponding principal axes of the other. This paper is a conceptual review of the main features of ξ0. We give a concise formula for calculating ξ0, but our main goal is to illustrate the behaviour of ξ0 geometrically. When the first of two moment tensors is fixed, the angle ξ0 between them becomes a function on the unit ball. The level surfaces of ξ0 can then be depicted in the unit ball, and they give insights into ξ0 that are not obvious from calculations alone. We also include a derivation of the known probability density of ξ0. The density is proportional to the area of a certain surface . The easily seen variation of with t then explains the rather peculiar shape of . Because the curve is highly non-uniform, its shape needs to be considered when analysing distributions of empirical ξ0 values. We recall an example of Willemann which shows that ξ0 may not always be the most appropriate measure of separation for moment tensor orientations, and we offer an alternative measure.