Volume in moment tensor space in terms of distance

Abstract

Suppose that we want to assess the extent to which some large collection of moment tensors is concentrated near a fixed moment tensor m. We are naturally led to consider the distri bution of the distances of the moment tensors from m. This distribution, however, can only be judged in conjunction with the distribution of distances from m for randomly chosen moment tensors. In cumulative form, the latter distribution is the same as the fractional volume V(ω) of the set of all moment tensors that are within distance ω of m. This definition of V(ω) assumes that a reasonable universe M of moment tensors has been specified at the outset and that it includes the original collection as a subset. Our main goal in this article is to derive a formula for V(ω) when M is the set [Λ]U of all moment tensors having a specified eigenvalue triple Λ. We find that V(ω) depends strongly on Λ, and we illustrate the dependence by plotting the derivative curves V'(ω) for various seismologically relevant Λs. The exotic and unguessable shapes of these curves underscores the futility of interpreting the distribution of distances for the original moment tensors without knowing V(ω) or V'(ω). The derivation of the formula for V(ω) relies on a certain Φ σz coordinate system for [Λ]U, which we treat in detail. Our underlying motivation for the paper is the estimation of uncertainties in moment tensor inversion.