Spectral theory of constrained second-rank symmetric random tensors

Abstract

The random principal eigenvalues and random eigenvector parameters have been routinely estimated from second-rank symmetric (SRS) random tensors and geophysically interpreted in the Earth Sciences. Statistical inference of random eigen-values and random eigenvector parameters has almost always been made as if they were normally distributed. The practical validity and applicability of the assumption of normal distributions for random eigenvalues and random eigenvector parameters has not yet been checked, however. Statistical inference of random eigenvalues and random eigenvector parameters should be based on their joint probability density function (pdf) derived from that of the original random tensor. We shall extend the work of Xu & Grafarend (1996a,b) to the case of constrained SRS random tensors in this paper. All the relevant Jacobians for n-D unconstrained and 3-D constrained SRS tensors have been obtained. We then propose three pdf models for original SRS random tensors, which cover the commonly used Gaussian and Laplace pdfs and include pdf models for positive definite random material tensors. The pdfs of the random eigenvalues and random eigenvector parameters have been worked out. It is shown that the pdfs of the random eigenvalues and random eigenvector parameters are significantly different from the commonly used Gaussian pdf model. Deviatoric stress tensors and double-couple seismic moment tensors have been simulated to show the applications of the developed theory. The simulations have additionally indicated that Fisher’s pdf model for directional data is not representative of the random rotations of constrained SRS random tensors.