On the parametrization of equilibrium stress fields in the Earth

Abstract

A new method for parametrizing the possible equilibrium stress fields of a laterally heterogeneous earth model is described. In this method a solution of the equilibrium equations is first found that satisfies some desirable physical property. For example, we show that the equilibrium stress field with smallest norm relative to a given inner product can be obtained by solving a static linear elastic boundary value problem. We also show that the equilibrium stress field whose deviatoric component has smallest norm with respect to a given inner product can be obtained by solving a steady-state incompressible viscous flow problem. Having found such a solution of the equilibrium equations, all other solutions can be written as the sum of this equilibrium stress field and a divergence-free stress tensor field whose boundary tractions vanish. Given n divergence-free and traction-free tensor fields, we then obtain a simple n-dimensional parametrization of equilibrium stress fields in the earth model. The practical construction of such divergence- and traction-free tensor fields in the mantle of a spherically symmetric reference earth model is described using generalized spherical harmonics. © 2010 The Authors Journal compilation © 2010 RAS.