Moment Tensors: Decomposition and Visualization

Synonyms

Dynamics and mechanics of faulting; Earthquake source observations; Focal mechanism; Seismic anisotropy; Theoretical seismology

Introduction

Elastic waves are generated by forces acting at the source and affected by the response of a medium to these forces. Mathematically, they are expressed using the representation theorem (Aki and Richards 2002, Eq. 2.41):

$$ {u}_i\left(\mathbf{x},t\right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{V}{\iiint}\;{f}_k\left(\boldsymbol{\upxi}, \tau \right)\; {G}_{ik}\left(\mathbf{x},t;\boldsymbol{\upxi}, \tau \right)\; d\tau\;dV\left(\boldsymbol{\upxi} \right)}}, $$

(1)

where ${u}_i={u}_i\left(\mathbf{x},t\right)$ is the observed displacement field, x is the position vector of an observer, t is time, and Green’s tensor ${G}_{ik}={G}_{ik}\left(\mathbf{x},t;\boldsymbol{\upxi}, \tau \right)$is the solution of the equation of motion for a point single force with time dependence of the Dirac delta...