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Defining the scalar moment of a seismic source with a general moment tensor

AWE Blacknest, Brimpton, Reading, RG7 4RS, England bowers{at}blacknest.gov.uk
Department of Applied Mathematics and Theoretical Physics University of Cambridge, Silver Street, Cambridge, CB3 9EW, England

Abstract

We compare several published definitions of the scalar moment M₀, a measure of the size of a seismic disturbance derived from the second-order seismic moment tensor M (with eigenvalues m₁ m₃ m₂). While arbitrary, a useful definition is in terms of a total moment, M^T₀ = M_I + M_D, where M_I = |M|, with M = (m₁ + m₂ + m₃)/3, is the isotropic moment, and M_D = max(|m_j – M|; j = 1, 2, 3), is the deviatoric moment. This definition is consistent with other definitions of M₀ if M is a double couple. This definition also gives physically appealing and simple results for the explosion and crack sources. Furthermore, our definitions of M^T₀, M_I and M_D are in accord with the parameterization of the moment tensor into a deviatoric part (represented by T which lies in [–1,1]) and a volumetric part (represented by k which lies in [–1, 1]) proposed by Hudson et al. (1989).

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