Journal of Seismology

, Volume 15, Issue 2, pp 411–427

The moment magnitude Mw and the energy magnitude Me: common roots and differences


  • Peter Bormann
    • Division 2: Physics of the EarthGFZ German Research Centre for Geosciences
    • Division 2: Physics of the EarthGFZ German Research Centre for Geosciences
    • International Seismological Centre

DOI: 10.1007/s10950-010-9219-2

Cite this article as:
Bormann, P. & Di Giacomo, D. J Seismol (2011) 15: 411. doi:10.1007/s10950-010-9219-2


Starting from the classical empirical magnitude-energy relationships, in this article, the derivation of the modern scales for moment magnitude Mw and energy magnitude Me is outlined and critically discussed. The formulas for Mw and Me calculation are presented in a way that reveals, besides the contributions of the physically defined measurement parameters seismic moment M0 and radiated seismic energy ES, the role of the constants in the classical Gutenberg–Richter magnitude–energy relationship. Further, it is shown that Mw and Me are linked via the parameter Θ = log(ES/M0), and the formula for Me can be written as Me = Mw + (Θ + 4.7)/1.5. This relationship directly links Me with Mw via their common scaling to classical magnitudes and, at the same time, highlights the reason why Mw and Me can significantly differ. In fact, Θ is assumed to be constant when calculating Mw. However, variations over three to four orders of magnitude in stress drop Δσ (as well as related variations in rupture velocity VR and seismic wave radiation efficiency ηR) are responsible for the large variability of actual Θ values of earthquakes. As a result, for the same earthquake, Me may sometimes differ by more than one magnitude unit from Mw. Such a difference is highly relevant when assessing the actual damage potential associated with a given earthquake, because it expresses rather different static and dynamic source properties. While Mw is most appropriate for estimating the earthquake size (i.e., the product of rupture area times average displacement) and thus the potential tsunami hazard posed by strong and great earthquakes in marine environs, Me is more suitable than Mw for assessing the potential hazard of damage due to strong ground shaking, i.e., the earthquake strength. Therefore, whenever possible, these two magnitudes should be both independently determined and jointly considered. Usually, only Mw is taken as a unified magnitude in many seismological applications (ShakeMap, seismic hazard studies, etc.) since procedures to calculate it are well developed and accepted to be stable with small uncertainty. For many reasons, procedures for ES and Me calculation are affected by a larger uncertainty and are currently not yet available for all global earthquakes. Thus, despite the physical importance of ES in characterizing the seismic source, the use of Me has been limited so far to the detriment of quicker and more complete rough estimates of both earthquake size and strength and their causal relationships. Further studies are needed to improve ES estimations in order to allow Me to be extensively used as an important complement to Mw in common seismological practice and its applications.


Moment and energy magnitudeGutenberg–Richter magnitude–energy relationshipSeismic momentRadiated seismic energyRadiation efficiency
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© Springer Science+Business Media B.V. 2010