# The moment magnitude *M*_{w} and the energy magnitude *M*_{e}: common roots and differences

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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

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## Abstract

Starting from the classical empirical magnitude-energy relationships, in this article, the derivation of the modern scales for moment magnitude *M*_{w} and energy magnitude *M*_{e} is outlined and critically discussed. The formulas for *M*_{w} and *M*_{e} calculation are presented in a way that reveals, besides the contributions of the physically defined measurement parameters seismic moment *M*_{0} and radiated seismic energy *E*_{S}, the role of the constants in the classical Gutenberg–Richter magnitude–energy relationship. Further, it is shown that *M*_{w} and *M*_{e} are linked via the parameter Θ = log(*E*_{S}/*M*_{0}), and the formula for *M*_{e} can be written as *M*_{e} = *M*_{w} + (Θ + 4.7)/1.5. This relationship directly links *M*_{e} with *M*_{w} via their common scaling to classical magnitudes and, at the same time, highlights the reason why *M*_{w} and *M*_{e} can significantly differ. In fact, Θ is assumed to be constant when calculating *M*_{w}. However, variations over three to four orders of magnitude in stress drop Δ*σ* (as well as related variations in rupture velocity *V*_{R} and seismic wave radiation efficiency *η*_{R}) are responsible for the large variability of actual Θ values of earthquakes. As a result, for the same earthquake, *M*_{e} may sometimes differ by more than one magnitude unit from *M*_{w}. 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 *M*_{w} 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, *M*_{e} is more suitable than *M*_{w} 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 *M*_{w} 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 *E*_{S} and *M*_{e} calculation are affected by a larger uncertainty and are currently not yet available for all global earthquakes. Thus, despite the physical importance of *E*_{S} in characterizing the seismic source, the use of *M*_{e} 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 *E*_{S} estimations in order to allow *M*_{e} to be extensively used as an important complement to *M*_{w} in common seismological practice and its applications.