The relation between M-E, M-L and M-w in theory and numerical simulations for small to moderate earthquakes

Abstract

This article addresses the question of how moment magnitude, Mw, local magnitude, ML, and energy magnitude, ME, of small to moderate earthquakes would scale with each other in an ideal elastic medium and why they scale differently in reality. It turns out that ML, in the way it is commonly determined, is a poor and inconsistent measure of earthquake size. For moderate to large events, the situation could be improved with a new magnitude, MA, that is not biased by the Wood-Anderson response. Mw is robust and can in principle be determined in a consistent way over the entire imaginable magnitude range, but it accounts only for the static component of the earthquake source. Could ME be an alternative? A simple conceptual analysis reveals the relation between radiated seismic energy, seismic moment, static stress drop, apparent stress, radiation efficiency and rupture velocity. Numerical simulations based on an extended-source model show that, to estimate the radiated seismic energy (and thus also ME), one needs to take into account the effects of rupture directivity and radiation pattern. Anelastic attenuation and the bandwidth of the recording instrument determine the magnitude limit below which pulse widths and corner frequencies remain nearly constant. Below this limit, all information about the dynamics of the source is lost and attempts to correct for this in the presence of noise and aleatory signal variability, even with well-calibrated attenuation models, are probably futile. We thus have to accept the fact that ME (just as ML) is a different measure of earthquake size for small earthquakes than for moderate to large earthquakes.