On the Relationship between Mw and ML for Small Earthquakes

Estimating the moment magnitudes (M-w) of a small earthquake is a challenging task. One viable option to measure its size is to calculate its local magnitude (M-L) and convert it to the physically based M-w. Unfortunately, to correctly perform such a conversion is not easy; moreover, even though many studies demonstrate that the equivalence between M-L and M-w is incorrect for small events, these two parameters are sometimes thought to be strictly equivalent, regardless of the earthquake's size. Using random vibration theory, we show that, below M-w similar to 4, the M-L of a small earthquake is proportional to the logarithm of its seismic moment, and the following relationship holds: M-wSMALL = 2/3M(LSMALL) + C' . We test our findings on a high-quality data set in the Upper Tiber Valley (northern Apennines, Italy), composed of events in the range of 0 <= M-L <= 3.8, for which we compute accurate estimates of M-L and M-w.