Estimating the moment magnitudes (Mw) of a small earthquake is a challenging task. One viable option to measure its size is to calculate its local magnitude (ML) and convert it to the physically based Mw. Unfortunately, to correctly perform such a conversion is not easy; moreover, even though many studies demonstrate that the equivalence between ML and Mw 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 Mw∼4, the ML of a small earthquake is proportional to the logarithm of its seismic moment, and the following relationship holds:
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≤ML≤3.8, for which we compute accurate estimates of ML and Mw.

Online Material: Details of the processing procedure, figures of the empirical regional attenuation functional, and source terms of 1191 events from the Alto Tiberina fault (ATF) data set and earthquake catalog.

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