On the Relationship betweenMwandMLfor Small Earthquakes

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 basedMw. Unfortunately, to correctly perform such
a conversion is not easy; moreover, even though many studies demonstrate that the equivalence
betweenML andMw 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:
EQ-TARGET;temp:intralink-;;55;536 MwSMALL
2
3
MLSMALL
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 ≤ 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.