Local magnitude, a moment revisited

A simple, theoretical analysis shows that both local magnitude M-L and seismic moment M-0 or equivalently moment magnitude M-w are, in principle, measures of basic properties of the earthquake source: ML is proportional to the maximum of the moment-rate function, whereas M-0 is proportional to its integral. Thus, in theory, this implies that M-L proportional to (2/3) log M-0 and M-L = M-w over the entire range for which M-L can be determined. In practice, observed differences between M-L and Mw are telling us something either about the physics of the earthquake source or about inadequacies in our wave-propagation model and in our ways of measuring M-L. If influences of propagation and instrument response were properly corrected for and if effects of radiation pattern and rupture directivity were averaged out, then systematic deviations of M-L relative to Mw could be interpreted in terms of changes in stress drop or rupture velocity. However, model calculations show that, because of the way attenuation along the path is usually corrected for, we have to expect that, in most cases, M-L for small events (M-w < 2) is systematically underestimated by as much as a whole unit. Moreover, for small events with few recordings, single-station scatter due to radiation pattern and directivity can be responsible for random errors that are also on the order of a whole unit. Thus systematic and random errors in the determination of M-L for small earthquakes are likely to be much greater than the variability of M-L with respect to Mw, which could be expected from variations in source properties. The extrapolation of constant offset corrections between regional M-L scales and Mw to smaller events, for which independent determinations of M-0 are usually lacking, is not advisable: in most cases the large random errors and systematic underestimation of M-L can contribute a significant bias to magnitude recurrence relations.