We illustrate the systematic difference between moment magnitude and local magnitude caused by underlying earthquake source physics using seismic moments submitted to the Statewide California Earthquake Center/U.S. Geological Survey Community Stress Drop Validation Study 2019 Ridgecrest data set. Although the relationship between seismic moment and moment magnitude (⁠M or Mw⁠) of log10(M0)∼1.5*M is uniformly valid for all earthquake sizes by definition (Hanks and Kanamori, 1979), the relationship between local magnitude ML and moment is itself magnitude dependent. For moderate events, ∼3 < M < ∼6, M and ML are coincident; for earthquakes smaller than M ∼3, ML∼1.0log10M0 (Hanks and Boore, 1984). This is a physical consequence of the corner frequency fc becoming larger than the upper frequency of observation and implies that ML and M differ systematically by a factor of 1.5 for these small events. Although this idea is not new, we propose a new, continuous relationship between local magnitude and moment, for magnitudes 2–6, which extrapolates to smaller and larger magnitudes, applicable to southern California specific to the Ridgecrest region. We make use of the abundance of seismic moments as submitted by many participants of the Community Stress Drop Study, compared to the Southern California Seismic Network (SCSN) catalog magnitudes. Overall, the seismic moments in the Community Study recover moment magnitude well, so we use our new ML–M0 to convert ML to M⁠, refining the SCSN operational MLr scale. This systematic difference of 50% in slope between local and moment magnitude at small magnitudes has implications for spectral stress‐drop estimates, earthquake ground‐motion modeling, as well as other magnitude scales and earthquake occurrence statistics.

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