Functional shape of the earthquake frequency-magnitude distribution and completeness magnitude

We investigated the functional shape of the earthquake frequency-magnitude distribution (FMD) to identify its dependence on the completeness magnitude M-c. The FMD takes the form N(m) proportional to exp(-beta m)q(m) where N(m) is the event number, m the magnitude, exp(-beta m) the Gutenberg-Richter law and q(m) a detection function. q(m) is commonly defined as the cumulative Normal distribution to describe the gradual curvature of bulk FMDs. Recent results however suggest that this gradual curvature is due to M-c heterogeneities, meaning that the functional shape of the elemental FMD has yet to be described. We propose a detection function of the form q(m) = exp(kappa(m - M-c)) for m < M-c and q(m) = 1 for m >= M-c, which leads to an FMD of angular shape. The two FMD models are compared in earthquake catalogs from Southern California and Nevada and in synthetic catalogs. We show that the angular FMD model better describes the elemental FMD and that the sum of elemental angular FMDs leads to the gradually curved bulk FMD. We propose an FMD shape ontology consisting of 5 categories depending on the M-c spatial distribution, from M-c constant to M-c highly heterogeneous: (I) Angular FMD, (II) Intermediary FMD, (III) Intermediary FMD with multiple maxima, (IV) Gradually curved FMD and (V) Gradually curved FMD with multiple maxima. We also demonstrate that the gradually curved FMD model overestimates M-c. This study provides new insights into earthquake detectability properties by using seismicity as a proxy and the means to accurately estimate M-c in any given volume.