A Note on Empirical Scaling Laws for Tremor Swarms in Subduction Zones

 Tectonic tremor swarms are commonly observed at depths near the brittle-ductile transition at convergent plate boundaries. Composed of many temporally overlapping low frequency earthquakes (LFEs), these swarms extend over distances of 5 to 500 km and persist over times ranging from 1 to 1000 hours. The largest swarms have been correlated with slow slip earthquakes and we assume here that smaller swarms also serve as proxies for slow slip events. Swarms are characterized by their area A, their duration T, their scalar seismic moment ????0 (and corresponding moment magnitude m), the number of their constituent LFEs ????????, and their along-strike propagation velocity ????. These parameters have been linked in the literature by the following five scaling relations: 1) the scalar moment of a swarm is proportional to its duration, ????0 ~ ????,  2) the number of swarms ???????? follows the Gutenberg-Richter (G-R) frequency-magnitude relation, ???????? = 10????−???????? with b =1, 3) the number of swarms is a power law function of their duration, ???????? ~ ????−2/3, 4) the number of swarms is a power law function of the number of events in a swarm, ???????? ~ ????????−2/3 , and 5) the along-strike velocity of a swarm scales with its duration ???? ~ ????−0.8. We demonstrate here that if scaling law (1) is correct then scaling law (3) is equivalent to the G-R distribution (2) with b = 1. If the moment is proportional the number of events in the swarm, ????0 ~ ????????, then scaling law (4) is also equivalent to the G-R distribution (2) with b = 1. Further, if ????̅ ~ ????01/6, as observed for repeating earthquakes on the San Andreas Fault, then scaling law (5) can be written as ????̅ ~ ???? where ????̅ is the average displacement and L is the along-strike fault length. The relation ????̅ ~ ???? implies that a slow earthquake behaves more like a crack than like a self-healing slip pulse often used to describe normal earthquakes, a result that is consistent with the observation of rapid tremor reversals. Finally, the emergent relation ????0 ~ ???????? provides a possible explanation for scaling law (1) ????0 ~ ????, and a fractal distribution of swarm sizes with dimension D = 1.6 leads to the observed G-R relation with b = 1. This fractal dimension characterizes the early stages of fragmentation, consistent with the idea that tremor is the seismic signature of the breakup and underplating of subducting oceanic crust.