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 𝑀₀ (and corresponding moment magnitude m), the number of their constituent LFEs 𝑁_𝑒, and their along-strike propagation velocity 𝑣. 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, 𝑀₀ ~ 𝑇,  2) the number of swarms 𝑁_𝑠 follows the Gutenberg-Richter (G-R) frequency-magnitude relation, 𝑁_𝑠 = 10^{𝑎−𝑏𝑚} with b =1, 3) the number of swarms is a power law function of their duration, 𝑁_𝑠 ~ 𝑇^−2/3, 4) the number of swarms is a power law function of the number of events in a swarm, 𝑁_𝑠 ~ 𝑁_𝑒^−2/3 , and 5) the along-strike velocity of a swarm scales with its duration 𝑣 ~ 𝑇^−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, 𝑀₀ ~ 𝑁𝑒, then scaling law (4) is also equivalent to the G-R distribution (2) with b = 1. Further, if 𝑑̅ ~ 𝑀₀^1/6, as observed for repeating earthquakes on the San Andreas Fault, then scaling law (5) can be written as 𝑑̅ ~ 𝐿 where 𝑑̅ is the average displacement and L is the along-strike fault length. The relation 𝑑̅ ~ 𝐿 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 𝑀₀ ~ 𝑁_𝑒 provides a possible explanation for scaling law (1) 𝑀₀ ~ 𝑇, 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.