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# Similar scaling laws for earthquakes and Cascadia slow-slip events

## Abstract

Faults can slip not only episodically during earthquakes but also during transient aseismic slip events1,2,3,4,5, often called slow-slip events. Previous studies based on observations compiled from various tectonic settings6,7,8 have suggested that the moment of slow-slip events is proportional to their duration, instead of following the duration-cubed scaling found for earthquakes9. This finding has spurred efforts to unravel the cause of the difference in scaling6,10,11,12,13,14. Thanks to a new catalogue of slow-slip events on the Cascadia megathrust based on the inversion of surface deformation measurements between 2007 and 201715, we find that a cubic moment–duration scaling law is more likely. Like regular earthquakes, slow-slip events also have a moment that is proportional to A3/2, where A is the rupture area, and obey the Gutenberg–Richter relationship between frequency and magnitude. Finally, these slow-slip events show pulse-like ruptures similar to seismic ruptures. The scaling properties of slow-slip events are thus strikingly similar to those of regular earthquakes, suggesting that they are governed by similar dynamic properties.

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## Data availability

The durations and moments estimated in this study are listed in Extended Data Table 1 and in the Source Data of Fig. 3. The slip model of Michel et al.15, which is used as input in this study is available at: ftp://ftp.gps.caltech.edu/pub/avouac/Cascadia_SSE_Nature/Data_for_Nature/.

## References

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## Acknowledgements

This study was funded by NSF award EAR-1821853. S.M. is currently supported by a postdoctoral fellowship from CNES. We thank J. Gomberg for discussion and for providing a revised version of the catalogue of tremor durations presented in ref. 10. We thank R. Burgmann for comments that helped to improve the study.

## Author information

Authors

### Contributions

S.M., A.G. and J.-P.A. designed the study, interpreted the results and wrote the manuscript; S.M. and A.G. performed the computations. J.-P.A. defined the scope of the study.

### Corresponding author

Correspondence to Sylvain Michel.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Peer review information Nature thanks Roland Burgmann, Ken Creager and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

## Extended data figures and tables

### Extended Data Fig. 1 Moment–duration and moment–area scaling laws for automatic measurements.

a, Relationship between the moment released by SSEs and their duration. The black dashed line shows the best linear fit. b, Comparison with the scaling laws for slow (red shading) and regular (green shading) earthquakes proposed by Ide et al.6. c, Relationship between the moment released by SSEs and their rupture area. The black dashed line shows the best linear fit. d, Comparison with the scaling laws of regular earthquakes (green shading). Stress-drop isolines are estimated from the circular crack model.

### Extended Data Fig. 2 SSEs duration estimations for the example of SSE 34.

a, The blue lines represent the moment rate evolution of each sub-fault participating in the SSE 34. The sub-faults moment rate is estimated using a zero-phase digital filtering on the rough $${\delta }_{{\rm{deficit}}}$$ using a 5-day window (effectively 9 days). The yellow line shows the number of tremors per day within the SSE rupture area. The solid red lines indicate the start and end times picked manually to estimate the minimum duration. They are determined by the timing of the first and last sub-faults with $${\dot{M}}_{{\rm{0deficit}}} < {\dot{M}}_{{\rm{0thresh}}}$$ (the threshold rate is represented by the horizontal black dashed line). The dashed red lines similarly indicate the start and end times of the SSEs picked to estimate the maximum duration. They are determined by the times of the first and last sub-faults when $${\dot{M}}_{{\rm{0deficit}}} < 0$$. The dotted green lines indicate the automatic time picks for the start and end of the SSEs15. b, The black dots show the cumulative moment release in excess of the moment release that would have accumulated at the interseismic rate (as the SSE are extracted from the time series corrected for long-term interseismic strain). The blue line is its smoothed version using the same filter as indicated in a. The red and green vertical lines and the yellow curve are the same as in a. To illustrate the methodology used to calculate the SSE moment release, M0, we indicate the values taken for the calculation based on the minimum duration by two horizontal solid black lines. c, The blue line indicates the SSE moment rate (sum of the moment rates of the SSE sub-faults). The horizontal black dashed line represents the sum of $${\dot{M}}_{{\rm{0thresh}}}$$ of all of the sub-faults. The red, yellow and green lines are the same as in a.

### Extended Data Fig. 3 Segment delimitation.

a, SSEs cumulative slip. The pink line indicates a representative line of the average along-strike location of SSEs given by Michel et al.15. b, Map indicating the number of times that a sub-fault has experienced an SSE. The black contours delimit the extent of each SSE. The dashed black lines in a and b correspond to the selection of segments.

### Extended Data Fig. 4 Comparison with slip models of a previously published study.

a, c, e, The cumulative slip models for SSEs 3, 7 and 10 of Michel et al.15. b, d, f, The cumulative slip models of the same SSEs estimated by Schmidt and Gao27. The magnitudes indicated in all panels are calculated by taking a shear modulus μ = 30 GPa.

## Supplementary information

### Supplementary Information

This file includes: an explanation on how Slow Slip Events (SSEs) duration and moment release measurements were estimated; a discussion on moment, duration and area estimation biases, and a comparison with SSEs from the literature; a comparison of tremors durations10 and SSE durations from geodesy (this study); the source time functions of the SSEs analyzed in this study followed by explanations on how the onset and end times of each event were estimated.

## Rights and permissions

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Michel, S., Gualandi, A. & Avouac, JP. Similar scaling laws for earthquakes and Cascadia slow-slip events. Nature 574, 522–526 (2019). https://doi.org/10.1038/s41586-019-1673-6

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• DOI: https://doi.org/10.1038/s41586-019-1673-6

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