# Omori law

## Omori law

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1. Omori law • Students present their assignments • The modified Omori law • Omori law for foreshocks • Aftershocks of aftershocks • Physical aspects of temporal clustering

2. Omori law: the modified Omori law Omori law (Omori, 1894): the modified Omori law (Utsu, 1961): and its cumulative form (for p=1): where t is time, N is earthquake count, C1, C2 and p are fitting coefficients. The decay exponent, p, is commonly referred to as the “p-value”. But why study aftershocks?

3. Omori law: Aftershocks around the world 1995 Mw 6.9 Kobe, Japan duration background

4. Omori law: Aftershocks around the world 1979 Mw 6.6 Imperial Valley, CA

5. Omori law: Aftershocks around the world 1989 Mw 7.1 Loma Prieta, CA

6. Omori law: Aftershocks of small mainshocks The traditional approach is to consider as mainshocks only earthquakes that are large and infrequent. Recent studies show that small-to-moderate earthquakes also enhance the seismicity in their vicinity. • Aftershocks of aftershocks also decay according to the modified Omori law.

7. Omori law: Aftershocks of small mainshocks • When analyzing spatio-temporal clustering with respect to small earthquakes, it is useful to construct a composite catalog of stacked aftershock sequences. • A recipe for analysing aftershocks of microearthquakes: • We consider each earthquake as a potential mainshock, and for each such mainshock compute the rupture dimensions. • Calculate lag-times and distances between each potential mainshock and all later earthquakes within the study area. • Stack mainshock-aftershock pairs with an inter-event distance that is less than twice the mainshock radius to get a “composite catalog”.

8. Omori law: Aftershocks of small mainshocks • Micro-earthquakes during “background activity” also trigger aftershocks that decay according to the modified Omori law.

9. Omori law : Remote aftershocks cumulative Omori law (See also Brodsky et al., 2000.) • The decay of remote aftershocks follows the modified Omori law!

10. Omori law : Remote aftershocks days since mainshock

11. Omori law: Aftershocks of aftershocks and the origin of remote aftershocks • The mainshock index quantifies the degree to which the triggering effect of a given aftershock is locally more important than the mainshock. The mainshock index of event i is defined as: • t is time measured from the mainshock time • t is the lag time between the mainshock and aftershock I • r is inter-event distance • R is the rupture radius

12. Omori law: Aftershocks of aftershocks and the origin of remote aftershocks Mainshock index for dummies

13. Omori law: Aftershocks of aftershocks and the origin of remote aftershocks Comparison with a mainshock index of a sequence decaying locally according to the Omori law: which has the properties:

14. Omori law: Aftershocks of aftershocks and the origin of remote aftershocks • i>1 is indicative of seismicity rate increase in the vicinity of the aftershock in question, suggesting that the triggering effect (but not the stress transfer) of that aftershock in that region is stronger than the triggering effect of the mainshock and the previous aftershocks. • Most (if not all) Landers remote aftershocks were not directly triggered by landers, but are aftershocks of previous aftershocks.

15. Omori law: Foreshocks • The increase in foreshock rate too follows an Omori law, with t being the time to the mainshock. From Jones and Molnar, 1979

16. Omori law: Physical aspects Implications of static-kinetic friction on earthquake timing: The “clock advance” does NOT depend on the time of the stress application.

17. Omori law: Physical aspects Implications of rate-and-state friction on earthquake timing: The “clock advance” depends on the time of the stress application.

18. Omori law: Physical aspects Implications of the friction law on temporal clustering:

19. Summary: • Not only aftershocks of large quakes, but also aftershocks of aftershocks decay according to the modified Omori law. • Micro-earthquakes during “background activity” also trigger aftershocks that decay according to the modified Omori law. • The decay of remote aftershocks follows the modified Omori law. • Most (if not all) Landers remote aftershocks were not directly triggered by the Landers earthquake, but are aftershocks of previous aftershocks. • The increase in foreshock rate too follows an Omori law, with t being the time to the mainshock. • Stress perturbation applied on a population of faults governed by static-kinetic friction cannot give rise to seismicity rate change.

20. Further reading: • Scholz, C. H., The mechanics of earthquakes and faulting, New-York: Cambridge Univ. Press., 439 p., 1990. • Ziv, A., On the Role of Multiple Interactions in Remote Aftershock Triggering: The Landers and the Hector Mine Case Studies, Bull. Seismol. Soc. Am., 96(1), 80-89, 2006.