Omori law The modified Omori law Omori law for foreshocks Aftershocks of aftershocks Physical aspects of temporal clustering
Omori law Why study aftershocks?
Omori law: the modified Omori law Omori law (Omori, 1894): N (t) = C 1 t the modified Omori law (Utsu, 1961): N (t) = C 1 ( C 2 + t), p and its cumulative form (for p=1): t # N(t) = N (t)dt = C 1 ln t & % +1(, $ C 2 ' 0 where t is time, N is earthquake count, C 1, C 2 and p are fitting coefficients. The decay exponent, p, is commonly referred to as the p-value.,
Omori law: Aftershocks around the world 1995 Mw 6.9 Kobe, Japan background duration
Omori law: Aftershocks around the world 1979 Mw 6.6 Imperial Valley, CA
Omori law: Aftershocks around the world 1989 Mw 7.1 Loma Prieta, CA
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.
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 its 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.
Omori law: Aftershocks of small mainshocks Micro-earthquakes during background activity also trigger aftershocks that decay according to the modified Omori law.
Omori law: Remote aftershocks The Mw7.4 Izmit (Turkey): N ( Izmit + 10 days) N Izmit - 100 days N 1985-2002 ( ) ( ) Mw5.8 Two weeks later
Omori law: Remote aftershocks N ( Izmit + 10 days) N Izmit - 100 days N 1985-2002 ( ) ( ) cumulative Omori law (See also Brodsky et al., 2000.) The decay of remote aftershocks follows the modified Omori law!
The decay of M7.4 Izmit aftershocks throughout Greece is very similar to the decay of M5.8 Athens aftershocks in Athens area (just multiply the vertical axis by 2). Omori law: Remote aftershocks
Omori law: Remote aftershocks N ( Landers + 10 days) N Landers - 100 days N 1985-2002 ( ) ( ) days since mainshock
Omori law: Remote aftershocks Figure from Kilb et al., 2000 ΔCFF(t) = Δσ S (t) µδσ N (t), The magnitude of static stress changes decay as disatnce -3. The magnitude of the peak dynamic stress changes decay as distance -1. At great distances from the rupture, the peak dynamic stresses are much larger than the static stresss.
Omori law: Remote aftershocks Instantaneous triggering No triggering Time Time
Omori law: Remote aftershocks Indeed, distant aftershocks are observed during the passage of the seismic waves emitted from the mainshock rupture. Izmit aftershocks in Greece. Brodsky et al., 2000
Omori law: Remote aftershocks Dynamic stress changes trigger aftershocks that rupture during the passage of the seismic waves. But the vast majority aftershocks occur during the days, weeks and months after the mainshock. Dynamic stress changes cannot trigger delayed aftershocks, i.e. those aftreshocks that rupture long after the passage of the seismic waves emitted by the mainshock. It is, therefore, unclear what gives rise to delayed aftershocks in regions that are located very far from the mainshock.
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 Dt is the lag time between the mainshock and aftershock I r is inter-event distance R is the rupture radius ( ) λ i = N Δt i < t 2Δt i,r < 2R i N( 0 < t Δt i,r < 2R i ).
Omori law: Aftershocks of aftershocks and the origin of remote aftershocks Mainshock index
Omori law: Aftershocks of aftershocks and the origin of remote aftershocks li>1 is indicative of seismicity rate increase in the vicinity of the aftershock in question, suggesting that the triggering effect of that aftershock in that region is stronger than the triggering effect of the mainshock and the previous aftershocks. l in north1
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: λ i Omori = 2Δt i Δt i Δt i 0 C 1 (C 2 + t i ) p dt C 1 (C 2 + t i ) p dt, which has the properties: For Δt 0, λ Omori i 1 and For Δt, λ Omori i 0.
Omori law: Aftershocks of aftershocks and the origin of remote aftershocks In conclusion, most (if not all) Landers remote aftershocks were not directly triggered by landers, but are aftershocks of previous aftershocks. Comparison with theoretical l Figure 6. Percentage of k k th (p 1) as a function of the threshold magnitude for earthquakes that occurred during the 100 days after the Landers earthquake within regions North1 (solid) and North2 (dashed). Earthquakes that occurred during the first 24 hr were excluded from this analysis.
Omori law: Aftershocks of aftershocks and the origin of remote aftershocks Hector Mine aftershocks
Omori law: Aftershocks of aftershocks and the origin of remote aftershocks Note that: The sequence consists of several sub-sequences, and the onset of activity migrated southward. Many of the quakes that occurred between 33N and 33.5N are aftershocks of a M4.3 that ruptured 10 minutes after the mainshock. M4.37 that occurred 2.4 days after the mainshocks triggered a burst of seismicity near latitude 33N. Hector Mine aftershocks Figure 8. Time-space diagram for the Hector Mine aftershocks in area South. The size of the circles is proportional to the earthquake magnitude. The vertical dashed lines indicate the timing of the three largest earthquakes.
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
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.
Omori law: Physical aspects Implications of the friction law on temporal clustering: Can t explain Can explain
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.
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.