8 8 - - 1 1 1 1 A A P P R R I I L L 2 2 0 0 1 1 3 3 J J O O I I N N T T C C R R M M - - I I M M P P E E R R I I A A L L C C O O L L L L E E G G E E S S C C H H O O O O L L A A N N D D W W O O R R K K S S H H O O P P Unifying scaling law of waiting times in earthquakes, simulations and a shattering glass J ORDI B ARÓ ,E DUARD V IVES Departament d’Estructura i Constituents de la Matèria, Facultat de Física, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Catalonia, Spain A BSTRACT : A sample of Vycor is compressed a constant pressure increasing rate (R=1.6 kPa/s in the example). Acoustic signals are recorded by a transducer in touch, revealing an avalanche process. Signals lack characteristic Energy scale for 5 decades, stationary and rate independent. Shattering avalanches fulfil Omori law, productivity law and all the dependence on driving rate is captured in the mean activity rate: A VALANCHE S IZE D ISTRIBUTION : [1] E. K.H. Salje et al., Phil. Mag. Lett. 91, 554-560 (2011) [2] J. Baró et al., Phys. Rev. Lett. 110, 088702 (2013) [3] T. Utsu, Pure Appl. Geophys. 155, 509 (1999) [4] F. Omori, J. College Sci., Imp. Univ. Tokyo, 7, 111–200 (1894) [5] T. Utsu, Y. Ogata. R.S.Matsuura , J. Phys. Earth 43, 1 (1995) [6] A. Helmstetter and D. Sornette, Phys. Rev. Lett. 94, 038501 (2005) [7] P. Bak, K. Christensen, L.Danon, T. Scanlon, Phys. Rev. Lett. 88 178501 (2002) [8] A. Corral, Phys. Rev. Lett. 92, 108501 (2003) [9] H. Bi, G. Börner, Y. Chu, Astron. Astrophys. 218 19-23 (1989) R EFERENCES : C OMPRESSION OF A P OROUS G LASS (V YCOR ) Statistical approach to Earth's seismological activity reveals: GUTTEMBERG-RICHTER LAW : [3] Scale Invariance in Energy with b ~ 1, ε ~ 1.66. OMORI AND PRODUCTIVITY LAW : [4,5] Production of aftershocks after a big event decaying in time and dependence on the magnitude of the triggering earthquake: with p ~ 1.0-1.4 , α ~ 0.8 . S TATISTICAL L AWS FOR E ARTH Q UAKES A FTERSHOCKS AND O MORI L AW: The Unifying Scaling Rule[7,8] states that all temporal statistical properties depend only on the mean activity rate <r i > of each set of measurements. Mean rates <r E,R > of avalanches with energy greater than E decrease with E -(ε-1) . <r E,R > is used to scale the rate distribution function from different time windows ∆t = 1, 5, 10, 25, 50, 100, 250, 500 s. The distribution of waiting times[7,8] between avalanches greater than a certain released energy E scales with the mean rate, the resulting distribution is close to the collapsing of Earthquakes and is fulfilled by a simulated ETAS model. We use Bi et al. [9] test to finally refuse the hypothesis of local Poissonity and prove the presence of clustering effects. Consider the {t i } events from a point process; the waiting time to their closest event δt i = min{t i −t i−1 ,t i+1 −t i }; and the consecutive waiting time δτ i =t i−2 if δti =t i −t i−1 and δτ i =t i +2 if δti = t i+1 −t i . Then we built the statistic variable. For a local Poisson process it can be shown that the variable Hi will be uniformly distributed and thus <H>=0.5. For a regular pattern (similar interevent times) H i will spread around <H>= 2/3. If some clustering phenomena is present in the process (like the one implemented in the ETAS Model) we find an excess of high and low values of H. A K-S test reveal clustering effects. U NIFIED S CALING L AW FOR WAITING T IMES C LUSTERING FROM B I TEST Despite the broad range of involved magnitudes it has long been suggested that acoustic emission experiments can draw profound analogies between earthquakes and mechanical failure of materials. The acoustic emission recorded during the compression of a porous silica glass (Vycor) show both scale-free invariance[1] and temporal clustering phenomena[2] similar to the laws governing seismological activity: The Gutenberg- Richter law[3], the modified Omori’s law[4], and the law of aftershock productivity[5] hold for a minimum of 5 decades, are independent of the compression rate, and keep stationary for all the duration of the experiments. We compare our results with the simulation of the Epidemic Type Aftershock (ETAS) Model[6]. The joint fulfilment of the so called unifying scaling rule[7,8] as well of the rejection of Poissonity by the Bi-test[9] increases our confidence on the similitude between both phenomena J. Baró, A. Corral, X. Illa, A Planes, E.K.H. Salje, W. Schranz, D. E. Soto-Parra, and E. Vives Phys. Rev. Lett. 110, 088702 (2013) 1 + υ = 2.07 1 - ξ = 0.55 [6] A simple marked point branching process used as a first approximation to the spatio-temporal clustering apparent in seismological data. Implements naturally both Omori and Productivity Laws by computing the point process rate: Where: · is the production of aftershocks caused by i. · is a time depending background rate: local Poisson. · is the characteristic branching ratio. CAUTION! Is not a microscopical model but a branching (single parent) process; mean field approach; limited at time "c"; lacks any way to balance energy; only work on subcritical regime (n 0 <1). E PYDEMIC TYPE A FTERSHOCKS (ETAS) M ODEL p=1 Finnancial Suport: