2. THE CASCADING LADDER OF AFTERSHOCKS 1.Earthquakes can trigger other earthquakes, regardless of their size. 2.The causality of “mainshock A triggered aftershock B” needs to be modified into “mainshock A triggered C1, which triggered C2, …, which triggered B”  Cascading 3.If indirect triggering is important in the overall aftershock budget, then direct triggering must be confined to spatial ranges and times shorter than the size of the total aftershock sequence. 4.The cascading effect is therefore dominated by small shocks. 2

3. NEW VERSATILE MODEL 1.Previous models have the limitations of being either heavily parameter- dependent or model-dependent. (however, some model-dependent stochastic declustering models relaxed the binary-linking (mean-field) approximation.) 2.The new “model-independent stochastic declustering” (MISD) is a rapidly converging algorithm with a small number of hypotheses: linearity and mean- field. 3

4. THE MODEL 4

5. THE ASSUMPTIONS 5

6. THE ALGORITHM I 6

7. THE ALGORITHM II 7 The a posteriori background rate is Convergence is reached when the weights (or the rates) do not substancially change during an iteration

8. TESTING THE MODEL 8

9. RESULTS I 9 BareDressed

10. RESULTS II 10 “This implies that short-lasting triggering mechanisms, acting at the time scales of few days, could be the key process, along with the cascading effect, in controlling earthquake dynamics.”

11. RESULTS II 11 “This implies that short-lasting triggering mechanisms, acting at the time scales of few days, could be the key process, along with the cascading effect, in controlling earthquake dynamics.” “This shows that cascading triggering drives the expansion of aftershocks zones.”

12. RESULTS III 12 “This demonstrates the importance of small shocks in controlling the regional seismicity.”

13. RESULTS IV 13

14. Conclusions 14 “Large earthquakes casually trigger aftershocks during a relatively short time span. However, they condition regional seismicity for a much longer time period and over larger, time-increasing distances, through the local triggering caused by their aftershocks.” The cascading effect is therefore dominated by small shocks. The cascading of aftershock triggering is characterized by a scale-invariant anatomy, making earthquake declustering an ill-defined problem. Interestingly, “if small and large earthquakes are dynamically similar, then the initiation process is scale-invariant and therefore the size of the earthquakes is inherently unpredictable.” (Kanamori and Brodsky, 2004), on the scaling relations of dynamic parameters.

15. 15