## 3 Method: Empirical Approach
Figure 2, relative to the _worldwide_ DS1, yields a single cluster in the upper right corner, while Figure 3, relative to the _Californian_ DS4, yields two different clusters. To understand how this distribution affects real data, I selected a narrow band along the abscissa (Figure 4) and recomputed both \(SP(\Delta t)\) and \(\lambda(\Delta t)\) using the normalized values from the whole dataset. The band width was chosen in such a way that there were at least 200 \(\Delta t\) samples in each band (with the exclusion of the last one, to the right, where its number is barely above 50). A selection of some significant results is shown in Figure 5, relative to the worldwide dataset (DS1), and Figure 6 for the Californian dataset (DS4). A complete presentation of all the available SPs can be found in Appendix (see File S2 in supplementary). Figure 4: Example of how events are selected to compute the new survival probability functions depending on the previous \(\Delta t\). Figure 5: Survival probability (**left**) and \(\lambda(\Delta t)\) (**right**) for DS1 along the narrow bands, selected as in Figure 4. Solid lines refer to empirical functions, while dashed lines refer to Poissonian ones. To clarify what we can see up to now, we must go back to the event triplets. The narrow band refers to the time interval between the first and second events of the triplet, while the survival probability function is relative to the inter-event time interval between the second and the third events. We note that, disregarding the time interval between the first and second events, whenever the third event occurs over a short temporal interval, the survival probability distribution is very different from the Poissonian one, even though it is more non-Poissonian if the time interval between the first and second events is very small. This, however, could simply mean that events tend to cluster and, to evaluate the occurrence probability of events, we must firstly de-cluster. However, there is another remarkable feature; when the time interval between the first and second events is small, even when the third event occurs over a very large temporal interval (e.g., several times the dataset average), its occurrence probability is noticeably different from a Poissonian one. Conversely, if the time interval between the first and the second event is larger than the dataset average, the occurrence of the third event for large inter-event time intervals becomes almost Poissonian. This has two very interesting consequences. The first is that, whenever the previous events are clustered, their influence has an extremely long range. The second and maybe even more important consequence is that, given this long-range temporal influence, de-clustering is plainly wrong; this is true also for large-magnitude events. The last feature which we must address is how to compute seismic hazard. By this, I mean, for instance, to evaluate the period of time that will yield a 10% occurrence probability. A simple answer would be that of Reference [3]. However, in a non-Poissonian approach, hazard would require continuous re-computation. Let us consider an example (see Figure 7). Assume that that the \(\Delta t\) of the second-to-last occurrence, with respect to the last event, is about 1.5. In this case, the appropriate survival probability function is represented by the solid line and, in that portion, the Poissonian probability would be represented by the dashed line. In this case, the two lines are almost indistinguishable. We can also suppose that no event occurred up to now, and that the time elapsed from the last event is \(t^{*}=1.20\overline{\Delta t}\). At this point, the future occurrence probability must still be one, such that the portion of the survival probability to the right of \(t^{*}=1.20\overline{\Delta t}\) must be scaled in such a way that, at \(t^{*}\), the new survival probability function will be 1. In the example, this means multiplying it by about 3.02. The new function is represented by the dotted line.