Generalized measures of clustering in a sandpile-based model of earthquakes

.

After quite some time, I was able to publish a single-author work again. My paper on a sandpile-based model of earthquakes was accepted in EPL (Europhysics Letters). 

• R.C. Batac, Clustering regimes in a sandpile with targeted triggering, EPL (Europhysics Letters) 135, 19003, DOI:10.1209/0295-5075/135/19003 (2021). 

This work is an extension of an idea that came to me during one of the discussions with my principal investigator during my postdoc work. Prof. Dr. Holger Kantz, himself an early researcher on discrete models of self-organized criticality (SOC), noted that the sandpile model, while exhibiting SOC, in fundamentally different from earthquakes, which are perhaps the most famous large-scale examples of SOC. Whereas earthquakes dissipate energy is bursts of correlated activity, the sandpile generates single-shot avalanches. When energy is depleted in a sandpile grid, it has to wait again before a new significant event can be generated. 

Because of this, for the latter part of my postdoc, I dealt with the problem of whether the sandpile, the first and paradigm model of SOC, could be modified to be able to capture the behavior of earthquakes. 

With a lot of trial and error, starting with the more complicated rules, I was able to come up with the solution! The solution, it turns out, is actually the simplest (least complicated) among the implementations I tried. By introducing a targeted triggering probability \(p\) that the next sandpile trigger will be directed towards the most susceptible site instead of a random location, I was able to recover the key signatures of seismicity in magnitude, space, and time. When I divided the tasks with my students, with one of them probing from the temporal and another from the spatial side, we literally had goosebumps upon realizing that the range of \(p\) values that correspond to realistic earthquake statistics—between 0.005 and 0.01%—emerge naturally. We published our results in the journal Nonlinear Processes in Geophysics [1]—after it was rejected in EPL. 

Four years hence, I revisited the problem, thereby effectively tying the lose ends of the work. In this work, I ran extensive simulations to probe what will happen to the model results when the parameter \(p\) is scanned over a broader range of values, not just for the ones reported in the previous work. Additionally, I used the generalized space-time metric \(\eta\), as introduced by Baiesi and Paczuski [2] and utilized by Zaliapin et al. [3] for real-world data. These are some of the key results of the model. 

Energy (Magnitude) distributions. Introducing \(p\) does not affect the power-law distribution of energies. This is an important result, because this is one of the first signatures of seismicity that associated it with SOC behavior. The very broad range of \(p\) values tested on the model produced the Gutenberg-Richter (GR) law of earthquakes magnitudes. This is a very strong indication that the model exhibits SOC—the resulting system response is independent of the details of the driving. 

Bimodal Space-Time Properties. As noted in the previous work [1], the correlated (independent) events tend to cluster (separate) in both space and time. The \(\eta\) metric shows bimodal distributions for the various \(p\) values investigated, except for \(p=0\) corresponding to the original sandpile (which, as Holger noted, produces avalanches that are depleted once and hence not bursty) and \(p=1\) (the other extreme, corresponding to purely correlated sequences). As \(p\) is increased, the modal value corresponding to shorter \(\eta\) values dominate, indicating the dominance of correlated events over independent events in the background. 

Relative density plots of the locations of “epicenters” generated by the model for \(p=0.05\%\).

Fractal Spatial Patterns? Another interesting result is the recovery of fractal-looking patterns of simulated earthquake epicenters in the model. While the original sandpile shows a uniform distribution of avalanche origins (akin to the epicenters), the introduction of nonzero \(p\) values tend to produce fault-like patterns over the grid (see Figure above). As a side note: This has not been investigated in detail; but if proven to be true, then this add another layer of validity to the model, as it captures this feature of real earthquakes as an emergent manifestation. 

As a final note: the review and manuscript preparation process itself is very tedious due to the lockdowns brought about by the pandemic. I even received a proof with handwritten notes from the typesetter, who was out of the office for the duration of the editing process. It took a while before the final manuscript came out. But when it did, it was all worth it. ■

References:

[1] R.C. Batac, A.A. Paguirigan Jr., A.B. Tarun, and A.G. Longjas, Sandpile-based model for capturing magnitude distributions and spatiotemporal clustering and separation in regional earthquakes, Nonlinear Processes in Geophysics 24(2), 179-187, DOI:10.5194/npg-24-179-2017 (2017).  

[2] M. Baiesi and M. Paczuski, Scale-free networks of earthquakes and aftershocks. Physical Review E 69, 066106, DOI:10.1103/PhysRevE.69.066106 (2004).

[3] I. Zaliapin, A. Gabrielov, V. Keilis-Borok, and H. Wong, Clustering analysis of seismicity and aftershock identification. Physical Review Letters 101(1), 018501, DOI:10.1103/PhysRevLett.101.018501 (2008).