A Simple Example
We now return to the question posed at the beginning of this paper of whether large earthquakes, which have been repeatedly found to have interval statistics that are exponentially (Poisson) distributed, nonetheless have information content about past and future events.
To show that this is not a contradiction, we consider the following simple model simulation:
The basic state variable curve \(\Theta_{\text{sim}}(t)\) is specified as the logistic function:
\(\Theta_{\text{sim}}(t)=\frac{1}{(1+exp(-t^{{}^{\prime}})}\) (4) where: \(t^{{}^{\prime}}=\frac{\text{Δt}}{\tau}+6\) and Δt is the time since the last large “earthquake.”
Failure (a “large earthquake”) occurs when \(\Theta_{\text{sim}}(t)\)= 0.995. At failure, we then set \(\Delta t=0\), and declare that a large “earthquake” has occurred.
After a “large earthquake” occurs, the next value of \(\tau\) is chosen from an exponential distribution whose mean is taken to be 25 “months”.
The future time window \(T_{W}\) = 40 “months” is used to evaluate nowcast skill.
We then progressively increase Δt by 1 “month” intervals until the next large “earthquake” occurs, at which point we repeat the process.
The results of a long simulation of 183 large “earthquakes” is shown in Figure 4. There we see that a short segment of the time series\(\Theta_{\text{sim}}(t)\) as shown in Figure 4a is generally similar to that shown in Figure 3a. Figure 3b shows that there is significant nowcast skill, equal to 0.88, with a skill index of 95.81%, meaning that the true signals can be differentiated from false signals with a high degree of reliability. By construction, however, we also see from Figures 4c and 4d that the interval statistics for \(\Theta_{\text{sim}}(t)\)conform to those of a Poisson interval distribution (exponential distribution). We can therefore say that for this model simulation, the existence of Poisson interval statistics does not imply a lack of predictability for\(\Theta_{\text{sim}}(t)\), similar to what we found with the California data set.ConclusionIn this paper we have analyzed earthquake catalogs to understand the information they contain. Interval statistics observed in catalogs are usually taken to indicate random (Poisson) events having no memory. However, we have shown that the temporal clustering, or variation in monthly rate of the small earthquakes, does contain important information. This temporal clustering is in the form of bursts of activity that can be modeled with invasion percolation networks (Rundle et al., 2020). We thus find that the process of de-clustering a catalog is shown to remove information content, and to increase the information entropy. The current results are consistent with the cluster analysis of Rundle and Donnellan (2020). We also note that this general decrease in monthly rate leading up to the next big earthquake might also be regarded as the “long tail” of the Omori aftershock distribution. We have used this idea to construct a state variable \(\Theta(t)\) by defining a 2-parameter filter, based on an exponential moving average (EMA) of small earthquake seismicity, together with an assumption about the minimum number of small earthquakes during a month-long interval. The interplay between seismic activation, for example aftershocks, and seismic quiescence, can be analyzed by standard methods. These methods are receiver operating characteristic (ROC), positive predictive value (PPV), and Shannon information entropy. We note that quiescence has been identified as a precursor to major earthquakes in previous research (Kanamori, 1981; Wyss and Haberman, 1988; Haberman, 1988; Main and Meredith, 1991; Huang et al., 2001; Chouliaras, 2009; Weimer and Wyss, 1994; Torman et al., 2010; Rundle et al., 2011; Katsumata, 2011; Nanjo, 2020). ROC analysis clearly shows that use of the optimized state variable\(\Theta(t)\) to describe the earthquake cycle in California has nowcast skill. Skill is the ability to distinguish between future time windows\(T_{W}\) that are likely to contain a large earthquake (true signal) and those that are not (false signal). The positive predictive value PPV can be interpreted as an indicator of the chance of a large earthquake during \(T_{W}\). Furthermore, the Shannon information content of both the ROC and PPV can be demonstrated to contain more information, or lower surprise value, than a random predictor. Or in other words the random predictor has higher information entropy than \(\Theta(t)\). To summarize, in reference to the original question posed by Gardner and Knopoff (1974) regarding earthquake predictability, we find the following. Their conclusion may apply to earthquake interval statistics where the ordering of temporal bursts and clustering (variation in monthly rate) has been lost through the de-clustering process, thereby increasing the information entropy in the catalog. But if small earthquakes are used to build a state variable, to which a threshold criterion is then applied, we find that there does exist information value in the resulting state variable \(\Theta(t)\). The original (non-declustered) catalog is thus found to contain significant information that can be used to compute and test earthquake probabilities without need to resort to models of stress accumulation and release, for example.
Acknowledgements . Research by JBR was supported under NASA grant NNX12AM22G, and by DoE grant DE- SC0017324 to the University of California, Davis. Portions of this research were also carried out at the Jet Propulsion Laboratory, California Institute of Technology under contract with NASA. Research by GCF was supported by grant NSF 2210266 CINES and is gratefully acknowledged.
Supplementary Material. Python code that can be used to reproduce the results of this paper can be found in the Supplemental material, or on the AGU ESSOAR preprint archive version of this paper.
Data. Data for this paper was downloaded from the USGS earthquake catalog for California, and are freely available there. The Python code mentioned above can be used to download these data for analysis.