Introduction
Are major earthquakes random events in time? Or possibly chaotic, with order in the chaos if we know where to look? These questions lie at the heart of the debate on whether earthquakes can be predicted or anticipated, and whether it is possible to quantitatively characterize the current state of earthquake hazard.
Many years ago, Gardner and Knopoff (1974) wrote a paper with the title: “Is the sequence of earthquakes in Southern California, with aftershocks removed, Poissonian?” Their abstract: “Yes.” The analysis they did was based on fitting the intervals between events to an exponential probability distribution, which is often called Poisson statistics. This type of statistics is well-known to apply to many types of random counting problems, from the arrivals of automobiles in parking lots, to neutron decay, to calls per hour at a call center, and many other applications.
Since that time, many other researchers have searched for temporal structure in earthquake intervals, with generally negative results (e.g., Scholz, 2019; comprehensive review by Rundle et al., 2021a and references therein). Exceptions do exist, such as are seen in Episodic Tremor and Slip and small repeating earthquakes (Rundle et al., 2021a; Rouet-Leduc, 2019), but this behavior does not generally apply to large damaging earthquakes.
In all of these studies, the fundamental question underlying these investigations can be phrased as: How much information does an earthquake catalog contain? This is the question that we consider in this paper.
To summarize our results: We find that there is skill in earthquake nowcasts, as measured by the Receiver Operating Characteristic (ROC) curve, used in machine learning to evaluate signal detection. Skill is defined as the ability to discriminate between true signals and false signals. We quantify this in terms of Shannon Information Entropy, using as probabilities the ROC curve and its associated Precision (Positive Predictive Value). We show that nowcasts of real data have lower entropy (higher information content) than random data. Using a simple simulation of a nowcast state variable curve with random (exponential) recurrence times, we show that Poisson recurrence does not imply a lack of predictability or skill using the state variable. The state variable time series resembles the long-hypothesized cycle of tectonic stress accumulation and release for major earthquakes. We conclude that the observation of Poisson recurrence statistics does not necessarily imply a lack of earthquake predictability.