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.