There are a handful of statistical models for earthquake recurrence interval distributions that are under widespread consideration by the seismological community.

The most commonly used is the exponential distribution. This is associated with a Poisson process, and is the distribution that results from earthquakes being distributed uniformly and randomly within some time interval. Consequently, the probability of an earthquake (or other event) occurring at any time does not change with time since the previous event (in other words, the hazard function is time-invariant); this leads to the characterization of the exponential recurrence distribution as “random”, “memoryless”, or “time-independent”. The exponential distribution is also the simplest to describe statistically, as it requires only one parameter (the mean rate parameter), which is the inverse of the statistical scale parameter. (The scale parameter of a distribution determines the dispersion of the distribution, while the shape parameter determines the shape or form of the distribution; a parameter that translates the distribution along the x axis is called a location parameter.) The standard deviation of a large number of samples generated from an exponential distribution is equal to the mean.

The other distributions that are in common usage are time-dependent distributions, meaning that the probability of an event occurring at any time since the previous event changes with the elapsed time since that event. This class of distributions includes the lognormal, Weibull, and Brownian passage-time distributions. Though these distributions differ in notable ways, particularly in the properties of the right tails at values greater than several times the mean , they share a general shape and, given suitable parameters, generated sample sets of small size may not be substantively different. In fact, the distributions are similar enough that it is difficult, if not impossible, to discriminate between them given realistic seismologic and paleoseismologic datasets . These distributions are described by both the scale and shape parameters.

The behavior of these distributions and of empirical datasets may be characterized by the regularity of the spacing between events (i.e., the recurrence intervals): these may be periodic, unclustered (i.e., “random”), or clustered. Assignment into these categories is typically done with a parameter known as the coefficient of variation, or CV=σ/μ, where σ is the standard deviation of the recurrence intervals, and μ is the mean recurrence interval.

Periodic earthquakes are those that occur more regularly than random, and have a CV<1 (i.e., σ<μ). These may be generated by any of the time-dependent distributions described above with suitable scale and shape parameters. (Note that in this paper, the use of the term “periodic” does not mean perfectly repeating as it might in the physics or mathematics literature; the behavior referred to is most accurately termed “quasi-periodic”, but that term will not be used in the interests of conciseness.)

Unclustered earthquakes occur as regularly as random, and have a CV=1. These may be generated by the exponential distribution (which can generate no other), or by any of the time-dependent distributions as well, given the appropriate parameters. Note that sample sets generated from these different distributions will not be identical: sequences with an exponential recurrence distribution will have many more pairs of events that are much more closely spaced together than the mean, and more pairs of events that are much more widely spaced than the mean, compared to a sequence generated from the lognormal distribution. Nonetheless, these will cancel out in the aggregate statistics, so that the standard deviations will be equal. A comparison of these may be seen in Fig. .

Clustered earthquake sequences have sets of very tightly spaced earthquakes that are widely separated (Fig. ), and have a CV>1. These may be generated from a hyperexponential distribution, which is the sum of multiple exponentials with different means, or from the time-dependent distributions described above, given the right parameters.

No consensus exists among earthquake scientists as to the most appropriate recurrence interval distribution. As is generally the case with propriety, the safest and probably most correct assumption is that it is context-dependent. Many studies of plate boundary faults such as the San Andreas conclude that major or “characteristic” earthquakes are periodic e.g.,. Conversely, many intraplate faults with low slip rates appear to show clustered earthquakes separated by long intervals of seismic quiescence e.g.,. However, one can find examples of studies indicating the opposite conclusions, even from the same study areas .

For each of the earthquake recurrence distributions, a 2 000 000-year-long time series of cumulative displacements is calculated, and then slip rates are estimated over time windows of different lengths.

The construction of the displacement histories is straightforward. From each recurrence distribution, a little over 2000 samples are drawn randomly. Then, these are combined with an equal number of displacement samples drawn randomly from the earthquake slip distribution. Finally, a cumulative displacement history is created for each series from a cumulative sum of both the recurrence interval samples (producing an earthquake time series) and displacement samples (producing a cumulative slip history). Years with no earthquakes are represented as having no increase in cumulative displacement. Then, the series is trimmed at year 2 000 000; it is initially made longer because the stochastic nature of the sample sets means that 2000 earthquakes may not always reach 2 000 000 years.

The displacement histories in Fig.  clearly show that given the stochastic nature of the samples, the cumulative displacements can diverge greatly from the mean. The magnitude of this divergence appears to be related to the CV of the recurrence interval distributions: the clustered series (CV=2) has by far the most divergence, both unclustered series (lognormal and exponential with CV=1) behave qualitatively similarly, and the periodic series (CV=0.5) tracks most closely with the mean. The divergences from the mean are driven by successive closely spaced earthquakes, perhaps with high displacements, or by long durations of quiescence. The clustered series in particular shows a pattern of many closely spaced events (clusters) leading to a much higher than average displacement accumulation rate, followed by very long episodes of dormancy in which regression to the mean occurs. From visual inspection, the dormant episodes appear to be composed of single or dual exceptionally long inter-event times. This of course is reflected in the great asymmetry of this distribution (Fig. ), with the very short mode and “fat” right tail.

Please note that in the construction of the cumulative displacement histories, all samples are independent. This means that the duration of any recurrence interval does not depend on the duration of the previous or subsequent interval (in other words, there is no autocorrelation in these series); the same applies to the displacement samples. It is currently unknown to what degree autocorrelation exists in real earthquake time and displacement series, or how much correlation is present between recurrence intervals and subsequent displacements. Autocorrelation in recurrence interval sequences is essentially unstudied, though on the basis of a preliminary unreviewed analysis , I suspect that it is as important as CV.

Furthermore, the magnitude of displacement is independent of the corresponding recurrence interval. The framework of elastic rebound theory in its most basic form should predict some correspondence between inter-event (loading) duration and slip magnitude, and this is included (implicitly or explicitly) in oscillator models incorporating complete stress or strain release in each earthquake e.g., or in any model where coseismic friction drops to zero, as this is functionally equivalent (because fa=τas/τan, where fa, τas, and τan are, respectively, friction at rupture arrest, shear stress at rupture arrest, and effective normal stress at rupture arrest; zero friction implies zero shear stress, or complete stress drop). Given a reasonably constant loading rate, complete shear stress or strain release implies some proportionality between the loading time and displacement. Nonetheless, this correspondence is not found in the more extensive paleoseismic datasets, such as those by (or the correlation may be negative as found by ), but the number of paleoseismic datasets of sufficient size and quality to identify these effects with statistical significance are few indeed.

Because this modeling strategy involves sampling independence, it is essentially a neutral model. If any correlation structure exists in the sample sets, it will affect the displacement histories in predictable ways. Negative autocorrelation in the sample sets, meaning that a long interval (or slip distance) is followed by a short interval (or slip distance) and vice versa, will cause a more rapid regression to the mean slip rate line, and decrease the scatter in the slip rate estimates. A positive correlation between recurrence (loading) intervals and slip magnitudes will have the same effect. Conversely, positive autocorrelation in either of the sample sets, or negative correlation between the recurrence intervals and slip magnitudes, will lead to slower regression to the mean line and therefore an increase in the scatter of the slip rate estimates.

The uncertainty in the estimated slip rates due to earthquake cycle variability is estimated by taking a function, R^, that calculates the mean slip rate within a time window t, and sliding it along the displacement series. R^ is calculated simply as R^(D0,D1,t)=D1-D0t, where D0 is the cumulative displacement at the beginning of the time window, D1 is the cumulative displacement at the end of the time window, and t is the length of the time window. The ^ symbol signifies an estimate rather than the true value R. This slip rate estimation method is intended to represent a neotectonic-style slip rate estimate in which the number of earthquakes that have contributed to the observed deformation is unknown, as are the durations of the open intervals that bound the time window (one of which precedes deposition of the marker unit, and one is the time since the most recent earthquake and the measurement time). By sliding R^ over the displacement series, a set of many samples of R^ is generated, so that we may analyze the distribution. The number of samples is n=N-t+1, where N is the length of the total series (2 000 000 years in this study).

A major goal of this study is to provide an answer to the following question. How long should slip rates be measured over in order to estimate a meaningful rate? This question will be answered by looking at the distribution in R^ as a function of t. Fifty values of t from 500 years to 100 000 years, logarithmically spaced, are used. Note that given μ of 1000 years, this translates to 0.5–100 mean numbers of earthquakes in the window.

The results of these calculations are shown in Fig.  for up to 60 mean earthquake cycles. It is clear that the total variability in the estimated slip rates is initially quite high when t is short (< 10 000 years or ∼10 earthquakes). Particularly when t is < 5000 years, the maximum rates are a factor of 3 or more greater than the true rate R, but the median rates are lower than R – this means it is more likely that fewer earthquakes are captured in the time window than naively expected given the mean recurrence, and that the time contained in the open intervals is a substantial fraction of the total time window. As the median is lower than R, most measurements over these short timescales will underestimate the mean rate, although not necessarily by much.

With longer t, between 10 000–20 000 years (or 10–20 earthquakes), the variation in the slip rate estimates stabilize to within ±100 % of the mean (Fig. ) for all distributions, though this happens most quickly in the periodic distribution, and most slowly in the clustered distribution. In fact, the only exception here is that the lower bound of the clustered distribution can stay at zero for more than 60 mean earthquake cycles. It is highly unlikely that any given recurrence interval will be this long; but given thousands of earthquakes over millions of years, the chance of such an event occurring at least once is far more likely. For rate estimates longer than several tens of mean earthquake cycles, the variation decreases very slowly but progressively with increasing window length.

Note that with a measurement time exceeding 5–10 mean earthquake cycles, the standard error (n/σD2+σr2, where σD is the standard deviation of displacement; and σr is the standard deviation of the recurrence interval) is a reasonable approximation for the standard deviation of the variability in the slip rates due to earthquake-cycle stochasticity, and shows broadly similar decreasing variation with increasing earthquake cycles. However, the standard error is symmetrical, though the variability displayed here is asymmetrical due to the asymmetry of the recurrence interval and displacement distributions.

The most pragmatic motivation for this study is to understand how much epistemic uncertainty in a slip rate measurement results from the aleatoric (or natural) variability in earthquake recurrence. However, the previous results have focused on describing the natural variability, and how much a measured rate may deviate from the “true” secular rate, i.e., R^/R. In these methods and results, there is no epistemic uncertainty because all quantities are known perfectly. Of course, in a real slip rate study, the measured value is known, but the true value is not. The epistemic uncertainty then is present, and can be quantified here by evaluating the true rate R relative to the measured rates R^, so that the distribution of R/R^ at a given t represents the epistemic uncertainty distribution about the measured value.

The epistemic uncertainty relative to the measured rate is shown in Fig.  for all distributions for the first 40 000 years (or 40 earthquakes), represented by the 5th, 25th, median, 75th, and 95th percentiles, and numerical results are given in Table . Several things are clear in these plots.

First, the variance in the distributions is quite large for the first several thousand years (or several mean earthquake cycles), but becomes much more compact after ∼15 mean earthquake cycles, as with the slip rate estimates in Fig. . The right tails (or upper bounds in Fig. ) in fact are infinite (or undefined) for the first few earthquake cycles, because in some fraction of the simulations R^ is zero.

Second, the distributions are asymmetrical, especially the 5–95 % interval. The 95th percentile is generally several times as far from the measured value as the 5th percentile, meaning that the true value of the slip rate may be much greater than the measured rate but not a commensurately small fraction of the measured rate.

Third, the median rate before convergence at 5μ is greater than the true rate, meaning that in most cases very short-term slip rate measurements will underestimate the true slip rate; this is a systematic bias. This is a particular concern for paleoseismological slip rate estimates, where there are rarely more than five events in any given trench, though this bias will be decreased if slip rate measurements are made from closed intervals only. However, the median is less than 2 times R^ following just two to three events, so the systematic bias is unlikely to be much greater than the measurement uncertainties in the age or offset of the events.

It is of both theoretical and practical interest to be able to evaluate whether fault slip rates may have changed over some time period, or between multiple sets of measurements. From a theoretical perspective, understanding under what conditions fault slip rates change can lead to much insight into fault processes such as growth e.g., and interaction e.g.,. Practically, if an older (or longer-term) slip rate is quite different from the contemporary rate, then its inclusion in a seismic hazard model may lead to inaccurate hazard estimation.

First, a necessary definition is given: a slip rate change in this discussion means a real change in R, not a change in the estimate R^, which leads to a change in the distribution of earthquake recurrence and/or displacement parameters with time (in statistical terminology, the recurrence and displacement distributions are then non-stationary). This sort of change may be associated with secular changes in fault loading, stemming from changing stress or strain boundary conditions.

Discerning a real slip rate change, rather than a change in R^ due to natural variability, requires consideration of the lengths of time over which the different slip rate measurements were made and the associated uncertainty. If the distributions defined by two estimates R1^ and R2^ and their empirical distributions (reflecting the number of earthquakes as well as the CVs of the underlying earthquake recurrence distributions) are known, then the null hypothesis that the two slip rates are drawn from the same stationary distribution can be tested with a Kolmogorov–Smirnov test.

However, it is unlikely that the values for the recurrence distributions and the number of earthquakes that have transpired are sufficiently known to make the calculation, unless the fault has received in-depth paleoseismic and neotectonic study. As formal hypothesis testing may not be possible given typical slip rate datasets, an informal way of gauging the likelihood of a slip rate change is to crudely estimate (“guesstimate”) the number of possible earthquakes and the recurrence distribution, and then use the closest values in Table  with propagated measurement uncertainty to evaluate the amount of overlap between the two slip rate estimates. If the overlap between the distributions is a small fraction of the total range of the distributions, then it is likely that a real slip rate change occurred. This is clearly not appropriate for a real hypothesis test, but it may aid researchers in developing ideas or intuition about the behavior of a given fault.