Because of the natural (aleatoric) variability in earthquake recurrence
intervals and coseismic displacements on a fault, cumulative slip on a fault
does not increase linearly or perfectly step-wise with time; instead, some
amount of variability in shorter-term slip rates results. Though this
variability could greatly affect the accuracy of neotectonic (i.e., late
Quaternary) and paleoseismic slip rate estimates, these effects have not been
quantified. In this study, idealized faults with four different,
representative, earthquake recurrence distributions are created with equal
mean recurrence intervals (1000 years) and coseismic slip distributions, and
the variability in slip rate estimates over 500- to 100 000-year measurement
windows is calculated for all faults through Monte Carlo simulations. Slip
rates are calculated as net offset divided by elapsed time, as in a typical
neotectonic study. The recurrence distributions used are quasi-periodic,
unclustered and clustered lognormal distributions, and an unclustered
exponential distribution. The results demonstrate that the most important
parameter is the coefficient of variation (

Fault slip rates are generally estimated by dividing measurements of the offset of
geologic marker features by the time over which that offset accumulated (it is not
currently possible to measure a slip rate directly, though the term “slip rate
measurement” may be used to compare to a simulated or modeled value). The uncertainty in
the resulting slip estimate is typically treated as

The magnitude of the perturbation to the slip rate estimate is, of course, a function
of the number of cumulative earthquakes that have contributed to the measured
offset (plus any
aseismic strain such as afterslip or creep). For older Quaternary markers that have
experienced tens to hundreds of major earthquakes, the effects will be minor; and for
bedrock geologic markers with kilometers of displacement, the earthquake cycle is likely
not worth accounting for. However, due to progressive erosion of geologic markers and the
challenge of dating many late Pliocene to early Quaternary units (which are too old for
radiocarbon and many cosmogenic nuclide systems), geologists often have no choice but to
choose late Pleistocene to Holocene markers to date. These units may also be more
desirable targets if the scientists are primarily concerned with estimating the
contemporary slip rate on a fault with a slip rate that may vary over Quaternary
timescales

Careful paleoseismologic and neotectonic scientists will take this into account in their
slip rate calculations if sufficient data are available, especially in the years after a
major earthquake

However, the recurrence intervals between successive earthquakes on any given fault
segment have some natural (i.e.,

The physical mechanisms responsible for the aleatoric variability in earthquake
recurrence intervals and displacements are still unclear, and the subject of active
investigation. Most earthquakes serve to release differential stresses caused by relative
motions of tectonic plates or smaller crustal blocks; relative plate velocities measured
over tens of thousands to millions of years from geologic reconstructions are similar
enough to those measured over a few years through GPS geodesy that sudden transient
accelerations and decelerations are unlikely

These stress perturbations may be “static” coseismic instantaneous stresses in the
elastic upper crust resulting from earthquake displacement

Though the physical mechanisms responsible and the statistical character of this natural variability remain under debate, its effects on the estimated slip rates may still be estimated given some common parameterizations.

In this study, the effects of the natural variability in earthquake recurrence intervals and per-event displacements on neotectonic slip rate estimates are investigated through Monte Carlo simulations. The study is geared towards providing useful heuristic bounds on the aleatoric variability and epistemic uncertainty of late Quaternary slip rate estimates for fault geologists, probabilistic seismic hazard modelers, and others for whom such uncertainties are important.

To study the effects of the natural variability in the earthquake cycle on estimated slip rates, long displacement histories of a simulated fault with different parameterizations of the earthquake recurrence distribution will be created. Then, the mean slip rate over time windows of various sizes will be calculated from each of the simulated displacement histories, and the distribution in these results will be presented, representing the natural variability in this quantity. The code used in the simulations is publicly available (Styron, 2018).

To isolate the effects of the earthquake cycle from other phenomena that may affect slip rate estimates, this study does not attempt to model erosion, nor does it consider any measurement uncertainty in the age or offset of the faulted geologic markers; these quantities are assumed to be perfectly known. Additionally, though natural variability in per-earthquake displacement is included in the model, it is minor and the same for all recurrence distributions; though it is a random variable in the simulations, it is not an experimental variable. Furthermore, though the model has one length dimension (fault offset), it is still best thought of as a point (0-dimensional) model, as there is no spatial reference or along-strike or down-dip variability, and hence the magnitude of each earthquake is undefined, and no magnitude–frequency distribution exists.

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

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

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

Periodic earthquakes are those that occur more regularly than random, and have a

Unclustered earthquakes occur as regularly as random, and have a

Earthquake recurrence distributions; “logn” represents lognormal; “exp” represents exponential. Colors for each distribution are the same in all figures.

Clustered earthquake sequences have sets of very tightly spaced earthquakes
that are widely separated (Fig.

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

Spacing of 15 simulated successive earthquakes from each recurrence distribution. Note that the gap between the last displayed earthquake and the right side of the plot does not represent a long recurrence interval.

This study will compare four recurrence interval distributions (Fig.

A

Please note
that

An

An

An

These distributions have been selected to represent a diversity of behaviors with a compact and tractable number of simulations, and particularly to explore how changes in CV as well as the shape of the distribution impact slip rate estimates.

All earthquake recurrence distributions share a single earthquake slip distribution
(Fig.

Earthquake slip distribution.

The choice of the lognormal distribution is for convenience, simplicity, and flexibility: it is a common well-known distribution and – should one be interested – can be easily given different shape and scale values to modify the CV or change the mean slip rate in the modeling code used in this paper.

However, it is not necessarily the most accurate representation of earthquake slip
variability.

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.

Simulated displacement histories for each of the recurrence
distributions, and the “true” mean line at 1 mm year

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

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

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.

Envelopes of estimated slip rates as a function of the mean number
of earthquakes (or thousands of years) over which the slip rate was
estimated. All slip rates have a true value of 1 mm year

The uncertainty in the estimated slip rates due to earthquake cycle
variability is estimated by taking a function,

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

The results of these calculations are shown in Fig.

With longer

Note that with a measurement time exceeding 5–10 mean earthquake cycles, the standard
error (

Epistemic uncertainty relative to the measured rate for each of the
recurrence distributions as a function of the mean number of earthquakes (or
thousands of years) over which the slip rates were measured.

The distributions in this study were chosen to have

NF is also equal to the mean recurrence interval

This normalization will obviously be more accurate if

The most pragmatic motivation for this study is to understand how much

The epistemic uncertainty relative to the measured rate is shown in Fig.

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

Epistemic uncertainty table showing the percentiles for the
slip rate variability (in mm year

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

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

First, a necessary definition is given: a slip rate change in this discussion means a
real change in

Discerning a real slip rate change, rather than a change in

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

This work seeks to evaluate the effect of natural (aleatoric) variability in
earthquake recurrence intervals on slip rate measurements. The study
simulates cumulative displacement during 2 000 000 earthquakes for faults
with stationary long-term slip rates of 1 mm year

The variability in slip rates calculated over time windows less than five
mean earthquake cycles is very large, but begins to stabilize
following

The most important factor in controlling the variability in slip rate
estimates is the coefficient of variation (CV); the different
distributions themselves are relatively unimportant. Faults with

The epistemic uncertainties around a measured slip rate are similarly large initially and then decrease with time. These uncertainties are initially biased, such that the measured slip rates typically underestimate the true slip rate for < 5 mean earthquake cycles, but this fades with time. However, the uncertainties remain asymmetric, with a strong right skew.

All code is available at

The supplement related to this article is available online at:

The author declares that there is no conflict of interest.

Michael Oskin and two anonymous reviewers are thanked for their reviews, which improved the content and presentation of the manuscript. Mark Allen is also thanked for his editorial work. Edited by: Mark Allen Reviewed by: Michael Oskin and two anonymous referees