Time-Dependent Renewal-Model Probabilities
When Date of Last Earthquake is Unknown
Edward H Field (USGS)
Thomas H. Jordan (USC)
Abstract
We derive time-dependent, renewal-model earthquake probabilities for the case
where the date of last event is unknown, and compare these to the timeindependent Poisson probabilities that are customarily used as an approximation
in this situation. For typical parameter values, the renewal-model probabilities
exceed Poisson results by more than 10% when the forecast duration exceeds
20% of the mean recurrence interval. We also derive probabilities for the case
where the last event is further constrained to have occurred before historical
record keeping began (the historic open interval). With respect to the latest
forecast model for California, we conclude that the Poisson approximation is
adequate for most applications, but that the historic open interval produces
important differences that should be accounted for.
Introduction
The primary scientific basis for computing long-term earthquake probabilities on a fault
has been Reid’s elastic-rebound theory (Reid, 1911), which posits that the likelihood of a large
earthquake goes down when and where there has been such an event, and increases with time as
regional tectonic stresses rebuild. For example, all previous Working Groups on California
Earthquake Probabilities (WGCEP, 1988, 1990, 1995, 2003, and 2007), which were
commissioned for official policy purposes, have applied some variant of this model (e.g., see
Field (2007) for a review of those through 2003). Such studies use a point-process renewal
model to compute earthquake probabilities (e.g., Lindh, 1983). For example, the 1988 and 1990
WGCEPs represented successive recurrence intervals using the Lognormal distribution, where the
average inter-event time was mostly obtained by dividing observed slip-per-event by the fault slip
rate. The two most recent WGCEPs, 2003 and 2007, used the Brownian Passage Time (BPT)
model (Matthews et al., 2002), which gives results that are generally indistinguishable from those
obtained using a Lognormal distribution. The aperiodicity of recurrence intervals, defined as the
standard deviation divided by the mean, has ranged between 0.2 and 0.7 in these studies.
Application of time-dependent renewal models requires information on the date of last
event. Where such knowledge is lacking, the tradition has been to apply the time-independent
Poisson model for earthquake probabilities (e.g., WGCEP 2007); the implicit assumption being
that Poisson probabilities are the same as what you get by applying a time-dependent renewal
model while accounting for all possible dates of the last event (e.g., using the total probability
theorem). We evaluate this assumption here, and also investigate the situation where we have a
known “historic open interval”, meaning the last event must have occurred prior to when humans
would have reported such an event.
Unknown Date of Last Event
Let 𝑓(𝑡) represent the probability density function (PDF) of recurrence intervals for a
chosen renewal model, where 𝑡 is relative to the date of the last event and greater than zero. The
probability of having an earthquake in the next ∆𝑇 years (the forecast duration), conditioned by
the fact that it has been T years since the last event, is given as
𝑇+∆𝑇
𝑃(𝑇 ≤ 𝑡 ≤ 𝑇 + ∆𝑇|𝑡 > 𝑇) =
∫𝑇
𝑓(𝑡)𝑑𝑡
∞
∫𝑇 𝑓(𝑡)𝑑𝑡
=
𝐹(𝑇+∆𝑇)−𝐹(𝑇)
,
1−𝐹(𝑇)
(1)
where 𝐹(𝑇) is the cumulative distribution function defined as
𝑇
𝐹(𝑇) = ∫0 𝑓(𝑡)𝑑𝑡.
ILLUSTRATE WITH A FIGURE? (e.g., FIGURE 1 IN THE OTHER PAPER?)
If we have zero knowledge of the date of last rupture, then we must consider all
possibilities. Let 𝑝(𝜏) represent the probability distribution for the time since last event, where 𝜏
is positive. Consider a perfectly periodic system where the PDF is a Dirac delta function with a
mean recurrence interval of  (actually, all recurrence intervals are  since the aperiodicity is
zero). For this case, we know the date of last event must be less than the recurrence interval
(T ), and that the probability distribution for 𝑝(𝜏) must be uniform (because we have no
information on what it might be). We can therefore write 𝑝(𝜏) for this special case as
𝑝0 (𝜏) =
1 − 𝐻(𝜏 − 𝜇)
𝜇
where the subscript 0 indicates that 𝛼 = 0, H is the Heaviside step function, and the denominator
ensures that this “boxcar” distribution has an area of 1.0. TOM –SHOULD WE WRITE: 𝑝(𝜏, 𝛼 =
0) RATHER THAN THE SUBSCRIPT? We can then represent 𝑝(𝜏) for the general case (nonzero aperiodicity) as a superposition of these special cases, where  is replaced with the possible
recurrence intervals, t, and we weight each boxcar by the probability of each recurrence-interval,
𝑓(𝑡), as follows:
∞
𝑝(𝜏) =
∫0 [1 − 𝐻(𝜏 − 𝑡)] 𝑓(𝑡)𝑑𝑡
∞
∫0 𝑡 𝑓(𝑡)𝑑𝑡
.
∞
Again, the denominator ensures a total probability of 1.0. From the fact that ∫0 𝑡 𝑓(𝑡)𝑑𝑡 = 𝜇
∞
and ∫0 𝑓(𝑡)𝑑𝑡 = 1.0, we can rewrite the above as
∞
𝑝(𝜏) =
1−∫0 𝐻(𝜏−𝑡)𝑓(𝑡)𝑑𝑡
𝜇
𝜏
=
1−∫0 𝑓(𝑡)𝑑𝑡
𝜇
𝑝(𝜏) =
1−𝐹(𝜏)
𝜇
(2)
Thus, the probability distribution for the time since last event, when the latter is unknown, is one
minus the cumulative distribution divided by the mean recurrence interval.
Using the total probability theorem, we can now combine equations (1) and (2) to give
the probability of an event occurring in the next ∆𝑇 years for the case where the date of last event
is completely unknown:
∞
𝑃(∆𝑇) = ∫ 𝑃(𝑇 ≤ 𝑡 ≤ 𝑇 + ∆𝑇|𝑇) 𝑝(𝑇)𝑑𝑇
0
∞ 𝐹(𝑇+∆𝑇)−𝐹(𝑇)
1−𝐹(𝑇)
) ( 𝜇 ) 𝑑𝑇
1−𝐹(𝑇)
= ∫0 (
1
∞
= 𝜇 ∫0 (𝐹(𝑇 + ∆𝑇) − 𝐹(𝑇)) 𝑑𝑇
𝑇 +∆𝑇
=
∆𝑇−∫𝑇 𝐻
𝐻
𝐹(𝑇)𝑑𝑇
𝜇
(3)
where the last line results from the fact that 𝐹(𝑇) is monotonically increasing with an asymptote
of 1.0.
We can now address the question of whether equation (3) gives the same result as the
probability of one or more events from the time-independent Poisson model, the latter of which is
1 − 𝑒 −∆𝑇/𝜇 .
(4)
Figure 1 compares these two probabilities, both plotted as a function of forecast duration
normalized by the mean recurrence interval (∆𝑇/𝜇), with the BPT model assuming an
aperiodicity of 0.3. The two probabilities are equal for very small forecast durations (equations
(3) and (4) are both equal to ∆𝑇/𝜇 in this limiting case). However, the difference increases with
forecast duration. In fact, “important” differences, defined as those greater than 10% based on
typical engineering studies, appear when the duration exceeds 20% of the mean recurrence
interval (a result that is also insensitive to aperiodicities between 0.2 and 0.7). Figure 1 also
implies that BPT probabilities are 35% greater for a 30-year forecast on a fault that has a mean
recurrence interval of 25 years (e.g., the Parkfield fault; Bakun et al., 2005); this value is 40% and
16% for aperiodicities of 0.2 and 0.7, respectively.
If we consider 50 years as the duration of interest in building codes, the difference
between BPT and Poisson probabilities becomes important when recurrence intervals are less
than 250 years (50 years divided by 0.2). Figure 2 shows the distribution of recurrence intervals
among the 2606 fault sections in the latest California model -- the 3rd Uniform California
Earthquake Rupture Forecast (UCERF3, Field et al., 2013), -- implying important differences for
about 9% of the fault sections (gray vertical line in Figure 2). These shorter recurrence-interval
faults are more likely to have a known date of last event, however, making the UCERF3%age
somewhat less (although those that remain are also some of the most hazardous faults).
EXPLORE IN MORE DETAIL?; NOT TRIVIAL WITH RESPECT TO PALEO SITE
CONSTRAINTS WHERE THE SPATAIL EXTENT OF RUPTURE IS UNKNOWN?
The BPT probabilities defined here are for the next event, while the Poisson probabilities
are for one or more events. This difference does not influence the conclusions here, other than
the discrepancies being slightly larger if the BPT probabilities were to include more than just the
next event.
Figure 1. Brownian Passage Time (BPT) earthquake probability (black line)
for the case where date of last event is unknown (equation (3)), assuming an
aperiodicity of 0.3, and plotted against forecast duration divided by the mean
recurrence interval. Also shown are the Poisson probabilities (gray line) of
having one or more events (1 − 𝑒 −∆𝑇/𝜇 ).
Figure 2. Cumulative distribution of recurrence intervals among the 2606
fault sections in the UCERF3 model (averaged over all logic-tree branches
for Fault Model 3.1). The dotted and dashed vertical lines mark recurrence
intervals of 250 and 1600 years, respectively.
Historic Open Interval
We now turn to the case where we have a known historic open interval, 𝑇𝐻 , such that the
last event must have occurred more than 𝑇𝐻 years ago (𝑇 ≥ 𝑇𝐻 ). The PDF for the date of last
event, condition on the fact that it was more than 𝑇𝐻 years ago, is obtained from equation (2) as
𝑝(𝜏|𝜏 ≥ 𝑇𝐻 ) =
𝑝(𝜏)
∞
∫𝑇 𝑝(𝑡)𝑑𝑡
=
𝐻
1−𝐹(𝜏)
∞
∫𝑇 [1−𝐹(𝑡)]𝑑𝑡
𝐻
where, the denominator ensures a total probability of 1.0. Combining this with equation (1) in the
total probability theorem gives
∞
𝑃(∆𝑇|𝑇 > 𝑇𝐻 ) = ∫𝑇 𝑃(𝑇 ≤ 𝑡 ≤ 𝑇 + ∆𝑇|𝑇) 𝑝(𝑇|𝑇 ≥ 𝑇𝐻 )𝑑𝑇
𝐻
∞
𝐹(𝑇+∆𝑇)−𝐹(𝑇)
1−𝐹(𝑇)
)( ∞
) 𝑑𝑇
1−𝐹(𝑇)
∫ [1−𝐹(𝑡)]𝑑𝑡
= ∫𝑇 (
𝐻
𝑇𝐻
∞
=
∫𝑇 (𝐹(𝑇+∆𝑇)−𝐹(𝑇))𝑑𝑇
𝐻
∞
∫𝑇 [1−𝐹(𝑡)]𝑑𝑡
𝐻
𝑇 +∆𝑇
=
∆𝑇−∫𝑇 𝐻
𝐹(𝑇)𝑑𝑇
(5)
𝐻
∞
∫𝑇 [1−𝐹(𝑇)]𝑑𝑇
𝐻
∞
As expected, this reduces to equation (3) when 𝑇𝐻 = 0 (because ∫0 [1 − 𝐹(𝑡)]𝑑𝑡 = 𝜇) I
VERIFIED THIS NUMERICALLY BUT NOT ANALYTICALLY.
To quantify the potential benefit of considering 𝑇𝐻 , Figure 3 shows the ratio of
probabilities from equation (4) to those from equation (3), plotted as a function of the historic
open interval normalized by the mean recurrence interval (𝑇𝐻 /𝜇). The BPT aperiodicity is again
0.3, and different curves are shown for different normalized forecast durations (∆𝑇/𝜇). As
expected, probabilities increase with greater historic open intervals because we are known to be
closer to failure compared to having no constraint on the date of last event. The difference
becomes smaller for larger normalized forecast durations, which is expected because both
probabilities approach 1.0 at infinite time.
For shorter forecast durations (e.g., ∆𝑇 ≤ 0.7𝜇 for the example in Figure 3) the
difference becomes “important” (≥10%) when the historic open interval exceeds about 10% of
the mean interval (𝑇𝐻 ≥ 0.1𝜇). If we take the historic open interval in California to be ~160
years, then the difference will be important when the mean recurrence interval is less that ~1600
years. According to Figure 2, about 50% of the UCERF3 fault sections fall into this category
(dashed vertical line marks 1600-year recurrence intervals).
Figure 3. BPT-model probabilities for the case where the historic open
interval is accounted for (equation (5)), divided by the probabilities for
when this information is ignored ( 𝑇𝐻 = 0), plotted against the open
interval divided by the mean recurrence interval ( 𝑇𝐻 /𝜇 ). The
alternative curves represent results for different normalized forecast
durations (∆𝑇/𝜇) as labeled. These results assume an aperiodicity of
0.3.
Again, this conclusions breaks down for longer forecast durations (∆𝑇 ≥ 0.7𝜇). For the
50-year forecasts of interest in engineering design, however, the breakdown is only for faults
where the recurrence interval is less than ~70 years; not only does this represent less than 1% of
UCERF3 fault sections (Figure 2), but these high-rate faults are also almost guaranteed to have a
date of last event given the historic open interval of ~160 years.
Conclusions
THIS REPEASE THE ABSTRACT, SO FEEL FREE TO BUTCHER
Correctly accounting for the unknown date of last event in a time-dependent renewal
model, as derived here, produces earthquake probabilities that differ from those implied by a
time-independent Poisson model. The difference increases with forecast duration, being zero for
very short forecasts, and becoming important (≥10%) when the forecast duration exceeds 20% of
the mean recurrence interval (at least for the range of renewal models typically applied). For the
latest California forecast model (UCERF3), this difference is important for less than 9% of the
fault sections, making this a relatively minor issue compared to other model approximations and
uncertainties.
Accounting for the historic open interval, the period of over which an event is known to
have not occurred, produces probabilities that are greater than those obtained when such
information is ignored. Although the difference depends on several variables, we find the
difference to be greater than 10% for about half of UCERF3 faults, implying that historic open
intervals are an important consideration when forecasting California earthquakes.
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