Introduction
Studies from the natural hazards literature indicate that certain processes
show evidence of nonstationary behavior through trends in magnitudes over
time. Such trends have been attributed
to changes in climate patterns, e.g., for wind speeds (de Winter et al.,
2013), wildfires (Liu et al., 2010), typhoons (Kim et al., 2015), and extreme
precipitation (Roth et al., 2014), and also linked directly to human
activities, e.g., increase in earthquakes from wastewater injection associated
with hydraulic fracturing (Ellsworth, 2013). Other natural hazards such as
floods resulting from streamflow (Di Baldassarre et al., 2010; Vogel et al.,
2011) and from sea level rise (Obeysekera and Park, 2012) may be a result of
a myriad of anthropogenic influences including climate change, land use
change, and even natural processes such as land subsidence. In addition to
evidence of magnitude changes of many natural hazards, recent reports
document a corresponding surge in human exposure to natural hazards (Blaikie
et al., 2014), along with a 14-fold increase in economic damages due to
natural disasters since 1950 (Guha-Sapir et al., 2004). Given evidence of
trends and the consequent expected growth in devastating impacts from natural
hazards across the world, new methods are needed to characterize their
probabilistic behavior and communicate event likelihood and the risk of
failure associated with infrastructure designed to protect society against
such events. The existing rich and evolving field of hazard function analysis (HFA)
is well suited to problems in which the probability of an event is
changing over time, yet to our knowledge has not been applied to natural hazards.
Summary of natural hazards employing Poisson–GP model.
Natural hazard
References
Evidence of nonstationarity in process
Nonstationary POT model formulated?
Floods
Todorovic (1978);
Bayazit (2015) for a review
Yes, in Strupczewski et al. (2001);
Madsen et al. (1997)
Villarini et al. (2012)
Earthquakes
Gutenberg and Richter (1954);
Ellsworth (2013) for a review
No
Utsu (1999)
Extreme rainfall
Sugahara et al. (2009);
Begueria et al. (2011); Tramblay et al. (2013);
Yes, in all references
Bonnin et al. (2011)
Roth et al. (2014); Sugahara et al. (2009)
Wildfires
Holmes et al. (2008)
Liu et al. (2010)
No
Extreme wind speeds
Palutikof et al. (1999) for review;
Young et al. (2011);
Young et al. (2011)
Jagger and Elsner (2006)
Pryor and Barthelmie (2010) for review
Wave height (proxy
Davison and Smith (1990)
Méndez et al. (2006); Young et al. (2011);
Yes, in Méndez et al. (2006);
for storm surge)
Ruggerio et al. (2010)
Ruggerio et al. (2010)
Daily max and
Waylen (1988)
Keellings and Waylen (2014)
Yes, but not derived in text
min temperatures
Ecological extremes
Katz et al. (2005)
Katz et al. (2005)
Yes
(sediment yield)
Probabilistic analysis of natural hazards normally takes one of two
approaches in fitting a probability distribution to hazard event data series.
As we summarize in Table 1, one common approach employs the peaks over
threshold (POT) data set, also commonly referred to as the partial duration
series (PDS), to characterize exceedances above some defined magnitude
(threshold) that occur over an interval of time. The second method is to fit
a probability distribution to the annual maximum series (AMS), common
practice in hydrology (Gumbel, 2012; Stedinger et al., 1993) and also
appropriate for earthquakes (Thompson et al., 2007) and many other processes
(Beirlant et al., 2006; Coles et al., 2001). Hydrologists have extensively
studied the theoretical relations between POT and AMS methods (Stedinger et
al., 1993; Todorovic, 1978) and compared the two for characterizing the
probabilities of flood events (Madsen et al., 1997). In general the POT
approach appears to provide a larger data set to draw from; however as the
threshold of exceedance used to define the POT series is lowered, the series
of maxima begin to exhibit temporal dependence which complicates the
probabilistic analysis considerably. Further complexities arise in POT
analyses due to subjectivity of the threshold and difficulty in confirmation
of independence between events (Stedinger et al., 1993).
Still the POT method is widely used for many natural phenomena either because
there is an intuitive choice for the threshold of exceedance, as in the case
of earthquakes where the magnitude of completeness is often selected, or
because the analyst wishes to maximize the use of data and does not always
understand the tradeoff between the length of the POT series and the inherent
increase in its temporal dependence structure. Due to the ubiquitous
application of the POT method, a substantial number of textbooks and articles
have studied methods for minimizing the difficulty of implementation, with
emphasis on the subjectivity of threshold selection and evaluation of
independence of events (Davison and Smith, 1990; Smith, 2003). This study
assumes a POT approach is taken for characterizing the magnitudes of natural
hazards, following many examples from the literature including extreme winds
(Palutikof et al., 1999), earthquakes (Pisarenko and Sornette, 2003),
wildfires (Schoenberg et al., 2003), and wave heights (Lopatoukhin et al.,
2000), among others. Future work will extend these analyses to AMS series (Read, 2016).
Application of general POT model in natural hazards
The generalized Pareto (GP) distribution, a generalization of the exponential
distribution, was first introduced as a limiting distribution for modeling
high level exceedances by Pickands III (1975) and later developed by Hosking and
Wallis (1987), who discuss its theory and an application for extreme floods.
Davison and Smith (1990) provide techniques for dealing with serially
dependent and seasonal data for modeling exceedances above a threshold with
the GP distribution. Hosking and Wallis (1987) discuss the fundamental
properties of the GP distribution. See Pickands III (1975) and Davison and
Smith (1990) for further theoretical background on the application of the GP
distribution for modeling POT series. In general, the GP distribution arises
for variables whose distributions are heavy-tailed, in cases where the
lighter-tailed exponential distribution does not provide sufficient
robustness (Hosking and Wallis, 1987).
The GP distribution has been widely applied to natural hazards (see Table 1)
and in many other fields including financial risk, insurance, and other
environmental problems (Smith, 2003) to characterize the magnitude of
exceedances above a threshold. Hosking and Wallis (1987), Stedinger et
al. (1993),
and others show that if the time between the peaks corresponding to a
POT series follow a Poisson distribution, and the POT magnitudes follow an
exponential distribution, then the AMS follow a Gumbel distribution.
Similarly, Hosking and Wallis (1987) and others show that if time between the
peaks of the POT series are Poisson and the POT magnitudes follow a
two-parameter GP (GP2) distribution, then the AMS follows a generalized extreme
value distribution. Table 1 lists many natural hazards problems that
apply the Poisson–GP model for their probabilistic analysis. Here we consider
a POT that follows the GP model (the exponential distribution is a special
case of the GP model when the limit of the shape parameter approaches 0;
see Davison and Smith (1990) for details). For example the Gutenberg–Richter
model developed for earthquake magnitudes is a two-parameter exponential model
(Gutenberg and Richter, 1954).
For the natural hazards listed in Table 1, the common approach is to assume
that the probability of exceedance (p) for a given magnitude event is
constant from year to year, i.e., stationary through time. Under the
assumption of stationarity, the theoretical relationships between POT and AMS
enable straightforward computation of summary and design metrics such as the
quantile or percentile of the distribution associated with a particular
average return period and/or reliability (Stedinger et al., 1993). When
evidence of nonstationarity, or a trend in either or both the frequency or
the magnitude of the exceedance events occurring through time is detected,
then p can no longer be assumed as a constant and the traditional
Poisson–GP (or other) model must be modified to account for dependence on
time and/or some other explanatory co-variate. Not adjusting the
probabilistic analysis for a positive trend when one is present can lead to
gross over-estimation of the expected return period and reliability of a
system, as shown for floods by several recent studies (Salas and Obeysekera,
2014; Read and Vogel, 2015). Furthermore, Vogel et al. (2013) document that
without a rigorous probabilistic analysis of trends in natural hazards, we
may overlook and fail to prepare for a wide range of societal outcomes which
may have occurred repeatedly in the past.
Introduction to hazard function theory and implications for
natural hazards
Despite the similarity in name, the application of the theory of HFA, also commonly referred to as survival analysis, is
practically absent from general literature in the field of natural hazards.
HFA is a well-established set of tools useful for conducting a
“time-to-event” analysis or for understanding the distribution of survival
(failure) times for a given process (e.g., survival rate of a chronic disease,
time until electrical burnout of a device, age-specific mortality rate). This
is precisely the concern of those modeling natural hazards that are changing
over time. Generally, HFA is comprised of three primary functions: (1) the
hazard function, h(t), which is defined as the failure rate, or as the
likelihood of experiencing a failure at a particular point in time; (2) the
survival function, ST(t), defined as the exceedance probability for the
random variable time (T) or in reliability engineering as the cumulative
distribution function (CDF) of T, FT(t), where ST(t) = 1 - FT(t);
and (3) the cumulative hazard function, H(t), interpreted as
the total number of failure events over a period of time.
Most applications of HFA are interested in computing design metrics based on
knowledge of h(t), ST(t), and H(t), e.g., the mean time to failure, or
the reliability of surviving a certain amount of time without at
least one failure event. In nearly all of the literature on HFA, the process
for defining these three functions begins in one of two ways: either by first
identifying an appropriate hazard function h(t), a possible path if
sufficient knowledge (or empirical evidence) of the failure process is known
(e.g., does the probability of failure increase or decrease, or is it constant
over time); or by estimating the survival function ST(t) by fitting a
set of survival time data to a distribution (Klein and Moeschberger, 1997).
Neither of these two approaches are particularly suited to natural hazards,
because until this paper there has not been any guidance on how to select an
appropriate a hazard function h(t) for a natural hazard random variable.
While reliance on empirical data to fit a survival function to data is a
common approach in other fields, often in regards to natural hazards our
data set is incomplete.
The theory of HFA is derived elsewhere and summarized in numerous textbooks
(Finkelstein, 2008; Kleinbaum and Klein, 1996; Klein and Moeschberger,
1997); hence we only summarize the relevant relationships among the hazard
rate function h(t), the probability density function (PDF) of the time to
failure fT(t), the CDF of the time of failure FT(t), its corresponding
survival function ST(t), as well as the cumulative hazard function H(t):
h(t)=fT(t)1-FT(t),ST(t)=1-FT(t)=exp-∫0th(s)ds,H(t)=-lnST(t)=∫0th(s)ds.
HFA has been applied in the fields of bio-statistics and medicine (Cox, 1972;
Pike, 1966) as well as many other disciplines including economics (Kiefer,
1988) and engineering (Finkelstein, 2008; Hillier and Lieberman, 1990). Very
little attention has been given to the use of HFA to natural hazards (Read,
2016; Katz and Brown, 1992; Lee et al., 1986). Concepts from hazard function
theory were applied to develop dynamic reliability models for characterizing
composite risk of hydrologic and hydraulic failures in conveyance systems in
the 1980s (Lansey, 1989; Tung, 1985; Tung and Mays, 1981).
The primary goal of this work is to use the theory of HFA to link the
probabilistic properties of T with properties of the probability
distribution for a nonstationary natural hazard event X. We begin by
explicitly relating h(t) to exceedance probability associated with the
event magnitudes for a general hazard, X, assuming the natural hazard POT
follows a GP distribution. We then derive ST(t), H(t) and
MTTF = E[T] for the case of the GP2 model.
We test the assumption that h(t) = pt by comparing theoretical
ST(t) to empirical survival time values derived from Monte Carlo
experiments. Since an exponential model is a special case of the GP model,
our results also apply to POT series which follow an exponential model.
Recall that the exponential distribution is of interest when the AMS is
Gumbel, which is applicable for many natural hazards. Since both the
exponential and GP2 models are widely used for representing natural event
magnitudes in a POT, the analysis presented here is relevant for a wide range
of nonstationary natural hazards. We hope to demonstrate that HFA can be a
useful methodology for characterizing nonstationary natural hazards, for
communicating natural hazard event likelihood under nonstationarity, and for
computing corresponding design metrics that reflect the changing behavior of
the both the magnitude and frequency of a natural hazard through time.
Hazard function analysis for nonstationary natural hazard magnitudes
To relate the properties of h(t) with the magnitudes of a particular
natural hazard (X), we first consider the stationary situation in which a
natural hazard event has CDF FX(x; θ), with generic parameter
set θ that does not change in time, and the exceedance probability
po associated with some threshold magnitude value xo is constant
through time. In the stationary case, the hazard failure rate, h(t), is
constant so that h(t) = po (Read, 2016), and the time to failure, T,
always follows an one-parameter exponential distribution (or if discrete, the
geometric distribution); computation of the average return period is readily
obtained from probability theory as the expected value of the exponential
variable E[T] = 1/p.
If the magnitudes of a natural hazard exhibit an increasing trend through
time, this indicates that the exceedance probability associated with a
particular design event is changing with time, which we denote as pt. In
such a situation, the expectation E[T] is no longer a sufficient statistic
for the distribution of T, and a more complex analysis is needed (Read and Vogel, 2015).
For a nonstationary natural hazard, the parameters of the CDF of X are no
longer constant and the CDF is now given as FX(x; θ(t)). For
a process with increasing hazard event magnitudes, this implies that the
exceedance probability is also increasing with time such that
pt = 1 - FX(x; θ(t)). Likewise, h(t) is no longer
constant and, as we will show, can be computed directly from a probabilistic
analysis of the natural hazard of interest by equating h(t) = pt. This
assumption is the primary difference between this study and other HFA
applications in the literature, as we explicitly link h(t) to the event
magnitudes of a natural hazard rather than use empirical evidence (fitting)
The advantage of this approach is to generalize across many types of natural
hazards such that future researchers can pursue HFA as an alternate
methodology for describing the probabilistic behavior of nonstationary
natural hazards.
GP2 model for magnitudes of natural hazards
As discussed earlier, the GP distribution is widely used in modeling the
magnitudes above a pre-defined threshold for a variety of natural hazards.
In this section we present the GP2 stationary and nonstationary models,
reviewing literature to support the selection of our nonstationary natural
hazard model formulation. We then use HFA theory to derive h(t), ST(t), and
H(t) for the GP2 nonstationary model and discuss the findings and
interpretations for each function.
Stationary GP2 model
We begin by describing the definitions for a random hazard variable X whose
POT event magnitudes follow a GP2 distribution. The stationary PDF and CDF,
introduced by Hosking and Wallis (1987), are
fx(x)=1α1-κxα1κ-1forκ≠0,Fx(x)=1-1-κxα1κforκ≠0,
where
α is the scale parameter and κ is the shape parameter. We use
the coefficient of variation, Cx = σx/μx,
to represent the variability of the system.
Note that when κ = 0 and Cx = 1, Eqs. (4) and (5) reduce to
the exponential distribution with a mean of α; this form corresponds
to a Gumbel distribution for the AMS. The first and second moments of X are
μx=α1+κ,σx2=α2(1+κ)2(1+2κ).
Combining Eqs. (6) and (7) for the GP2 model yields
Cx=12κ+1.
The quantile function for the GP2 distribution for a design event,
xo, associated with exceedance probability, p, is written by rearranging
Eq. (5) as
xo=ακ1-pκ.
These equations serve as the foundation for developing a nonstationary GP2
model, discussed in the next section.
Nonstationary GP2 model
Although we could not locate any previous research mathematically linking HFA
and nonstationary natural hazards, there are numerous papers that employ a
nonstationary GP model for the POT magnitudes of specific natural hazards
(shown in Table 1). We briefly review those models to provide context for the
trend model adopted here. Literature on nonstationary GP models for specific
natural hazards has employed a variety of parameterizations. For example,
Roth et al. (2012, 2014) considered models of changes in the POT threshold
over time and Strupczewski et al. (2001) modeled the arrival time
distribution of the POT with time-varying Poisson parameters. Strupczewski et
al. (2001) also modeled the changes in the magnitudes of the POT events over
time by modeling changes in the GP parameters over time as we do here.
Nearly all previous studies that employed nonstationary POT models in the
context of natural hazards adopt some form of the Poisson–GP model, and many
whose concerns regard increasing magnitudes have been specific to extreme
rainfall. With respect to extreme daily rainfall, most have built
nonstationary GP2 models assuming a trend in the scale parameter (α),
either modeled linearly (Beguería et al., 2011; Sugahara et al., 2009)
or log-linearly (Tramblay et al., 2013). Roth et al. (2014) note that
modeling a trend in the threshold level itself indicates a comparable trend
in the scale parameter. Tramblay et al. (2013) used time-varying co-variates
in the Poisson arrivals (occurrence of seasonal oscillation patterns) and in
the magnitudes (monthly air temperature) to model heavy rainfall in southern
France and found improvement from the stationary model. As pointed out by
Khaliq et al. (2006), Tramblay et al. (2013), and others, it is less common
to vary the shape parameter (κ) through time due to difficulty with
precision and a lack of evidence on model improvement with a time-varying
shape parameter.
Studies from other natural hazards are consistent with those in extreme
rainfall for nonstationary Poisson–GP model formulations, though with more
examples of time variation in the shape parameter. For example, Strupczewski
et al. (2001) used linear and parabolic trends in both α and κ
to model flood magnitudes; others have explored linear models in κ
for extreme winds (Young et al., 2011) and in sediment yield (Katz et al.,
2005). For wave height, several assumed a trend in the location parameter
either as linear (Ruggiero et al., 2010) or log-linear (Méndez et al.,
2006) formulations. Renard et al. (2006) used a Bayesian approach to explore
step change and linear trend models in α for general purpose with an
application to floods. The Bayesian framework was also used by Fawcett and
Walshaw (2015) to present a new hybridized method for estimating more precise
return levels for nonstationary storm surge and wind speeds.
Derivation of nonstationary GP2 hazard model
Our approach is to derive the primary HFA functions fT(t), FT(t),
ST(t), h(t), and H(t) using the nonstationary CDF, FX(x; θ(t)), for a GP2
random variable X. To develop a nonstationary GP model, we employ an
exponential trend model in the scale parameter αx(t), so that
αx(t)=αoexp(βt).
The model in Eq. (10) is equivalent to a model of the conditional mean of
the natural hazard X and has been found to provide an excellent
representation of changes in the mean annual flood for flood series at
thousands of rivers in the United States (Vogel et al., 2011) and in the
United Kingdom (Prosdocimi et al., 2014). This model is described by Khaliq
et al. (2006) and was also used in Tramblay et al. (2013) for extreme rainfall.
We assume that the shape parameter κ is constant through time as
consistent with previous studies discussed earlier. This assumption implies
that Cx is fixed (Eq. 10), or that the variability of the system is
assumed constant over the time period, defined at t = 0, and thus the
standard deviation changes in step with the mean (parameterized by α).
Again, there is reasonable evidence that this is the case for floods (see
Vogel et al., 2011; Prosdocimi et al., 2014).
Following Vogel et al. (2011), Prosdocimi et al. (2014), and Read and Vogel (2015),
we replace the trend coefficient β in Eq. (10) with the
more physically meaningful magnification factor M to represent the ratio of
the magnitude of the natural hazard quantile at time period (t + d)
to the natural hazard quantile at time t. For the model developed here,
the magnification factor, M, can be derived by combining the GP2 quantile
function in Eq. (9) and the trend model in Eq. (10), inserting into
the expression below:
M=xo(t+d)xo(t)=1λexpβ(t+d)ln(pt)1λexpβtlnpt=exp[β⋅d].
Thus M reflects the change in the magnitude of the natural hazard over
time. So for example, M = 2 and d = 10 indicates that the magnitudes
of the natural hazard have increased twofold over d, for all values of p.
In this section we derive h(t), ST(t), H(t), and fT(t) for the nonstationary GP2
model. First recall that the hazard function is equal to the exceedance
probability through time for a natural hazard event series, h(t) = pt.
Using the relationships above we derive an expression for h(t) dependent only on
fundamental parameters M, po, and Cx.
Consider that in Eq. (9) the design event is fixed as xo and
associated with po and αo at time t = 0. To derive the
expression of interest, h(t) = pt = 1 - FX(xo; θ(t)), we
insert the quantile function in Eq. (9) and trend model in Eq. (10)
into Eq. (5), replacing β with M by combining with Eq. (11). The
result yields the hazard function:
h(t)=pt=1-FCxo;t=1-1-po1-Cx22Cx2Mt/d2Cx21-Cx2,
where κ is replaced with Cx after rearranging Eq. (8) and d is the time
step associated with M. Combining the theoretical relationships in Eqs. (1)–(3)
with Eq. (12) leads to expressions for ST(t), H(t), and fT(t) solved by
numeric integration.
ST(t)=exp-∫0tbdsH(t)=∫0tbds.
Finally, the PDF of the time to failure distribution for the GP2
nonstationary model is
fT(t)=b⋅exp-∫0tbds,
where b = 1-1-p1-Cx22Cx2Ms/d2Cx21-Cx2.
For the case of the one-parameter exponential, these functions simplify to
h(t) = poM-t/d, ST(t) = exp-∫0tpoM-s/dds,
HT(t) = ∫0tpoM-s/dds, and
fT(t) = ddt exp∫0tpoM-s/dds
(see Read (2016) for a complete derivation).
Impacts of nonstationarity on probabilistic analysis of natural hazards using HFA
In this section we explore how HFA can characterize the behavior of
nonstationary natural hazards whose PDS magnitudes follow a GP2 model. Our
results are exact (within the limitations of numerical integration) because
they result from the derived analytical equations in Eqs. (12)–(15) for the HFA
functions fT(t), FT(t), and ST(t), h(t), and H(t)
corresponding to a natural hazard X which follows a GP2 model. With no loss
in generality, we assume the mean of the GP2 natural hazard of unity. We
investigate the impact of small and large trends (corresponding to
magnification factors, M, ranging from 1 to 1.25 with d = 10) for a range
of physical systems characterized by a range in variability corresponding to
a range in the coefficient of variation of X, Cx, from 0.5 to 1.5
corresponding to a range in the GP2 shape κ between -0.28 to 1.5)
for three event sizes (po = 0.01, 0.002, 0.001). We verified our
assumption that h(t) = pt for the GP2 case presented here by
simulating failure times from a GP2 random variable for a range of po,
M, and Cx values and comparing to the theoretical ST(t) curves.
Figure 1 presents the hazard function h(t), examining how a particular
design event po = 0.002 (500-year event) is influenced by the
magnitude of the trend and system variability: (a) increasing variability,
Cx = 0.75, 1.25, 1.5 for a set M = 1.1; and (b) increasing
trend values, M = 1.1, 1.25, 1.5 for a set Cx = 0.75. Note
that in the stationary case, pt = po = 0.002 results in
a constant horizontal line, and as the magnitude of the trends increases (as
M increases), the hazard rate h(t) tends toward unity earlier in time.
Even from this initial relatively simplistic investigation we find that the
hazard functions exhibit complex shapes, with some exhibiting inflection
points and others without an inflection. Another important point is that the
variability of the hazard magnitudes (characterized by the shape of the PDF
of X) impacts the rate at which h(t) increases, so that less variable
hazards tend to have higher hazard rates than more variable hazards. This
point is perhaps initially counter-intuitive, and our interpretation is that
if a hazard is more consistent (with less variability), a larger trend
ensures exceedance more so than a less consistent system that has a wider
range of small and large events. This finding is relevant for planning
purposes as it indicates which systems may be greater impacted by nonstationarity.
Hazard function h(t) for the nonstationary GP2 model and
po = 0.002 for (a) a range of variability
(Cx = 0.75, 1.25, 1.5), given M = 1.1, and (b) a range of
trend values (M = 1.1, 1.25, 1.5), given Cx = 0.75.
Typically in HFA work, the survival function ST(t) is presented as a
primary figure in understanding risk of failure and likelihood of
experiencing an exceedance event within a given period of time. Since
ST(t) also represents the relationship between system reliability and
time and because many fields employ the concept of reliability to protect
against natural hazards, the ST(t) function is also relevant for
planning purposes in this context (see Read and Vogel (2015) for further
discussions relating to flood management and design). Figure 2 illustrates
ST(t) for a po = 0.002 event with a fixed Cx = 0.75
representing a slightly less variable system and a range of increasing
trends (M = 1.02, 1.1, 1.25) compared with stationary conditions
(M = 1). Points represent simulated failure times using Monte Carlo
analysis and lines are theoretical ST(t) from Eq. (13). The agreement
between the simulations and theoretical lines verifies the assumption that
h(t) = pt. Additionally, Fig. 2 shows that even a small trend
significantly reduces the system reliability compared with our expectations
under stationary conditions. For example, the reliability of a structure
designed to protect against a 500-year event under stationary conditions
after 50 time periods is quite high (ST(t) = 0.90); however, as M
increases, the reliability decreases significantly, approaching 0 for
M = 1.1 and 1.25 at t = 50. This suggests that if one was
designing infrastructure to withstand a particularly large magnitude event
over a planning period, under nonstationary conditions, the design would need
to be significantly larger, and it may not even be possible to design a
structure to achieve the same reliability as expected under stationary conditions.
Survival function ST(t) for the nonstationary GP2 model,
po = 0.002 and Cx = 0.75, for a range of trend values
(M = 1, 1.02, 1.1, 1.25); lines represent theory (Eq. 13) and points
are empirical values from Monte Carlo simulation.
A unique tool offered by HFA that can provide advancements in planning for
nonstationary natural hazards is the cumulative hazard function H(t), which
represents the total hazard over a given amount of time (Wienke, 2010). For
example, if po = 0.002, as expected under stationary conditions,
H(t) = 1 for t = 500, or we will experience, on average, one
exceedance event every 500 years. However, if a trend with a magnification
factor M = 1.1 (d = 10) is introduced in the same system, the
time it takes for H(t) = 1 is about 36 time periods, or another view,
H(t) = 333 events for t = 500. Figure 3 illustrates these
interpretations for two exceedance event sizes, fixed Cx = 0.75:
(a) po = 0.002, showing the number of time periods until H(t) = 1;
and (b) po = 0.001, showing the total number of events over time. In
Fig. 3a, we note that the stationary M = 1 line corresponds with the
H(t) = 1 for t = 500 as expected, and that as M increases, the
time until an exceedance event occurs dramatically decreases (note the log
x axis scale). Figure 3b depicts a similar story but illustrates an
alternate interpretation: the total number of exceedance events over a time
period for the rarer po = 0.001 event, where H(t) ranges from 1
for t = 1000 as expected under stationary conditions to
H(t) = 10 + events in fewer than 50 time periods with a large M.
Cumulative hazard function H(t) for the nonstationary GP2 model,
with a fixed Cx = 0.75 for a range of trend values (M = 1,
1.02, 1.1, 1.25); panels show two different size exceedance events:
(a) po = 0.002 and (b) po = 0.001.
When one wishes to communicate the risk of failure and event likelihood, the
cumulative hazard function is a useful metric for describing total risk (or
reliability) over a certain planning horizon. While our analysis assumes that
the trend would increase over the entire time period, perhaps a “worst
case” scenario, our results show that in the presence of an increasing trend in the
POT series, we may experience far more exceedance events than expected under
stationary conditions. Ignoring such trends may result in significant
increased damages and losses from under-design of infrastructure or
insufficient planning in populated areas.
After computing ST(t) and H(t) we can easily use Eq. (1) to determine the
PDF of the time to failure distribution for the nonstationary GP2 model by
Eq. (15). Since we are interested in the behavior of fT(t) due to
trends on a range of physical systems and for extreme events, we plot
fT(t) for a fixed Cx = 0.75 in Fig. 4, for a range of
increasing trends (M = 1, 1.02, 1.1, 1.25) and two event sizes
(po = (a) 0.01 and (b) 0.001). We note that the shape of fT(t)
evolves from the expected exponential curve under stationary conditions, to a
more symmetric or normally distributed shape as M increases. These results
complement those by Read and Vogel (2015) who show similar behavior for a
nonstationary two-parameter lognormal model of an AMS series of floods.
PDF of the time to failure distribution for the nonstationary
GP2 model, with a fixed Cx = 0.75 for a range of trend values
(M = 1, 1.02, 1.1, 1.25); panels show three exceedance event sizes
increasing in extremity: (a) po = 0.01 and (b) po = 0.001.
Similarly, in Fig. 5 we fix the trend at M = 1.05 and explore the
behavior of fT(t) over a realistic range of Cx values (0.5, 0.75,
1.5), for the same two event sizes (po = (a) 0.01 and (b) 0.001). As
consistent with Fig. 1 showing h(t) for various Cx values, the shape
of fT(t) in less variable systems (lower Cx) is more impacted by a
trend than a more variable system, as indicated by the sharp peaks and shift
in timing of the peaks (Fig. 5).
Our investigation of the behavior of fT(t) for the nonstationary GP2
model indicates that the shape and timing of the distribution changes with
both the magnitude of the trend and variability of natural hazard. We also
note that fT(t) exhibits complex patterns under nonstationary
conditions, e.g., fT(t) is less impacted in shape/timing by M for
smaller events (po), and that the presence of a trend leads to a range
of shapes (approaching normal for large positive M) of the time to failure
distribution. This more complicated behavior implies that under the premise
of nonstationarity, we can no longer assume the failure time distribution is
exponential in shape and that the MTTF is equal to 1/p. In fact, the mean
of the distribution of T is no longer a sufficient statistic as is the case
under stationary conditions. We are the first to document such changes in the
context of natural hazards for the GP2 distribution and anticipate that
others will continue to do so for other events and distributions that exhibit
nonstationarity. Using the derivations of h(t), ST(t), H(t), and
fT(t) that we have provided here, one can use knowledge of the system
(M, Cx) and existing design metrics (po and reliability
standards) in combination with HFA to better understand and characterize
natural hazards as they change through time.
Summary and conclusions
We have presented a general introduction to the probabilistic analysis of
nonstationary natural hazards using the well-developed field of HFA. We cite numerous sources of evidence which suggests
that the magnitudes of many natural hazards are increasing, thus requiring
new tools for conducting frequency analysis. To the authors' knowledge, the
analysis and discussion presented here provides the first formal
probabilistic analysis that draws a mathematical linkage between the
magnitude of a natural hazard event (X) with the distribution of failure
times (T) for the occurrence of some event exceeding a design threshold (xo).
Through the lens of HFA, we have presented an alternative
methodology for characterizing the likelihood of a natural hazard whose event
magnitudes follow a GP2 distribution and are increasing over time. By
explicitly linking properties of the time to failure T with the exceedance
probability p of a natural hazard (X), we have derived the primary hazard
analysis equations: the hazard function h(t), the survival (reliability)
function ST(t), the cumulative hazard function H(t), and the PDF of
the time to failure distribution fT(t) corresponding to a POT series of
natural hazards which follow the GP2 distribution. We demonstrate that the
assumption h(t) = pt is valid for the case presented by comparing
Monte Carlo simulated failure times from a GP2 random variable with
FX(xo; θ(t)) with HFA theory. We parameterize this GP2 model
such that it only depends on the design exceedance probability at time
t = 0, po, the known system variability Cx, and the
magnification factor M, and we use this to explore the impact of positive
trends on the reliability or survival ST(t) until an exceedance event.
PDF of the time to failure distribution for the nonstationary
GP2 model, with a fixed M = 1.05 for a range of trend values
(Cx = 0.5, 0.75, 1.50); panels show three exceedance event sizes
increasing in extremity: (a) po = 0.01 and (b) po = 0.001.
Findings of this investigation suggest that under nonstationary conditions,
medium and large events could occur with much greater frequency than under
stationary conditions (no trend). We find that the total number of hazards as
characterized by H(t) within a given planning period may substantially
increase in the presence of a positive trend in the POT magnitudes of a
natural hazard. Perhaps most importantly, under nonstationary conditions, the
distribution of the time to failure is no longer exponentially distributed,
and instead takes on a distribution with different shapes, depending on the
variability of the hazard and the magnitude of the trend. As trend magnitudes
increase for a range of event sizes (po), the shape of the distribution
of the survival time approaches normality and exhibits a sharp peak with a
heavy upper tail. We also find that variability impacts the shape and timing
of this peak in fT(t), such that less variable systems (lower Cx)
are more affected by larger M values, i.e., produce a more pronounced peak
and a greater shift in timing.
The implications of these findings for planning and design for nonstationary
natural hazards are significant. Given a historic (or future) increasing
trend in the magnitudes of a particular hazard, we should prepare to experience
exceedance events much more frequently (Obeysekera and Salas, 2016). Through
exploration of ST(t) for various trend factors, we also note that
determining the reliability of a system over time is more complicated given
uncertainty in the magnitude of the trend and how it will manifest.
In either case, continuing to assume stationary conditions when computing
system reliability for design purposes, when a positive trend in the POT
magnitudes has been observed historically, may pose a significant risk to the
populations and infrastructure in that region. Thus we recommend that design
practices should be reviewed and adapted for cases where nonstationary
behavior of natural hazards is evident in order to avoid under-design (also
see Vogel et al., 2013).
Overall, we have shown that HFA provides a set of tools for understanding the
probabilistic behavior of nonstationary natural hazards for application to a
wide range of natural phenomena. We intend for this analysis to inform future
work on modeling nonstationary natural hazards with HFA, for example by
developing other models that may include co-variates (Villarini et al.,
2013), extensions to AMS series, and also exploring the impact of decreasing
trends. We expect that additional research on this topic will contribute to
the emerging conversation on planning for nonstationary natural hazards and
shed light on innovative methods to determine best practices for
infrastructure design. Results of this work further support the need for a
risk-based decision analysis framework for selecting a design event under
nonstationarity (Rosner et al., 2014). Such a framework can provide guidance
in choosing infrastructure that minimizes the risk of under-design
(protection) and over-design (excess spending) through probabilistic decision analysis.