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Abstract

100 years from its original definition by Fuller [1914], the probabilistic concept of return period is widely used in hydrology as well as in other disciplines of geosciences to give an indication on critical event rareness. This concept gains its popularity, especially in engineering practice for design and risk assessment, due to its ease of use and understanding; however, return period relies on some basic assumptions that should be satisfied for a correct application of this statistical tool. Indeed, conventional frequency analysis in hydrology is performed by assuming as necessary conditions that extreme events arise from a stationary distribution and are independent of one another. The main objective of this paper is to investigate the properties of return period when the independence condition is omitted; hence, we explore how the different definitions of return period available in literature affect results of frequency analysis for processes correlated in time. We demonstrate that, for stationary processes, the independence condition is not necessary in order to apply the classical equation of return period (i.e. the inverse of exceedance probability). On the other hand, we show that the time-correlation structure of hydrological processes modifies the shape of the distribution function of which the return period represents the first moment. This implies that, in the context of time-dependent processes, the return period might not represent an exhaustive measure of the probability of failure, and that its blind application could lead to misleading results. To overcome this problem, we introduce the concept of Equivalent Return Period, which controls the probability of failure still preserving the virtue of effectively communicating the event rareness. This article is protected by copyright. All rights reserved.
WATER RESOURCES RESEARCH, VOL. ???, XXXX, DOI:10.1002/,
100 years of Return Period: Strengths and
limitations
E. Volpi,1A. Fiori,1S. Grimaldi,23 F. Lombardo,1and D. Koutsoyiannis4
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Corresponding author: E. Volpi, Department of Engineering, University of Roma Tre, Via Vito
Volterra, 62, 00146 Roma, Italy (elena.volpi@uniroma3.it)
1Department of Engineering, University of
Roma Tre, Via V. Volterra, 62, 00146
Rome, Italy
2Department for Innovation in Biological,
Agro-food and Forest systems (DIBAF),
University of Tuscia, Via San Camillo De
Lellis snc, 01100 Viterbo, Italy
3Honors Center of Italian Universities
(H2CU), Sapienza University of Rome, Via
Eudossiana 18, 00184 Roma, Italy
4Department of Water Resources, Faculty
of Civil Engineering, National Technical
University of Athens, Heroon Polytechneiou
5, 15780 Zographou, Greece
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: 100 YEARS OF RETURN PERIOD X - 3
Abstract. 100 years from its original definition by Fuller [1914], the prob-1
abilistic concept of return period is widely used in hydrology as well as in2
other disciplines of geosciences to give an indication on critical event rareness.3
This concept gains its popularity, especially in engineering practice for de-4
sign and risk assessment, due to its ease of use and understanding; however,5
return period relies on some basic assumptions that should be satisfied for6
a correct application of this statistical tool. Indeed, conventional frequency7
analysis in hydrology is performed by assuming as necessary conditions that8
extreme events arise from a stationary distribution and are independent of9
one another. The main objective of this paper is to investigate the proper-10
ties of return period when the independence condition is omitted; hence, we11
explore how the different definitions of return period available in literature12
affect results of frequency analysis for processes correlated in time. We demon-13
strate that, for stationary processes, the independence condition is not nec-14
essary in order to apply the classical equation of return period (i.e. the in-15
verse of exceedance probability). On the other hand, we show that the time-16
correlation structure of hydrological processes modifies the shape of the dis-17
tribution function of which the return period represents the first moment.18
This implies that, in the context of time-dependent processes, the return pe-19
riod might not represent an exhaustive measure of the probability of failure,20
and that its blind application could lead to misleading results. To overcome21
this problem, we introduce the concept of Equivalent Return Period, which22
D R A F T September 22, 2015, 2:23pm D R A F T
X-4 : 100 YEARS OF RETURN PERIOD
controls the probability of failure still preserving the virtue of effectively com-23
municating the event rareness.24
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: 100 YEARS OF RETURN PERIOD X - 5
1. Introduction
“The storm event had a return period of 30 years” or “this dam spillway was designed25
for a 1000-year return period discharge” are two classical statements that one could read26
or hear everyday. High-school students could read them in newspapers, housewives could27
hear them at the market or hydrologists could write them in a technical report. This simple28
example recalls that the return period is the most ubiquitous statistical concept adopted29
in hydrology but also in many other disciplines (seismology, oceanography, geology, etc...).30
It appears that the concept of return period was first introduced by Fuller [1914] who31
pioneered statistical flood frequency analysis in the USA. Return period finds wide pop-32
ularity mainly because it is a simple statistical tool taken from engineering practices33
[Gumbel, 1958]. For example, engineers who work on flood control are interested in the34
expected time interval at which an event of given magnitude is exceeded for the first time,35
which gives a definition of the return period. Another common definition is the average of36
the time intervals between two exceedances of a given threshold of river discharge. From37
a logical standpoint, the first definition is as justifiable as the second one; they generally38
differ, even though they become practically indistinguishable if consecutive events are39
independent in time. Both are used in hydrology [Fern´andez and Salas, 1999a, b] and, in40
this paper, we will show how they may affect the frequency analysis applications under41
certain conditions.42
The return period is inversely related to the probability of exceedance of a specific43
value of the variable under consideration (e.g. river discharge). For example, the annual44
maximum flood-flow exceeded with a 1% probability in any year is called the 100-year45
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flood. Therefore, a T-year return period does not mean that one and only one T-year46
event should occur every Tyears, but rather that the probability of the T-year flood being47
exceeded is 1/T in every year [Stedinger et al., 1993].48
The traditional methods for determining the return period of extreme hydrologic events49
assume as key conditions that extreme events (i) arise from a stationary distribution, and50
(ii) are independent of one another. The hypotheses of stationarity and independence51
are commonly assumed as necessary conditions to proceed with conventional frequency52
analysis in hydrology [Chow et al., 1988]. Recently, the former assumption has been53
questioned by several researchers [e.g. Cooley, 2013; Salas and Obeysekera, 2014; Du54
et al., 2015; Read and Vogel, 2015]. However, we endorse herein the following important55
statement by Gumbel [1941] about the general validity of stationarity assumption. “In56
order to apply any theory we have to suppose that the data are homogeneous, i.e. that no57
systematical change of climate and no important change in the basin have occurred within58
the observation period and that no such changes will take place in the period for which59
extrapolations are made. It is only under these obvious conditions that forecasts can be60
made”. The reader is also referred to Koutsoyiannis and Montanari [2015] and Montanari61
and Koutsoyiannis [2014], where it can be noted that many have lately questioned the62
stationarity assumption, but careful investigation of claims made would reveal that they63
mostly arise from the confusion of dependence in time with nonstationarity.64
The purpose of this paper is to investigate the properties of return period when the65
independence condition is omitted. In hydrology, indeed, dependence has been recognized66
by many scientists to be the rule rather than the exception since a long time [e.g. Hurst,67
1951; Mandelbrot and Wallis, 1968]. The concept of dependence in extreme events relates68
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to the fact that the occurrence of a high or low value for the variable of interest (e.g.69
river discharge) has some influence on the value of any succeeding observation. Leadbetter70
[1983] found that the type of the limiting distribution for maxima is unaltered for weakly71
dependent occurrences of extreme events. We demonstrate that, under general depen-72
dence conditions, the classical relationship between the return period and the exceedance73
probability is again unaltered. On the other hand, we investigate the impact of the de-74
pendence structure on the shape of the distribution function of which the return period75
represents the first moment.76
Based on the papers by Fern´andez and Salas [1999a], Sen [1999], and Douglas et al.77
[2002] we first summarize in Section 2 the available definitions of return periods (aver-78
age occurrence interval - and - average recurrence interval) specifying the mass function79
equations and the related return period formulae. Moreover, in Section 2.2 and 2.3 the80
independent and time-dependent cases are analyzed in detail, while an Appendix provides81
the proof that the widely used return period equation (average recurrence interval) is not82
affected by the dependence structure of the process of interest. However, in Section 2.383
it is pointed out that the time-dependence influences the shape of the interarrival time84
distribution function and the probability of failure.85
Two illustrative examples, i.e. using a two-state Markov process and an autoregressive86
process, are described in Section 3 and results are discussed in Section 4 in order to in-87
vestigate further the theoretical premises depicted in Sections 2.2 and 2.3. Besides, to88
overcome the difficulties that arise from the application of the return period concept in a89
time-dependent context, we propose in Section 4.1 the adoption of an Equivalent Return90
Period (ERP ), which resembles the classical definition of return period in the case of in-91
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dependence while it is able to control the probability of failure under the time-dependence92
condition. The ERP can be useful to avoid introducing the concept of probability of fail-93
ure in engineering practice. Indeed, the latter may not be as simple to understand as the94
return period, which is a well-established concept in applications, routinely employed by95
practitioners.96
Concluding remarks discuss the obtained results by stressing caution against using the97
concept of return period blindly given that multiple definitions exist. However, we confirm98
the virtue of return period showing that the classical formulation is insensitive to the time-99
dependence condition.100
2. Return period and probability of failure
2.1. Mathematical framework
Let Z(τ) be a stochastic process that characterizes a natural process typically evolving101
in continuous time τ. As observations of Z(τ) are only made in discrete time, it is assumed102
here that the observations are made at constant time intervals ∆τ, and this interval is103
considered the unit of time. Hence, we consider the corresponding discrete-time process104
that is obtained by sampling Z(τ) at spacing ∆τ, i.e. Zj=Z(jτ) where j(= 1,2, ...)105
denotes discrete time. For convenience, herein we express discrete time as t=jj0,106
where j0is the current time step; therefore the discrete-time process is indicated as Zt
107
and t= 0 denotes the present. We assume that Ztis a stationary process [Papoulis, 1991];108
thus, it is fully described up to the second order properties by its marginal probability109
function and its autocorrelation structure. Generally, in this paper we use upper case110
letters for random variables or events, and lower case letters for values, parameters, or111
constants.112
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We are interested in the occurrence of possible excursions of Ztabove/below a high/low113
level (threshold) z, which may determine the failure of a structure or system. In particular,114
we define a dangerous event as A={Z > z}, which is an extreme maximum; anyway,115
Acould be any type of extreme event, i.e. maximum or minimum. In the following we116
denote by pthe probability of the event B={Zz}, which is the complement of A; the117
probability of the event Ais given by 1 p= Pr {Z > z}= Pr A.118
In hydrological applications, it is usually assumed that the event Awill occur on average119
once every return period T, where Tis a time interval and, for annual observations (i.e.,120
τ= 1 year), a number of years. In other words, the average time until the threshold z121
is exceeded equals Tyears [Stedinger et al., 1993], such as122
T
τ= E [X] =
t=1
t fX(t) (1)
where Xis the number of discrete time steps to the occurrence of an event A,fX(t) =123
Pr {X=t}is its probability mass function (pmf) and E[.] denotes expectation. The124
definition of the return period leads to the formulation of the so-called probability of125
failure R(l) (also known in literature as ”risk”, even if it does not account for damages)126
which measures the probability that the event Aoccurs at least once over a specified127
period of time: the design life l(e.g. in years) of a system or structure, where l/τis a128
positive integer. Mathematically, we have129
R(l) = Pr{Xl/τ}=
l/τ
t=1
fX(t) (2)
Thus, the probability of failure is nothing else than the distribution function FX(t) com-130
puted at t=l/τ.131
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As mentioned in the Introduction, two different definitions of the return period are132
available in the hydrological literature [see, e.g., Fern´andez and Salas , 1999a and Douglas133
et al., 2002]. The return period Tmay be defined as:134
(i) the mean time interval required to the first occurrence of the event A,135
(ii) the mean time interval between any two successive occurrences of the event A.136
Definition (i) assumes that an event Aoccurred in the past (at t < 0); the discrete time137
elapsed since the last event Ato the current time step t= 0 is defined as elapsing time and138
it is denoted here as te; the sketch in Figure 1 illustrates the variables used in the present139
analysis. In this work, we assume that time tecan be either deterministically known or140
unknown and investigate implications of both conditions on the analytical formulation of141
the return period. Under definition (i), the return period is based on the waiting time142
(W), i.e. the number of time steps between t= 0 and the next occurrence of A(see143
Figure 1). The sum of the waiting time and the elapsing time is denoted as interarrival144
time N=W+te.145
If we assume that teis unknown, the probability mass function of the waiting time146
is given by the joint probability of the sequence of events (B1, B2, ..Bt1, At) (see, e.g.,147
Fern´andez and Salas , 1999a)148
fW(t) = Pr (B1, B2, ..Bt1, At) (3)
where At(Bt) is the event A(B) occurred at time t. Instead, if teis determin-149
istically known, the pmf of the waiting time is given by the joint probability of150
the sequence of events (B1, B2, ..Bt1, At) conditioned to the realization of the events151
(Ate, Bte+1, ..B1, B0) occurred at t0, i.e.152
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fW|te(t) = Pr (B1, B2, ..Bt1, At|Ate, Bte+1, ..B1, B0) (4)
=Pr (Ate, Bte+1, ..B1, B0, B1, B2, ..Bt1, At)
Pr (Ate, Bte+1, ..B1, B0)
Definition (ii) assumes that an event Ahas just occurred at t= 0. In such a case te= 0153
and the waiting time Wis identical to the interarrival time N. The pmf of the interarrival154
time fNis therefore a special case of equation (4), for te= 0, i.e.155
fN(t) = Pr (B1, B2, ..Bt1, At|A0) (5)
=Pr (A0, B1, B2, ..Bt1, At)
Pr A0
Note that Figure 1 depicts a more general case than the one represented by equation (5).156
In the Figure, we assume that two successive occurrences of the dangerous event Aare157
at times teand t. Then, Nis the time elapsed between the two. As stated above, the158
specific case expressed by equation (5) can be obtained by setting te= 0. Moreover, we159
stress here that the relation N=W+tein the Figure holds only in the case the elapsing160
time teis known, i.e. when we account for the conditional waiting time W|te.161
It is interesting to note that the probability distributions of the unconditional (W, equa-162
tion (3)) and conditional (W|te, equation (4)) waiting time are interrelated. In Appendix163
A we derive some useful relations between the return periods TW,TW|teand TN.164
Substituting fW(equation (3)), fW|te(equation (4)) or fN(equation (5)) to fXin (1)165
and (2), we obtain the expressions of the return periods TW,TW|teand TNand of the166
corresponding probabilities of failure RW,RW|teand RN, respectively. In general, the167
probability mass functions given by equations (3) to (5) are expected to have different168
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shapes, leading to different values of the return period of the event A. In the following,169
we illustrate and discuss the differences among the above definitions when varying the170
correlation structure of the process Zt; specifically, we study first the independent case,171
which is customary in hydrological applications, and then the more general case with some172
positive correlation in time (persistent case).173
2.2. Independent case
If Ztis a purely random process, then its random variables are mutually independent174
and their joint probability distribution equals the product of marginal ones. Therefore, we175
may write e.g. Pr (B0, B1, B2, ..Bt1, At) = Pr B0Pr B1... Pr At. Substituting in equations176
(3), (4) and (5) the products of the marginal exceedance or non-exceedance probabilities177
and thanks to the stationarity assumption (that implies Pr At= 1 pand Pr Bt=p178
for any t), we can derive the same geometric distribution in all cases. Therefore, fW=179
fW|te=fN=f, with180
f(t) = pt1(1 p) (6)
It follows from equation (1) that the return period T(T=TW=TW|te=TN) is given181
by182
T
τ=1
1p(7)
while the variance of the pmf (6) is v=p/ (1 p)2. From equation (6), it also follows183
that the probability of failure given by equation (2) becomes184
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R(l) = 1 1τ
Tl/τ
= 1 pl/τ(8)
where again R=RW=RW|te=RN.185
Thus, for the independent case all the definitions of return period collapse to the same186
expression (7). This result, which is well known in the literature [e.g. Stedinger et al.,187
1993], builds on the fact that in the independent case the occurrence of an event at any188
time t0 does not influence what happens afterwards.189
2.3. Persistent case
Although independence of Ztis usually invoked for the derivation of equation (7) [e.g.190
Kottegoda and Rosso, 1997, p. 190], it is possible to show that the mean interarrival time191
TNis equal to (7) also in case of processes correlated in time; the general proof, which192
is given here for the first time, is illustrated in detail in Appendix B. The same property193
was shown by Lloyd [1970] for the particular case of a Markov chain process. As shown194
in Appendix B, equation (7) for the mean interarrival time holds true, regardless of the195
type of the correlation structure of Zt.196
Even though the dependence structure of the process Ztdoes not affect the expected197
value of N(i.e., TN), we show that this is not the case with its pmf fN(see equation198
(5)). Let us consider a process characterized by a positive correlation in time. If a199
dangerous event Aoccurs at t= 0, then the conditional probability of occurrence of200
another dangerous event at t= 1 will be greater than 1 p(independent case); this201
yields that the probability mass function fN(t) will have a larger mass for t= 1 and a202
lower mass elsewhere with respect to the independent case (equation (6)). Hence, while203
the mean value remains the same, the variance of the interarrival time Nis larger than204
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that of the independent case and it increases with the temporal correlation. This implies205
that the probability of failure RN(following equation (2)) is strongly affected by the206
time-dependence structure of the process.207
Conversely, the return periods TWand TW|tedo account for the temporal correlation of208
Zt. Recalling that (1 p) = 1/E [N] (see equations (7) and (1)), it can be shown that209
(see Appendix A, equation (A8))210
TW
TN
=1
2E [N2]
E [N]2+1
E [N](9)
Equation (9) shows that TWis greater than or equal to TN. It is easy to check that211
TW=TNfor independent processes, in line with the discussion reported under Section212
2.2. When the process is correlated in time, the term E [N2]/E [N]2is expected to increase213
with the autocorrelation of the process, thus resulting in the inequality TW> TN. Hence,214
the mean waiting time is generally larger than the mean interarrival time for temporally215
correlated processes.216
In the following Sections we will examine the pmfs of the waiting times Wand W|teand217
the interarrival time N, as well as their average values (TW,TW|teand TN), as functions218
of the temporal correlation of the process. To this end, we make use of two different219
illustrative examples, the first is based on a Markov chain, while the second uses an220
AR(1) model. For convenience - and without loss of generality - ∆τis set equal to one.221
3. Illustrative examples
3.1. Example 1: two state Markov-dependent process
We consider here a stochastic process Ztwhich is based on a Markov chain Yt. This222
process is considered here since it allows to easily derive the analytical expressions of223
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the probability mass functions of the waiting and interarrival times, as done in previous224
literature works by Lloyd [1970], Rosbjerg [1977] and Fern´andez and Salas [1999a]. The225
Markov chain Ythas two states, which here represent the events At={Zt> z}and226
Bt={Ztz}with probability 1 pand p, respectively. For the Markov property, the227
probability of a state at a given time tdepends solely on the state at the previous time228
step t1, e.g. Pr (Bt|Bt1...B0) = Pr (Bt|Bt1). Applying the chain rule to the Markov229
property (e.g. Papoulis, 1991, p. 636), it follows that the joint probability of a sequence230
of states, e.g. Pr (B1, B2, ..Bt) = Pr{Z1z, Z2z, ..., Ztz}, can be written as231
Pr (B1) Pr (B2|B1)... Pr (Bt|Bt1) = Pr{Z1z}Pr{Z2z|Z1z}... Pr{Ztz|Zt1232
z}.233
The process Ztdescribed above is indicated in the following as two state Markov-234
dependent process and denoted by 2Mp. For each value of p(i.e. of z)Ztis fully charac-235
terized by the marginal probabilities of the states Aand B(1 pand p) and by the tran-236
sition probability matrix, M= [[Pr (At+1|At),Pr (At+1|Bt)],[Pr (Bt+1|At),Pr (Bt+1|Bt)]]237
where Pr (At+1|At) + Pr (Bt+1|At) = 1 and Pr (At+1 |Bt) + Pr (Bt+1|Bt) = 1. We denote238
by qthe joint probability of non-exceedance of the threshold value zfor two successive239
events, i.e. q= Pr (Bt+1 , Bt) for any t; it ensues that M= [[1 (pq)/(1 p),1240
q/p],[(pq)/(1 p), q/p]].241
The probability mass function of the unconditional waiting time fW(equation (3))242
becomes243
fW(t) = 1p(t= 1)
pq
pt21q
p(t2) (10)
with mean given by244
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TW= 1 + p2
(pq)(11)
and variance var [W] = p2(pp2+q)/(pq)2. After substituting equation (10) in (2),245
the probability of failure in a period of length lis given by246
RW(l) = 1 pq
pl1
(12)
while the pmf of the conditional waiting time fW|te(equation (4)) for te>0 reduces to247
fW|te(t) = q
pt11q
p(13)
with mean248
TW|te=p
(pq)(14)
and variance var [W|te] = pq/(pq)2. The probability of failure based on the conditional249
waiting time is given by250
RW|te(l) = 1 q
pl1
(15)
Equation (14) shows how for the 2Mp model the mean waiting time distribution is not251
affected by the value of te. This builds upon the fact that the conditional non-exceedance252
probability at tdepends only on that at t1, due to the property of the Markov chain.253
Finally, the pmf of the interarrival time N(equation (5)) assumes the following expres-254
sion255
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fN(t) = 1(pq)/(1 p) (t= 1)
(pq)
(1p)q
pt21q
p(t2) (16)
while its mean is given by equation (7) with ∆τ= 1 (following the general proof given in256
Appendix B), and the variance is equal to var [N] = p(p2p2+q)/[(p1)2(pq)]. The257
probability of failure in a period of length lis given by258
RN(l) = 1 pq
1pq
pl1
(17)
The joint probability qmay assume values in the range [max(2p1,0) , p]: the lower259
and upper bounds correspond to perfect negative and positive correlations in time, respec-260
tively; in the independent case, q=p2. We consider here only processes positively corre-261
lated (i.e. persistent), as it is commonly the case in hydrology (e.g. rainfall and discharge);262
thus, q[p2, p]. Furthermore, we assume that Ztis a standard Gaussian process and that263
the joint probability qis ruled by a bivariate Gaussian distribution; under the latter264
assumption, qcan be described in terms of the lag-1 autocorrelation coefficient ρ. Specifi-265
cally, qis computed as q= Pr {Zt+1 z, Ztz}=z
−∞ z
−∞ fZ(ztzt+1;0,Σ2)dzt+1dzt
266
where fZis the probability density function of the bivariate Gaussian distribution267
N2(Z;0,Σ2) with zero mean and Σ2={{1, ρ},{ρ, 1}}, with ρ[0,1]. Note that ρ268
denotes the correlation in the parent process Ztand not that between the events exceed-269
ing the threshold, i.e. A={Z > z}. The correlation between the extremes is ruled by270
the shape of the parent bivariate distribution, which is assumed here to be Gaussian; the271
latter assumption implies that the correlation between the events Ais negligible to null272
for high threshold values, since the Gaussian process is asymptotic independent.273
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3.2. Example 2: AR(1) process
We now assume that Ztfollows an AR(1) process (first-order autoregressive process), i.e.274
Zt=ρZt1+αtwhere ρis the lag-1 correlation coefficient and αt∼ N 0,1ρ2, such275
that the process is characterized by a multivariate Gaussian distributions Nt(Z;0,Σt)276
with Z={Z1, Z2..Zt}and Σt=ρ|ik|,i, k = 1..t. We assume again ρ[0,1].277
Even if conceptually simple and similar to the 2Mp (see e.g. Saldarriaga and Yevjevich,278
1970), AR(1) is rather different in terms of the pmfs fW,fW|teand fN. Both the processes279
are based on the Markov property; however, in AR(1) the Markov property applies to the280
continuos random variable Zand not to the state Y={Zz}. It means that in AR(1)281
the joint probability fZ(z1, z2..zt) can be expressed as fZ(z1)fZ(z2|z1)..fZ(zt|zt1), while282
the same simplification cannot apply to the joint probability of a sequence of states, e.g.283
Pr (B1, B2, ..Bt) = Pr{Z1z, Z2z, ..., Ztz}, as for 2Mp. The joint probability of284
any sequence can be estimated by proper integration of the joint pdf of the multivariate285
Gaussian distribution Nt. This entails that the pmfs fW,fW|teand fN, given by equations286
(3), (4) and (5) respectively, cannot be simplified as in the case of 2Mp, but they can be287
written as288
fW(t) = z
−∞ z
−∞
.. +
z
fZ(z1, z2..zt;0,Σt)dz1dz2..dzt(18)
fW|te(t) = +
zz
−∞ .. +
zfZ(ztezte+1..zt;0,Σt+te)dztedzte+1..dzt
+
zz
−∞ .. z
−∞ fZ(ztezte+1..z0;0,Σte)dzte..dz0
(19)
while fNcan be derived from the latter under the assumption te= 0. Finally, substituting289
the previous expressions in (1) and (2) we get the corresponding return periods and290
probabilities of failure.291
D R A F T September 22, 2015, 2:23pm D R A F T
: 100 YEARS OF RETURN PERIOD X - 19
Interestingly enough, unlike the 2Mp, fW|te(19) depends on te, i.e. the elapsing292
time. This relies on the fact that the conditional non-exceedance probability at t, i.e.293
Pr (Bt|Bt1...B0), generally depends on the whole sequence of previous events for AR(1),294
while it only depends on that at t1 for the 2Mp. In such a sense, AR(1) is more295
correlated than 2Mp.296
4. Results and discussion
We start this Section by discussing the effects of temporal correlation on the probability297
mass functions fW(equation (10)) and fN(equation (16)), and the related return periods298
TW,TN(equation (11) and (7) with ∆τ= 1, respectively) for the two state Markov-299
dependent process (2Mp).300
Figure 2 illustrates TWand TNas functions of the independent return period T(i.e. of301
the non-exceedance marginal probability p) for several values of the correlation coefficient302
ρ. It is seen that TNequals T, being independent of ρas demonstrated in Appendix B;303
for ρ= 0 (black line) it is always TN=TW=T= (1 p)1. Conversely, the mean304
waiting time TWincreases with ρ(equation (9)); TWis always greater than the mean305
interarrival time TN, which thus represents a lower bound for the return period (Figure306
2a). Specifically, for values of Taround 5, TWis roughly eight times larger than TNfor307
ρ= 0.99 and about twice for ρ= 0.75; for small and very large values of T(i.e. for308
small and high values of the threshold z, respectively) TWtends to the independent limit309
T= (1 p)1(Figure 2b).310
As discussed in Section 2.3, although TN=Tfor any ρ, the pmf fN(as well as fW) may311
be significantly influenced by the correlation structure of the Ztprocess. The distribution312
functions of Wand Nare illustrated in Figure 3, for various values of ρand p= 0.9.313
D R A F T September 22, 2015, 2:23pm D R A F T
X - 20 : 100 YEARS OF RETURN PERIOD
The mean values for each distribution (i.e. the return periods normalized with respect to314
τ= 1) are denoted by the vertical dashed lines. The broadness of both distributions315
increases with ρ, as also indicated by the increase of their variance and skewness (not316
shown).317
Figure 3a shows that the distribution function computed at TW, which corresponds to318
the probability of failure in the period TW(see equation (2)), is independent of ρtaking319
approximately the value 0.63 for high values of p[Stedinger et al., 1993].320
On the other hand, FNchanges dramatically when increasing temporal correlation ρ.321
This may result in very high values of the probability of failure for the same TN, even322
for small time intervals t(Figure 3b). Thus, although the return period TNremains the323
same for correlated and independent processes (all the vertical dashed lines corresponding324
to the different values of ρcollapse into a unique line, depicted in black), the probability325
that the threshold zis exceeded in the period TNcan be much larger for the former than326
for the latter (up to about 0.9 for the limit case ρ= 0.99).327
We now illustrate and discuss the probability functions for W,W|teand Nfor the AR(1)328
process, as well as the corresponding mean values, as functions of the lag-1 autocorrelation329
coefficient ρ. Results are compared to those obtained for the previously analyzed 2Mp330
case.331
The probability mass functions fW(equations (18)) and fN(equation (19) for te= 0)332
for AR(1) are similar to those for 2Mp, even if they are characterized by a much larger333
dispersion, and thus they are not shown here. Their averages TWand TNare depicted334
in Figure 4, as function of the independent return period T, for ρ= 0.75 and ρ= 0.99.335
TWand TNfor AR(1) (continuos lines) are also compared to those pertaining to the 2Mp336
D R A F T September 22, 2015, 2:23pm D R A F T
: 100 YEARS OF RETURN PERIOD X - 21
(dashed lines). The mean waiting times TWfor the two models are similar, although TW
337
is generally larger for AR(1); since the two processes have the same ρ, this result is a338
direct consequence of the stronger correlation of AR(1) with respect to 2Mp, as explained339
in previous Section. Larger differences are expected for even more persistent processes,340
i.e. processes characterized by a longer range persistence with respect to the AR(1).341
As mentioned in the previous Section, the stronger correlation of AR(1) also influences342
the mean conditional waiting time TW|te, which depends on the elapsing time tein contrast343
to that of 2Mp. TW|te(te) is illustrated in Figure 5 for p= 0.9 and for a few values of the344
correlation coefficient ρ. For each value of ρ,TW|teis by definition equal to the mean inter-345
arrival time TNfor te= 0 (see equation (4)); TW|teincreases with tetending to an asymp-346
totic value that is greater than TW(dashed lines). This behaviour arises from the fact that347
the conditional non-exceedance probability (B1, B2, ..Bt1, At|Ate, Bte+1, ..B1, B0) (eq.348
4) depends on the whole sequence of previous events. However, as tebecomes very high349
the previous dangerous event Atehas occurred too distant in time to significantly affect350
the realization of the next event at time t; the latter is mainly controlled by a sequence351
of antecedent events whose length strictly depends on the shape of the autocorrelation352
function of the underling process Zt. Due to the exponential shape of the AR(1) auto-353
correlation function, i.e. ρt(t) = ρt,TW|teis expected to approach the asymptotic value354
when tebecomes larger than the integral scale of the process, λ(ρ) = 1/(1 ρ).355
Conversely, TW|tefor 2Mp maintains a constant value for any te>0 since the con-356
ditional joint probability in equation (4) Pr(B1, B2, ..Bt1, At|Ate, Bte+1, ..B1, B0) de-357
pends only on the state at t= 0, due to the Markov property of the Ytchain (as already358
D R A F T September 22, 2015, 2:23pm D R A F T
X - 22 : 100 YEARS OF RETURN PERIOD
discussed in Section 3.1); moreover, being influenced by a longer sequence of safe events359
(B), both TW|teand TWof AR(1) are larger than those of 2Mp (results not shown).360
We finally explore how the probabilities of failure RW(TW), RW|teTW|teand RN(TN)361
behave as functions of the correlation coefficient ρ; results are summarized in Figure 6 for362
the processes 2Mp and AR(1) and compared to the independent case. For both processes,363
the probability of failure based on the interarrival time (N) may assume values much364
larger than the independent case; RN(TN) significantly increases with the autocorrelation365
of the process ρ, (compare e.g. 2Mp for ρ= 0.75 and ρ= 0.99) and, more generally, with366
the correlation structure of the process (compare AR(1) and 2Mp for the same value of ρ).367
On the contrary, when we consider the waiting time W(conditional and unconditional),368
the probability of failure is less than the independent case. This reduction is significant369
when we account for the elapsing time te, thus when we add information about the last370
dangerous event occurred in the past. Note that Figure 6 specifically refers to the cases371
te= 10 for AR(1) while it is representative of any te>0 for 2Mp. As for AR(1),372
RW|teTW|tereduces with respect to the independent case when teis much larger than373
the integral scale of the process, i.e. te> λ when ρ= 0.75 (λ= 4) (Fig. 6a); conversely,374
when the event Ahas happened in the recent past (when ρ= 0.99, we have te< λ with375
λ= 100), the conditional waiting time for high phas a behaviour which approaches that376
of the interarrival time (i.e. with higher probability of failure than the independent case,377
as in Figure 6b).378
4.1. Equivalent Return Period (ERP)
The return period is a means of expressing the exceedance probability. Despite being379
a standard term in engineering applications (in engineering hydrology in particular), the380
D R A F T September 22, 2015, 2:23pm D R A F T
: 100 YEARS OF RETURN PERIOD X - 23
concept of return period is not always an adequate measure of the probability of failure381
and has been sometimes incorrectly understood and misused [Serinaldi, 2014]. The results382
discussed in previous Section strengthen the above message, extending it to correlated Zt
383
fields (with Markovian dependence); for the cases examined here, the statistics of the384
waiting or interarrival time show negligible differences with respect to the independent385
case for small values of ρ, while they are strongly affected by the autocorrelation when386
ρ&0.5 (see Figures 2 and 5). Consequently, using directly the probability of failure387
in engineering practice could be a better choice under the latter condition. However,388
although more effective and appropriate, the probability of failure may not be as simple389
to understand as the return period, which is already an established concept in applications390
and routinely employed by practitioners.391
To overcome this problem, we introduce the concept of ”equivalent” return period392
(ERP ). Its aim is to retain the relative simplicity of the return period concept and393
extend it to temporally correlated hydrological variables; for correlated processes, ERP394
is defined to be the period that would lead to the same probability of failure pertaining to395
a given return period Tin the framework of classical statistics (independent case). Hence,396
ERP resembles the classical definition of return period in the case of independence, thus397
preserving its simplicity and strength in indicating the event rareness; in addition it is398
able to control the probability of failure under the time-dependence condition.399
ERP can be defined starting from the concept of interarrival time (N) or waiting400
time (W). Practitioners should adopt the most appropriate definition according to the401
circumstances, the task and the data available. If the time teelapsed since the last402
dangerous event is known, it could be adopted the definition based on the conditional403
D R A F T September 22, 2015, 2:23pm D R A F T
X - 24 : 100 YEARS OF RETURN PERIOD
waiting time, or that based on the interarrival time in the case te= 0; the latter could404
be the case where an existing structure failed because of an event Aand the immediate405
construction of another structure is needed (as discussed by Fern´andez and Salas [1999a]).406
In the case we are accounting for the interarrival time (N), ERP can be calculated407
assuming RN(ERP ) = R(T) where RNis the probability of failure based on the inter-408
arrival time (equation (2) for fX=fN), while R(T) is given by equation (8) for l=T.409
For the 2Mp RNis given by equation (17) (where ∆τ= 1) when l=ERP ; thus, the410
analytical formulation of ERP can be easily derived as411
ERP = 1 + ln 1p
pq+1
1pln p
ln q
p
(20)
For the AR(1), RNcan be numerically computed by substituting equation (19) in (2).412
In the case of more complex models for the simulation of hydrological quantities, ERP413
could be computed directly by numerical Monte Carlo simulations.414
Figure 7 depicts the behaviour of ERP as function of T, for both the AR(1) (continuous415
lines) and 2Mp processes (dashed lines; equation (20) with p= 1 1/T ). The figure416
shows that the values of ERP and Ttend to coincide asymptotically; this is especially417
so for small correlation coefficients. For a given T, the value of ERP is always smaller418
(sometimes much smaller) than T; differences increase with the correlation coefficient ρ419
and with the correlation structure of the process (compare AR(1) to 2Mp). Recalling that420
T= 1/(1 p), Figure 7 can be used either to determine ERP when the p-th quantile zis
421
known (i.e., for a given event A={Z > z}that will be exceeded with probability 1 p)422
in risk assessment problems, or to determine the design variable (i.e. the threshold z) in423
terms of ponce the ERP is fixed in design problems; in the latter case we choose ERP424
D R A F T September 22, 2015, 2:23pm D R A F T
: 100 YEARS OF RETURN PERIOD X - 25
and then calculate the design variable z, such that the probability of failure is equal to425
that we should have in the independent case.426
We emphasize that results shown here are obtained under several assumptions, such as427
the type of temporal correlation, bivariate Gaussian distribution, etc.; this implies that, for428
example, a different distribution may result in larger differences between the independent429
and time-correlated conditions (due, e.g., to asymptotic dependence). Hence, further work430
is needed to generalize the above results.431
5. Conclusions
The return period is a critical parameter largely adopted in hydrology for risk assess-432
ment and design. It is defined as the mean value of the waiting time to the next dangerous433
event (TW) or the interarrival time between successive dangerous events (TN). As shown434
in previous literature, both definitions lead to the same result in the case of time inde-435
pendence of the underlying process. However, in cases of time-persistent processes the436
two definitions lead to different expressions. Hence, we reexamine herein the above defi-437
nitions in the context of temporally correlated processes; furthermore, by making use of438
two illustrative examples we discuss the effects of the temporal correlation ρof the parent439
process on the return period and the probability of failure. The examples proposed here440
are based on a two state Markov-dependent process (2Mp), and an AR(1) process; even441
if the two processes share the Markov property, they are characterized by rather different442
time distributions.443
The main conclusions drawn in this paper are listed below.444
We provide a unitary framework for the estimation of the return periods TW,TN
445
and the related probabilities of failure RW,RNin the context of persistent processes:446
D R A F T September 22, 2015, 2:23pm D R A F T
X - 26 : 100 YEARS OF RETURN PERIOD
we provide general relationships for the probability functions of the waiting time W(un-447
conditional and conditional on the time teelapsed since the last dangerous event) and448
the interarrival time N. The choice between Wand Nin applications depends on the449
available information on past events and the type of structure.450
We demonstrate that the mean interarrival time TNis not affected by the time-451
dependence structure of the process, e.g. the correlation coefficient ρ. Thus, the well452
known formula for independent processes is valid for any process, temporally correlated453
or not.454
Although TNis not affected by ρ, for persistent processes the corresponding proba-455
bility of failure can be much larger than that pertaining to the independent case, which is456
itself not negligible. Hence, the mean interarrival time TNcan easily provide a biased and457
wrong perception of the risk of failure, especially in the presence of temporally correlated458
hydrological variables.459
On the other hand, the mean waiting times effectively account for the correlation460
structure of the hydrological process. TWis always larger than the mean interarrival461
time TN, which acts as a lower bound. If the time tefrom the last dangerous event is462
deterministically known, we can use that information to condition the waiting time Wto463
the next occurrence.464
The return periods TWand TW|tetypically increase with the correlation ρ. Specif-465
ically, they depend on the overall correlation structure of the process, as highlighted by466
comparing results for 2Mp and AR(1); in the case of processes characterized by a longer467
range persistence with respect to the AR(1), we may expect even stronger differences.468
D R A F T September 22, 2015, 2:23pm D R A F T
: 100 YEARS OF RETURN PERIOD X - 27
The analyses carried out here provide some further insight into the overall meaning469
and significance of the return period, especially in view of hydrological applications, but470
also in other geophysical fields. Despite being a simple and easy to implement metric, the471
return period should be used with caution in the presence of time-correlated processes.472
Indeed, the probability of failure depends on the whole shape of the probability function,473
which in turn may strongly depend on ρ, and the return period is just the first order474
moment; the latter may not be relevant when in presence of asymmetric and skewed475
distributions, like e.g. some of those displayed in Figure 3.476
To partially overcome the above limitations, we propose to adopt in the time-477
dependent context the Equivalent Return Period (ERP ), which preserves the virtue of the478
classical return period of effectively communicating the event rareness. ERP resembles479
the classical definition of return period in the case of independence, while it is able to480
control the probability of failure under the time-dependence condition.481
We conclude with a note on the practical implications of the present analysis. Results482
shown here highlight that the independence condition is not necessary for the application483
of the classical return period equation; notwithstanding this, practitioners should take484
care of the time-persistence structure of the process when estimating risk from data, to485
correctly evaluate the probability of failure (e.g. through E RP ). However, it is interesting486
to stress that the differences between the correlated and uncorrelated case are small to487
negligible when ρ.0.5. Thus, the temporal correlation of the process may be safely488
disregarded in such cases, as far as the return period is concerned.489
Acknowledgments. We thank the Editor and the three anonymous Reviewers for490
their thoughtful comments. The research has been partially funded by the Italian Min-491
D R A F T September 22, 2015, 2:23pm D R A F T
X - 28 : 100 YEARS OF RETURN PERIOD
istry of University and Research through the projects PRIN 2010JHF437 and PRIN492
20102AXKAJ. No data was used in producing this manuscript.493
D R A F T September 22, 2015, 2:23pm D R A F T
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Appendix A: General relationships between fW,fW|teand fN
Since we can write that Pr (Ate, Bte+1 , ..B1, B0) = Pr (B0, B1, ..Bte1, Ate) =494
fW(te+ 1), the probability mass function of the conditional waiting time, fW|te(t), can495
be expressed as function of fWand fNas in the following496
fW|te(t) = Pr (Ate, Bte+1, ..B1, B0, B1, B2, ..Bt1, At)
Pr (Ate, Bte+1, ..B1, B0)(A1)
=Pr (Bte+1, ..B1, B0, B1, B2, ..Bt1, At|Ate) Pr Ate
Pr (Ate, Bte+1, ..B1, B0)
=(1 p)
fW(te+ 1)fN(t+te)
By making use of the simple identity Pr(C) = Pr(AC) + Pr(BC), which is valid for any497
events Aand C(with Balways denoting the complement of A), fWcan be expressed as498
function of fN
499
fW(t) = Pr (B1, ..Bt1, At) (A2)
= Pr (A0, B1, ..Bt1, At) + Pr (B0, B1, ..Bt1, At)
= Pr (B1, ..Bt1, At|A0) Pr A0+ Pr (B0, B1, ..Bt1, At)
=fN(t) (1 p) + fW(t+ 1)
by solving equation (A2) for fNand substituting the resulting expression in (A1) we500
obtain501
fW|te(t) = 1
fW(te+ 1) [fW(t+te)fW(t+te+ 1)] (A3)
Since fNis a special case of fW|te, when te= 0 equation (A3)502
D R A F T September 22, 2015, 2:23pm D R A F T
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fN(t) = 1
1p[fW(t)fW(t+ 1)] (A4)
Moreover, if we exploit the recursive property of equation (A2), we can write503
fW(2) = fW(1) (1 p)fN(1) (A5)
fW(3) = fW(2) (1 p)fN(2)
=fW(1) (1 p)fN(1) (1 p)fN(2)
fW(4) = fW(3) (1 p)fN(3)
=fW(1) (1 p)fN(1) (1 p)fN(2) (1 p)fN(3)
...
thus obtaining504
fW(t+ 1) = fW(1) (1 p)
t
k=1
fN(k) (A6)
= (1 p)1
t
k=1
fN(k)
= (1 p) [1 FN(t)]
= (1 p)FN(t)
where we used fW(1) = Pr A1= 1 pand the survival function of N, i.e. FN(t) =505
1FN(t) = 1 t
k=1fN(k) =
k=t+1fN(k). The relationship between fW|teand fNis506
obtained by substituting equations (A4) and (A6) into (A3)507
fW|te(t) = fN(t+te)
FN(te)(A7)
D R A F T September 22, 2015, 2:23pm D R A F T
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We adopt equation (A6) to derive the analytical expression of the return period TWas508
function of fN
509
TW
τ=
t=1t(1 p)FN(t1) = (1 p)
t=1tF N(t1) (A8)
= (1 p)
t=1t
k=tfN(k) = (1 p)
k=1fN(k)k
t=1t
= (1 p)
k=1
k(k+ 1)
2fN(k)
= (1 p)
k=1
k2
2fN(k) +
k=1
k
2fN(k)
=(1 p)
2EN2+ E [N]
Finally, substituting equation (A7) into (1) we obtain TW|teas function of fN
510
TW|te
τ=
t=1tfN(t+te)
FN(te)(A9)
=1
FN(te)
t=1 [(t+te)fN(t+te)tefN(t+te)]
=1
FN(te)
k=te+1kfN(k)te
k=te+1fN(k)
=1
FN(te)
k=te+1kfN(k)teFN(te)
=
k=te+1
kfN(k)
FN(te)te
Appendix B: Mean interarrival time, TN
Substituting equation (5), which is of general validity, in (1) we have511
D R A F T September 22, 2015, 2:23pm D R A F T
X - 32 : 100 YEARS OF RETURN PERIOD
TN
τ=
t=1
t fN(t) = 1 Pr{N= 1}+ 2 Pr{N= 2}+... (B1)
= Pr (A1|A0) + 2 Pr (B1, A2|A0) + 3 Pr (B1, B2, A3|A0) + ...
=1
Pr A0
[Pr (A0, A1) + 2 Pr (A0, B1, A2) + 3 Pr (A0, B1, B2, A3) + ...]
=1
1p[Pr (A0, A1) + 2 Pr (A0, B1, A2) + 3 Pr (A0, B1, B2, A3) + ...]
By making use again of the identity Pr(CA) = Pr(C)Pr(CB), where Balways denotes512
the opposite event of A, we obtain513
TN
τ=1
1p[(Pr A0Pr (A0, B1)) + 2 (Pr(A0, B1)Pr (A0, B1, B2)) (B2)
+3 (Pr (A0, B1, B2)Pr (A0, B1, B2, B3)) + ..]
=1
1p[Pr A0+ Pr (A0, B1) + Pr (A0, B1, B2) + Pr (A0, B1, B2, B3) + ..]
Using once more the same identity, we find514
TN
τ=1
1p[(1 Pr B0) + (Pr B1Pr (B0, B1)) (B3)
+ (Pr (B1, B2)Pr (B0, B1, B2)) + ..]
=1
1p
which proves to be valid because of stationarity, i.e. Pr B0= Pr B1, Pr (B0, B1) =515
Pr (B1, B2), etc.516
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Sciences Journal,44 (6), 871–878, doi:10.1080/02626669909492286.566
Serinaldi, F. (2014), Dismissing return periods!, Stochastic Environmental Research and567
Risk Assessment, pp. 1–11, doi:10.1007/s00477-014-0916-1.568
Stedinger, J. R., R. M. Vogel, and E. Foufoula-Georgiou (1993), Frequency analysis of569
extreme events, in Handbook of Hydrology, edited by D. Maidment, chap. 18, McGraw-570
Hill, New York.571
D R A F T September 22, 2015, 2:23pm D R A F T
X - 36 : 100 YEARS OF RETURN PERIOD
List of Figures572
1. Illustrative sketch of the quantities involved in the definitions of the return period:573
excursions of the Ztprocess above/below a threshold level zdefining the dangerous (At)574
and safe (Bt) events.575
2. Two state Markov-dependent process (2Mp): return periods TWand TNas function576
of Tfor several values of the correlation coefficient ρin absolute value (a) and normalized577
with respect to the independent value T(b). Note that TN=Tfor every value of ρ, while578
TW=TN=Tfor ρ= 0 (black line).579
3. Two state Markov-dependent process (2Mp): distribution functions of the waiting580
time, FW(a) and of the interarrival time, FN(b) for p= 0.9 and for several values of the581
correlation coefficient ρ; the averages of the distributions (return periods) are indicated by582
the vertical dashed lines. For the sake of clarity, the distribution functions of the discrete583
random variables Wand Nare represented as continuous functions.584
4. Return periods TWand TNas function of Tand for two values of the correlation585
coefficient ρfor the AR(1) process (continuous lines) compared to the two state Markov-586
dependent process (2Mp, dashed lines). Note that TN=Tfor every value of ρ, while587
TW=TN=Tfor ρ= 0 (black line).588
5. AR(1) process: mean conditional waiting time TW|te(continuous lines) as function589
of the elapsing time tefor p= 0.9 and for several values of the correlation coefficient ρ;590
the corresponding mean unconditional waiting times TW(dashed lines) are depicted as591
reference.592
6. Probabilities of failure RW(TW) (continuous lines), RW|teTW|te(dot-dashed lines)593
or RN(TN) (dashed lines) as functions of pfor both AR(1) (a, b) and 2Mp (c, d); graphs594
D R A F T September 22, 2015, 2:23pm D R A F T
: 100 YEARS OF RETURN PERIOD X - 37
refer to the cases ρ= 0.75 (a, c) and ρ= 0.99 (b, d). Note that RW|teTW|teof 2Mp is595
valid for any te>0. Results are compared to the independent case (black line).596
7. Equivalent Return Period (ERP ), based on the interarrival time N, as function of597
the independent return period Tfor several values of the lag-1 correlation coefficient ρ;598
curves for ρ < 0.75 are not shown because the differences between ERP and Tare small599
to negligible.600
D R A F T September 22, 2015, 2:23pm D R A F T
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