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A Critical Review of Probability of Extreme Rainfall: Principles and Models

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Abstract

Probabilistic modelling of extreme rainfall has a crucial role in flood risk estimation and consequently in the design and management of flood protection works. This is particularly the case for urban floods, where the plethora of flow control cites and the scarcity of flow measurements make the use of rainfall data indispensable. For half a century, the Gumbel distribution has been the prevailing model of extreme rainfall. Several arguments including theoretical reasons and empirical evidence are supposed to support the appropriateness of the Gumbel distribution, which corresponds to an exponential parent distribution tail. Recently, the applicability of this distribution has been criticized both on theoretical and empirical grounds. Thus, new theoretical arguments based on comparisons of actual and asymptotic extreme value distributions as well as on the principle of maximum entropy indicate that the Extreme Value Type 2 distribution should replace the Gumbel distribution. In addition, several empirical analyses using long rainfall records agree with the new theoretical findings. Further- more, the empirical analyses show that the Gumbel distribution may significantly underestimate the largest extreme rainfall amounts (albeit its predictions for small return periods of 5-10 years are satisfactory), whereas this distribution would seem as an appropriate model if fewer years of measurements were available (i.e., parts of the long records were used).
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Acriticalreviewofprobabilityofextreme
rainfall:principlesandmodels
Chapter·February2007
DOI:10.1201/9780203945988.ch7
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A CRITICAL REVIEW OF PROBABILITY OF EXTREME
RAINFALL: PRINCIPLES AND MODELS
Demetris Koutsoyiannis
Department of Water Resources, Faculty of Civil Engineering, National Technical
University of Athens, Heroon Polytechneiou 5, GR-157 80 Zographou, Greece
(http://www.itia.ntua.gr/dk/, dk@itia.ntua.gr)
Abstract
Probabilistic modelling of extreme rainfall has a crucial role in flood risk estimation
and consequently in the design and management of flood protection works. This is
particularly the case for urban floods, where the plethora of flow control Vites and the
scarcity of flow measurements make the use of rainfall data indispensable. For half a
century, the Gumbel distribution has been the prevailing model of extreme rainfall.
Several arguments including theoretical reasons and empirical evidence are supposed
to support the appropriateness of the Gumbel distribution, which corresponds to an
exponential parent distribution tail. Recently, the applicability of this distribution has
been criticized both on theoretical and empirical grounds. Thus, new theoretical
arguments based on comparisons of actual and asymptotic extreme value distributions
as well as on the principle of maximum entropy indicate that the Extreme Value Type
2 distribution should replace the Gumbel distribution. In addition, several empirical
analyses using long rainfall records agree with the new theoretical findings. Further-
more, the empirical analyses show that the Gumbel distribution may significantly
underestimate the largest extreme rainfall amounts (albeit its predictions for small
return periods of 5-10 years are satisfactory), whereas this distribution would seem as
an appropriate model if fewer years of measurements were available (i.e., parts of the
long records were used).
2
1. Introduction
The design and management of flood protection works and measures requires reliable
estimation of flood probability and risk. A solid empirical basis for this estimation can
be offered by flow observation records with an appropriate length, sufficient to
include a sample of representative floods. In practice, however, flow measurements
are never enough to support flood modelling. Particularly, in urban floods the control
points are numerous and the flow gauge sites scarce or non existing at all (for example
in Athens, a city with a history extended over several millennia, traversed by the
Kephisos and Ilisos Rivers and other urban streams, no flow gauge with systematic
measurements has ever operated). The obvious alternative is the use of hydrological
models with rainfall input data and the substitution of rainfall for streamflow empiri-
cal information. Notably, even when flow records exist, yet rainfall probability has
still a major role in hydrologic practice; for instance in major hydraulic structures, the
design floods are generally estimated from appropriately synthesised design storms
(e.g. U.S. Department of the Interior, Bureau of Reclamation, 1977, 1987; Sutcliffe,
1978).
However, from the birth time of science, which is typically located in the era of the
Ionian philosophers (6th century BC), it is known that the empirical evidence alone
never suffices to form a comprehensive and consistent picture of natural phenomena
and behaviours. A theory, based on reasoning, is required to interpret empirical obser-
vations and draw such a picture. Such a theory has been sought for more than 26 cen-
turies, since the formulation of the first logical explanations of hydrometeorological
phenomena by Anaximander (c. 610- c. 547 BC) and Anaximenes (585-525 BC) of
Miletus, who studied the formation of clouds, rain and hail (Koutsoyiannis and Xan-
thopoulos, 1999; Koutsoyiannis et al., 2006). However, still the state of affairs
regarding understanding and description of these phenomena and their behaviours
may be not satisfactory.
Some of the questions in seeking a fundament for a theory are philosophical ques-
tions; for instance the concepts of infinite vs. finite and of determinism vs. indetermi-
nism, including the notions of probability and entropy. It is necessary to briefly dis-
cuss these questions because they greatly influence our perception of hydrometeo-
rological phenomena including rainfall and flood.
The history of infinite goes back to the 6th century BC, with Anaximander, who
regarded infinite as the cosmological principle, and continues with Zeno of Elea (c.
490- c. 430 BC) and his famous paradoxes, and later with Aristotle (384-328 BC) who
introduced the notion of potential infinite, as opposed to the actual or complete infi-
nite. The Aristotelian potential infinite “exists in no other way, but … potentially or
by reduction” (Physics, 3.7, 206b16). It is generally claimed that the problem of
mathematical infinite was tackled in the late 19th century. According to Bertrand Rus-
sell, Zeno’s paradoxes “after two thousand years of continual refutation, … made the
foundation of a mathematical renaissance (Russell, 1903). Furthermore, “for over two
thousand years the human intellect was baffled by the problem [of infinity]… The
definite solution to the difficulties is due to Georg Cantor” (Russell, 1926; see also
Crossley et al., 1990 and Priest, 2002).
In hydrometeorology, however, the concept of infinity is still not understood and this
situation has led to fallacies of upper bounds in precipitation and flood, the well-
3
known concepts of the probable maximum precipitation (PMP) and probable maxi-
mum flood (PMF) (World Meteorological Organization, 1986). These contradictory
concepts are still in wide use, even though merely the Aristotelian notion of potential
infinite would suffice to abandon them. To quote, for example, Dingman (1994, p.
141) “conceptually, we can always imagine that a few more molecules of water could
fall beyond any specified limit.” This thinking is absolutely consistent with the Aris-
totelian potential infinite.
Criticisms of the PMP and PMF concepts must have started from the 1970s; among
them, one of the neatest was offered by Benson (1973):
“The ‘probable maximum’ concept began as ‘maximum possible’ because it was
considered that maximum limits exist for all the elements that act together to pro-
duce rainfall, and that these limits could be defined by a study of the natural proc-
esses. This was found to be impossible to accomplish – basically because nature is
not constrained to limits ... At this point, the concept should have been abandoned
and admitted to be a failure. Instead, it was salvaged by the device of renaming it
‘probable maximum’ instead of ‘maximum possible’. This was done, however, at a
sacrifice of any meaning or logical consistency that may have existed originally ...
The only merit in the value arrived at is that it is a very large one. However, in
some instances, maximum probable precipitation or flood values have been
exceeded shortly after or before publication, whereas, in some instances, values
have been considered by competent scientists to be absurdly high … The method
is, therefore, subject to serious criticism on both technical and ethical grounds –
technical because of a preponderance of subjective factors in the computation
process, and because of a lack of specific or consistent meaning in the result; ethi-
cal because of the implication that the design value is virtually free from risk.”
More recently, the particular hypotheses and methodologies elements of the different
approaches for estimating PMP have been also criticized. The so-called statistical
approach to PMP, based on the studies of Hershfield (1961a, 1965) has been revisited
recently (Koutsoyiannis, 1999) and it was concluded that the data used by Hershfield
do not suggest the existence of an upper limit. To formulate his method, Hershfield
compiled a huge and worldwide rainfall data set (a total of 95 000 station-years of
annual maximum rainfall belonging to 2645 stations, of which about 90% were in the
USA), standardized each record and found the maximum over the 95 000 standardized
values, which he asserted PMP. Clearly then, the PMP hypothesis is based on the
incorrect interpretation that an observed maximum in precipitation is a physical upper
limit; had the sample size been greater, the estimated PMP value would been greater,
too.
The situation is perhaps even worse with the so-called moisture maximization
approach of PMP estimation (World Meteorological Organization, 1986), which
seemingly is more physically based than the statistical approach of Hershfield. This is
the most representative and widely used approach to PMP, and is based on the
“maximization” of the observed atmospheric moisture content (i.e. to a maximum
observed value) and on the assumption that if the moisture content were maximum,
then the rainfall depth would be greater than observed by a factor equal to the ratio of
the maximum over the observed rainfall depth. Applying this “maximization” proce-
dure for all observed storms, the PMP value is assumed to be the maximum over all
maximized depths.
4
Clearly, then, the approach suffers twice by the incorrect interpretation that an
observed maximum is a physical upper limit. This fallacy is used for first time to
determine the maximum moisture content (formally, the maximum dew point,
assuming that the observed maximum in a record of about 50 years is a physical limit;
obviously, had the record length been 100 or 200 years the observed maximum dew
point would most likely be higher). This logic is also used for a second time to deter-
mine the PMP as the maximum of observed maximized values. Papalexiou and Kout-
soyiannis (2006) have demonstrated the arbitrariness of the approach and its enor-
mous sensitivity to the observation records (e.g. a missing rainfall observation could
result in 25% reduction of the PMP value). The arbitrary assumptions of the approach
extend beyond the confusion of maximum observed quantities with physical limits.
For example, the logic of moisture maximization at a particular location is unsup-
ported given that a large storm at this location depends on the convergence of atmos-
pheric moisture from much greater areas.
In conclusion, it is surprising that the contradictory PMP and PMF concepts are
regarded by many as concepts more physically based than a probabilistic approach to
extreme rainfall and flood. This is particularly the case because the PMP and PMF
concepts are greatly based on probabilistic or statistical assumptions, which in addi-
tion are rather misrepresentations of physical phenomena and indicate confused inter-
pretation of probability. In turn, as will be discussed in the next section, these very
concepts may have also affected probabilistic approaches of hydrological processes,
in an attempt to make them consistent with the unsupported assumption of an upper
bound.
This situation harmonizes with a dominant logic in hydrometeorology that probability
does not offer a physical insight and is not related to understanding of physical phe-
nomena, but rather it is only an unavoidable modelling tool. In contrast, understanding
and insights are regarded as pertinent to deterministic thinking and to mechanistic
explanations of phenomena. This logic, however, ought to have been abandoned at the
end of the 19th century, after the development of statistical thermophysics and later of
the quantum physics which rely upon the concepts of probability and statistics and
depart from mechanistic physics. More recently, the study of chaotic dynamical sys-
tems and the astonishing results that the evolution of even the simplest nonlinear
systems is unpredictable after a short lead time, have demonstrated the ineffectiveness
of deterministic thinking. In this respect, even a faithful follower of determinism is
inevitably forced to accept probabilistic description of phenomena for practical prob-
lems. However, when using probabilistic descriptions the gain may be greater if these
descriptions are not regarded as incomprehensible mathematical models but rather as
insightful physical descriptions.
The notion of indeterminism is at least as old as Heraclitus (c. 535 - 475 BC) and the
notion of probability is the extension (quantifying transformation) of the Aristotelian
idea of “potentia” (Popper, 1982, p. 133). The mathematical formalism of probability
is much older than the recent notion of chaotic systems albeit its concrete fundament
was offered in the mid 20th century by Kolmogorov (1933). The notion of probability
may imply indeterminism from the outset (all events are possible, usually with differ-
ent probabilities, but eventually one occurs) and may differ from the deterministic
thinking (only one event is possible but it may be difficult to predict which one).
The notion of probability in synergy with the notion of infinite can remove paradoxi-
5
cal impressions related to upper bounds of physical quantities such as rainfall: The
probability that rainfall exceeds any positive number x decreases toward zero as x
decreases, becomes inconceivably small for very high x and becomes precisely zero
for x = . So, there is no need to assume such controversial concepts as PMP. This
was explained half a century ago by the famous statistician Feller (1950), using
another example, the age of a person:
“The question then arises as to which numbers can actually represent the life span
of a person. Is there a maximal age beyond which life is impossible, or is any age
conceivable? We hesitate to admit that man can grow 1000 years old, and yet cur-
rent actuarial practice admits no bounds to the possible duration of life. According
to formulas on which modern mortality tables are based, the proportion of men
surviving 1000 years is of the order of magnitude of one in 101036 a number with
1027 billions of zeros. This statement does not make sense from a biological or
sociological point of view, but considered exclusively from a statistical standpoint
it certainly does not contradict any experience. There are fewer than 1010 people
born in a century. To test the contention statistically, more than 101035 centuries
would be required, which is considerably more than 101034/ lifetimes of the earth.
Obviously, such extremely small probabilities are compatible with our notion of
impossibility. Their use may appear utterly absurd, but it does no harm and is con-
venient in simplifying many formulas. Moreover, if we were seriously to discard
the possibility of living 1000 years, we should have to accept the existence of a
maximum age, and the assumption that it should be possible to live x years and
impossible to live x years and two seconds is as unappealing as the idea of unlim-
ited life.”
In hydrometeorology, the introduction and development of the concepts of probability
and statistics have been closely related to the study of extreme rainfall and flood and
were greatly determined by the design needs of flood protection works. Empirical
ideas similar to the modern probability concepts had been formulated in hydrology
about a century ago (for instance, the hydrological frequency curves known as “dura-
tion curves”; Hazen, 1914). At about the same time, great mathematicians were
developing the theoretical foundation of probability of extreme values (von Bort-
kiewicz, 1922a, b; von Mises, 1923; Fréchet, 1927; Fisher and Tippet, 1928; Gne-
denco, 1941). Around the 1950s the empirical and theoretical approaches converged
to form the branch of hydrology now called hydrologic statistics, whose founders
were Jenkinson (1955), Gumbel (1958) and later Chow (1964). However, as already
stated above, based on the PMP example, the current state of knowledge is not satis-
factory and several important questions still wait for answers. For instance, Klemeš
(2000) argues that “The distribution models used now, though disguised in rigorous
mathematical garb, are no more, and quite likely less, valid for estimating the prob-
abilities of rare events than were the extensions ‘by eye’ of duration curves employed
50 years ago.” Obviously, however, the probabilistic approach to extreme values of
hydrological processes signifies a major progress in hydrological science and engi-
neering as it quantifies risk and disputes arbitrary and rather irrational concepts and
approaches.
The most important questions that have not received definite answers yet are related
in one or another manner to the notion of infinite. These questions concern the
asymptotic distribution of maxima, a distribution that assumes a number of events
6
tending to infinity, and are focused on the distribution tails, i.e. the behaviour of the
distribution function as the hydrological quantity of interest tends to infinity.
Thus, if one is exempted from the concept of an upper limit to a hydrological quantity
and adopts a probabilistic approach, one will accept that the quantity may grow to
infinity with decreasing probability of exceedence. In this case, as probability of
exceedence tends to zero, there exists a lower limit to the rate of growth which is
mathematically proven. This lower limit is represented by the Gumbel distribution,
which has the “lightest” possible tail. So, abandoning the PMP concept and adopting
the Gumbel distribution can be thought of as a step from a finite upper limit to infin-
ity, but with the slowest possible growth rate towards infinity. Does nature follow the
slowest path to infinity? This question is not a philosophical one but has strong engi-
neering implications. If the answer is positive, the design values for flood protection
structures or measures will be the smallest possible ones (among those obtained by the
probabilistic approach), otherwise they will be higher. These questions are studied in
this article with the help of some recent works.
2. Basic concepts of extreme value distributions
It is recalled from probability theory that, given a number n of independent identically
distributed random variables, the largest (in the sense of a specific realization) of them
(more precisely, the largest order statistic), i.e.:
X := max {Y1, Y2, …, Yn} (1)
has probability distribution function
Hn(x) = [F(x)]n (2)
where F(x) := P{Yi x} is the common probability distribution function of each of Yi.
Herein, F(x) will be referred to as parent distribution. If n is not constant but rather
can be regarded as a realisation of a random variable with Poisson distribution with
mean ν, then the distribution of X becomes (e.g. Todorovic and Zelenhasic, 1970;
Rossi et al., 1984),
H΄
ν(x) = exp{–ν[1 – F(x)]} (3)
Since ln[F(x)]n = n ln {1 – [1 – F(x)]} = n {–[1 – F(x)] – [1 – F(x)]2 – …} n [1 –
F(x)], it turns out that for large n or large F(x), Hn(x) H΄
n(x). Numerical investigation
shows that even for relatively small n, the difference between Hn(x) and H΄
n(x) is small
(e.g., for n = 10, the relative error in estimating the exceedence probability 1 – Hn(x)
from (3) rather than from (2) is about 3% at most).
In hydrological applications concerning the distribution of annual maximum rainfall
or flood, it may be assumed that the number of values of Yi (e.g., the number of
storms or floods per year), whose maximum is the variable of interest X (e.g. the
maximum rainfall intensity or flood discharge), is not constant. Besides, the Poisson
model can be regarded as an acceptable approximation for such applications. Given
also the small difference between (3) and (2), it can be concluded that (3) should be
regarded as an appropriate model for the practical hydrological applications discussed
in this article.
7
The exact distributions (2) or (3), whose evaluation requires the parent distribution to
be known, have rarely been used in hydrological statistics. Instead, hydrological
applications have made wide use of asymptotes or limiting extreme value distribu-
tions, which are obtained from the exact distributions when n tends to infinity. Gum-
bel (1958) developed a comprehensive theory of extreme value distributions.
According to this, as n tends to infinity Hn(x) converges to one of three possible
asymptotes, depending on the mathematical form of F(x). Obviously, the same limit-
ing distributions may also result from H΄
ν(x) as ν tends to infinity. All three asymp-
totes can be described by a single mathematical expression introduced by Jenkinson
(1955, 1969) and become known as the general extreme value (GEV) distribution.
This expression is
H(x) = exp
1 + κ
x
λ ψ
–1/κ
, κx κλ(ψ – 1/κ) (4)
where ψ, λ > 0 and κ are location, scale and shape parameters, respectively; ψ and κ
are dimensionless whereas λ has same units as x. (Note that the sign convention of κ
in (4) may differ in some hydrological texts). Leadbetter (1974) showed that this
holds not only for maxima of independent random variables but for dependent random
variables, as well, provided that there is no long-range dependence of high-level
exceedences.
When κ = 0, the type I distribution of maxima (EV1 or Gumbel distribution) is
obtained. Using simple calculus it is found that in this case, (4) takes the form
H(x) = exp[–exp (–x/λ + ψ)] (5)
which is unbounded from both from above and below (– < x < +).
When κ > 0, H(x) represents the extreme value distribution of maxima of type II
(EV2). In this case the variable is bounded from below and unbounded from above
(λψλ/κ x < +). A special case is obtained when the left bound becomes zero (ψ =
1/κ). This special two-parameter distribution is
Η(x) = exp
λ
κx
1/κ
, x 0 (6)
In some texts, (6) is referred to as the EV2 distribution. Here, as in Gumbel (1958),
the name EV2 distribution is used for the complete three-parameter form (equation
(4)) with κ > 0. Distribution (6) is referred to as the Fréchet distribution.
When κ < 0, H(x) represents the type III (EV3) distribution of maxima. This, how-
ever, is of no practical interest in hydrology as it refers to random variables bounded
from above (– < x λψλ/κ). As discussed in the introduction, many regard an
upper bound in hydrological quantities as reasonable. Even Jenkinson (1955) regards
the EV3 distribution as “the most frequently found in nature, since it is reasonable to
expect the maximum values to have an upper bound”. However, he leaves out rainfall
from this conjecture saying “to a considerable extent rainfall amounts are ‘uncon-
trolled’ and high falls may be recorded”. In fact, he proposes the EV2 distribution for
rainfall (note that he uses a different convention, referring to EV2 as type I). In a
recent study, Sisson et al. (2006), even though detecting EV2 behaviour of rainfall
8
maxima, attempt to incorporate the idea of a PMP upper bound within an EV2 model-
ling framework (see also Francés, this volume).
The simplicity of the above mathematical expressions is remarkable. This extends to
the inverse function x(H) xH that is used to estimate a distribution quantile for a
given non-exceedence probability H. This is
xH = (λ/κ) [exp(κ zH) – 1] + λψ (7)
where zH is the so called Gumbel reduced variate, defined as
zH := –ln(–ln H) (8)
For the Gumbel distribution, (7) takes the special form
xH = λ(zH + ψ) (9)
which implies a linear plot of xH versus zH (a plot known as the Gumbel probability
plot). For the Fréchet distribution, (7) takes the form
xH = λψ exp(κ zH) (10)
which implies a linear plot of ln xH versus zH (a plot referred to as the Fréchet prob-
ability plot).
The close relationship between the distribution of maxima H(x) and the tail of the
parent distribution F(x) allows for the determination of the latter if the former is
known. The tail of F(x) can be represented by the distribution of x conditional on
being greater than a certain threshold ξ, i.e. Gξ(x) := F(x|x > ξ), for which:
1 Gξ(x) = 1 – F(x)
1 – F(ξ) , x ξ (11)
If one chooses ξ so that the exceedence probability 1 – F(ξ) equals 1/ν, the reciprocal
of the mean number of events in a year (this is implied when the partial duration
series is formed from a time series of measurements, by choosing a number of events
equal to the number of years of record), and denote G(x) the conditional distribution
for this specific value, then:
1 G(x) = ν[1 – F(x)] (12)
Combining equation (12) with equation (3) it is obtained that:
G(x) = 1 + ln H΄
ν(x) (13)
If H΄
ν(x) is given by the limit distribution H(x) in equation (4), then it is concluded that
for κ > 0:
G(x) = 1 –
1 + κ
x
λ ψ
–1/κ
, x λψ (14)
which is the generalized Pareto distribution. Similarly, for κ = 0:
9
G(x) = 1 – exp(–x/λ + ψ), x λψ (15)
which is the exponential distribution. For the special case ψ = 1/κ:
G(x) = 1 –
λ
κx
1/κ
, x λ/κ (16)
In this way, a one to one correspondence between the type of the extreme value distri-
bution and the type of the tail of the parent distribution is established. The EV1 distri-
bution (κ = 0, equation (5)) corresponds to an exponential parent distribution tail
(equation (15)), else known as short tail, or light tail. The EV2 distribution (κ > 0,
equation (4) including the special case (6)) corresponds to an over-exponential parent
distribution tail (equation (14) including the special case (16)), else known as hyper-
exponential tail, Pareto tail, power-law tail, algebraic tail, long tail, heavy tail and fat
tail.
From the distribution functions H(x) and G(x), two return periods can be defined as
follows:
T := δ / [1 – G(x)], T΄ := δ / [1 – Η(x)] (17)
where δ is the mean interarrival time of an event that is represented by the variable X.
In both cases X represents annual values, so δ = 1 year; δ is most commonly omitted
but here we kept it for dimensional consistency, given that the return period has units
of time, typically expressed in years.
Equation (16) is precisely a power law relationship between the distribution quantile x
and the return period T:
x = (λ/κ)(T/δ)κ (18)
In the generalized Pareto case (equation (14)), the corresponding relationship is
x = (λ/κ)[(T/δ)κ – 1 + κψ] (19)
whereas in the exponential case the corresponding relationship is
x = λ [ln(T/δ) + ψ)] (20)
3. The dominance of the Gumbel distribution
Due to their simplicity and generality, the limiting extreme value distributions have
become very widespread in hydrology. In particular, EV1 has been by far the most
popular model. In hydrological education is so prevailing that most textbooks contain
the EV1 distribution only, omitting EV2. In hydrological engineering studies, espe-
cially those analysing rainfall maxima, the use of EV1 has become so common that its
adoption is almost automatic, without any reasoning or comparison with other possi-
ble models. Sometimes, it is also suggested, or even required, by the guidelines or
regulations of several organizations, institutes and country services. Historically,
several reasons have been contributed to the prevailing of the Gumbel distribution:
10
Theoretical reasons. Most types of parent distributions functions that are used in
hydrology, such as exponential, gamma, Weibull, normal, lognormal, and the EV1
itself (e.g. Kottegoda and Rosso, 1997) belong to the domain of attraction of the
Gumbel distribution. In contrast, the domain of attraction of the EV2 distribution
includes parent distributions such as Pareto, Cauchy, log-gamma (also called log-
Pearson type 3), and the EV2, which traditionally are not in very common use in
hydrology, particularly in rainfall modelling.
Simplicity. The mathematical handling of the two-parameter EV1 is simpler than that
of the three-parameter EV2.
Accuracy of estimated parameters. Obviously, two parameters are more accurately
estimated than three. For the former case, mean and standard deviation (or second L-
moment) suffice, whereas in the latter case the skewness is also required and its esti-
mation is extremely uncertain for typical small-size hydrological samples.
Practical reasons. Probability plots are the most common tools used by practitioners,
engineers and hydrologists, to choose an appropriate distribution function. As
explained earlier, EV1 offers a linear Gumbel probability plot of observed xH versus
observed zH (which is estimated in terms of plotting positions, i.e. sample estimates of
probability of non-exceedence). In contrast, a linear probability plot for the three-
parameter EV2 is not possible to construct (unless the shape parameter κ is fixed).
This may be regarded as a primary reason of choosing EV1 against the three-parame-
ter EV2 in practice. For the two parameter EV2 (Fréchet) distribution, a linear plot
(ln xH versus zH) is possible as discussed earlier. However, empirical evidence shows
that, in most cases, plots of xH versus zH give more straight-line arrangements than
plots of ln xH versus zH.
From a practical point of view, the choice of an EV1 over an EV2 distribution may be
immaterial if small return periods T are considered. For instance, in typical storm
sewer networks, designed on the basis on return periods of about 5-10 years, the dif-
ference of the two distributions is negligible; besides, in such return periods even
interpolation from the empirical distribution would suffice. However, for large T (>
50 years), for which extrapolation is required, EV1 results in probability of
exceedence of a certain value significantly lower than EV2. That is, for large rainfall
depths, EV1 yields the lowest possible probability of exceedence (the highest possible
T) in comparison to those of EV2 for any value of κ. For T > 1000, the return period
estimated by EV1 could be orders of magnitude higher than that of EV2 (see Figure 3
and its discussion in section 5).
This should be regarded as a strong disadvantage of EV1 from the engineering point
of view. Normally, this would be a sufficient reason to avoid the use of EV1 in engi-
neering studies. Obviously, this disadvantage of EV1 would be counterbalanced only
by strong empirical evidence and theoretical reasoning. In practice, the small size of
common hydrological records (e.g. a few tens of years) cannot provide sufficient
empirical evidence for preferring EV1 over EV2. This will be discussed further in
section 5. In addition, the theoretical reasons, exhibited above, are not strong enough
to justify the adoption of the Gumbel distribution. This will be discussed in section 4.
11
4. Theoretical justification of the distribution type
As discussed above, the rainfall process at fine time scales (hourly, daily) has been
modelled by distributions belonging to the domain of attraction of EV1 such as
gamma or Weibull. However, the adoption of these distributions is rather empirical,
not based on theoretical reasoning. Thus, the above theoretical justification of the
EV1 distribution is inconsistent. In contrast, recently three arguments have been for-
mulated that favour the EV2 over the EV1 distribution, which are summarized below.
Argument 1: Asymptotic vs. actual distribution. What matters in hydrological
applications, is the actual distribution of maxima, i.e. Hn(x) or H΄
n(x) as given in (2) or
(3), respectively. The asymptotic distribution H(x) for n provides a useful indica-
tion of the behaviour in the tails but not necessarily a model for practical use. It has
been observed (Koutsoyiannis, 2004a) that the convergence of Hn(x) to H(x) may be
enormously slow. This is demonstrated in Figure 1, which depicts Gumbel probability
plots of the exact distribution functions of maxima Hn(x) for n = 103 and 106 for a
parent distribution function that is Weibull (F(y) = 1 – exp(–yk)) with shape parameter
k = 0.5. The parent distribution belongs to the domain of attraction of the Gumbel
limiting distribution, so the Gumbel probability plot tends to a straight line as n .
However, even for n as high as 106 the curvature of the distribution function is appar-
ent. Obviously, in hydrological applications, such a high number of events within,
say, a year, is not realistic (it can be expected that the number of storms or floods in a
location will not exceed the order of 10-102). Thus, the limiting distribution for n
may be not useful. The slow convergence in this case should be contrasted with fast
convergence in other limiting situations; for example the distribution of the sum of a
number of variables to the normal distribution, according to the central limit theorem,
is very fast, so that about 10-30 events suffice to obtain an almost perfect approxima-
tion to the normal distribution.
Let us assume that the Weibull distribution (which belongs to the domain of attraction
of EV1) with shape parameter smaller than 1 (e.g. k = 0.5 as in the example of Figure
1) can be a plausible parent distribution of storms and floods at a fine time scale,
which is known to be positively skewed and with J-shaped density function. Accord-
ingly, as observed in Figure 1, the probability plot of the exact distribution of maxima
should be a convex curve, rather than a straight line, which indicates that, for a rela-
tively small n, a three-parameter EV2 distribution may approximate sufficiently the
exact distribution. Thus, even if the parent distribution belongs to the domain of
attraction of the Gumbel distribution, an EV2 distribution can be a choice better than
EV1.
Argument 2: Change of domain of attraction due to parameter changes. In argu-
ment 1 it was assumed that the random variables Yi whose maximum values are stud-
ied are independent and identically distributed ones. However, it is more plausible to
assume that different Yi have the same type of distribution function Fi(y) but with
different parameters. The statistical characteristics (e.g., averages, standard deviations
etc.) and, consequently, the parameters of distribution functions exhibit seasonal
variation. In addition, evidence from long geophysical records shows that there exist
random fluctuations of the statistical properties on multiple large time scales (e.g.,
tens of years, hundreds of years, etc.).
In this respect, it has been shown theoretically that a gamma parent distribution,
12
which belongs to the domain of attraction of EV1, switches to the EV2 domain of
attraction if its scale parameter varies randomly following another gamma distribution
function (Koutsoyiannis, 2004a). This point was also made by Katz et al. (2005) for
an exponential parent distribution, which is a special case of the gamma distribution
function. In addition, it was demonstrated using Monte Carlo simulations (Koutsoy-
iannis, 2004a) that a gamma parent distribution function with constant shape parame-
ter and scale parameter shifting between two values, which are sampled at random
with specified probabilities, results in an actual (for n = 5) extreme value distribution
which is closely approximated by an EV2 distribution, whereas the EV1 distribution
departs significantly from the simulated actual distribution.
Argument 3: Principle of maximum entropy. The principle of maximum entropy is
a well established mathematical and physical principle, defined on grounds of prob-
ability theory, that can infer the detailed structure or behaviour of a system from
rough (macroscopical) information of the system. For a stochastic system, the princi-
ple can determine the distribution function of the system states, from assumed macro-
scopical constraints (e.g. moments) of the system. The classical definition of entropy
φ, known as the Boltzmann-Gibbs-Shannon entropy, is
φ := Ε[–ln f(Y)] = –
f(y) ln f(y) dy (9)
where f(y) := dF(y)/dy denotes the probability density function of the parent variable
and E[.] denotes expectation.
In a recent study, Koutsoyiannis (2005a) has shown that the principle of maximum
entropy can predict and explain the distribution functions of hydrological variables
using only two “macroscopic” statistical properties of observed time series (equality
constraints), the mean µ and the standard deviation σ, as well as the inequality con-
straint that the variables under study are non-negative quantities. For variables with
high variation (σ/µ > 1) the classical entropy φ fails to apply with these constraints. In
this case, a generalized definition of entropy, due to Tsallis (1988, 2004) should be
used instead. This is
φq =
1 –
0
[f(x)]q dx
q – 1 (17)
and precisely reproduces φ when q = 0. Maximization of φq with the aforementioned
constraints results in Pareto tail of the parent distribution with shape parameter κ =
(1 – q)/q. Now, there is sufficient empirical evidence that at small time scales rainfall
exhibits high variation (σ/µ > 1). In this case, maximization of Tsallis entropy yields
power-type (Pareto) distribution.
5. Empirical justification of the distribution type of extreme rainfall
In seeking empirical evidence to justify the distribution type, one must be aware of
bias in statistical estimations and error probability in statistical tests that emerge from
typical hydrological samples. In fact, estimation bias and error probability are very
large and this explains why the inappropriateness the EV1 distribution was not under-
13
stood for so many years. Specifically, typical annual maximum rainfall series with
record lengths 20–50 years completely hide the EV2 distribution and display EV1
behaviour. This was initially demonstrated by Koutsoyiannis and Baloutsos (2000)
using an annual series of maximum daily rainfall in Athens, Greece, extending
through 1860–1995 (136 years). This series was found to follow EV2 distribution, but
if smaller parts of the series were analysed, the EV1 distribution seemed to be an
appropriate model.
A systematic Monte Carlo simulation study to address this problem has been done in
Koutsoyiannis (2004a). Some of the results, concerning the estimation bias, are
depicted in Figure 2. A negative bias, defined as estimated κ minus true κ, is apparent,
for both the moments and L-moments estimators. It can be observed that for true κ =
0.15 (a value that is typical for extreme rainfall, as will be discussed later) and for a
record length of 20 years the bias of the method of moments is –0.15, which means
that the estimated κ will be zero! Even for a record length of 50 years the negative
bias is high (b = –0.12), so that κ will be estimated at 0.03, a value that will not give
good reason for preferring EV2 to EV1.
The situation is improved if L-moments estimators are used as the resulting bias is
much lower. However the method of L-moments is relatively new (Hosking et al.,
1985; Hosking, 1990) and its use has not been very common so far. In addition, even
the method of L-moments is susceptible to type II error (no rejection of the null false
hypothesis of an EV1 distribution against the true alternative hypothesis of EV2 dis-
tribution) with a high probability. As demonstrated in Koutsoyiannis (2004a) for κ =
0.15 and record length 20 years the frequency of not rejecting the EV1 distribution is
80%. Even for record length 50 years this frequency is high: 62%.
The results of this analysis show that (a) only long records (e.g. 100 years or more)
could provide evidence of the distribution type of extreme rainfall, and (b) even with
these records, the estimation of the shape parameter κ of the GEV distribution is
highly uncertain, and an ensemble of many records should be used to obtain a reliable
estimate.
In this respect, Koutsoyiannis (2004b) compiled an ensemble of annual maximum
daily rainfall series from 169 stations of the Northern Hemishpere (28 from Europe
and 141 from the USA) roughly belonging to six major climatic zones. All series had
lengths from 100 to 154 years, the top three (in terms of length) being Florence,
Genoa and Athens, with record lengths 154, 148 and 143 years respectively. The
empirical distribution of one of the stations (Athens, Greece) is shown in Figure 3, on
Gumbel probability plot, along with the theoretical EV2 and EV1 distributions fitted
by several methods. The plot clearly shows that (a) the EV2 distribution fits the
empirical one better than the EV1 distribution; for the highest observed daily rainfall
(~150 mm), EV2 and EV1 assign return periods of ~200 and ~1000 years (differing
by a factor of 5), respectively; for a rainfall depth of ~220 mm, EV2 and EV1 assign
return periods of ~1000 and ~100 000 years (differing by two orders of magnitude),
respectively. These observations demonstrate how important the correct choice of the
theoretical model is and how much the EV1 distribution underestimates the return
period of extreme rainfall.
In addition, a PMP value, estimated by Hershfield’s method is also plotted in Figure
3. As discussed above, this value should not be regarded as an upper bound of rainfall
14
but just as a value with high return period. It turns out from Figure 3 that the return
period of this PMP values is around 50 000 years. It may be useful to mention that the
aforementioned critical revisit (Koutsoyiannis, 1999) of Hershfield’s data set, on
which his method was based, revealed that Hershfield’s PMP should be regarded as a
rainfall value with return period of about 65 000 years.
These findings are representative of a general behaviour of all 169 rainfall records. In
fact, in more than 90% of the records the estimated κ by the methods of maximum
likelihood and L-moments were positive. The small percentage of non-positive κ in
the remaining records is fully explained as a statistical sampling effect. This provides
sufficient support for a general applicability of the EV2 distribution worldwide. Fur-
thermore, the ensemble of all samples were analysed in combination and it was found
that several dimensionless statistics, including the coefficient of variation of the
annual maximum series, are virtually constant worldwide, except for an error that can
be attributed to a pure statistical sampling effect. This enabled the formation of a
compound series of annual maxima, after standardization by mean, for all 169 sta-
tions. The empirical distribution of the compound series is shown in Figure 4, on
Gumbel probability plot, along with the theoretical EV2 and EV1 distributions fitted
by several methods. The plot clearly shows that the EV2 distribution fits the empirical
one whereas the EV1 distribution is totally inappropriate. The compound series also
supported the estimation of a unique κ for all stations, which was found to be 0.15.
The same data set was revisited in Koutsoyiannis (2005a) in a framework investigat-
ing the applicability of the maximum entropy principle in hydrology. In this case,
instead of series of annual maxima, the series-above-threshold were constructed for
168 out of 169 records (in the Athens case only the annual maximum values were
available, and thus the construction of a series-above-threshold was not possible). All
series were standardized by their mean and merged in one sample with length 17 922
station-years. The empirical distribution of this sample is depicted in Figure 5 (double
logarithmic plot), where values lower than 0.79 are not shown, as this number is the
lowest value of the merged series-above-threshold. In addition, several theoretical
distribution functions are also plotted. Among these, the Pareto distribution is
obtained by the maximum entropy principle for coefficient of variation σ/µ = 1.19.
The agreement of the Pareto distribution with the empirical one is remarkable. The
Pareto distribution is precisely consistent with the EV2 distribution of the annual
maximum, as justified in section 2. The shape parameter of the Pareto distribution, as
obtained by the maximum entropy principle, is 0.15, the same value with the one
obtained by fitting the EV2 distribution in the compound series of annual maximum
rainfall.
Additional empirical evidence with same conclusions is provided by the aforemen-
tioned Hershfield’s (1961a) data set. Koutsoyiannis (1999) showed that this is con-
sistent with the EV2 distribution with κ = 0.13. The plot of Figure 6 (EV2 probability
plot with fixed κ = 0.15, which is further explained in section 7) indicates that the
value κ = 0.15 can be acceptable for that data set too. This enhances the trust that an
EV2 distribution with κ = 0.15 can be thought of as a generalized model appropriate
for mid latitude areas of the north hemisphere.
Additional empirical evidence with same orientation was provided by Chaouche
(2001) and Chaouche et al. (2002). Chaouche (2001) exploited a data base of 200
rainfall series of various time steps (month, day, hour, minute) from the five conti-
15
nents, each including more than 100 years of data. Using multifractal analyses he
showed that (a) an EV2/Pareto type law describes the rainfall amounts for large return
periods; (b) the exponent of this law is scale invariant over scales greater than an
hour; and (c) this exponent is almost space invariant.
Other studies have also expressed scepticism for the appropriateness of the Gumbel
distribution for the case of rainfall extremes and suggested hyper-exponential tail
behaviour. Thus, Wilks (1993), who investigated empirically several distributions
which are potentially suitable for describing extreme rainfall, using rainfall records of
13 stations in the USA with lengths ranging from 39 to 91 years, noted that EV1 often
underestimates the largest extreme rainfall amounts and suggested an update and
revision of the Technical Paper 40 (Hershfield, 1961b), a widely used climatological
atlas of United States that was compiled fitting EV1 distributions to annual extreme
rainfall data. Coles et al. (2003) and Coles and Pericchi (2003) concluded that infer-
ence based on the Gumbel model to annual maxima may result in unrealistically high
return periods for certain observed events and suggested a number of modifications to
standard methods, among which is the replacement of the Gumbel model with the
GEV model. Mora et al. (2005) confirmed that rainfall in Marseille (a raingauge
included in the study by Koutsoyiannis, 2004b) shows hyper-exponential tail behav-
iour. They also provided two regional studies in the Languedoc-Roussillon region
(south of France) with 15 and 23 gauges, for which they found that a similar distribu-
tion with hyper-exponential tail could be fitted; this, when compared with previous
estimations, leads to a significant increase in the depth of rare rainfall. On the same
lines, Bacro and Chaouche (2006) showed that the distribution of extreme daily rain-
fall at Marseille is not in the Gumbel law domain. Sisson et al. (2006) highlighted the
fact that standard Gumbel analyses routinely assign near-zero probability to subse-
quently observed disasters, and that for San Juan, Puerto Rico, standard 100-year
predicted rainfall estimates may be routinely underestimated by a factor of two.
Schaefer et al. (2006) using the methodology by Hosking and Wallis (1997) for
regional precipitation-frequency analysis and spatial mapping for 24-hour and 2-hour
durations for the Wahington State, USA, found that the distribution of rainfall
maxima in this State generally follows the EV2 distribution type.
6. The distribution tails in other hydrological processes
The theoretical arguments presented in section 4 that support the EV2 over the EV1
distribution are not related merely to rainfall but rather to any process with high vari-
ability. Thus, it could be expected that other processes should also exhibit a similar
behaviour.
This is the case for flood runoff. In fact, as demonstrated by Koutsoyiannis (2005b, c)
and Gaume (2006), there are theoretical reasons by which we can conclude that the
type of extreme value distribution in rainfall and runoff will be the same. If rainfall
follows the EV1 distribution, then it can be shown that runoff will also follow the
EV2 distribution. Conversely, if runoff follows the EV2 distribution, then rainfall
should necessarily follow the EV2 distribution. Perhaps, the EV2 distribution in flood
is easier to verify empirically (due to magnification of variability of extremes) and
thus, the EV1 distribution has not been as standard in flood modelling as is in rainfall
modelling. Thus, the log-gamma model, which belongs to the domain of attraction of
EV2 has more frequently used in flood modelling. For instance, this model is the
16
federally adopted approach to flood frequency in the USA (US Water Resources
Council, 1982). But when flood frequency is estimated from rainfall, which is mod-
elled using the EV1 model, then the flood frequency becomes necessarily consistent
to the EV1, as explained above. Several more recent studies have also supported a
three-parameter GEV over an EV1 distribution for floods (Farquharson et al., 1992;
Madsen et al., 1997).
Similar results have been provided by fractal/multifractal analyses. Thus, Turcotte
(1994) studied flood peaks over threshold in 1200 stations in the United States and
concluded that they follow a fractal law, which essentially is described by equation
(18). Pandey et al. (1998) established power-law distributions for daily mean stream-
flows in 19 river basins in the USA. Similarly, Malamud and Turcotte (2006) exam-
ined six river basins from different climatic regions and hydrologic conditions in the
USA and concluded in power law distributions using either flood peaks over threshold
or all daily mean streamflows, also considering in some cases paleoflood data.
Naturally, other hydrological processes driven by runoff are anticipated to follow
long-tail distributions, too. However, it may be more difficult to verify empirically the
type of distribution tail in such cases, because instrumental records are typically much
shorter. Nevertheless, reconstructions of time series are possible in some other cases,
for instance, in sediment yield time series from sediment deposits. Thus, Katz et al.
(2005) were able to detect long tail behaviour in the annual sediment yield time series
Nicolay Lake on Cornwall Island, Canada. In addition, Katz et al. (2005) provide an
excellent review of the tail behaviours of several ecological variables.
7. Practical issues for the application of the EV2 distribution
As discussed in section 3, the simplicity and the two-parameter form of the EV1
distribution are strong points that made it prevail in hydrology. However, if the shape
parameter of the EV2 distribution is fixed (in extreme rainfall κ = 0.15, as discussed
in section 5) the general handling of the distribution becomes as simple as that of the
EV1 distribution. For example, the estimation of the remaining two parameters
becomes similar to that of the EV1 distribution. That is, the scale parameter can be
estimated by the method of moments from:
λ = c1σ (21)
where c1 = κ/Γ(1 – 2κ) – Γ2(1 – κ) or c1 = 0.61 for κ = 0.15, while in the EV1 case c1
= 0.78. The relevant estimate for the method of L-moments is:
λ = c2λ2 (22)
where λ2 is the second L-moment and c2 = κ/[Γ(1 – κ)(2κ – 1)] or c2 = 1.23 for κ =
0.15, while in the EV1 case c2 = 1.443. The estimate of the location parameter for
both the method of moments and L-moments is:
ψ = µ/λ – c3 (23)
where c3 = [Γ(1 – κ) – 1]/κ or c3 = 0.75 for κ = 0.15, while in the EV1 case c3 = 0.577.
17
If, in addition to λ and ψ, the shape parameter is to be estimated directly from the
sample (which is not advisable but it may be useful for comparisons) the following
approximate equations can be used (Koutsoyiannis, 2004b):
κ = 1
3 1
0.31 + 0.91Cs + (0.91Cs)2 + 1.8 (24)
κ = 8c – 3c2, c := ln2
ln3 2
3 + τ3 (25)
where Cs and τ3 are the regular and L skewness coefficients, respectively. The former
corresponds to the method of moments and the resulting error is smaller than ±0.01
for –1 < κ < 1/3 (–2 < Cs< ). The latter corresponds to the method of L-moments and
the resulting error is smaller than ±0.008 for –1 < κ < 1 (–1/3 < τ3 < 1).
The construction of linear probability plots is also easy if κ is fixed. It suffices to
replace in the horizontal axis the Gumbel reduced variate zH = –ln(–lnH) (equation
(8)) with the GEV reduced variate zH = [(–lnH)κ – 1]/κ. An example of such a plot is
depicted in Figure 6.
8. Resulting intensity-duration-frequency curves
The construction of rainfall intensity-duration-frequency (IDF) relationships or curves
is one of the most common practical tasks related to the probabilistic description of
extreme rainfall. Unfortunately, however, the construction is typically performed by
empirical procedures (e.g. Chow et al., 1988). Even the terms “duration” and “fre-
quency” in IDF are misnomers; in fact, “duration” should read “timescale” (in order
not to be confused with the duration of a rainfall event) and “frequency” should read
“return period”. Thus, the IDF relationships are mathematical expressions of the rain-
fall intensity i(d, T) averaged over timescale d and exceeded on a return period T.
The recent theoretical advances in the probabilistic description can support a more
theoretically based, mathematically consistent, and physically sound approach. A few
assumptions are needed to support such an approach, namely:
1. The separability assumption, according to which the influences of return period
and timescale are separable (Koutsoyiannis et al., 1998), i.e.,
i(d, T) = a(T) / b(d) (26)
where a(T) and b(d) are mathematical expressions to be determined.
2. The similarity assumption, according to which the distribution of average rain-
fall intensity conditional on being wet is statistically similar for all time scales
(Koutsoyiannis, 2006).
3. A stochastic description of rainfall intermittency, which, as suggested by Kout-
soyiannis (2006) should be a generalization of a Markov chain process that
results applying the maximum entropy principle to the rainfall occurrence
process.
18
4. A probabilistic distribution of the rainfall depth at any scale, which as discussed
above should be of Pareto/EV2 type.
Based on assumptions 1-3, Koutsoyiannis (2006) showed that the function b(d) can be
approximated for relatively short timescales by the expression (here written in slightly
different form)
b(d) = (1 + d/θ)η (27)
where θ > is a parameter with units same as the timescale d and η is a dimensionless
parameter with values in the interval (0, 1). This resembles an expression historically
established with empirical considerations. The approximate character of (46) as well
as that of assumptions 1 and 2 should be underlined. At the same time, it should be
noted that (46) is more accurate than a pure power law of b(d), which has been sug-
gested by modern fractal approaches. Particularly, (46) implies a decrease of rainfall
intensity on small timescales, as compared to what is predicted by a power law. This
is very important for the design of urban drainage networks that have small concen-
tration times.
Furthermore, assumption 3 combined with (19) results in
i(d, T) = (λ/κ)[(T/δ)κ – 1 + κψ] (28)
By comparison of (28) with (26), we conclude that only the scale parameter λ should
be a function of timescale d and particularly that λ ~ (1 + d/θ) η. We easily then
deduce that the final form of the IDF will be
i(d, T) = λ΄ (T/δ)κψ΄
(1 + d/θ)η (29)
where ψ΄ := 1 – κψ and λ΄ := (λ/κ) (1 + d/θ)η, which should be constant, independent
of d, Notice that (29) is dimensionally consistent and that the return period T refers to
the parent distribution (and thus it can take values smaller than δ = 1 year, but neces-
sarily greater than δψ΄1/κ). Also, notice that the numerator of (29) differs from a pure
power law that has been commonly used in engineering practice. By virtue of (13) and
(17), (29) can be easily converted in terms of the return period of the distribution of
maxima and takes the form
i(d, T) = λ΄ [–ln(1 – δ/T΄)]κψ΄
(1 + d/θ)η (30)
In the latter case, obviously T΄ should be greater than δ = 1 year. All parameters are
precisely the same in both (29) and (30). Consistent parameter estimation techniques
for these relationships have been discussed in Koutsoyiannis et al. (1998).
9. Conclusions
Historically, the modelling of rainfall has suffered from several fallacies, such as the
existence of an upper bound (PMP), and empirical practices that do not have theoreti-
cal support. Rational thinking and fundamental scientific principles, formulated since
the birth of science in ancient Greece, can help combat such fallacies.
19
Probability, statistics and stochastic processes have offered a better alternative in
perceiving and modelling of the rainfall process. However, even the probabilistic
approaches have suffered from misconceptions and bad practices that have resulted in
underestimation of rainfall variability and uncertainty. Among them is the wide appli-
cation of the Gumbel or EV1 distribution, which has been the prevailing model for
rainfall extremes despite the fact that it yields unsafe (the smallest possible) design
rainfall values.
More recent studies have provided theoretical arguments and general empirical
evidence from many rainfall records worldwide, which suggest a long distribution tail
and favour the EV2 distribution of maxima. Simultaneously, they explain that the
broad use of the EV1 distribution worldwide is in fact related to statistical biases and
errors due to small sample sizes, rather than to the real behaviour of rainfall maxima,
which should be better described by the EV2 distribution. Similar behaviours have
been also detected in other hydrological processes such as streamflow and sediment
transport.
The new methodological framework is more theoretically consistent, and more
mathematically and physically sound (justified by the physico-mathematical principle
of maximum entropy). Simultaneously, it is very simple so as to allow its easy
implementation in typical engineering tasks such as estimation and prediction of
design parameters, including the construction of IDF curves. The new framework
imposes also some requirements for stochastic models of rainfall, many of which are
currently not consistent with the long tail behaviour of the rainfall distribution.
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22
n = 10
6
n = 10
3
n =1
0
0.2
0.4
0.6
0.8
1
-20246810
Gumbel reduced variate
Standardised distribution quantile 1
Figure 1: Gumbel probability plots of exact distribution function of maxima Hn(x) for n = 103 and 106,
also in comparison with the parent distribution function F(y) H1(y), which is Weibull with shape
parameter k = 0.5. The distribution quantile has been standardised by x0.9999 corresponding to zH = 9.21
(from Koutsoyiannis, 2004a).
-0.25
-0.2
-0.15
-0.1
-0.05
0
0 0.05 0.1 0.15 0.2 0.25 0.3
Shape parameter, κ
Estimaton bias 1
20
30
50
100
150
L moments
estimator
moments
estimator
Sam
le size
Figure 2: Bias in estimating the shape parameter κ of the GEV distribution using the methods of
moments and L-moments (from Koutsoyiannis, 2004a).
23
0
50
100
150
200
250
300
350
400
450
-2024681012
Gumbel reduced variate
Rainfall depth (mm)
Empirical
EV2/L-moments
EV2/Max likelihood
EV2/Mom ents
EV1/L-moments
1.01
1.2
2
5
10
20
50
100
200
500
1000
2000
5000
10000
20000
50000
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Estimated PMP value
Figure 3: Empirical distribution and theoretical EV2 and EV1 distributions fitted by several methods
for the annual maximum daily rainfall series of Athens, National Observatory, Greece (Gumbel prob-
ability plot; from Koutsoyiannis, 2004b). The PMP value (424.1 mm) was estimated by Koutsoyiannis
and Baloutsos (2000).
0
1
2
3
4
5
6
7
8
-2024681012
Gumbel reduc ed variate
Rescaled rainfall depth
Empirical
EV2/Least squares
EV2/Moments
EV2/L-moments
EV2/Max likelihood
EV1/L-moments
1.01
1.2
2
5
10
20
50
100
200
500
1000
2000
5000
10000
20000
50000
100000
Return
p
eriod,
y
ears
Figure 4: Empirical distribution and theoretical EV2 and EV1 distributions fitted by several methods
for the unified record of all 169 annual maximum rescaled daily rainfall series (18 065 station-years;
from Koutsoyiannis 2004b).
24
0.1
1
10
0.1 1 10 100 1000 10000 100000
T (years )
x
Empirical Pareto
Exponential Truncated Normal
Normal
Figure 5: Plot of daily rainfall depth from the unified standardized sample above threshold, formed
from data of 168 stations worldwide, vs return period, in comparison to Pareto, exponential, truncated
normal and normal distributions (adapted from Koutsoyiannis, 2005a).
0
2
4
6
8
10
12
14
16
-2 3 8 13 18 23 28 33
GEV reduced variate
Hers hfield-standardised rainfall depth
Empirical
κ = 0.15
κ = 0.13 (Koutsoyiannis, 1999)
1.01
1.2
2
5
10
20
50
100
200
500
1000
2000
5000
10000
20000
50000
100000
Return
p
eriod,
y
ears
Figure 6: Empirical distribution of standardized rainfall depth k = (Xµ)/σ for Hershfield’s (1961a)
data set (95 000 station years from 2645 stations), as determined by Koutsoyiannis (1999), and fitted
EV2 distributions with κ = 0.13 (Koutsoyiannis, 1999) and κ = 0.15 (Koutsoyiannis, 2004b) (EV2
probability plots with fixed κ = 0.15).
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... It is widely recognised that heavy rainfall at a daily time-scale has a crucial role in flood risk estimation and consequently in design and management of flood protection works (Koutsoyiannis, 2007). Due to the availability of daily rainfall data across the world, precipitation at daily scale represents one of the most investigated variables in hydrology (De Michele, 2019). ...
... The Fréchet distribution is a special case of the Generalized Extreme Value Distribution (GEV, type II), defined in [0, ∞). Its CDF is provided by Eq. 5 and it is defined by the scale parameter β > 0 and the shape parameter γ > 0. Koutsoyiannis (2007) shows that the Fréchet distribution, in its form provided by Eq. 5, is characterised by an heavy tail: moreover, following the classification provided by Ouarda et al. (1994), Fréchet belongs to the case in which the tail has the same behaviour of the Pareto distribution. ...
... For the Gumbel distribution the shape parameter is γ = 0, and both the scale and the location are β,θ > 0. Koutsoyiannis (2007) has shown that the Gumbel distribution, in its form provided by Eq. 8, is characterised by a light tail. Therefore, unlike the other distributions where the shape parameter is an unknown quantity, in Eq. 8, γ is defined and equal to zero. ...
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Study region We investigate samples from two Italian regions, i.e. Lazio and Sicily, located in central and south Italy, respectively, and characterized by two diverse climates. Study focus In engineering practice, the study of maxima daily rainfall values is commonly dealt with light-tailed probability distribution functions, such as the Gumbel. The choice of a distribution rather than another may cause estimation errors of rainfall values associated to specific return periods. Recently, several studies demonstrate that heavy-tailed distributions are preferable for extreme events modelling. Here, we opt for six theoretical probability distribution functions and evaluate their performance in fitting extreme precipitation samples. We select the samples with two common methods, i.e. the Peak-Over-Threshold and the Annual Maxima. We assess the best fitting distribution to the empirical samples of extreme values through the Ratio Mean Square Error Method and the Kolmogorov-Smirnov test. New hydrological insights for the region The assessment of the best fitting distribution to daily rainfall of the two different areas investigated here leads to interesting remarks. Despite the diversity of their climate, results suggest that heavy-tailed distributions describe more accurately empirical data rather than light-tailed ones. Therefore, extreme events may have been largely underestimated in the past in both areas. The proposed investigation can prompt the choice of the best fitting probability distribution to evaluate the design hydrological quantities supporting common engineering practice.
... The PDF of Gumbel distribution function is defined as where and are location and scale parameters, respectively. The corresponding CDF of the Gumbel distribution is given by Koutsoyiannis (2007) For a given return period T, the quantile estimate of the Gumbel distribution can be obtained using the inverse of the CDF given in Eq. 2.2 as follows: ...
... Our results are closed in terms of Normal distribution. The empirical analyses performed by Koutsoyiannis (2007) highlight that the Gumbel distribution may significantly underestimate the largest extreme values. Meanwhile, light-tailed Gumbel and Weibull distributions give the worst fit among all the competing models in the study, which fits no grid at all. ...
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The temperature in the mountains has been increasing at an unprecedented rate in the global warming era. As a result, it is necessary to evaluate suitable models that could provide precise maximum temperature estimates. This paper explores the goodness-of-ft of the two-parameter bell-shaped, light-tailed, and heavy-tailed distribution functions for modeling the annual maximum temperature in the Northwest Himalayan region of India. The distributions under consideration are Gamma, Gum�bel, Lognormal, Normal, and Weibull. The method of maximum likelihood estimation is used for parameter estimation along with Akaike information criteria for model selection. Gridded data from Climate Research Unit, UK, was obtained at the 525 grids of the region. This study shows that Normal distribution gives the best ft followed by Lognormal and Gamma distributions, and these three models jointly ft all the grids in the region. Furthermore, we estimate the 5, 10, 20, 50, 100, and 500 years return level of annual maximum temperature starting from 2017. The future projections reveal that, on average, the region will face 1.28(1.25 − 1.32) ◦C, 1.64(1.60 − 1.67) ◦C, 1.93(1.89 − 1.97) ◦C, 2.26(2.22 − 2.31) ◦C, 2.49(2.44 − 2.54) ◦C, and 2.94(2.88 − 3) ◦C temperature rise by the years 2022, 2027, 2037, 2067, 2117, and 2517, respectively. In comparison to the middle of the region, the higher and lower belts of the region will be severely impacted
... Les erreurs relatives sont calculées dans un contexte opérationnel caractérisé par une utilisation exclusive de la loi de Gumbel dans les travaux de génie civil [7,8,19]. La simplicité du calcul de la distribution de Gumbel et le tracé sur l'échelle linéaire de probabilité sont des éléments qui ont contribué à sa popularité auprès des utilisateurs [20,21]. Ainsi, pour les différents quantiles estimés, le biais (%) entre la loi de Gumbel et les lois théorique et pratique est calculé à l'aide de la Relation (8). ...
... En effet, une utilisation exclusive de la loi de Gumbel au profit des lois théoriques et pratiques montre que la loi de Gumbel sous-estime les quantiles de pluie maximale journalière par rapport aux approches théoriques et pratiques. Cette observation avait été déjà faite par plusieurs études [21,34,35]. Ainsi donc, l'estimation des quantiles de pluie journalière avec la loi de Gumbel dans un projet sous dimensionnerait les ouvrages tout en minimisant le coût et le niveau de sécurité. ...
... Some of the reviewed literature include [8][9][10][11][12][13][14][15][16], etc. and all the reviewed literature adopted Gumbel extreme type 1 (EV1) the default distribution in the development of IDF curves and models in Nigeria. But Gumbel EVI may significantly underestimate the largest extreme rainfall amount (albeit their predictions for small return periods of 5-10 years are satisfactory) [17]. Consequently, the applicability of Gumbel (EV1) has often been criticized both on theoretical and empirical grounds. ...
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... This led to major improvement works on the Bruton dam (Pether, 2010). However, general advice on the use of higher estimates of both design rainfall and PMP has not (Lowing & Law, 1995;Austin et al., 1995;Koutsoyiannis, 2007;Fontaine & Potter, 1989;Nathan et al., 2016). The overall consensus is that the annual exceedance probability (AEP) of PMP is somewhere between 250,000 -2 million years. ...
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L‐moments are expectations of certain linear combinations of order statistics. They can be defined for any random variable whose mean exists and form the basis of a general theory which covers the summarization and description of theoretical probability distributions, the summarization and description of observed data samples, estimation of parameters and quantiles of probability distributions, and hypothesis tests for probability distributions. The theory involves such established procedures as the use of order statistics and Gini's mean difference statistic, and gives rise to some promising innovations such as the measures of skewness and kurtosis described in Section 2, and new methods of parameter estimation for several distributions. The theory of L‐moments parallels the theory of (conventional) moments, as this list of applications might suggest. The main advantage of L‐moments over conventional moments is that L‐moments, being linear functions of the data, suffer less from the effects of sampling variability: L‐moments are more robust than conventional moments to outliers in the data and enable more secure inferences to be made from small samples about an underlying probability distribution. L‐moments sometimes yield more efficient parameter estimates than the maximum likelihood estimates.
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This paper aims to show the benefit of a regional approach for the estimation of rare daily rainfall. The studied region is Languedoc-Roussillon (south of France), where recent exceptional storms necessitate the revision of the statistical distributions, particularly their asymptotic tails over extreme values. The example of a large single-site time series of maximum daily rainfall at Marseille (1864-2002), very close to the studied region, shows a hyper-exponential behaviour for extreme events. At the regional scale, the homogenization process of daily maximum rainfall has been performed by considering that the coefficients of variation of the yearly maximum daily rainfall are stationary over the study zone. Two regional sample studies have been carried out on 15 and 23 gauges, randomly distributed in space, and a similar distribution could be fitted to both samples. As in the case of Marseille, the regional distribution shows a hyper-exponential asymptotic behaviour at the extreme values. The obtained regional distribution provides a systematic method for computation of rare daily rainfall that may be applied in every part of the studied region and, when compared with previous estimations, leads to a significant increase in the depth of rare rainfall.