A metastatistical approach to rainfall extremes

Advances in Water Resources 79 (2015) 121–126
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Advances in Water Resources
journal homepage: www.elsevier.com/locate/advwatres
A metastatistical approach to rainfall extremes
Marco Marani a,b,⇑, Massimiliano Ignaccolo a,c
a
Division of Earth & Ocean Sciences, Nicholas School of the Environment, Department of Civil and Environmental Engineering, Pratt School of Engineering, Duke University,
Durham, NC, USA
b
Department of Civil, Environmental, and Architectural Engineering, University of Padova, Padova, Italy
c
DIIAR, Politecnico di Milano, Milano, Italy
a r t i c l e
i n f o
Article history:
Received 28 August 2013
Received in revised form 1 March 2015
Accepted 3 March 2015
Available online 13 March 2015
Keywords:
Extreme events
Generalized Extreme Value distribution
Metastatistics
a b s t r a c t
The traditional statistical theory of extreme events assumes an asymptotic regime in which the number
of events per year is large enough for a limiting Generalized Extreme Value distribution to apply. This has
been shown not to be applicable to many practical cases. We introduce here a Metastatistical Extreme
Value (MEV) approach which is defined in terms of the distribution of the statistical parameters describing ‘‘ordinary’’ daily rainfall occurrence and intensity. The method does not require an asymptotic
assumption, and naturally accounts for the influence of the bulk of the distribution of ordinary events
on the distribution of annual maximum daily rainfall. Building on existing observations showing the distribution of daily rainfall to be Weibull right-tail equivalent, the MEV approach is then specialized to yield
a compact and easily applicable formulation. We apply this formulation to Monte Carlo experiments
based on Weibull statistics derived from the 3-century long rainfall time series observed in Padova
(Italy). We find an excellent agreement between MEV estimates and the ‘observed’ frequency of occurrence of extreme events in the synthetic time series generated. GEV and Gumbel estimates, on the contrary, exhibit systematic errors. Tests with different rates of occurrence of rainfall events show slight
improvements of the GEV and Gumbel estimation bias when the number of events/year is increased.
However, a constant bias in GEV and Gumbel estimates is seen for (synthetic) climates where the number
of events and the distribution of intensities is varied stochastically. The estimation root mean square
error is also larger for the GEV and Gumbel distributions than for the MEV approach. Hence, GEV and
Gumbel quantile estimates are more likely to be further away from the actual value than MEV estimates.
Finally, the application of the new MEV approach to subsets of the long Padova time series identifies
marked variabilities in rainfall extremes at the centennial time scale.
Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction
The definition and estimation of extreme rainfall events is of
central importance in the analysis of past and projected rainfall
regimes, as well as in the design of any water resources management and flood control infrastructure. For a given event duration
of interest (here we will focus on the important case of daily duration), extreme value analysis usually studies the distribution of
yearly maxima, y, either directly or by considering the distribution
of rainfall values over a high threshold [1]. Under the assumptions
that (i) rainfall intensity may be assumed independent and identically distributed (i.i.d.) and (ii) the number of events per year tends
to infinity, the classical Extreme Value Theory (EVT) identifies a
Generalized Extreme Value (GEV) distribution of yearly maxima
⇑ Corresponding author at: Division of Earth & Ocean Sciences, Nicholas School of
the Environment, Department of Civil and Environmental Engineering, Pratt School
of Engineering, Duke University, Durham, NC, USA.
http://dx.doi.org/10.1016/j.advwatres.2015.03.001
0309-1708/Ó 2015 Elsevier Ltd. All rights reserved.
[2–6], which has been and still is widely applied [[7–12], e.g.]. It
is important to emphasize here that the GEV is not an exact distribution of yearly maxima, and that the actual extreme value distribution may converge to a GEV distribution only as the number of
events/year is ‘‘large enough’’, a potentially problematic concept as
the number of events/year (wet days in the present case) is
necessarily limited. However, little work has addressed the conditions under which the actual distribution may be considered to be
close to the limiting GEV form [13,14, e.g.] or how the possible
variability of the rainfall depth distribution (i.e. a violation of the
i.i.d. hypothesis e.g. due to seasonality), can affect the resulting
extreme value distribution [15]. These analyses show that indeed
the actual extreme value distribution of rainfall may in practice
be quite far from the asymptotic GEV form.
We propose here a non-asymptotic approach to the definition
and evaluation of an extreme value distribution based on a metastatistical approach (also referred to as superstatistics [16], compound distributions [17], or doubly stochastic processes [18] in
122
M. Marani, M. Ignaccolo / Advances in Water Resources 79 (2015) 121–126
different contexts). Our approach avoids the need of assuming an
infinite collection of events, i.e. it avoids the asymptotic assumption, and allows for interannual variability to be accounted for.
The manuscript is organized as follows. In Section 2 we describe
the data used in our analyses. In Section 3 we briefly summarize
the classical extreme event theory and introduce the new
Metastatistical Extreme Value formulation (MEV). A Results section compares GEV and MEV performances, and a Discussion and
Conclusion section closes the paper.
2. Data
As noted, the distribution describing the n-sample maximum
will strictly be a GEV distribution, independent of the specific value
n, only for ‘large enough’ values of the number of wet days. When
the number of wet days n is not large enough for the asymptotic
regime of the EVT to apply (e.g. this has been shown to be the case
in practice for Weibull variates [13,21]) one must use Eq. (1).
However, a useful approximation of Hn ðyÞ that does not require
n ! 1 can be obtained by considering U n , the expected largest
value of the variable X in n realizations. Because U n is on average
exceeded once every n realizations of X [22,23]:
WðU n Þ ¼
We analyze extremes in the daily rainfall time series observed
in Padova (Italy) over a span of almost three centuries, as well as
on synthetic data derived from its statistical properties. The
Padova dataset, comprised of 275 complete years of daily observations is described in detail elsewhere [19,20], and provides an
exceptionally long record, particularly suitable to test estimates
of high return period extremes.
3. Methods
We first briefly summarize the EVT, as typically used in hydrology, and then introduce a metastatical approach to the definition of
extreme value distributions.
1
n
ð4Þ
(note that a Weibull plotting position estimate, WðU n Þ ¼ 1=ðn þ 1Þ,
could also be used with no consequence of substance). Using this
result we can rewrite the cumulative probability for the n-sample
maximum Y n as
WðyÞ n
Hn ðyÞ ¼ ½FðyÞn ¼ ½1 WðyÞn ¼ 1 nWðU n Þ
ð5Þ
For y > U n (i.e. for an extreme value larger than the average maximum value in the observations) the term WðyÞ=WðU n Þ < 1.
Therefore, for large values of y, i.e. for extremes, we can use the
Cauchy approximation: ð1 zÞn ffi 1 n z ffi expðn zÞ, valid for
z 1. Hence, Eq. (5) can be approximated as:
3.1. Extreme value theorem
WðyÞ
Hn ðyÞ ¼ exp WðU n Þ
We use the random variable X > 0 to indicate daily rainfall
depth, f ðxÞ being its probability density function, FðxÞ ¼ PðX 6 xÞ
its cumulative distribution function, and WðxÞ ¼ 1 FðxÞ the exceedance probability. Notice that having considered X > 0, no probability atom at X ¼ 0 need be considered to represent the finite
probability of zero rainfall. The maximum of n realizations of the
stochastic variable X; Y n ¼ maxðx1 ; x2 ; . . . ; xn Þ, is also a random variable, often termed an n-maximum or a maximum with cardinality
n of the ‘‘parent’’ stochastic variable. In hydrologic practice n will
be the number of wet days in a given year, itself a discrete random
variable. If the events generating the n realizations of X are
independent, the cumulative distribution, Hn ðyÞ, of Y n may be
expressed as
Eq. (6) is sometimes referred to as the ‘‘penultimate’’ approximation [24,22], the ‘‘ultimate’’ approximation being Eq. (2), only
valid when n is very large. The error associated with the penultimate
approximation can be quantified through the relative error eðyÞ ¼
f½expðWðyÞ=WðU n Þ ½1 WðyÞ=ðnWðU n ÞÞn g=½1 WðyÞ= ðnWðU n ÞÞn .
For y ¼ U n [24]: eðU n Þ ¼ ðexpð1Þ ½1 1=nn Þ=½1 1=nn . For example, for n ¼ 50 the relative error is eðU 50 Þ ¼ 0:01. Note that for values
y > U n , of greatest applicative interest, the relative error is smaller
than eðU n Þ, as WðyÞ < WðU n Þ. The penultimate approximation has
been used in the evaluation of extreme values in some geophysical
contexts, such as in modelling wind power [24,23] or of drought
severity [25], but very rarely has it been applied to rainfall extremes
[26,15].
Hn ðyÞ ¼ ½FðyÞn
3.2. The case of Weibull variates
ð1Þ
Upon definition of a renormalized variable Sn ¼ ðY n bn Þ=an (an > 0
and bn being constants), the EVT establishes that [2–4]
n
lim PðSn < sÞ ¼ lim Hn ðsÞ ¼ lim ½Fðan s þ bn Þ ¼ HðsÞ
n!1
n!1
n!1
ð2Þ
The limiting distribution HðsÞ in Eq. (2), depending on the tail
behaviour of WðyÞ, can only be one of three distributions: (i) the
Gumbel distribution (Extreme Value 1 – EV1, or double exponential), when the tail of WðyÞ decreases faster than a power law; (ii)
the Frechét distribution (EV2), when the tail of WðyÞ behaves as a
power law for large values of x; and (iii) the Weibull distribution
(EV3), when x has a finite upper limit [2–4].
In terms of the non renormalized variable y, the three asymptotic types, EV1–EV3, can be thought of as special cases of a single
Generalized Extreme Value distribution [6]:
y l1=k
HGEV ðyÞ ¼ exp 1 þ k
r
þ
ð3Þ
where ðÞþ ¼ maxð; 0Þ; l is the location parameter, r > 0 is the
scale parameter, and k is a shape parameter. The limit k ¼ 0 corresponds to the EV1 distribution, k > 0 to the EV2 distribution, and
k < 0 to the EV3 distribution.
ð6Þ
Daily rainfall as been shown to be accurately modelled as a
Weibull variate [27]. Hence we consider here the important case
of WðxÞ ¼ expððx=CÞw Þ. Under these assumptions, the yearly
maximum daily rainfall depth, i.e. the maximum depth over the
n wet days occurred in a generic year, is distributed as:
w n
y
Hn ðyÞ ¼ 1 exp C
ð7Þ
Hence, the penultimate approximation takes the following forms:
w w
y
y
ffi exp exp w þ ln n
Hn ðyÞ ffi 1 n exp w
C
C
ð8Þ
Note that these expressions (and later results in this Section) are
valid also for distributions that are only right-tail equivalent to a
Weibull distribution [24,27] (two distributions F 1 and F 2 are
right-tail equivalent if ð1 F 1 ðxÞÞ=ð1 F 2 ðxÞÞ ! 1 when x ! þ1).
3.3. A metastatistical approach
The cumulative probability, Hn ðyÞ, of the n-maximum Y n
depends on the number of wet days, n, and on the parameters,
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M. Marani, M. Ignaccolo / Advances in Water Resources 79 (2015) 121–126
100
Y
10-2
10-4
10-5
0.1
1
10
100
IV
(a)
I
II
III
IV
(b)
40
15
I
II
III
IV
(c)
12
9
6
3
1.4
I
II
III
IV
(d)
80
h
dh f ðn; hÞ Hn ðy; n; hÞ
ð9Þ
where f ðn; hÞ is the joint probability distribution (discrete in n) of
the number of wet days in a year and of the parameter vector.
The symbol d
h denotes the differential dh1 dh2 dhk . We refer
to fðyÞ in Eq. (9) as the Metastatistical Extreme Value (MEV) distribution. This formulation is practically appealing. The number of
days with non-zero rainfall clearly shows a wide interannual variability (see below for an analysis of such variability in the test case
of the Padova time series). Moreover, nonstationarities in the rainfall-generating processes could be accounted for through suitablydefined distributions of h in Eq. (9).
However we restrict our analysis here to the stationary (or periodic) case, in which we can assume past observations to be representative of future realizations. In this case, and without the need
to make assumptions on the specific form of f ðn; hÞ; fðyÞ can be estimated based on T years of rainfall observations as:
T
1X
Hn ðy; nj ; hj Þ
T j¼1 j
ð10Þ
which may be formally obtained from Eq. (9) with
P
f ðn; hÞ ¼ 1T Tj¼1 dðn nj ; h hj Þ (dð; Þ = a multi-dimensional Dirac’s
Delta), accounting for the empirical frequency distribution of the
observed maxima. If one assumes daily precipitation to be
Weibull-distributed, the MEV expression is:
"
!#
T
1X
ywj
fMEV ðyÞ ffi
exp exp wj þ ln nj
T j¼1
Cj
ð11Þ
In other cases one may assume the probability distribution of
daily rainfall to be time-invariant (the physical rainfall-generating
mechanisms remain unchanged) and to be right-tail equivalent to
a Weibull distribution [27], such that only the number of rainy
days changes from year to year according to a discrete distribution
gðnÞ. By substituting the first expression in Eq. (8) into Eq. (9) we
have:
w w
X
y
y
¼ 1 hni exp
gðnÞ 1 n exp w
C
Cw
n
1
’ exp exp w yw þ C w ln hni
C
fðyÞ ¼
ð12Þ
where hni indicates the expected value of the number of wet days.
C
w
h ¼ ðh1 ; ::; hk Þ, of the distribution of the parent variable x. To make
this dependence explicit we now adopt the notation Hn ðy; n; hÞ. In
the realistic context in which both n and h are random variables,
we suggest that a meaningful definition of the probability, here
denoted as fðyÞ, that a yearly maximum Y be smaller than y, is
the expected value of Hn ðy; n; hÞ, computed over all possible
realizations of n and h:
XZ
120
80
Fig. 1. Daily rainfall survival probability, WðxÞ, for the 1841–2006 period (squares)
and for each single year within the same period (solid lines). The vertical dashed
line indicates the threshold used to left-censor the data as in [27].
fðyÞ ffi
III
160
1841 - 2006
x (mm)
n
II
40
10-3
n
Ψ (x)
10
fðyÞ ¼
I
120
-1
0.9
0.4
1730
1780
1830
1880
year
1930
1980
0
0.2 0.4
frequency
Fig. 2. The annual maxima (Y, panel (a)), the number of wet days per year (n, panel
(b)), the Weibull scale parameter (C, panel (c)) and shape parameter (w, panel (d))
obtained for the Padova time series. We indicate: the shape and scale parameter
values obtained from the entire dataset (black line), the values obtained using 1year (red line) and 10-year (blue line) sliding windows. Panels on the right portray
the empirical frequency distributions from the time series in the left panels. (For
interpretation of the references to colour in this figure legend, the reader is referred
to the web version of this article.)
The case in which also w and C are random variables of course
requires the full specification of the joint distribution function
f ðn; hÞ.
4. Results
We first characterize the statistics of daily precipitation in the
Padova dataset. We later use this information to generate synthetic
rainfall time series with known, and realistic, statistical properties.
Using such synthetic time series we compare rainfall extremes as
estimated with the MEV approach and via the GEV formulation.
Finally, we apply the MEV approach to the Padova data set.
4.1. The Padova time series and the Weibull approximation
The Padova time series includes four long sub-intervals of
uninterrupted observations: 1725–1764, 1768–1807, 1841–1880
and 1887–2006 [20]. For each interval we calculate the daily rainfall exceedance probability WðxÞ and find a wide interannual variability, particularly in the tail of the distribution (Fig. 1).
Next, we analyze the yearly maximum, Y, and the yearly number of wet days, n. The yearly maxima (Fig. 2(a)) display a large
variability, with periods in which the values of Y are significantly
smaller than for the rest of the time series (e.g. see the ’800).
Overall, the mode of Y is in the range 40–65 mm and the pdf has
a positive skewness. The number of wet days also seems to exhibit
a marked inter-annual variability and, qualitatively, a temporal
correlation structure (Fig. 2(b)). The number of events ranges
between about 40 days yr1 and 160 days yr1, with a mode
between 100 days yr1 and 120 day yr1.
We also analyze the distribution of the scale (C) and shape (w)
parameters obtained from fitting a Weibull distribution (with a
least-square method) to non-overlapping moving windows of
length 1 yr and 10 yrs. The fit starts by left-censoring rainfall
observations with a 10 mm-threshold [27]. This procedure filters
out small ordinary rainfall values, affected by higher relative
M. Marani, M. Ignaccolo / Advances in Water Resources 79 (2015) 121–126
4.2. MEV, GEV, and Gumbel
-ln(-ln(ζ(y)))
1,000 yr
100 yr
-ln(-ln(ζ(y)))
1,000 yr
100 yr
-ln(-ln(ζ(y)))
We compare here the MEV approach with the GEV and the traditional Gumbel approaches by means of synthetic data sets. The
synthetic data sets are constructed by drawing daily rainfall values
from a Weibull distribution, as justified by previous work analyzing rainfall data at the global scale [27]. The parameters of the
Weibull distribution used are obtained from the Padova time series
to ensure the synthetic values generated are representative of real
daily rainfall. The goal of these simulations is to produce realistic
time series for which extreme value properties are known, such
that the effectiveness of different extreme value estimation
approaches can be evaluated.
We consider three different statistical set ups, for each of which
we generate a large number (1000) of synthetic time series. The A
setup is a ‘‘homogeneous’’ case, in which the number of wet days/
year is n ¼ 100 days/year, and C and w are equal to the values
obtained from the 1957–2006 time interval of the Padova time series (C ¼ 7:3 mm, w ¼ 0:82). The B setup is homogeneous in n and
heterogeneous, or time-variant, in C and w : n ¼ 100 days/year
while C and w are drawn, with uniform probability, from the pairs
of values observed in the Padova time series. Setup C is timevariant in n and homogeneous in C and w : C ¼ 7:3 mm,
w ¼ 0:82, while n is drawn from a uniform distribution between
n ¼ 21 days/year and n ¼ 50 days/year. This latter experiment also
1,000 yr
100 yr
(a)
Constant n, C, and w
ζ GEV
o ζ obs
ζ GUM
ζ MEV
(b)
Constant n - Random C and w
(c)
Random n - Constant C and w
0
100
200
y (mm)
300
Fig. 3. Doubly logarithmic (Gumbel) plots of different estimates of the extreme
value distribution fðyÞ. Different panels refer to different synthetic data sets: A
(panel a), B (panel b), C (panel c). The solid lines indicate the MEV estimate fMEV ðyÞ
(blue), the GEV estimate fGEV ðyÞ (red), and the Gumbel estimate fGUM ðyÞ (green).
Open circles indicate the empirical frequency distribution averaged across different
realizations, fobs ðyÞ. (For interpretation of the references to colour in this figure
legend, the reader is referred to the web version of this article.)
explores the effects of a small number of rainfall events per year on
the convergence properties of the actual extreme value distribution to the GEV distribution.
For each setup we generate 1000 realizations of N yr ¼ 50 years
of synthetic rainfall, to reproduce typical (though optimistic) sample sizes encountered in practice, and we estimate the distribution
of extremes using the MEV, GEV, and Gumbel approaches. For each
realization: (1) we estimate C and w from the synthetic time series
via least square fit; (2) We extract the sequence of the N yr annual
maxima Y j R ; (3) We use Eq. (11), to obtain 1000 estimated
MEV probability distributions; (4) We estimate, via Maximum
Likelihood, the corresponding 1000 GEV and Gumbel distributions;
(5) We compute the ensemble average distributions,
fMEV ðyÞ; fGEV ðyÞ, and fGUM ðyÞ as the means of the distributions of Y
computed over the 1000 synthetic time series.
To test the predictive performance of the estimated distributions fðyÞ; fGEV ðyÞ, and fGUM ðyÞ, we compare them against the
observational frequency, fobs ðyi Þ, computed for the annual maxima
from sequences of N ¼ 106 years of synthetic rainfall generated
according to the above definitions for each setup A, B, and C. The
choice of using 106 values ensures that extremes with return
period of 100 and 1000 years are appropriately represented by
the usual plotting-position estimate fobs ðyi Þ ¼ ni =ðN þ 1Þ (ni being
the number of annual maxima in the sample smaller or equal
to yi ).
The results (Fig. 3) show that in all cases the distribution
obtained with the MEV approach coincides with the observational
frequencies. This indicates that use of Eq. (11) provides an accurate
representation of the extreme value distribution even when the
Weibull distribution generating daily values changes over time
(setups B and C). The GEV distribution, on the other hand, consistently overestimates the precipitation value associated with a
given return period, while the Gumbel approach consistently
underestimates it. It is interesting to note that these discrepancies
between GEV and Gumbel distributions and the observed one are
very marked also for the time-invariant case, when the sample
used for fitting the extreme value distributions more accurately
represents the underlying population and convergence to the limiting GEV distribution should be more rapid.
Mean of % estimation error
errors, and focuses only on significant events, of chief interest here.
We also note that all that is needed for Eq. (12) or Eq. (10) to be
applicable is that the distribution of daily values be right-tail
equivalent to a Weibull distribution. Hence the analysis of the
upper range of values of X suffices for our purposes.
C and w show significant changes over the observation period
(e.g. compare the 1841–1880 and the 1887–2006 periods in particular), suggestive of different rainfall regimes taking place across
centuries (see Fig. 2(c) and (d)). Visual inspection of the values of C
and w suggests precipitation processes to be relatively homogeneous particularly within sub-intervals 1768–1807, and 1887–
2006. Sub-interval 1841–1880 exhibits more widely different
fluctuations of the values of n; C, and w.
Root Mean Square % Error
124
40%
(a)
100 yrs GEV
1000 yrs
MEV
20%
100 yrs GUM
1000 yrs
100 yrs
1000 yrs
0%
-20%
-40%
80%
(b)
60%
40%
20%
0%
20
40
60
80
100 120 140
number of wet days/year
160
180
200
Fig. 4. The relative errors for daily rainfall intensity corresponding to a return
period of 100 years (solid red line for the GEV approach and solid green line for the
Gumbel approach) and 1000 years (dotted red line for the GEV approach and dotted
green line for the Gumbel approach). Panel (a) shows results for the homogeneous
test case, A: the number of events/year and the parameters of the Weibull
distribution are kept fixed for the whole synthetic time series. Panel (b) shows
results for the heterogeneous case B: the Weibull parameters used to generate the
daily rainfall sequences are extracted randomly every two years. (For interpretation
of the references to colour in this figure legend, the reader is referred to the web
version of this article.)
M. Marani, M. Ignaccolo / Advances in Water Resources 79 (2015) 121–126
125
period extreme events in actual applications, as wide interannual
variability, such as that evidenced in Fig. 1, and possibly systematic
changes, are likely the norm.
4.3. MEV estimates of high return period events
Fig. 5. MEV extreme rainfall distributions obtained for different sub-intervals of the
Padova time series. The hatched areas represent confidence intervals of (vertical)
width equal to two standard deviations. The standard deviation is estimated using
the asymptotic value of 5.1% in Fig. 4(b).
In order to further characterize the error associated with the
GEV and Gumbel estimates, we consider two more test cases in
which we generate 1000 replicates of synthetic time series of daily
rainfall. The first synthetic test case (A2) is generated, as for test A
above, by assuming a fixed number of wet days, n, and a Weibull
distribution of daily rainfall with constant parameters. This is the
homogeneous test case. The second test case (B2) is obtained, similarly to test B above, with a fixed number of wet days n, by changing the parameters of the Weibull distribution every 2 years
(chosen with uniform probability from the set of Weibull parameters obtained from the Padova time series). For both test experiments we progressively increase the fixed number of wet days
from n ¼ 10 events/year to n ¼ 200 events/year, to address the
accuracy of the asymptotic assumption embedded in the GEVGumbel approaches. From each synthetic time series realization
we estimate fGEV ðyÞ and fGUM ðyÞ (using ML) and derive the daily
rainfall intensity corresponding to a return period of 100 years
(1000 years) yGEV;100 (yGEV;1000 and yGUM;100 yGUM;1000 ). These values
are compared with the values obtained from the observed frequency distribution, by computing relative errors such as
(yGEV;100 -yth;100 Þ=yth;100 Þ 100 (yth;100 being the 100th yearly maximum value in an list of ‘‘observations’’ sorted in ascending order).
Finally, the mean relative error is computed over all the 1000 replicates of the synthetic time series for both test cases. Results show
(Fig. 4) that both in the homogeneous A2 case (panel (a)) and in the
inhomogeneous case B2 (panel (b)), the relative errors decrease as
the cardinality increases, but that a constant bias always remains.
In the homogeneous case the GEV approach systematically overestimates the 100-yr extreme rainfall intensity by 5% even for
large numbers of wet days. The Gumbel approach systematically
underestimates the 100-yr extreme rainfall intensity by about
5%. For the 1000-years return period intensities, the GEV approach
severely overestimates actual extreme events (minimum relative
error is 30% for n = 200 events/year) whereas the Gumbel
approach yields underestimation errors of about 10%. It is interesting to see that, while a noticeable dependence on the number of
wet days exists in the homogeneous case, estimation errors are larger in the non-homogeneous case, with relative errors almost
independent of the number of wet days for n > 20 events/year. In
the non-homogeneous case the GEV and Gumbel approaches give
similar absolute values of the relative errors (about 10%, but with
opposite signs) for the 100-year return period. Estimations errors
for the 1000-year return period are about +50% for the GEV estimate and about 20% for the Gumbel approach. These findings
have significant implications for the estimation of high-return
It is interesting, also in view of practical applications, to compare extreme value distributions obtained through the MEV
approach for different periods from the Padova time series (see
[20] for similar considerations using the traditional GEV approach).
The observed differences (Fig. 5) are significant when compared
with the uncertainty associated with MEV estimates, represented
by a confidence interval equal to two times the standard deviation
of the estimation error (hatched area in Fig. 5). The MEV distribution adapted on the 1841–1880 period gives the smallest
extreme events: the estimated 1000 yr rainfall depth for 1841–
1880 is h1000 110 mm. The estimated 1000 yr rainfall based on
data from 1725 to 1764 is h1000 130 mm, while the estimate from
1887 to 2006 is h1000 160 mm, very similar to the estimate
obtained for 1768–1807 (distribution not shown here because
almost superimposed on the one from 1887 to 2006). In this case,
the estimate of a 1000-yr return period design rainfall based on
data from the 1841–1880 period would be seriously underestimating the extreme events which occurred in the subsequent century.
Similar conclusions can be reached if one considers the 100-year
return period, for which estimates vary from about 90 mm for
1841–1880 to about 120 mm for 1887–2006. Overall, the interannual differences in extremes estimated using the MEV distribution
are coherent with those obtained by a conventional GEV analysis
[20], even though the range covered by such estimates is not quite
as large. These results show that, even in times in which anthropogenic climatic changes may safely be assumed not to affect rainfall regimes (e.g. consider the 1725–1764 vs. the 1841–1880
periods), extreme events exhibit a wide long-term variability, such
that a simplistic interpretation of the past may lead to grossly erroneous inferences on the future [20].
5. Summary and conclusion
We have introduced here a metastatistical approach to daily
rainfall extreme event evaluation. In its general formulation (Eqs.
(9) and (10)) the approach is valid for an arbitrary (and, in particular, arbitrarily small) value of the yearly number of rainfall events,
as it does not rely on an asymptotic approximation as the classical
EVT. Both the number of wet days per year and the parameters controlling rainfall intensity are also allowed to vary stochastically, as
observed in practice for a long rainfall time series (see Figs. 1 and 2).
Global observations and inferences show that daily rainfall can
be considered, to a good approximation, to be Weibull right-tail
equivalent. The specialization of the general MEV expression to
the Weibull case yields a compact expression (Eq. (11)), which
can be easily applied by estimating the Weibull parameters
(C j ; wj ) and the number of wet days (nj ) on a yearly basis. Monte
Carlo experiments based on Weibull-distributed synthetic daily
rainfall show an excellent agreement between MEV-estimated
extremes and ‘‘observed’’ occurrence of extreme events. GEV and
Gumbel approaches, on the contrary, are shown to be affected by
systematic errors and a large root mean square error.
Experiments using a varying number of wet days/year show a mild
dependence of GEV and Gumbel bias on n and show a significant
tendency of the GEV approach to overestimate actual extremes
and of the Gumbel method to underestimate them. When n; C,
and w are allowed to vary stochastically, the GEV and Gumbel bias
seems to be insensitive to n, and no improvement in the estimation
accuracy is seen for (synthetic) climates with a larger number of
126
M. Marani, M. Ignaccolo / Advances in Water Resources 79 (2015) 121–126
rainy days/year. We hypothesize this to be a symptom of the
sensitivity of the traditional methods to violations of the asymptotic hypothesis: when n cannot be considered to be ‘‘large
enough’’, the specific value of n matters, particularly if it is timevarying. The MEV method, on the contrary, does not exhibit a
detectable bias for any value of the number of wet days, and is here
proposed as a general approach to extreme value estimation.
MEV estimates for the long rainfall time series recorded in
Padova (Italy) illustrate the wide variability of extremes and the
magnitude of the estimation errors that can arise when the past
is assumed to be a faithful representation of the future. Very significant deviations of actual extreme events from estimated ones
occur across centuries [20, see also]. The MEV approach explicitly
identifies and separates the roles of conceptually and physically
meaningful sources of variability: the stochastic occurrence of
rainfall events and the probability distribution of daily rainfall
intensities. We also suggest that the use of a time-varying metastatistical characterization of these processes, i.e. a time-varying
distribution f ðn; C; w; hðtÞÞ, though not yet pursued here, could provide a natural way to treat changing extremes. In particular, the
metastatistical parameters h could be linked to dominant largescale climatic patterns, such as ENSO and NAO, to provide a clear
connection between changing extremes, endogenous climatic
oscillations, and anthropogenic climate change.
Acknowledgements
This research was funded by NSF-EAR-13-44703 ‘‘The Direct
and Indirect Effects of Plantation Forestry Expansion on Usable
Water in the Southeastern US’’. We also acknowledge support by
the Nicholas School of the Environment and the Pratt School of
Engineering at Duke University.
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