A probabilistic approach to the concept of Probable Maximum Precipitation
7th Plinius Conference on Mediterranean Storms
European Geosciences Union (EGU), Rethymnon, Greece, 5-7 October 2005
Topic 3: Meteorological, hydrological and geological risks, disaster management and mitigation strategies
S.M. Papalexiou and D. Koutsoyiannis
Department of Water Resources, School of Civil Engineering, National Technical University of Athens
Method overview
Abstract
The concept of Probable Maximum Precipitation (PMP) is based on the assumptions that: (a) there exists an upper physical limit of the
precipitation depth over a given area at a particular geographical location at a certain time of year, and (b) that this limit can be estimated based
on deterministic considerations. The most representative and widespread estimation method of PMP is the so-called moisture maximization
method. This method maximizes observed storms assuming that the atmospheric moisture would hypothetically rise up to a high value that is
regarded as an upper limit and is estimated from historical records of dew points. In this study, it is argued that fundamental aspects of the
method may be flawed or illogical. Furthermore, historical time series of dew points and “constructed” time series of maximized precipitation
depths (according to the moisture maximization method) are analyzed. The analyses do not provide any evidence of an upper bound either in
atmospheric moisture or maximized precipitation depth. Therefore, it is argued that a probabilistic approach is more consistent to natural
behaviour and provides better grounds for estimating extreme precipitation values for design purposes.
Statistical analysis of daily dew points
From the statistical theory of extremes to practice
40
50
0.00
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Shape parameter c
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Normal skew ness
N.O.A .
De Bilt
Den Helder
Groningen
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A ug
Sep
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Nov
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Figure 2: L-skewness of daily dew points.
0.20
Estimated values
0.05
Netherlands stations
Normal
GPA
Weibull 3
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Scale parameter
Location parameter
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Jan
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Figure 11: Box plot of maximum daily rainfall depths per month.
120
100
60
80
100
120
Maastricht
Figure 15: Correlation coefficients of maximized rainfall
depths with related factors.
n=5
Gum bel
0.15
0.10
0.05
N.O.A. station
Netherlands stations
Average
Gumbel
Weibull max n=5
Weibull max n=30
Weibul 3P
0.00
-0.20
-0.15
-0.10
-0.05
0.00
0.05
L-skew ness
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GEV
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Apr
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99.99
As concluded by the L-moment ratio
diagram (Figure 18), the General
Extreme Value distribution (GEV)
describes appropriately the
empirical distribution of the annual
maximum daily values of rainfall.
The GEV model was fitted on the
sample with three different methods.
0.15
0.10
Average L-skew ness of monthly
maximum rainf all depths
0.05
Average L-skew ness of monthly
maximized rainfall depths
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Figure 12: Box plot of maximized rainfall depths per month.
Ε.Α.Α.
De Bilt
Den Helder
Groningen
Maastricht
Figure 13: Average L-skewness of monthly maximum
and maximized daily rainfall depths.
Rainfall depht (mm)
21
7
9
11
13
15
17
19
0.25
0.20
Gum bel
0.15
Gumbel
GEV
E.A.A
De Bilt
Den Helder
Groningen
Maastricht
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0.00
0.00
0.05
0.10
0.15
0.20
0.25
L-skew ness
0.30
0.35
0.40
0.45
The estimation of the PMP,
according to the fitted model, was
associated to a return period or,
equivalently, a probability of
exceedance. It was concluded that
this probability is not negligible.
The analysis was also conducted
using the Gumbel model, which
obviously underestimates the
exceedance probability of the PMP
(Figure 19).
99.99
99.9
99
Figure 18: L-moment ratio diagram for annual maximum
daily rainfall depths.
% probability of exceedance
500
PMP
Gumbel L-moments
GEV L-moments
GEV Least square error
GEV Maximum likelihood
450
400
Rainfall depth (mm)
The sample of the 120 maximized rainfall depths was probabilistically approached,
and in comparison with the respective maximum monthly and annual rainfall
depths. The above data regarding the National Observatory of Athens are illustrated
in Figure 16. Probably, if a longer rainfall record was available, the estimation of
450
the PMP would be higher.
E.D. of annual maximum
400
GEV MLM of maximum annual
Furthermore, if the distribution
E.D. of maximized monthly
350
GEV MLM of monthly maximized
of maximized rainfalls is the
E.D. of monthly maximum
300
GEV MLM of monthly maximum
only component taken under
250
consideration, then the
200
estimation of PMP comes out as
150
an uncertain estimation of the
100
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first order statistic.
350
Estimated PMP
300
250
200
150
100
50
0
2
5
10 20
50 100 200
Return period (years)
500 1000 2000 500010000
Figure 19: Probability plot of annual maximum daily
rainfall values of NOA station on Gumbel paper.
0
140
2
5
10 20
50 100 200
Return period (years)
500 1000 2000 500010000
Figure 16: Probability diagram of annual maximum, monthly
maximum and monthly maximized daily rainfall depths.
The PMP estimation method was applied in regards to the monthly maximum daily
dew point, for a wide range of return periods. The assumed model of the monthly
350
maximum daily dew point was
300
the one derived by the three250
parameter Weibull distribution.
The estimation of the PMP, as
200
illustrated in Figure 17, is an
150
ascent function of the monthly
N.O.A.
100
De Bilt
maximum daily dew point.
Den Helder
PMP (mm)
correlation coefficient
Groningen
Dec
n=30
0.20
90
0.00
Jan
The maximized sample of rainfall depths was examined in respect with the
factors that may affect them. The maximized rainfalls seem to be directly
related to the observed
1.0
rainfalls, while there is
0.8
some doubt about their
0.6
correlation with the
0.4
maximum precipitable
0.2
water and no evidence of
0.0
any correlation with the
-0.2
hmax-h
hmax-Wmax
daily precipitable water
-0.4
hmax-W
(Figure 15).
-0.6
Den Helder
Nov
A probabilistic approach for the annual maximum
daily rainfall
Raifall num ber
De Bilt
Oct
0.25
99
0.20
Figure 14: Maximized rainfall depths of N.O.A station and
related factors.
E.A.A.
Average
Sep
As the condition of independence of random
variables is not valid, the parameter n was
inspected to be and proved, indeed, lower than
the theoretically expected value (Figure 7).
99.99
80
Rainfall depht (mm)
40
Aug
0.30
140
Figure 14 illustrates the values of maximized rainfall depths of the station of
National Observatory of Athens in descending order, the respective daily
recorded rainfall depths, the respective daily precipitable water and the
maximum monthly precipitable water. The 120 maximized rainfall depths
290
that are illustrated in
Maximized rainfall dephts
Figure 14 are the overall
Respective daily rainfall depht
240
Respective maximum monthly precipitable w ater
of the 10 maximum
Respective daily precipitable w ater
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rainfalls of each month. It
140
is evident that the
estimated PMP point is
90
located in a very uncertain
40
area of the curve, where
the slope is very high.
-10
20
Jul
Figure 7: Estimated values for parameter n.
The estimation method of the PMP was applied to the five stations in Netherlands and Greece. The maximized precipitation time series were analysed in comparison with
the observed ones. It was concluded that the maximization process causes, sometimes, a disproportional increase to the range of the values of observed rainfall and that the
maximized samples present a higher skewness than the observed ones, especially when the sample L-skewness values are low (Figures 11-13).
0
Jun
90
0.25
160
0
0
May
99
0.30
20
10
Apr
99.99
0.35
40
20
Mar
90
14
60
30
Feb
Figure 10: Probability plots of monthly maximum daily dew points of NOA station on theoretical maximum distribution paper.
L-skewness
Rainfall depth (mm)
Rainfall depth (mm)
100
Jan
99
180
110
Dec
99.99
20
200
120
Nov
90
Application of the PMP estimation method
140
130
Oct
99
12
Figure 9: Probability plots of daily dew points of NOA station on three parameter Weibull paper.
150
Sep
The L-moment ratio
diagram (Figure 8)
illustrates that the
theoretical maximum
distribution derived
from the parent threeparameter Weibull
distribution is more
appropriate than the
classic ones.
0.01
where LSE is the least square error between the theoretical
and the empirical data.
This strategy helped to better fit the theoretical maximum
distribution derived by the parent distribution.
0.1
10
Aug
99.99
4
0.01
5
Jul
0.01
LSEtot = LSE[F(x)] + [LSE[Hn(x)]]2
2
-
Jun
Figure 8: L-moment ratio diagram of maximum daily dew
points for each month.
25
0.5
-5
May
The condition of independence of random
variables is not valid, as proved through the high
values of autocorrelation coefficients (Figure 6).
Given the uncertainties related to the estimation of the
parameters of the parent distribution, as well as the
deviation of parameter n from its theoretical value, we
implemented a “parallel” optimization approach, by
simultaneously fitting the theoretical models F(x) and Hn(x)
to the empirical distributions of daily (Figure 9) and
monthly maximum daily (Figure 10) dew points. The
objective function is written as:
Dec
99.99
-10
Apr
Jan
Aug
99.99
0.01
-
-10
20
0.5
-5
25
-5
0.1
0.01
-
-10
99
25
51
Mar
Figure 6: Monthly average autocorrelation coefficient of daily
dew points.
Model fitting to the NOA sample: A
parallel optimisation approach
0.01
-15
99.99
0.5
0.1
0.01
0.1
41
Feb
2
10
5
2
0.5
0.1
31
Groningen
Maastricht
Figure 5: Parameters of the Weibull distribution, estimated by
the maximum likelihood method.
0.1
0.01
-15
20
Jan
100 synthetic samples of
3000 values, with a priori
statistical structure, were
generated (Figure 7); next,
their parameters were reevaluated using various
methods. The Monte Carlo
simulation approach proved
the uncertainty related to the
estimation of parameters of
the Weibull distribution.
10
0.5
0.1
0.1
0.01
-5
Dec
Apr
99.99
99
-10
Nov
Sam ple num be r
99.99
-15
Oct
De Bilt
Den Helder
0
0.01
Jan
99
Sep
-30
-0.05
-0.20
90
0.1
Aug
-10
N.O.A . Station
A verage
GNO
Pearson 3
0.00
N.O.A.
5
Maastricht
0.02
L-kurtosis
0.10
99
0.01
Jul
20
99.99
99.99
Jun
40
Figure 3: L-moment ratio diagram of daily dew points.
30
20
May
0.15
L-s k e w ne s s
70
50
Apr
30
Through the frequency
analysis of the samples, the
three-parameter Weibull
distribution was considered
as the most appropriate
theoretical model for the
empirical distribution of the
daily dew points (Figure 3).
Groningen
0.05
Jun
De Bilt
Den Helder
0.1
May
N.O.A.
0.2
A pr
Mar
The thee parameter Weibull distribution was fitted
to the samples of daily dew points, using four
different estimation methods. The variance of the
estimated parameter is obvious (Figure 5).
Maastricht
A verage
Mar
Feb
15
10
0.3
Figure 4: Estimated parameters of the three parameter Weibull
for NOA station.
-0.25
Feb
0.4
0.0
Jan
Jan
0.5
0.1
Location param eter α
-40
-0.15
20
0.6
0.5
L-skewness
-30
-0.20
0.7
0.2
-0.05
-0.10
25
0.8
20
Estimated values
0.05
30
0.9
30
Figure 1: Box plot of daily dew points of NOA station.
The values of L-skewness of
daily dew points were, in
general, negative. Moreover,
during the summer months
they were higher than in
winter months (Figure 2).
1.0
Scale parameter b
40
Dec
1
Nov
2
Oct
5
Sep
10
A ug
30
Jul
20
Jun
50
May
70
A pr
Autorrelation coefficient
Mar
95
90
-20
Feb
Parameter n
-10
Jan
L-kurtosis
0
L-kurtosis
10
Dew Points ( οC)
20
The Gumbel distribution, which is the most common probabilistic model for hydrological extremes, proved inadequate for describing the empirical distribution of
the monthly maximum daily dew points. Therefore, we attempted to apply the fundamentals of the theory of extremes in a direct manner. According to it, given a
number of n independent identically distributed random variables, the largest of them (more precisely, the largest order statistic), i.e. X = max (Y1, …, Yn) has
probability distribution function Hn(x) = [F(x)]n, where F(x) = P(Yi ≤ y) is the common probability distribution function, known as the parent distribution of Yi.
The frequency analysis for the daily dew points indicated that the three-parameter Weibull model is a sufficient probabilistic model for describing the empirical
distribution of them; hence F(x) can be used as the parent distribution. Consequently, the theoretical maximum distribution of the monthly maximum daily dew
point is described by Hn(x), where n stands for the days of each month.
Daily dew point time series, taken
from four stations in Netherlands
and one in Greece (the National
Observatory of Athens, NOA),
were analyzed. A common remark
for all stations is that the range of
dew points during the summer
months is much shorter, compared
to the winter ones (Figure 1).
30
In this study, the most representative and widespread estimation method of PMP, the so-called moisture
maximization method is applied and examined. The method is based on the following formula:
W
hm = m h
W
where hm is the maximized rainfall depth; h is the observed precipitation; W is the precipitable water in the
atmosphere during the day of rain, estimated by the corresponding daily dew point Td; and Wm is the
maximized precipitable water, estimated by the maximum daily dew point Td,m of the corresponding month.
The term Td,m is estimated either as the maximum historical value from a sample of at least 50 years length, or
as the value corresponding to a 100-years return period, for samples smaller than 50 years.
50
Groningen
Maastricht
0
100
1000
10000
100000
Return pe riod of m onthly m axim um dew point (years)
1000000
Figure 17: PMP estimations for various return periods of
monthly maximum average daily dew point.
Conclusions
1. There is no evidence for a physical upper bound regarding the dew points.
2. The estimation of the PMP, based on the moisture maximization concept, is
considerably uncertain and was proved too sensitive against the available data.
3. The study proved that the existence of an upper limit on precipitation, as implied
by the PMP concept, is statistically inconsistent. Moreover, such a limit cannot be
specified in a deterministic way, as the method implies; in reality, from a statistical
point-of-view, this “limit” tends to infinite.
4. According to the probabilistic analysis on the annual daily maximum rainfall
depths, the hypothetical upper limit of the PMP method corresponds to a small,
although realistic, exceedance probability. For example, this probability for the
NOA sample is 0.27%, a value that would not be acceptable for the design of a
major hydraulic structure, such as a dam.
5. A probabilistic approach, based on the GEV model, seems to be the most
consistent tool for studying hydrological extremes.