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Advances in Geosciences, 7, 51–54, 2006

SRef-ID: 1680-7359/adgeo/2006-7-51

European Geosciences Union

© 2006 Author(s). This work is licensed

under a Creative Commons License.

Advances in

Geosciences

A probabilistic approach to the concept of Probable Maximum

Precipitation

S. M. Papalexiou and D. Koutsoyiannis

Department of Water Resources, School of Civil Engineering, National Technical University of Athens, Greece

Received: 8 November 2005 – Revised: 12 December 2005 – Accepted: 14 December 2005 – Published: 24 January 2006

Abstract. The concept of Probable Maximum Precipitation

(PMP) is based on the assumptions that (a) there exists an up-

per 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 esti-

mation method of PMP is the so-called moisture maximiza-

tion method. This method maximizes observed storms as-

suming 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 pa-

per, it is argued that fundamental aspects of the method may

be ﬂawed or inconsistent. 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 ev-

idence of an upper bound either in atmospheric moisture or

maximized precipitation depth. Therefore, it is argued that

a probabilistic approach is more consistent to the natural be-

haviour and provides better grounds for estimating extreme

precipitation values for design purposes.

1 Introduction

The Probable Maximum Precipitation (PMP) is deﬁned as

“theoretically the greatest depth of precipitation for a given

duration that is physically possible over a given size storm

area at a particular geographical location at a certain time

of year” (WMO, 1986). Even though, the PMP approach

has been widely proposed and used as design criterion of

major ﬂood protection works (Schreiner and Reidel, 1978;

Collier and Hardaker, 1996), severe criticism has been made

by hydrologists not only to the concept of the PMP, which

practically assumes a physical upper bound of precipitation

amount, but also to the fact that this limit can be estimated

Correspondence to: S. M. Papalexiou

(sp@itia.ntua.gr)

based on deterministic considerations (Benson, 1973; Kite,

1988; Dingman, 1994; Shaw, 1994; Koutsoyiannis, 1999).

The main scope this paper is to apply the PMP estimation

method and to make a probabilistic analysis of its results. An

additional objective was to ﬁnd an appropriate probabilistic

model capable of describing the empirical distribution of the

monthly maximum daily dew points, in order to be used in

the application of the method. Last, but not least, an exclu-

sively probabilistic approach was applied to the annual max-

imum rainfall depths.

The methodology was applied to four stations in Nether-

lands (De Bilt, Den Helder, Groningen, Maastricht) and

the station of the National Observatory of Athens, Greece

(NOA).

2 Method overview

Techniques used for estimating PMP have been listed by

Wiesner (1970), as follows: (1) the storm model approach;

(2) the maximisation and transposition of actual storms; (3)

the use of generalised data or maximised depth, duration and

area data from storms; these are derived from thunderstorms

or general storms; (4) the use of empirical formulae deter-

mined from maximum depth duration and area data, or from

theory; (5) the use of empirical relationships between the

variables in particular valleys (only if detailed data are avail-

able); (6) statistical analyses of extreme rainfalls.

In this study, the most representative and widely used esti-

mation method of PMP, the so-called moisture maximization

method, is examined. The method is based on the simple

formula

hm=Wm

Wh, (1)

where hmis the maximized rainfall depth, his the observed

precipitation, Wis the precipitable water in the atmosphere

during the day of rain, estimated by the corresponding daily

dew point Td, and Wmis the maximized precipitable water,

52 S. M. Papalexiou and D. Koutsoyiannis: The concept of Probable Maximum Precipitation

0.00

0.05

0.10

0.15

0.20

0.25

-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30

L-skewness

L-kurtosis

NOA st atio n Netherlands station

s

Average Gumbel

Weibull max n=5 Weibull max n=30

Weibul 3P GEV

Gum b e l n=5

n=30

Fig. 1. L-moment ratio diagram of maximum daily dew points for

each month.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

NOΑDe Bilt Den Helder Groningen Maastricht

Stations

L-skewness

Ave rage L-skew ness of monthly

max imu m rai nf all dep th s

Ave rage L-skew ness of monthly

maximized r ainfall depths

Fig. 2. Average L-skewness of monthly maximum and maximized

daily rainfall depths.

estimated by the maximum daily dew point Td,m of the cor-

responding month.

The term Td,m is estimated either as the maximum histor-

ical 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 (WMO, 1986).

3 Statistical analysis of dew points

The Gumbel distribution (Gumbel, 1958), which is the

most common probabilistic model for hydrological extremes,

proved inadequate for describing the empirical distribution

of the monthly maximum daily dew points (Papalexiou,

2005). Therefore, the probability theory of maxima was ap-

plied. According to that theory, given a number of ninde-

pendent 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,(2)

where F(x)=P(Yi≤y)is the probability distribution func-

tion of each Yi, referred to as the parent distribution.

-10

40

90

140

190

240

290

0 20 40 60 80 100 120 140

Raifall number

Rainfall depht ( mm)

maximized r ainfall depht (mrd)

daily rainf all depht correspon ding to mrd

maximum monthly precipitable water cor responding to mrd

daily prec ipitable w ater cor responding to mrd

PMP

Fig. 3. Maximized rainfall depths of the NOA station and related

quantities.

The frequency analysis for the daily dew points indicated

the three-parameter Weibull model as a sufﬁcient probabilis-

tic model for describing their empirical distribution (Papalex-

iou, 2005). As a result, it can be used as parent distribution,

so that the theoretical maximum distribution of the monthly

maximum daily dew point can be described by Eq. (2), where

nstands for the days of each month.

Since the condition of independence of random variables

does not hold, as shown from the high values of autocorre-

lation coefﬁcients, the exponent nwas expected and proved,

indeed, lower than the number of days in a month (Papalex-

iou, 2005). Moreover, given the uncertainties related to the

estimation of the three parameters of the parent distributions,

as well as the uncertainty of the value of the exponent n, we

implemented a “parallel” optimization approach, by simulta-

neously ﬁtting the theoretical models F(x)and Hn(x)to the

empirical distributions of daily and monthly maximum daily

dew points. The objective function is written as

LSET ot al =LSE (F(x)) +[LSE (Hn(x))]2,(3)

where LSE is the least square error between the theoretical

and the empirical distributions. This strategy helped to better

ﬁt the theoretical maximum distribution derived by the parent

distribution.

The L-moment ratio diagram (Vogel and Fennessey, 1993;

Stedinger et al., 1993) shown in Fig. 1, illustrates that the

theoretical maximum distribution derived from the parent

three-parameter Weibull distribution is more appropriate than

the asymptotic ones (Gumbel, Generalized Extreme Value or

GEV).

4 Application of the PMP estimation method

The maximized precipitation time series were analysed in

comparison to the observed ones. It was concluded that the

maximization process causes sometimes a disproportional

increase in the range of the values of recorded rainfall and

that the maximized samples exhibit higher skewness than

S. M. Papalexiou and D. Koutsoyiannis: The concept of Probable Maximum Precipitation 53

1000050002000100050020010050201052

0

50

100

150

200

250

300

350

400

450

Return period (years)

Rainfall depht (mm)

E.D. of annual maximum

GEV MLM of max imu m ann ual

E.D. of max imiz ed mo nth ly

GEV MLM of mon thl y ma ximi zed

E.D. of monthly maximum

GEV MLM of mon thl y ma ximu m PM P

Fig. 4. Probability plot of annual maximum, monthly maximum

and monthly maximized daily rainfall depths of NOA (on Gumbel

probability paper).

0

50

100

150

200

250

300

350

100 1000 10000 100000 1000000

Return period of monthly maximum dew point (years)

PMP (mm )

NOA

De Bi lt

Den He lder

Groningen

Maastricht

Fig. 5. PMP estimations for various return periods of monthly max-

imum daily dew point.

the recorded ones (Fig. 2), especially when the sample L-

skewness values are low.

Figure 3 illustrates the values of maximized rainfall depths

of the station of NOA in descending order, the concurrent

daily-recorded rainfall depths, the concurrent daily precip-

itable water and the maximum monthly precipitable water.

The 120 maximized rainfall depths that are illustrated in

Fig. 3 (the highest is the estimated PMP), are the result of the

merging of the 10 maximized rainfall events of each month,

produced by the 10 maximum rainfall events of the respective

month. It is evident that the estimated PMP point is located

in a very uncertain area of the curve, where the slope is very

high.

In addition, as shown in Fig. 3, if the record was shorter

or had a missing data point (the most left in Fig. 3) the PMP

value would be downgraded from 240 to about 190mm.

Moreover, the sample of the 120 maximized rainfall

depths was analyzed in the same probabilistic manner as the

maximum monthly and annual rainfall depths were analyzed.

Figure 4 depicts the relevant data of NOA. The ﬁtted distri-

butions (GEV) to the maximum monthly and annual rainfall

depths, as Fig. 4 suggests, have no upper bound, so it is very

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

L-skewness

L-kurtosis

Gumbel

GEV

NOA

De Bi lt

Den Helde r

Groningen

Maa s t r ic h t

Gum b e l

Fig. 6. L-moment ratio diagram for annual maximum daily rainfall

depths.

% probability o f excee dence

0.01

0.02

0.05

0.1

0.2

0.5

1

2

5

10

20

30

50

70

90

95

99

99.9

99.99

2 5 10 20 50 100 200 500 1000 2000 500010000

0

50

100

150

200

250

300

350

400

450

500

Return period (years)

Rainfall depth ( mm)

PMP

Gu mbel L -mo ment s

GEV L- momen ts

GEV Least s quare error

GEV Max imum l ikel iho od Estimated PMP

Fig. 7. Probability plot of annual maximum daily rainfall values of

NOA station on Gumbel probability paper.

likely that if a longer rainfall record were available, the esti-

mate of the PMP would be higher.

Furthermore, if solely the distribution of maximized rain-

falls is taken under consideration, then the estimate of PMP

is the outcome of an extremely uncertain estimation of the

ﬁrst order statistic.

Finally, the PMP estimation method was applied with re-

gard to the monthly maximum daily dew point for a wide

range of return periods, admitting that the WMO suggestion

to use a 100 year return period is arbitrary and this return

period could be well assumed greater. The distribution of

the monthly maximum daily dew point was the one derived

by the three-parameter Weibull parent distribution. The es-

timate of the PMP, as illustrated in Fig. 5, is an increasing

function of the monthly maximum daily dew point.

5 A probabilistic approach for the annual maximum

daily rainfall depth

If one abandons the effort to put an upper limit to precipi-

tation and the deterministic thinking behind it, the next step

is to model maximum rainfall probabilistically. As the L-

moment ratio diagram of the annual maximum daily rainfall

54 S. M. Papalexiou and D. Koutsoyiannis: The concept of Probable Maximum Precipitation

(Fig. 6) suggests, the GEV distribution describes sufﬁciently

the empirical distribution of the annual maximum daily rain-

fall depths.

The GEV model was ﬁtted on the historical data with three

different methods, the method of Least Square Error (Pa-

poulis, 1990), the method of L-Moments (Hosking, 1990)

and the method of Maximum Likelihood (Fisher, 1922). Ac-

cording to the ﬁtted probabilistic model, the estimate of the

PMP value is associated with a return period or probability

of exceedence. It can be concluded that this probability is

not negligible. The above analysis was also conducted us-

ing the Gumbel model, which obviously underestimates the

exceedence probability of the PMP (Fig. 7).

6 Conclusions

In the above analysis, no evidence for an upper bound of dew

point and of precipitation was found. The estimation of the

PMP based on the moisture maximization concept is consid-

erably uncertain and was proven to be too sensitive against

the available data.

The study showed 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 asserts; in reality, from a

statistical point-of-view, this “limit” tends to inﬁnity.

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 not negligible,

exceedence probability. For example, this probability for the

Athens area is 0.27%, a value that would not be acceptable

for the design of a major hydraulic structure.

A probabilistic approach, based on the GEV model,

seems to be a more consistent tool for studying hydrological

extremes.

Edited by: V. Kotroni and K. Lagouvardos

Reviewed by: A. Koussis

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