ArticlePDF Available

Abstract and Figures

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 paper, 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 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 the natural behaviour and provides better grounds for estimating extreme precipitation values for design purposes.
Content may be subject to copyright.
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
A probabilistic approach to the concept of Probable Maximum
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
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
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
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
-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30
NOA st atio n Netherlands station
Average Gumbel
Weibull max n=5 Weibull max n=30
Weibul 3P GEV
Gum b e l n=5
Fig. 1. L-moment ratio diagram of maximum daily dew points for
each month.
NOΑDe Bilt Den Helder Groningen Maastricht
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
where F(x)=P(Yiy)is the probability distribution func-
tion of each Yi, referred to as the parent distribution.
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
Fig. 3. Maximized rainfall depths of the NOA station and related
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
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
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
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).
100 1000 10000 100000 1000000
Return period of monthly maximum dew point (years)
PMP (mm )
De Bi lt
Den He lder
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
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
Fig. 6. L-moment ratio diagram for annual maximum daily rainfall
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
Edited by: V. Kotroni and K. Lagouvardos
Reviewed by: A. Koussis
Benson, M. A.: Thoughts on the design of design floods, in Floods
and Droughts, Proc. 2nd Intern. Symp. Hydrology, pp. 27-33,
Water Resources Publications, Fort Collins, Colorado., 1973.
Collier, C. G and Hardaker, P. J.: Estimating probable maximum
precipitation using a storm model approach, J. Hydrol., 183,
277–306, 1996.
Dingman, S. L.: Physical Hydrology, Prentice Hall, Englewood
Cliffs, New Jersey, 1994.
Fisher, R. A.: On the mathematical foundations of theoretical statis-
tics. Philosophical Transactions of the Royal Society of London,
Series A, 222, 309–368, 1922.
Gumbel, E. J.: Statistics of Extremes, Columbia University Press,
New York, 1958.
Hosking, J. R. M.: L-moments: analysis and estimation of distribu-
tions using linear combinations of order statistics, J. Roy. Statist.
Soc. B, 52, 105–124, 1990.
Kite, G. W.: Frequency and Risk Analyses in Hydrology, Water
Resources Publication, Littleton, Colorado, 1988.
Koutsoyiannis, D.: A probabilistic view of Hershfield’s method for
estimating probable maximum precipitation, Water Resour. Res.,
35(4), 1313–1322, 1999.
Papalexiou, S. M.: Probabilistic and conceptual investigation of
the probable maximum precipitation, Postgraduate Thesis, 193
pages, Department of Water Resources, Hydraulic and Maritime
Engineering – National Technical University of Athens, Athens,
September 2005, (in Greek).
Papoulis, A.: Probability and statistics, Prentice-Hall, 1990.
Schreiner, L. C. and Reidel, J. T.: Probable maximum precipitation
estimates. United States east of 105th meridian, Hvdrometeoro-
logical Report 51, U.S. National Weather Service, Washington,
D.C., 1978.
Shaw, E. M.: Hydrology in Practice, 3rd edition, Chapman & Hall,
London, 1994.
Stedinger, J. R., Vogel, R. M., and Foufoula-Georgiou, E.: Fre-
quency analysis of exreme events, in: Handbook of Hydrology,
edited by: Maidment, D. R., McGraw-Hill, Chapter 18, 1993.
Vogel, R. M. and Fennessey, N. M.: L-moment diagrams should
replace product moment diagrams, Water Resour. Res., 29(6),
1745–1752, 1993.
Wiesner, C. J.: Hydrometeorology, Chapman and Hall, London,
World Meteorological Organization (WMO): Manual for Estima-
tion of Probable Maximum Precipitation, Operational Hydrology
Report 1, 2nd edition, Publication 332, World Meteorological
Organization, Geneva, 1986.
... Koutsoyiannis (2004) matched a probability distribution function Z. Şen (PDF) with the current FF group in a region and found that Generalized Extreme Value (GEV) PDF was the best. Koutsoyiannis and Papalexiou (2006) proposed monograms for the FF estimation. In this paper, PDF will be matched to FFs after the temporal SFF calculations. ...
... On the other hand, Lee and Singh (2020) stated the uncertainty components related to non-stationarity in probable maximum precipitation in the Brazos River basin. Koutsoyiannis and Papalexiou (2006) and Micovic et al. (2014) noted uncertainties in all versions of PMP calculations. The source of uncertainty comes from the FF itself and the arithmetic mean and standard deviation values in addition to the excessive precipitation record (Micovic et al. 2014). ...
... Hershfield (1961Hershfield ( , 1965 assumed in his original work that K = 15, but later realized that FF decreases within the annual maximum precipitation. Koutsoyiannis and Papalexiou (2006) assumed that the FF varies between 5 and 20 depending on the arithmetic mean and duration of precipitation data, and Hershfield's proposal is within this range. ...
Full-text available
PMP has two different estimation methods, namely statistical and hydro-meteorological approaches. The statistical method is based on the calculation of frequency factor (FF) by taking into account the arithmetic mean and standard deviation parameters. The classical probable maximum precipitation (PMP) is based on the (FF) calculated from the annual daily maximum precipitation (ADMP) time series records, which excludes the maximum recording. The classical method returns an FF value without any uncertainty. This paper suggests a successive FF (SFF) method that leads to a series of SFFs, starting with the first three records, and then scanning the entire time series. The probabilistic operation of the SFF sequence presents the uncertainty components in FF based on a set of preset exceedence probability levels and their corresponding return periods. The application of the methodology is presented for three ADMP records from Turkey, Algeria and Arabian Peninsula, which represent humid, semi-arid and arid regions, respectively. The arithmetic mean of the SSF values for the meteorology stations in each country was calculated as 3.07, 2.75 and 3.45, respectively. However, predetermined exceedence probability amounts are presented in the form of tables and graphics. It was concluded that the classical FF calculation provides a single value without any exceedence probability assessment, whereas the SFF method provides FF values with a range of exceedence probability levels.
... A similar study (Papalexiou and Koutsoyiannis 2006) of PMP estimates using the moisture maximization method yields a low, but not negligible, probability of exceedance. The definition of PMP takes into account the fact that the water depth generated by the total surface area of a storm differs from the water depth received over a basin of the same size, but whose shape does not coincide with the storm (Hansen et al. 1982). ...
... Several authors have discussed the controversies of taking the PMP as an upper bound, the procedure and the assumptions of PMP estimates. However, Papalexiou and Koutsoyiannis (2006) argued that there is no justification for taking PMP as an ''upper bound'', where they argued that the length of the historical data and the choice of a probability distribution function for maximum precipitable water influence the setting of the PMP value, and thus, an upper bound for precipitation is unobtainable. Abbs (1999) questioned the basic underlying assumptions of the moisture maximization approach. ...
... This technique is widely used and recognized by government agencies for the design of major water control structures in North America (Schreiner and Reidel 1978;Hansen et al. 1982;Ohara et al. 2011;Abbasnezhadi et al. 2020). Some fundamental aspects of this approach have been argued to be inconsistent (Papalexiou and Koutsoyiannis 2006;Rouhani and Leconte 2016). ...
Full-text available
The design of hydraulic infrastructure requires careful evaluation of extreme precipitation events. This paper presents an estimation of extreme precipitation events based on the Probable Maximum Precipitation (PMP) concept. The PMP approach is useful in determining probable maximum flood (PMF), which is required for the design of large hydraulic structures. Therefore, in this study, 24-h PMP estimates were performed through 43 rainfall stations located in the Cheliff watershed in Algeria. This estimation was implemented based on moisture maximization and Hershfield statistical method. The 24-h PMP values vary between 109.2 and 741.6 mm for the first approach and between 151.5 and 369.4 mm for the second approach. Using the moisture maximization approach, the 24-h PMP values obtained are approximately double those based on the Hershfield statistical method, with return periods ranging from 1000 to 28 10⁶ years for the majority of stations in the Cheliff basin.
... The moisture Maximization Method is too sensitive to available data. Papalexiou and Koutsoyiannis (2006) argued that the method is flawed in its fundamental aspects. It is obvious each method has its advantages and disadvantages. ...
This chapter describes most of the PMP estimation methods, their advantages, and their disadvantages. Also, we included 2 examples from Malaysia and Bangladesh. We anticipate that this will be the most user-friendly and detailed description of PMP estimation methods.
... 12) framework for assessing the peak breach discharge, instead of using deterministic application. This approach is commonly applied to several dam-related design aspects such as peak-over-threshold variables (Koutsoyiannis, 2021); precipitation (Papalexiou and Koutsoyiannis, 2006), streamflow and flood inundation (Dimitriadis et al., 2016), dam breach (Bellos et al., 2020). ...
Conference Paper
Full-text available
A comparative analysis, among different predictive breach models, is introduced using the most updated dam failure database provided by Hood et al (2019). The performance of each model is estimated using an appropriate statistical index by comparing the measured peak breach flow with the modelled peak breach flow. Having applied a regression analysis, a new formula is proposed, which exhibits a very good performance against the observed peak flow statistical sample. An alternative equation with six parameters which has a probabilistic background is proposed. Based on the analysis, three models-those of McDonald & Langridge-Monopolis, Pierce et al, and the proposed regression model-exhibit good performance against observed data. A hypothetical breach assessment case study has been carried out for a typical UK Pennine dam and a discussion is presented on the high uncertainty of the predictive breach models. The paper concludes that simplified breach formulae should always be applied with caution due to the limited number of dam failures which have been used to develop these predictive models. The limitations of the existing predictive models are discussed, and the foundation of a probabilistic breach analysis is proposed.
... Hydro-meteorological estimation approaches can usually be divided into various methods, such as (a) the storm model approach, (b) the generalized method, (c) the moisture maximization method, and (d) the storm transportation method. More details about these methods were mentioned in Rezacova et al. (2005), Rakhecha and Singh (2009), WMO (2009), Collier and Hardaker (1996), Beauchamp et al. (2013), Rakhecha et al. (1995), Papalexiou and Koutsoyiannis (2006), Casas et al. (2011), Micovic et al. (2015), Rouhani and Leconte (2016). However, based on the comparison of studies, there are no generally recommended approaches for PMP estimation (WMO 2009). ...
Full-text available
Due to the impacts of climate change on probable maximum precipitation (PMP) and its importance in designing hydraulic structures, PMP estimation is crucial. In this study, the effect of climate change on 24-h probable maximum precipitation (PMP24) was investigated in a part of the Qareh-Su basin located in the Southeast of Caspian Sea. So far, there have been no estimates of the hydrometeorological PMP values under climate change conditions in the study area. For this purpose, the climatic data were applied during the years 1988–2017. To generate future data, the outputs of the CanESM2 (Second Generation Canadian Earth System Model) model as a general circulation model (GCM) under optimistic (RCP2.6), middle (RCP4.5), and pessimistic (RCP8.5) emission scenarios, and statistical downscaling model (SDSM) were used in the near (2019–2048) and the far (2049–2078) future periods. The PMP24 values were estimated using a physical method in the baseline and future periods under the three scenarios. The PMP24 value was estimated about 143 mm for the baseline period, using a physical approach. These values were 98, 105, and 109 for the near-future and 129, 122, and 126 mm for the far-future period. The results showed that the physical approach's PMP24 values tend to fall at 14–38%. Overall, the PMP24 values decrease in the future, and the rate of PMP decrease in the near-future was more than the rate of the far-future. The spatial distribution maps of PMP24 in the baseline and future periods showed that the PMP24 values decreased from west to east.
... In cases of extreme rainfall and flood assessment studies, the PMP and subsequent probable maximum flood (PMF) methods play significant role (Hershfield 1961;Rakhecha et al. 1992;Koutsoyiannis 1999;Koutsoyiannis and Papalexiou 2006). However, climatic trends move so slowly that their impact on PMP is small compared to other uncertainties in estimating these extreme values. ...
Full-text available
Flood magnitude, frequency and intensity are bound to increase in many parts of the world due to global warming and its consequent effect as climate change impacts. The main purpose of this paper is to apply the classical probable maximum precipitation and probable maximum flood methodologies leading to a new concept of risk level charts, which provide hydrograph time to peak probable maximum discharge after the beginning of precipitation, base time and peak discharge values. Dimensionless hydrograph methodology is employed for flood hydrograph analysis. The applications of probable maximum precipitation and probable maximum flood methodologies are presented for Algerian meteorology stations' annual maximum daily precipitation amounts from 23 different locations at Hodna drainage basin in the northeastern of Algeria. Classical probable maximum precipitation frequency factor is obtained for each meteorology station record, which are then converted to pointwise probable maximum flood amounts that are helpful to construct practically applicable flood charts. A new relationship is provided between probable maximum precipitation and the frequency factor for the study area. The efficiency factor is calculated for each station to understand whether there is a further possibility for extreme precipitation, and consequent flood occurrences.
... What is atmospherically possible was embodied in the idea of a maximum possible precipitation, that later evolved into a probable maximum precipitation (PMP) (Benson, 1973;Hershfield, 1961). It is, however, difficult to define and its evaluation is subject to multiple assumptions and different sources of epistemic uncertainty, especially in what to assume about the potential advection of moisture and vapour (see, for example, Koutsoyiannis, 1999;Papalexiou & Koutsoyiannis, 2006;Micovic et al., 2015). ...
Full-text available
This paper provides a historical review and critique of stochastic generating models for hydrological observables, from early generation of monthly discharge series, through flood frequency estimation by continuous simulation, to current weather generators. There are a number of issues that arise in such models, from uncertainties in the observational data on which such models must be based, to the potential persistence effects in hydroclimatic systems, the proper representation of tail behaviour in the underlying distributions, and the interpretation of future scenarios. This article is protected by copyright. All rights reserved.
... From this perspective, Koutsoyiannis (1999) suggested assigning a return period to PMP values obtained using Hershfield's statistical approach (Hershfield 1961(Hershfield , 1965. Papalexiou and Koutsoyiannis (2006) performed a similar kind of analysis on PMP estimates obtained from moisture maximization method (or storm maximization approach). ...
Full-text available
Estimates of probable maximum precipitation (PMP) and corresponding probable maximum flood (PMF) are necessary for planning, design, and risk assessment of flood control structures whose failure could have catastrophic consequences. For PMP estimation, multifractal approach (MA) is deemed to be better than conventional approaches, which are based either on statistical concepts or physical aspects. The MA yields physically meaningful PMP estimates by attempting to capture scale-invariant multiplicative cascade mechanism inherent in rainfall. This paper attempts to gain insights into the performance of MA by comparing PMP estimates obtained using the approach with those resulting from the use of two widely used empirical approaches (storm maximization approach (SMA) and Hershfield method (HM)) on two flood-prone river basins (Mahanadi and Godavari) in India. The results indicate that rainfall data of the two river basins exhibit multifractal properties, and the use of MA has an advantage over HM and SMA in estimating PMP corresponding to longer durations (>3 days). PMP estimates obtained using HM are generally lower (higher) than those obtained using SMA for 1-day (higher) duration. PMP maps are prepared for the two Indian river basins corresponding to 1-day to 5-day durations. Further, PMP estimates obtained based on the PMP maps are provided for 18 catchments in the Mahanadi basin and 53 catchments in the Godavari river basin.
... There are various suggestions for limiting the moisture maximisation ratios in PMP estimation. The concept of limiting the maximisation ratio is to maintain the original dynamics of a particular storm event (Hansen et al., 1988) and setting upper boundaries may be recommended in order to not produce exaggerated PMP values (Lee et al., 2016), however, there is no scientific justification for limiting the moisture maximisation ratio (Rouhani and Leconte, 2016) and a study by Papalexiou and Koutsoyiannis (2006) showed no evidence of upper bounds in atmospheric moisture or precipitation. In Australia, upper limits have been chosen based on the analysis of actual storm events where the largest ratios were observed (Bureau of Meteorology, 2003;Minty et al., 1996;Walland et al., 2003). ...
Full-text available
Study region South Africa Study focus The Probable Maximum Precipitation (PMP) is the theoretical upper limit of extreme rainfall and is widely used by engineers and hydrologists to determine the Probable Maximum Flood (PMF) which is critical for the design and risk management of high-hazard hydraulic structures. In South Africa, the PMP was last estimated over 50 years ago using approximately 30 years of data. Since then numerous severe rainfall events have occurred which exceed presently available PMP estimates. Despite this, the outdated estimates are currently used in professional practice. In addition, modernised methods have been developed and applied globally to estimate the PMP, highlighting the need to update the PMP estimates for South Africa. This paper aims to present updated and modernised PMPs for the country. New hydrological insights for the region Using updated rainfall records and a storm maximisation and transposition approach, this paper provides new 1-day PMP estimates for the country. The importance of revising the PMP to include new extreme rainfall events is highlighted as the majority of the events used in this study occurred after the publication of the previous estimates. The use of extended rainfall records and modernised methods produces generally larger PMP estimates compared to the previous estimates and continual revision of the PMP is recommend to include new extreme rainfall events.
Full-text available
A simple alternative formulation of the Hershfield’s statistical method,for estimating probable maximum precipitation (PMP) is proposed. Specifically, it is shown that the published Hershfield’s data do not support the hypothesis that there exists a PMP as a physical upper limit, and therefore a purely probabilistic treatment of the data is more consistent. In addition, using the same data set, it is shown that Hershfield’s estimate of PMP may be obtained using the Generalized Extreme Value (GEV) distribution with shape parameter given as a specified linear function of the average value of annual maximum precipitation series, and for return period of about 60 000 years. This formulation substitutes completely the standard empirical nomograph,that is used for the application of the method. The application of the method,can be improved when long series of local rainfall data are available that support an accurate estimation of the shape parameter of the GEV distribution. 2
It is well known that product moment ratio estimators of the coefficient of variation Cν, skewness γ, and kurtosis κ exhibit substantial bias and variance for the small (n ≤ 100) samples normally encountered in hydrologic applications. Consequently, L moment ratio estimators, termed L coefficient of variation τ2, L skewness τ3, and L kurtosis τ4 are now advocated because they are nearly unbiased for all underlying distributions. The advantages of L moment ratio estimators over product moment ratio estimators are not limited to small samples. Monte Carlo experiments reveal that product moment estimators of Cν and γ are also remarkably biased for extremely large samples (n ≥ 1000) from highly skewed distributions. A case study using large samples (n ≥ 5000) of average daily streamflow in Massachusetts reveals that conventional moment diagrams based on estimates of product moments Cν, γ, and κ reveal almost no information about the distributional properties of daily streamflow, whereas L moment diagrams based on estimators of τ2, τ3, and τ4 enabled us to discriminate among alternate distributional hypotheses.
An approach to estimating probable maximum precipitation (PMP) using a simple, totally objective, storm model of convective systems is outlined. PMP hyetographs are compared with observations for severe events, and the variation of storm model PMP with storm duration is compared with PMP values derived using the UK Flood Studies Report (FSR) (NERC, 1975, Department of the Environment, London) approach. The storm model generates PMP values very similar to the FSR for storm durations less than 11 h, but for durations between 11 and probably 24 h the FSR PMP values are exceeded. It is suggested that longer duration events may be associated with mesoscale convective systems (MCSs), which have only been recognised in the UK since the FSR was published.
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.
Centre of Location. That abscissa of a frequency curve for which the sampling errors of optimum location are uncorrelated with those of optimum scaling. (9.)