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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

Content uploaded by Demetris Koutsoyiannis

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... The most widely used statistical estimation method developed by Hershfield (1961Hershfield ( , 1965a, has been adopted by the WMO (2009b) as one of the standard methods for PMP estimation. This method takes into account actual historical data at a particular location and expresses it in terms of statistical parameters (Koutsoyiannis, 1999). The procedure as developed by Hershfield (1961Hershfield ( , 1965a is based on the general frequency equation by Chow (1951) in the form of ...

... It is obvious each method has its advantages and disadvantages. So, it can be concluded that there is lack of a unique method for determining the upper bound of rainfall assuming it really exists (Koutsoyiannis, 1999). Therefore, attention should be given to selecting a particular method for a particular region. ...

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.

... The Hershfield method and its variants, as well as the multifractal approach, are common statistical methods [7]. It has the advantages of taking real historical data in the area of interest into consideration, describing it in terms of statistical characteristics, and being simple to apply [8,9]. ...

... These methods are especially helpful when doing rapid estimations or when additional meteorological data such as wind records and dew point are absent [14]. The statistical approach developed by Hershfield is frequently employed today [7][8][9][10]. ...

Extensive hydrological analysis is carried out to estimate floods for the Batu Dam, a hydropower dam located in the urban area upstream of Kuala Lumpur, Malaysia. The study demonstrates the operational state and reliability of the dam structure based on hydrologic assessment of the dam. The surrounding area is affected by heavy rainfall and climate change every year, which increases the probability of flooding and threatens a dense population downstream of the dam. This study evaluates the adequacy of dam spillways by considering the latest Probable Maximum Precipitation (PMP) and Probable Maximum Flood (PMF) values of the concerned dams. In this study, the PMP estimations are applied using comparison of both statistical method by Hershfield and National Hydraulic Research Institute of Malaysia (NAHRIM) Envelope Curve as input for PMF establishments. Since the PMF is derived from the PMP values, the highest design flood standard can be applied to any dam, ensuring inflow into the reservoirs and limiting the risk of dam structural failure. Hydrologic modeling using HEC-HMS provides PMF values for the Batu dam. Based on the results, Batu Dam is found to have 200.6 m3/s spillway discharge capacities. Under PMF conditions, the Batu dam will not face overtopping since the peak outflow of the reservoir level is still below the crest level of the dam.

... Furthermore, the concept of PMP was first introduced by the American Meteorological Society (AMS 1959) and later refined by others (Hansen et al. 1982;Swain et al. 2006;WMO 2009aWMO , b, 2011. It has been pointed out that the use of PMP in no way implies zero risk in reality (Koutsoyiannis 1999). According to the National Research Council (1994), the return period for PMP in the United States is between 10 5 and 10 9 years. ...

... Fallot et al. (2017) found similar probabilities for PMPs calculated in Switzerland by meteorological models. Koutsoyiannis (1999) fitted a generalized extreme value distribution (GEV) to frequency factors obtained from maximum precipitation at 2645 stations. He found that the highest value of Hershfield 15, corresponds to a return period of 60,000 years. ...

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.

... As for the PMP methodology, Salas et al. (2014) criticized flood-related water structure design and assessment practices for the many uncertainties involved in their determination and the fact that such estimates are constrained by an insurmountable upper bound and perceived as zero exceedence probability. Such a zero-exceedence probability upper limit is unrealistic because there are documented cases where recorded floods exceed estimated probable maximum floods (PMFs) (Benson 1973;Dawdy and Lettenmaier 1987;Koutsoyiannis 1999). Therefore, an exceedence probability-based design approach is advocated by some hydrologists. ...

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.

... Additional uncertainty comes from the perception that estimated PMP and PMF are upper bounds that may not be exceeded and as such have zero risk; as there is presumed no likelihood of occurrence beyond this level. An upper bound with zero risk is not a realistic assumption, especially as evidence of precipitation and flood events exceeding the estimated PMFs have been found Koutsoyiannis, 1999). Furthermore, Ishida et al. (2018) found significant increase in PMP estimation under 13 future projections over the American River Watershed in northern California. ...

Estimation of probable maximum flood (PMF) is a crucial process in water resources management and in the design of large hydraulic structures. However, there are uncertainties in the estimation of hydrologic conditions that contribute to extreme floods. In particular, this is the case in snow-dominated regions, as surface air temperature and wind speed are understood to have a substantial effect on the magnitude of a flood during a storm event. Motivated by the development of a new approach to investigate and estimate reliable PMF values and in an attempt to resolve the uncertainty issues, this study introduces a physically based modeling approach. For the case study, seven watersheds located in the Sierra-Nevada mountain range of California, including Cosumnes, Mokelumne, Stanislaus, Tuolumne, Merced, Upper San Joaquin, and Upper Kings were selected. The hydroclimate model was first implemented over the physical boundaries of the study region, and then utilized to simulate possible maximum flood conditions with input from 10 extreme precipitation scenarios. The study results provide evidence of a nonlinear atmospheric-hydrologic system; the extreme 72-h basin-averaged precipitation depth was found not to be linearly proportional to 72-h flow volume equivalent depth. It can also be concluded that a large precipitation depth may not be the sole reason for a large flood event. Temperature and other atmospheric variables also contribute significantly to the production of snowfall and liquid water available for runoff, and to the resulting hydrologic response, such as the flood peak discharge and volume. K E Y W O R D S atmospheric boundary condition shifting, integrated water vapor transport, probable maximum precipitation and probable maximum flood, watershed environmental hydrology model, weather research and forecasting model

... 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. ...

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.

... However, there are several known issues in this method. For example, due to the lack of consideration of the physical processes associated with precipitation, the PMP value estimated by GEV analysis is difficult to be viewed as the physical upper limit of precipitation (Koutsoyiannis, 1999). The GEV analysis is also sensitive to the outliers in the precipitation data, which can lead to misestimated PMP values (Nobilis et al., 1991). ...

The west coast of Canada is strongly affected by the extreme precipitation events triggered by frequent atmospheric river (AR) activities over the eastern North Pacific. Across the region, assessing the probable maximum precipitation (PMP), can provide valuable information for resilience building of the coastal communities that are vulnerable to hydrological risks. In this study, a 3-km convection-permitting regional climate model is used to physically estimate the PMP in Vancouver. This technique maximizes the effect of AR-related water vapor transport by spatially adjusting the lateral boundary conditions (LBCs) of the model simulations for the selected AR-related extreme precipitation events. The PMP in Vancouver is identified among the simulations driven by the spatially adjusted LBCs that are corresponding with the AR-induced "worst-case scenario," i.e., landfalling ARs hit Vancouver with optimal landfalling location and transport direction. Results suggest that the PMP in Vancouver, in terms of the maxima of the regionally averaged 72-h total precipitation for the historical extreme precipitation events, is up to 790 mm, which is 130% greater than the historical peak precipitation for the period 1980∼2017. On average, all the PMP simulations shows an overall increase by 81% in precipitation by relative to historical simulations. In addition, the PMP simulations suggested an overall decrease in snowfall by 12% due to the warmer near-surface air temperature; however, a pronounced increase in freezing rain is seen. The precipitation increase for the estimated PMP relative to the historical extreme precipitation is closely associated with the increased atmospheric moisture transport and the changes in the atmospheric dynamic factors when the AR effects are maximized. These include the enhanced low-tropospheric ascent and moisture transport convergence, which can induce stronger depletion of atmospheric moistures as indicated by the increased precipitation efficiency.

... The statistical estimation method refers to Hershfield's K m -value method (Hershfield, 1961). This method is based on Chow's general frequency equation and derived from the modified frequency analysis of AM series or precipitation quantile, which belongs to quasi statistical methods (Koutsoyiannis., 1999;Liao et al., 2020). The formula of the PMP statistical estimation method is: X m = X n + K m S n and K m = (X m − X n− 1 )/S n− 1 , where the frequency factor K m can be estimated by samples, X n− 1 and S n − 1 are the mean value and the standard deviation, respectively, of the annual maximum series with the largest term removed. ...

In recent years China has witnessed massive economic losses and fatalities by natural disasters caused by extreme precipitation including floods and landslides. Alleviating the adverse effects of extreme precipitation in modern societies relies on building protection infrastructure. In turn, cost-effective and robust infrastructure design requires the identification of the statistical properties and laws that govern extreme precipitation. China is a vast country with diverse climates and terrains, and as such, different analysis and research methods have been applied in its regions. In this paper, we provide a comprehensive review and synthesis of techniques and methods used in research and engineering practice for extreme precipitation analysis across the country. Specifically, we focus on: (1) annual maximum methods, (2) peaks over threshold methods, (3) probable maximum precipitation estimates, and (4) non-stationary analysis of precipitation extremes. Research on extreme precipitation in China is generally based on the above four approaches and this review aims to provide a detailed timeline of the evolution and application of these methods. Finally, we stress ideas for further research on frequency analysis of extreme precipitation in response to climate change and human activities in a changing environment.

... Rise of atmospheric moisture Abbs (1999) Non-linear relationshio between dewpoint and temperature Chen & Bradley (2006) No return period for PMP Koutsoyiannis (1999), Papalexiou & Koutsoyiannis (2006) Uncertain Gumbel distribution Papalexiou & Koutsoyiannis (2013) Climate change effects Somerset UK (Clark, 2014) which now has a 295 year historic flood record, the FEH 6-hour storm with 100% runoff only yields a flood of 327 m 3 ·s −1 which was almost equalled in 1768 (Clark, 1999). Furthermore, the flood of 1917 with a rainfall intensity of 25 mm·hr −1 gave a peak discharge of about 175 m 3 ·s −1 while the Martinstown storm of 1955 which took place 45 km to the south if transposed would have caused a flood that exceeded 330 m 3 ·s −1 . ...

Focal point of this work is the estimation of the distribution of maxima without the use of classic extreme value theory and asymptotic properties, which may not be ideal for hydrological processes. The problem is revisited from the perspective of non-asymptotic conditions, and regards the so-called exact distribution of block-maxima of finite-sized k-length blocks. First, we review existing non-asymptotic approaches/models, and also introduce an alternative and fast model. Next, through simulations and comparisons (using asymptotic and non-asymptotic models), involving intermittent processes (e.g., rainfall), we highlight the capability of non-asymptotic approaches to model the distribution of maxima with reduced uncertainty and variability. Finally, we discuss an alternative use of such models that concerns the theoretical estimation of the multi-scale probability of obtaining a zero value. A useful finding when the scope is the multi-scale modeling of intermittent hydrological processes (e.g., intensity-duration-frequency models). The work also entails step-by-step recipes and an R-package.

This study provides a stepwise approach to the temporal and spatial distribution of probable maximum precipitation (PMP) estimates derived from Hydrometeorological Report No. 51, 'Probable Maximum Precipitation Estimates - United States East of the 105th Meridian'. Included are discussions of the shape and orientation of isohyetal patterns for major rainfalls of record. An elliptical isohyetal pattern with a ratio of major to minor axes of 2.5 to 1 is recommended, and a procedure is outlined for obtaining appropriate isohyet values.-from STAR, 21(12), 1983

This article is both an extension and revision of a paper in which a large number of 24 hour rainfall data was analyzed for determining an enveloping rainfall. A series of annual maximum rainfalls for seven durations ranging from 5 min to 24 hrs. were selected for the following reasons: they are closer to the required results than any other statistic series; they are amenable to probability analysis because the items in the series are independent of each other and occur in a random fashion; and, they can be extracted from publications that are readily available.

The search for regularities in hydrologic relationships is discussed against the background of the general types of predictive models used in science. The various approaches to the study of water are compared and contrasted. The ideas discussed are illustrated by examples from the development of techniques in flood hydrology and by personal conclusions on the sources for new hypotheses in flood hydrology and the possibility of their verification.