Estimation of probable maximum precipitation 24-h (PMP 24-h) through statistical methods over Iran Open Access

The need for water resources to develop based on their application in different areas such as industry and agriculture is highly important. A considerable cost is spent on water resources management through dam construction and water reservoirs every year. Dam overflow due to the probable maximum flood (PMF) is designed based on the most probable precipitation by probabilistic analysis and recorded data. The probable maximum precipitation (PMP) refers to the maximum amount of precipitation that may occur meteorologically over a specified period in a particular area (WMO 1986). Also, the maximum 24-hour PMP values are used to design open channels, bridges, roads and railways, urban drainage channels and airport drainage, flood control, and many other hydraulic structures (Desa et al. 2001). Since maximum rainfall is one of the necessary and important preconditions for maximum flood occurrence in a given area, estimating the probable maximum precipitation is important in flood risk assessment. Flooding is the temporary covering of the land surface by the outflow of water from the natural river bed (Wallingford 2005), such that it submerges ordinarily dry areas of land. Flooding is one of the most significant and most catastrophic natural disasters in terms of casualties and financial damages, with more damages than other disasters such as drought and famine. According to United Nations statistics, about one-third of economic disasters are caused by floods, and two-thirds of the world's population is directly and indirectly affected by the relevant consequences (Pilon 2004).

In some countries, the PMF calculated by estimating the PMP is considered the flood safety of large dams with significant downstream hazards and no acceptable risk of casualties. In Iran, PMF studies as a significant flood safety measure are of great importance and sensitivity. In general, estimation of maximum flood potential is usually done for the following three engineering applications:

• Design high-risk dams in demographic areas in a way that will lead to significant damages.

• Operational planning and dam curve planning during floods.

• Floodplain and floodplain areas where nuclear power plants are built.

As mentioned, the PMP in estimating the PMF was used to design the large dam overflow as initial data. If overflow cannot pass the maximum flood with high safety, it may be that water overflow damages the walls or leads to erosion of the dam bases. Regarding the issues mentioned earlier, international dam organizations, particularly the National Committee for Large Dams, have strongly recommended that overflow capacity be taken into account based on the PMF estimated through PMP calculated by statistical analysis and synoptic methods. Therefore, estimation of PMP should be based on the international standards and recommendations of international organizations. In this study, the analysis has been done countrywide to compare the conditions of different climatic regions of the country in terms of estimating the PMP, using the first Hershfield method and the corrected Desa method.

Ros et al. (2008) estimated the PMF in the Kenyire basin in Southeast Asia. In this study, PMP estimation was used to calculate PMF, and then the rainfall-runoff model (HEC-HMS) was used to convert rainfall to runoff. PMF value for the basin was estimated at 214,000 M³/S.

Fattahi et al. (2010) calculated the probable maximum precipitation by two synoptic and statistical methods in the southwestern parts of Iran, and they compared the results of two methods. Results showed that PMP in all stations was about twice that of the synoptic method, and the least difference was observed between Abadan and Bushehr stations. Casas et al. (2010) estimated the PMP in a range of 5 minutes to 30 hours in the Barcelona area. They used two different methods of physical analysis of hurricanes and the statistical method of Hershfield, which showed a remarkable similarity between the results. Tingsanchali & Tanmanee (2012), for hydrological evaluation in Mae Sruai River Basin, used the Hershfield statistical method and synoptic analysis of heavy rainfall to calculate PMP at 1, 2, and 3-day ranges. Shirdeli (2012) estimated the 24-hour probable maximum precipitation in Zanjan province. The obtained results showed good consistency between the corrected Hershfield method and 24-hour observed rainfall in all stations of the study area compared to Hershfield's first approach.

In another study of PMP estimation by Sidek et al. (2013) in Batang-Padang Watershed of Malaysia, the highest ratio of 1, 3, and 5-day PMP values to maximum observed rainfall were 2.3, 1.94, and 1.82, respectively. This finding indicated that the above ratios tend to decrease with increasing the period length (from 1 to 5 days). Finally, the temporal pattern for ranges of 1,3, and 5-days was based on the observed rainfall threshold to produce PMF.

In a study on PMP and the effects of climate change, Kunkel et al. (2013) found that on a global scale, future PMP values are likely to be increased due to high moisture content and the possibility of injecting more moisture into storms will be increasing.

Chavan & Srinivas (2015) estimated the 1-day probable maximum precipitation by the synoptic method for the Mahanadi River basin and prepared PMP maps for this basin.

In an investigation conducted by Heidarpour (2016) to estimate PMF probability level in Golestan watershed from three approaches, (1) extrapolation of probability distribution functions, (2) use of rainfall-runoff models, and (3) use of experimental equations and relationships. In the second approach, the PMP was first calculated using statistical and synoptic methods (storm maximization and generalized PMP methods), and the estimated PMP values were converted to PMF values through the calibration and validation model (HEC-HMS).

Similarly, various studies have been conducted in different parts of the world, such as studies conducted by Ramak et al. (2017), Le Thuy et al. (2019), Sarkar & Maity (2020) in Iran, Vietnam, and India, respectively.

Toride et al. (2019) estimated PMP with realistically maximized storms in the Pacific Northwest region dominated by atmospheric rivers (ARs) using numerical weather models (NWMs), found that among the 20 most severe storms during 1980–2016, the AR event during 5–9 February 1996 produces the most significant 72-h basin-average precipitation. Clark & Dent (2021), by Estimates of 24-Hour Probable Maximum Precipitation for the British Isles, concluded that values of 24-hour PMP in Britain range from 600 mm in upland areas of the Lake District to 400 mm in parts of East Anglia. Sheikh Hefzul Bari (2021) estimated PMP for Bangladesh, indicating that higher values of PMP are found in the eastern part of the country whereas lower in the western part. However, maximum PMP (811 mm) is found in Bhola, located in the coastal region.

The statistical method for estimating the PMP when meteorological data such as dew point temperature and wind data series are not available could be based on storm data. In general, the evaluation of statistical estimates of PMP in many studies shows that the estimated values derived from the statistical method are higher than the values of the synoptic method. The main reason for this can be sporadic storm events in the statistical period, and such data will affect the mean and standard deviation calculations so that the magnitude of the impact will vary depending on the short- or long-term statistical period. In other words, the frequency distribution of the extreme rainfall is skewed to the right, so correction of such rare events is recommended in statistical estimations of PMP.

This method is recommended for watersheds less than 1,000 square kilometers (WMO 1986). This method can also be used with corrections for larger basins, which are in accordance with the Journal of the World Meteorological Organization 332:

• – Calculate the mean value (P_m) and standard deviation (SD_m) for the n-year statistical period

• – Calculate the mean value (P_mm) and standard deviation (SD_mm) by omitting outlying data or the maximum event observed during the n-year statistical period.

• – Calculate (Pm)/(Pmm) and (SDm)/(SDmm) ratios.

• – Due to the length of the statistical period (n), the correction coefficient for mean value and standard deviation is extracted from the graphs presented in the journal.

• – Modified mean values and the standard deviation of PMP obtained for each station.

• – If station values are used for estimation in the basin, surface reduction coefficients should be corrected for the basin area to consider regional climatic conditions and topographic effects in estimating the PMP.

This method has been used by Ghahraman (2008) for Atrak catchment, Paimazd (2002) for eastern basins of Hormozgan province and Tajbakhsh & Al-Ansari (2019) in the northeast of Iran, and in comparison with the first Hershfield method, they recommended the Desa-Hershfield method (Corrected Hershfield) in estimating point-based probable maximum precipitation.

Eliason (1991, 1994, 1997) developed the concept of the multi-station method based on the Gamble Type I distribution tail and assumed a PMP value for the entire region, assuming the stations in a region were homogeneous.

Briefly, this study aims to compare two statistical methods of calculating the probable maximum precipitation, including the original Hershfield method and Hershfield-Desa corrected method, along with considering spatial variations of PMP for the entire country. In this study, 116 stations were selected from synoptic stations throughout the country, with a maximum 24-hour rainfall over a 30-year statistical period (1981–2020), covering the study area completely. The characteristics of the locations, including latitude, longitude, maximum 24-hour annual precipitation, and long-term mean annual precipitation, were obtained from the Meteorological Organization's Statistics and Information Center. Then the database was created in Excel and GIS software, and the selected stations were mapped according to their geographical location. The 24-hour maximum rainfall data set has been used for selected stations, both Hershfield methods were used to calculate PMP, and then statistical variables such as coefficient of variation, skewness, standard deviation, and mean of the statistical period were obtained. In the next step, comparing the results of both methods, the PMP interpolated maps and maximum observed precipitation was mapped using ArcGIS software and interpolation method.

• – Using kriging to interpolate results in GIS:

Kriging is a statistical interpolation method in which data at points of interest are calculated using the weighted average of data observed elsewhere. This method is similar to the trend surface method, and it has been excepted the weights calculated based on the distance between points with known data and points of interest and their correlation. The control parameters of trend level weights and inverse distance intervals produce different data estimates for interpolation, and kriging is used to correct these defects. The kriging method determines the optimal weights for interpolation by using a variogram.

At first, the 24-hour maximum precipitation values of the selected stations were sorted in descending order with their statistical properties. Then according to the methods mentioned above, using the mean value and standard deviation of maximum 24-hour precipitation in the statistical period, and after eliminating maximum rainfall event and extracting the correction coefficients according to the respective graphs and the values mentioned above, corrected PMPs of stations were calculated using the Hershfield method (first approach) and the Hershfield-Desa method (second approach). The values of variability coefficient and skewness were also calculated for both methods.

Since the frequency coefficient K is less than 15 in the first Hershfield method, the PMP values calculated based on existing studies are generally estimated to be high, as shown in Table 1. Long-term precipitation during the study period accounts for more than half the number of selected stations higher than 70%, even at some stations in the country's southern areas such as Chabahar, Bandar Abbas, and Bandar-Lengeh, it has reached 319, 301, and 285% respectively. Nevertheless, in the country highland and mountainous areas, this ratio is below 50%, and in stations like Qazvin, Rasht and Bandar-Anzali are 46, 35, and 33%, respectively. Also, the ratio of PMP to maximum 24-hour annual precipitation in this method is between 2.54 to 4.02, which confirms such rainfall events are improbable to happen due to the long-term climatic characteristics of the country and main air masses which enter into Iran.

In the Hershfield-Desa method, the frequency factor of each station was calculated using Equation (2). This value was obtained between 2.02 and 6.66, and then the maximum K_m (frequency factor) was extracted from these values to calculate the PMP values. The PMP ratio calculated by this method to maximum 24-hour annual precipitation was obtained between 1.02 and 1.31, and also the PMP ratio to long-term mean precipitation value during the study period ranged from at least 13.22 up to 101.9%, as one-third of the number of selected stations, mainly in the southern and low-rainfall regions of the country, showed this ratio more than 50 percent.

Comparison of the PMP calculated values through the above methods shows that in the first approach, the computational values are mostly very high compared to the maximum 24-hour rainfall, and in the second approach concerning the first method, it is shown that these values are almost moderated at most of the stations. However, it still seems higher at some stations, such as Gorgan, Abadan, and Kermanshah, with a coefficient of more than 1.2.

Therefore, for further study, the selected stations' statistical characteristics in both methods were calculated and presented in Tables 2 and 3.

In the tables, all the mentioned stations' statistical characteristics are arranged in descending order according to their coefficient of variation. In this study, the minimum and maximum coefficients of variation in the first method ranged between 26.61 and 63.79% and in the Desa method from 23.73 to 59.07%. The lowest coefficient of variation for the stations located in the northern or highlands of the country includes stations such as Bandaranzali, Rasht, Qazvin, Khorramabad, and Zanjan, which confirmed better reliability of precipitation events in these areas. In other words, the precipitation changes in the mountainous and wet climate zones are not very significant, and according to the calculated skewness coefficient, the distribution was close to normal, and the empirical mean value does not change much compared to the theory mean. However, stations such as Bandar Abbas, Abadeh, Bandar Lengeh, Chabahar, and Bam had the highest coefficients of variation and skewness of more than 50% and 0.7, respectively, indicating that the maximum 24-hour precipitation variations of these stations were highly significant. Hence, it seems that the frequency factor (Km) obtained in the second approach (Hershfield-Desa) for all stations is equal to the maximum Km which obtained 6.66, should be modified for the stations whose coefficient of variation is relatively large. Thus, the frequency coefficient (K_m) can be adjusted to 5.71 in the above stations, which had the highest coefficient of variation of maximum 24-hour precipitation among the selected stations.

Results derived from the calculations of probable maximum precipitation in 45 synoptic stations of the country based on both two methods, including Hershfield (first approach) and Hershfield-Desa (second approach), are shown in Table 1. After plotting the PMP interpolation maps based on the above two methods and also plotting 24-hour annual rainfall maps for the country, the average of the total PMP values based on the Hershfield-I method was estimated to be 285.85 mm that compared to the Hershfield-Desa method with 100.12 mm, it is more than twice. The probable maximum precipitation values based on the two methods with the maximum observed 24-hour rainfall, 113 mm throughout the country, indicated that the Hershfield-Desa method is relatively more realistic.

Comparison of spatial variability of maximum 24-hour annual rainfall observed with PMP values calculated through the two mentioned methods, in general, indicated an increase in both estimated and observed maximum precipitation values in the northern regions, including the Caspian Sea coast and northern side of Alborz mountain chain as well as some parts of the southern and western regions. However, large parts of the country, including the interior, northwest, northeast, and eastern parts, illustrated relatively low values.

The geographic variability of the observed and estimated maximum rainfall cannot be the same for different regions. There are higher annual average precipitations in northern regions and some parts of the western Zagros mountain range, and consequently, such areas show the higher 24-hour maximum rainfall. However, despite not having a high annual average rainfall, the country's southern regions showed relatively higher 24-hour maximum rainfall due to short-term storm rainfall events and extreme rainfall variability.

In fact, due to the country's climatic characteristics, the arid southern climate, which mainly comprises the southern shores of the country, and parts of the southwestern mountain slopes, including the Zagros Mountains, which show warm semi-arid climate, had the highest 24-hour PMP. This situation indicates high variability of precipitation regimes in these areas. One of the main reasons could be proximity to the moisture source of the Persian Gulf and Oman Sea, as well as topographic status and location of the Zagros Mountains, which is located in the trajectory of the western and southwestern rainy systems that increase consequently the probability of heavy rainfall occurrence particularly in the southwestern areas of the country. Of course, the mentioned mechanism can also be addressed for the semi-humid climate of the west of the country, such as parts of the western border basin, including parts of southwestern Azerbaijan, Kurdistan, and Kermanshah, which have had the highest probable maximum precipitation. Also, the higher amount of probable maximum precipitation in the south-east coast of Iran, such as Chabahar, can be due to tropical cyclones' collision with the Mokran region, which usually leads to heavy rainfall in spring and autumn and additionally the seasonal monsoon rainfall, which is active in the area usually since early summer, will also be influential.

The very humid climate of the northern parts of the country and the Caspian Sea shores show the highest amount of observation and probable 24-hour precipitation across the country due to the Caspian Sea moisture source, which plays a particular role in the humidity feeding of the atmospheric systems originating from Siberian, Scandinavian, and Mediterranean regions and then entering to the study area. This condition, combined with topography and the western-eastern position of the Alborz Mountain Range, which is short from the Caspian Sea, especially in the western parts, like a quiet barrier, could lead to heavy rainfall in this region, especially along the western coasts.

The PMP, which is calculated using the maximum precipitation data observed over a specified period, is performed to estimate the PMF. Estimating possible floods for design and construction of irrigation and water reservoir projects such as dam construction are considered essential and in terms of dam safety, such as overflow capacity and dam failure, is of great importance. The purpose of this study is to compare the outputs of two main methods of statistical estimation of 24-hour PMP throughout the country using annual observation data of 24-hour maximum rainfall at 45 available synoptic stations between 1981 and 2020.

By using 24-hour maximum precipitation values, corrected PMPs of stations were calculated by the Hershfield method (first approach) and the Hershfield-Desa method (second approach). The values of variability coefficient and skewness were estimated for both methods as well. As the frequency coefficient K is below 15 in the first Hershfield method, the PMP values generally estimated to be high as presented in Table 1. Long-term precipitation during the study period accounts for more than half the number of selected stations higher than 70%. Also, the ratio of PMP to maximum 24-hour annual precipitation in this method was obtained from 2.54 up to 4.02, which implies that due to the long-term climatic background of the country, such rainfall events are implausible to happen. While the frequency factor of each station by the first method and also PMP values were obtained between 2.02–6.66, and between 1.02 to 1.31, respectively. Comparison of the PMP calculated values through the both methods reveals that in the first approach, the computational values are mostly very high compared to the maximum 24-hour rainfall, and in the second approach concerning the first method, it is shown that these values are almost moderated at most of the stations.

In the first Hershfield approach, the values of K (frequency factor) and 24-hour PMP were significantly more fluctuant than the Hershfield-Desa method, as the ratio of 24-hour PMPs obtained through the first Hershfield to the highest 24-hour observed rainfall is between 2.54 and 4.03, and to the annual mean of long-term precipitation was about 33 to 320% at the study stations. In comparison, these ratios in the Hershfield-Dasa method were significantly moderated to 1.02 to 1.3 and 13% to 101%, respectively. In both above methods, the estimated PMP values about the maximum observed rainfall and the mean long-term precipitation in the southern coasts of the country showed a significant difference compared to the high-rainy and mountain stations in the northern part of the country, which is due to the high variability of precipitation in the southern parts of the country. In terms of spatial variability, the maximum 24-hour rainfall observed, and PMP values estimated by both Hershfield methods indicated higher values in the country's north coasts, western areas, and southern coasts compared to other regions. However, there is a significant difference between the estimated PMP values in the Hershfield method with the maximum rainfall observed in the north (Ramsar Station), and this difference is about 754 mm. Although in the Desa method in most parts of the country, the results are almost more consistent with the observation.

Estimating the PMP close to the observation over the country through various methods is essential in selecting design floods for different climatic regions of the country, which is one of the significant problems in the design of water management projects. Flood discharge design is selected based on economic criteria, human casualties, and hydrological factors. In some cases, the safety of the system and its associated damages may not be significant, and in some cases the aquatic structure is of most importance, so it must withstand the most significant floods and remain stable. According to the results, the southern areas of the country with significant differences between estimated PMPs and its long-term average observations indicated a high challenge in PMF estimation and design flood selection. The risk of the dam and underwater structures is being increased in the country's southern areas and calls for more detailed studies in the future.

The authors received no financial support for the research, authorship, and/or publication of this article.

Not applicable.

This material is the authors' own original work that has not been previously published elsewhere, and also the paper is not currently being considered for publication elsewhere.

Data cannot be made publicly available; readers should contact the corresponding author for details.

The authors declare there is no conflict.