## Abstract

This study is the first study that worked on the temporal and spatial distributions of annual rainfall (*P*_{year}) and maximum 24-h rainfall (*P*_{max24h}) in the Semnan province. For this purpose, different statistical distributions were used to estimate the temporal *P*_{year} and *P*_{max24h} in the Semnan province. Six synoptic stations across the province were studied and all stations had complete *P*_{year} and *P*_{max24h} data. Different return periods were studied. The goodness fit test of statistical distributions for *P*_{year} showed that about 67% of the stations follow the Generalized Pareto (GP) distribution. Considering the *P*_{max24h}, 50% of the stations follow the GP distribution, and for the ratio of *P*_{max24h} to *P*_{year}, 50% of stations follow the Generalized Extreme Value (GEV) distribution. The spatial distribution of *P*_{year} and *P*_{max24h} showed that in all return periods, by moving to the southeast of the province, precipitation amounts decreased. While moving toward the Shahmirzad station, the amounts of *P*_{year} and *P*_{max24h} increased. Also, there was a logical relationship between the *P*_{year} and *P*_{max24h}. Consequently, the minimum value and the maximum value of the *R*^{2} coefficient in different return periods were equal to 0.992 and 0.980, respectively.

## HIGHLIGHTS

Statistical analysis was effective in calculating the highest average annual rainfall and the highest average 24-h load.

The use of geostatistical methods and statistical distributions shows a good performance in predicting the amount of precipitation.

The use of statistical distributions and geostatistical methods can be appropriate in locating the establishment of a meteorological station.

## INTRODUCTION

Predicting precipitation under the temporal and spatial distributions has great importance in water engineering, hydraulic structures, and water resources management (Krajewski & Smith 2002; Gräler *et al.* 2013; Volpi & Fiori 2014; Ghazvinian *et al.* 2020d, 2021; Ugbaje & Bishop 2020; Dadrasajirlou *et al.* 2023). The importance of this probabilistic forecast has increased in recent years due to the phenomenon of climate change and land-use change (Dore 2005; Fujita 2008; Flannigan *et al.* 2016; Konapala *et al.* 2020; Hu *et al.* 2021).

The spatio-temporal distribution of precipitation has shown a straightforward relationship for providing accessible water resources for humans and ecosystems (Milly *et al.* 2005; Oki & Kanae 2006; Mishra *et al.* 2011). Therefore, changes in precipitation due to human-caused climate change in the 21st century may result in changes in water accessibility (WA) that have implications for both humans and the biosphere (Haddeland *et al.* 2014; Schewe *et al.* 2014). Studies have examined several simultaneous changes in the *P*_{year} and *P*_{max24h} in these variables (Holdridge 1947; Thornthwaite 1948; Kottek *et al.* 2006). Furthermore, the characteristics of annual rainfall and 24-hour rainfall, even in arid regions, are the main basis for WA studies in existing climate classifications (Feng *et al.* 2013; Rajah *et al.* 2014). Also, changes in *P*_{year} cause drought, increase water stress in human society, and have adverse effects on agriculture and food production (Vaziri *et al.* 2018). Thus, 24-h rainfall impacts flood events.

Geostatistical methods are based on spatial variation functions (Eftekhari *et al.* 2021). Spatial variation functions are generally fitted using parametric methods, such as Kriging and inverse distance weighted (IDW). IDW and Kriging consider the influence of covariates on the predicted variable and use the spatial autocorrelation of the predicted variable and its cross-correlation information with related covariates, but they have different modeling methods and computational processes. St-Hilaire *et al.* (2003) evaluated the impact of the density of the network of water and meteorological stations on the estimation of annual and daily precipitation of rainfall and runoff events in the northern United States of America. The results showed that increasing the number of stations has an effect on the quality of estimating the spatial distribution of precipitation in the region.

Cheng *et al.* (2008) evaluated the rain gauge network using geostatistical methods in order to estimate the regional average of precipitation and point estimation in areas without stations. Variogram analysis showed that hourly rainfall has higher spatial changes than annual rainfall.

Therefore, forecasting *P*_{year} and *P*_{max24h} can help in planning and managing water resources. On the other hand, the phenomenon of climate change has increased the occurrence of marginal precipitation. In addition, due to the increase in the urban area, the runoff coefficient in catchments has increased. These have caused the catchments to be more sensitive to rainfall and, as a result, produce more floods than in previous years. This issue leads to an increasingly irreparable loss of life and property. In rainfall forecasting, a lot of research has been conducted with temporal and spatial distribution approaches and intelligent models (Nasseri *et al.* 2008; Mekanik *et al.* 2013; Solgi *et al.* 2016; Azad *et al.* 2019; Zhang *et al.* 2020). Also, studies such as Bayat *et al.* (2019) optimized rainfall estimation based on spatial variations of rainfall. In recent years, extensive scientific studies have been conducted to predict precipitation by considering the temporal and spatial distribution in different parts of the world.

*P*_{max24h} is used to calculate the maximum flow in the design of high-risk structures such as dams and nuclear power plants. Also, by using *P*_{max24h}, the maximum possible flood can be calculated. The calculation of *P*_{max24h} is based on the extreme and exceptional conditions of factors such as the continuity of rainfall, time distribution of rainfall, and infiltration, which requires a lot of data and information.

Geo-statistics is a branch of statistics that investigates natural phenomena that are both random and structural (Çadraku 2021). Geo-statistics originated from mineral resource assessment, but it has been widely used in many fields such as climate, hydrology, environment, and ecology (Çadraku 2022).

Spatial interpolation is the traditional way to transform point-wise rainfall into areal rainfall. It is a process of tapping and utilizing the spatial autocorrelation of rainfall and spatial intercorrelation between rainfall and related explanatory variables under a specific mathematical framework. The spatial distribution of precipitation and temporal distribution plays a critical role in the implementation of development projects. Using the spatial distribution of rainfall can help to determine the risk of floods or landslides in an area.

Fitting theoretical distributions using real data can be used to obtain the probability of occurrence or return period of natural phenomena. By choosing the appropriate statistical distribution, a better estimation of the return period of extreme phenomena with a low occurrence probability can be made, thus increasing the efficiency and reliability of water projects. Precipitation prediction in basins and rivers with different return periods that have statistical information can be conducted using theoretical statistical distributions (Rao & Hamed 2000). The advantage of matching the frequency distribution of existing data with a statistical distribution is developing the existing limited statistics to understand the course of events in the future. There is no consensus among hydrologists on using a particular distribution function. But choosing the appropriate statistical distributions whose results are close to reality (have the lowest error) is possible.

Therefore, recently a considerable number of research works in spatial distribution have been published. Husak *et al.* (2007) have studied the monthly rainfall distribution for drought monitoring. They used the Gamma distribution approach, and the results showed it has a good performance in 98% of stations. Xia *et al.* 2012 used rainfall data from 27 meteorological stations in China's Huai River Basin to study the trend of marginal rainfall in the region. Based on their study, three statistical distributions, namely GEV, GP, and Gamma were used. The GEV distribution was more accurate according to the Kolmogorov–Smirnov (K–S) test.

Bhavyashree & Bhattacharyya (2018) used 20 different probability distributions to fit the maximum daily rainfall in Karnataka, India. Their paper results revealed that probabilistic distributions such as Gamma, Pearson 5, and Weibull performed better.

Alam *et al.* (2018) examined the best probability distribution for predicting maximum monthly rainfall in Bangladesh. The results of their research indicated that in 36% of the stations, the GEV distribution and in 26% of the stations, the Pearson type 3 and Log-Pearson Type 3 distributions had better fit. The probabilistic distribution analysis of marginal precipitation in Mumbai, India, was implemented by Parchure & Gedam (2019) They considered 26 meteorological stations in their research, and the GEV distribution showed better results in 29% of the stations. Frechet and GP probability distributions also performed better in 27 and 22% of stations, respectively.

Li *et al.* (2020) increased the temporal coverage of precipitation data for reanalysis using Bayesian probabilistic methods. The results showed that the proposed method has a high performance. Lemus-Canovas *et al.* (2019) used a spatial regression model to investigate the spatial distribution of precipitation in the eastern Pyrenees. Consequently, the results revealed that there is a correlation between the estimated and observed precipitation with the coefficient of explanation (*R*^{2} = 0.8).

Khan *et al.* (2019) studied the spatial distribution of rainfall in the peninsular region of Malaysia from 1951 to 2007. The results showed that the amount of annual precipitation did not change significantly. The trend of regional precipitation had significant changes, which could be due to the increase in design periods with an average of 4.8 and 4.9 days per decade. Iqbal *et al.* (2019) studied the spatial distribution of precipitation and rainfall trends from 1951 to 2007 in the Himalayas in Pakistan. The results indicated changes in precipitation in summer between 0.25 and 1.25 mm per year. Also, the number of marginal rainfalls and dry days increased. Their research pointed out that increases in these two parameters have led to increased flood and drought.

Vélez *et al.* (2019) studied the spatial distribution of daily rainfall concentrations at 20 Puerto Rican stations from 1971 to 2010. The results showed that the index value of annual and daily rainfall concentration varies from southeast to northwest.

In general, the Semnan province has a great variety of climates. Also, according to the studies, forecasting the temporal and spatial distribution of precipitation is of great importance in the implementation of development projects and the development of economic infrastructure. On the other hand, the study of the temporal and spatial distribution of precipitation requires a large amount of observational data, making such research works difficult in the region of the Semnan province. In such cases, the use of probabilistic distributions can help study temporal precipitation changes. Therefore, in the present study, the temporal and spatial distribution of *P*_{year} and *P*_{max24h} in the Semnan province have been analyzed. For this purpose, probabilistic distributions and different spatial interpolation methods have been used. Then, the amount of *P*_{year} and *P*_{max24h} and its spatial distribution with 10-, 25-, 50-, 100-, and 200-year return periods were predicted. Also, the relationship between *P*_{year} and *P*_{max24h} was estimated in different return periods, which is helpful when the *P*_{max24h} data were missed.

## MATERIALS AND METHODS

### Study area

*et al.*2009). The Semnan province is in the range of 34° and 13′ to 37°20′ north latitude and from 51° and 51′ to 57° and 3′ east longitude. This province has an area of 97,491 km

^{2}, which includes about 9% of the total area of Iran. Thus, the elevation of this province differs from 645 to 3,885 meters. The aspect slope of this Semnan province is northeast. However, in the northern region of this province, the aspect slope is eastern (Figure 1). The purpose of Figure 1 is to show the difference in the height of stations relative to each other and the location of the studied stations on different slopes. Longitude, latitude, and altitude for the Semnan station are 52°25′, 35°35′, and 1,127, respectively; for the Shahroud station are 54°55′, 36°22′, and 1,325.2, respectively; for the Damghan station are 54°19′, 36°08′, and 1,155.4, respectively; for the Garmsar station are 52°21′, 35°14′, and 899.9, respectively (Ghazvinian

*et al.*2020b, 2020c; Dehghanipour

*et al.*2021; Karami

*et al.*2021; Karami & Ghazvinian 2022). Shahmirzad and Meyami stations have longitude 53°21′ and 55°37′, and latitude 35°46′ and 36°24′, respectively, and finally have a height from the free surface of 1,969 and 1,081 m. The topographic information of the studied area was available and the DEM and slope maps were provided by using ArcGIS software.

*P*

_{year}of less than 50 mm and a

*P*

_{year}of large areas of the province to less than 200 mm. The dry season almost has no significant rainfall in the Semnan province and corresponds to the summer and warm months of the year and the rainy season coincides with the winter and cold days of the year. Figure 2 shows the monthly mean average in the Semnan province. According to Figure 2, 77% of the precipitation amount occurs from January to June (winter and spring).

### Data used

In the present research, precipitation information from Semnan, Shahroud, Damghan, Garmsar, Shahmirzad, and Meyami stations has been used to study the temporal and spatial variations of rainfall in the Semnan province. The rainfall data period of Semnan, Shahroud, Damghan, and Garmsar is from 1986 to 2018. This period for Shahmirzad and Meyami stations is from 2001 to 2018 (Table 1).

. | P_{year} (mm). | P_{max24h}(mm). | ||||
---|---|---|---|---|---|---|

Stations . | Mean . | STD . | CV . | Mean . | STD . | CV . |

Semnan | 136.59 | 41.76 | 0.31 | 21.85 | 7.38 | 0.34 |

Shahrood | 148.17 | 57.63 | 0.39 | 18.56 | 9.68 | 0.52 |

Dameghan | 100.92 | 33.04 | 0.33 | 16.47 | 5.46 | 0.33 |

Garmsar | 112.91 | 38.55 | 0.34 | 19.06 | 6.74 | 0.35 |

Shahmirzad | 182.21 | 108.99 | 0.60 | 24.41 | 14.56 | 0.60 |

Meyami | 124.8 | 52.17 | 0.42 | 18.88 | 8.68 | 0.46 |

. | P_{year} (mm). | P_{max24h}(mm). | ||||
---|---|---|---|---|---|---|

Stations . | Mean . | STD . | CV . | Mean . | STD . | CV . |

Semnan | 136.59 | 41.76 | 0.31 | 21.85 | 7.38 | 0.34 |

Shahrood | 148.17 | 57.63 | 0.39 | 18.56 | 9.68 | 0.52 |

Dameghan | 100.92 | 33.04 | 0.33 | 16.47 | 5.46 | 0.33 |

Garmsar | 112.91 | 38.55 | 0.34 | 19.06 | 6.74 | 0.35 |

Shahmirzad | 182.21 | 108.99 | 0.60 | 24.41 | 14.56 | 0.60 |

Meyami | 124.8 | 52.17 | 0.42 | 18.88 | 8.68 | 0.46 |

STD, standard deviation; CV, coefficient of variation.

A time series is predictable if it has long-term memory. This approach is based on whether all the expected phenomena have occurred in the available time series or not (Karamouz *et al.* 2012).

Hurst (1951), by studying the water level of the Nile River, presented a test for extreme events, now referred to as the Hurst index. The steps of this method are as follows:

At the beginning, the data scale is normalized using Equation (1). By considering a runoff time series of we have:

*z*is zero, the last value of

*Y*(

*Y*), will always be zero. Therefore, the adjusted domain will be equal to Equation (3):

_{n}*Y*values is zero, the maximum values will always be greater than or equal to 0 and the minimum values less than or equal to 0. Therefore, the adjusted range is always non-negative. Therefore, the Hurst index is defined as Equation (4):where

*R*is the amplitude of the changes,

*S*is the time series standard deviation,

*a*is a fixed number,

*n*is the number of observations, and

*H*is the Hurst index. By drawing the left term of Equation (4) versus log(

*n*), the Hurst index was estimated. According to Hurst's findings, if the Hurst index value is 0.5, it indicates a normal independent process. If it is between 0.5 and 1, the time series is long enough for modeling, and the closer this index is to 1, there is no need to extend the time series data.

### HEC-SSP

HEC-SSP software was introduced in 2008 by the U.S. Army Corps of Engineers for statistical analysis of hydrological data (Brunner & Fleming 2010). This software has different sections, such as data definition and analysis (Harris *et al.* 2008; Root & Papakos 2010). Primarily, in the present study, the *P*_{year} and *P*_{max24h} were entered in the data definition section. Then, in the analysis and distribution fitting analysis section, different statistical distributions were fitted to the data. In the end, the best statistical distribution was selected, and the amount of precipitation was measured with different return periods for each station. It is necessary to mention that *P*_{year} is calculated based on the summing of daily rainfall in a year. Also, the *P*_{max24h} was the maximum daily rainfall in a year.

### Statistical distributions

*et al.*2011). The K–S test has been used to measure the adherence of a sample of a particular distribution (Massey 1951; Wilks 1995; Simolo

*et al.*2010). The statistic of this test represents the most significant difference between the expected and actual frequencies (as an absolute value) measured in different categories (Equation (5)). Gado

*et al.*(2021), Raziei (2021), Zhang & Li (2020) and Zhao

*et al.*(2020) have used the K–S criterion to determine the best statistical distribution:where indicates the actual cumulative relative frequency and shows the expected cumulative relative frequency (Jahan

*et al.*2019).

^{2}test (Equation (6)) is a good measure of the quality of the data fitting process in the probabilistic distribution used for the data (Coronado-Hernández

*et al.*2020):where

*χ*

^{2}is the value of the chi-square test,

*R*is the recorded value,

_{i}*M*is the modeled value.

_{i}Distribution . | Formulas . | Parameters . | Eq. number . |
---|---|---|---|

Normal | (7) | ||

Log normal | (8) | ||

GEV | (9) | ||

GP | (10) | ||

Exponential | (11) | ||

Triangular | (12) |

Distribution . | Formulas . | Parameters . | Eq. number . |
---|---|---|---|

Normal | (7) | ||

Log normal | (8) | ||

GEV | (9) | ||

GP | (10) | ||

Exponential | (11) | ||

Triangular | (12) |

### Spatial interpolation methods

Interpolation is for finding unknown values of some points based on the known values of other points. The most important application of these methods is estimating the spatial distribution based on the discrete points. Kriging, IDW, and RBF methods are some of the most potent interpolation methods used in many studies (Agung Setianto & Tamia Triandini 2013; Gong *et al.* 2014; Arifin *et al.* 2015; Ikechukwu *et al.* 2017; Liu *et al.* 2020). Therefore in the present study, the mentioned methods were used for spatial interpolation of *P*_{year} and *P*_{max24h} in the Semnan province. In this study, the cross-validation (CV) method was used. The CV method is best known for widespread use in interpolation applications (Hancock & Hutchinson 2002). The main benefit of the CV approach is that it is a clearly defined and user-independent process. The CV approach is not acceptable for surfaces that have an insufficient number of observed input points (Jeffrey *et al.* 2001).

#### Radial basis function

*N*is the number of interpolation points.

#### Inverse distance weighted

*et al.*2020). As positions get closer to the desired point, their weights will be considered more compared to farther ones. Against the Kriging method, this method is not considering the assumptions about the spatial relationship among the data (it does not have a variogram). It only assumes that as points are getting nearer to the estimation point, they will be considered similar (Noor

*et al.*2022). In this method, the inverse distances are considered with a numerical power between 1 and 5, but this number will be regarded as 2 in most cases (Wienhöfer

*et al.*2023). In IDW as distance increases, the weight decreases rapidly. Therefore, the interpolating calculation results in local answers, and since the weights are never zero, there is no discontinuity in the calculation (Chutsagulprom

*et al.*2022). But the disadvantage of this approach is that, unlike the Kriging, the map does not produce an estimation error. Another demerit is that this method does not take the shape of the samples into account; consequently, the weights are given to two or more specimens arranged in clusters next to each other with an approximately same direction and equal distance to the desired point, and will be considered similar to the weight given to a single sample that is at the same distance but in a different direction from the desired point (Chutsagulprom

*et al.*2022; Noor

*et al.*2022). The mentioned problem in the IDW method is not likely to happen in the Kriging approach due to its clustering properties. Since IDW assesses the weight by considering the inverse of the number of samples (1/

*n*), in the maps prepared, unlike Kriging, the minimum and maximum variables estimated can be seen at the location of the prototypes. The general equation of two-dimensional interpolation using IDW approach in Equation (14) (Arifin Arifin

*et al.*2015):where is the estimated value at position ,

*N*stands for the number of known positions adjacent to , refers to assigned weight to the known values in , is the Euclidean distance between each point in the positions and ,

*p*relates to the power that is affected by weight on

*w*.

#### Kriging

*et al.*2014). In other words, two points close to each other probably have similar values, while two points far apart are less similar in their values. The initial purpose of developing the Kriging method was to use mining based on central samples. The model of spatial diversity can be shown as Equation (15):

*Z*(

*s*) is the predicted value in the desired locations,

*μ*(

*s*) is the defining function and describes the trend component of

*Z*(

*s*), and

*ε*(

*s*) represents random residues, this parameter varies locally but is spatially dependent. Conventional Kriging is the simplest form of this method and is based on a constant

*μ*(s) (Goovaerts 1997). Spatial autocorrelation using a semi-variogram is represented by Equation (16) (Goovaerts 1997):

In the above equation, *Z*(*s _{i}*) and

*Z*(

*s*+

_{i}*h*) are values that are at a distance

*h*from each other, and is a set of values of observational distances. Theoretically, the observed values and the values calculated by the Kriging method should be the same. But in reality this does not happen due to the presence of a nugget effect (indicating measurement error or changes in the microscale). Therefore, the values calculated by the Kriging method will largely depend on the appropriate choice of variogram model (Gong

*et al.*2014). The most important main feature of the Kriging method is that the spatial smoothing is estimated through variograms. Also, the uncertainty estimates for the predicted values are presented as variance. A semi-variogram model is required to determine the expectations at which Kriging variance is calculated.The necessary calculations in this method can be performed in a defined local neighborhood (Goovaerts 1997), but we used global Kriging in this work, using all observations for spatial prediction.

*c*is a partial threshold value and therefore represents the variance of a random field. Also, parameter a expresses the distance of autocorrelated observations. Initialization parameters are essential for the geostatistical process, but the variogram model fitting process occurs automatically by minimizing the total weight of the square errors (Equation (19)):with

*γ*(

*h*), the value according to the parametric model is minimized.

### SURFER

SURFER (Keckler 1994) is very suitable and flexible software for drawing two-dimensional and three-dimensional models. This software has several versions. The latest version is a combination of CAD and SURFER 8 software. This software is very useful in drawing two-dimensional and three-dimensional isometric maps. Three important applications of this software in natural geography are as follows: (1) drawing topographic maps in different scales, (2) drawing isometric weather maps (isothermal, isobars, isohyetal), and (3) designing a three-dimensional model of topographic and geological maps in different scales.

### Genetic algorithm

Genetic algorithm (GA) is an evolutionary algorithm for optimization based on extensive and effective search in large spaces based on genes and chromosomes (Ehteram *et al.* 2018). This search consists of three steps. In the first stage, the initial population, consisting of a set of chromosomes, is formed. In the second stage, the value of each member is measured using the definition of the objective function. In the final step, genetic operators produce new members, which include producing offspring from selected parents and mutating the members, and finally, gradual evolution is performed (Goldberg 1989). The selection stage is based on the fitting degree of members and some of the most suitable chromosomes are selected for reproduction. Finally, genetic operators are implemented on members, and their genetic codes are modified and synthesized (Zhang *et al.* 2012). In this study, for the GA, the number of family members was 300, the number of offspring was 240, the number of mutant members was 90, the mutation rate was 0.04, and the number of iterative steps was 200. To select parents from family members, the Roulette Wheel Selection method was used.

### Evaluation criteria

*et al.*(2017) and Zabihi

*et al.*(2012) researches. In Equations (20)–(22),

*N*is the number of data, is the estimated value, is the measured value, is the estimated mean and is the mean of the measured values (Ghazvinian

*et al.*2019; Dadrasajirlou

*et al.*2022; Karami

*et al.*2022; Ghazvinian

*et al.*2020a):

## RESULTS AND DISCUSSION

### Fitting of statistical distributions

Table 3 shows the results of the study of the adequacy of *P*_{year} and *P*_{max24h} based on the Hurst method. According to this table, all the data used in the present study in all stations are adequate.

Stations . | ||||||
---|---|---|---|---|---|---|

. | Semnan . | Shahrood . | Dameghan . | Garmsar . | Shahmirzad . | Meyami . |

P_{year} | 0.62 | 0.64 | 0.58 | 0.53 | 0.73 | 0.66 |

P_{max24h} | 0.65 | 0.61 | 0.60 | 0.56 | 0.65 | 0.61 |

Stations . | ||||||
---|---|---|---|---|---|---|

. | Semnan . | Shahrood . | Dameghan . | Garmsar . | Shahmirzad . | Meyami . |

P_{year} | 0.62 | 0.64 | 0.58 | 0.53 | 0.73 | 0.66 |

P_{max24h} | 0.65 | 0.61 | 0.60 | 0.56 | 0.65 | 0.61 |

The results of the goodness fit test for statistical distributions for the *P*_{year} of the studied stations in the Semnan province are presented in Table 4. According to this table, GP distribution for Semnan, Damghan, Garmsar, and Meyami stations; Gumbel distribution for the Shahroud station; and exponential distribution for the Shahmirzad station had the best fitting results. 66.66% of the studied stations followed the GP distribution, and about 16.67% followed the Gumbel distribution and Exponential distribution. This result may be due to skewness in the distribution of *P*_{year} at the studied stations.

Stations . | Distribution . | K–S . | χ^{2}
. | R^{2}
. | Distribution . | K–S . | χ^{2}
. | R^{2}
. | Distribution . | K–S . | χ^{2}
. | R^{2}
. |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Semnan | GP | 0.074 | 0.140 | 0.945 | GEV | 0.077 | 0.230 | 0.883 | Ln-Normal | 0.112 | 0.273 | 0.814 |

Triangular | 0.074 | 0.144 | 0.932 | Normal | 0.083 | 0.241 | 0.853 | Log10-Normal | 0.112 | 0.726 | 0.785 | |

Uniform | 0.075 | 0.228 | 0.926 | G-Logistic | 0.103 | 0.268 | 0.827 | Exponential | 0.386 | 0.781 | 0.763 | |

Shahrood | Gumbel | 0.129 | 0.133 | 0.952 | Logistic | 0.158 | 0.246 | 0.901 | GP | 0.180 | 0.621 | 0.853 |

GEV | 0.158 | 0.187 | 0.938 | Normal | 0.180 | 0.261 | 0.886 | Exponential | 0.395 | 0.630 | 0.829 | |

Dameghan | GP | 0.084 | 0.238 | 0.932 | Normal | 0.109 | 0.911 | 0.897 | G-Logistic | 0.132 | 1.038 | 0.832 |

Uniform | 0.085 | 0.328 | 0.925 | Ln-Normal | 0.118 | 0.963 | 0.883 | Exponential | 0.373 | 1.083 | 0.813 | |

GEV | 0.105 | 0.545 | 0.908 | Log10-Normal | 0.118 | 0.982 | 0.864 | Empirical | 0.397 | 1.248 | 0.804 | |

Garmsar | GP | 0.095 | 0.157 | 0.966 | G-Logistic | 0.112 | 0.403 | 0.918 | Ln-Normal | 0.186 | 0.7289 | 0.881 |

Triangular | 0.096 | 0.178 | 0.954 | Normal | 0.112 | 0.642 | 0.903 | Log10-Normal | 0.186 | 0.810 | 0.869 | |

GEV | 0.099 | 0.294 | 0.938 | Uniform | 0.119 | 0.666 | 0.892 | Exponential | 0.315 | 1.316 | 0.850 | |

Shahmirzad | Exponential | 0.097 | 0.006 | 0.965 | GEV | 0.142 | 0.011 | 0.928 | Normal | 0.167 | 0.104 | 0.894 |

Triangular | 0.126 | 0.009 | 0.951 | Uniform | 0.150 | 0.091 | 0.903 | GP | 0.254 | 0.111 | 0.879 | |

Meyami | GP | 0.105 | 0.001 | 0.973 | Normal | 0.130 | 0.005 | 0.931 | Log10-Normal | 0.210 | 0.017 | 0.897 |

GEV | 0.118 | 0.002 | 0.953 | Uniform | 0.130 | 0.007 | 0.923 | Exponential | 0.210 | 0.098 | 0.885 | |

G-Logistic | 0.126 | 0.004 | 0.942 | Ln-Normal | 0.147 | 0.013 | 0.906 | Empirical | 0.336 | 0.101 | 0.867 |

Stations . | Distribution . | K–S . | χ^{2}
. | R^{2}
. | Distribution . | K–S . | χ^{2}
. | R^{2}
. | Distribution . | K–S . | χ^{2}
. | R^{2}
. |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Semnan | GP | 0.074 | 0.140 | 0.945 | GEV | 0.077 | 0.230 | 0.883 | Ln-Normal | 0.112 | 0.273 | 0.814 |

Triangular | 0.074 | 0.144 | 0.932 | Normal | 0.083 | 0.241 | 0.853 | Log10-Normal | 0.112 | 0.726 | 0.785 | |

Uniform | 0.075 | 0.228 | 0.926 | G-Logistic | 0.103 | 0.268 | 0.827 | Exponential | 0.386 | 0.781 | 0.763 | |

Shahrood | Gumbel | 0.129 | 0.133 | 0.952 | Logistic | 0.158 | 0.246 | 0.901 | GP | 0.180 | 0.621 | 0.853 |

GEV | 0.158 | 0.187 | 0.938 | Normal | 0.180 | 0.261 | 0.886 | Exponential | 0.395 | 0.630 | 0.829 | |

Dameghan | GP | 0.084 | 0.238 | 0.932 | Normal | 0.109 | 0.911 | 0.897 | G-Logistic | 0.132 | 1.038 | 0.832 |

Uniform | 0.085 | 0.328 | 0.925 | Ln-Normal | 0.118 | 0.963 | 0.883 | Exponential | 0.373 | 1.083 | 0.813 | |

GEV | 0.105 | 0.545 | 0.908 | Log10-Normal | 0.118 | 0.982 | 0.864 | Empirical | 0.397 | 1.248 | 0.804 | |

Garmsar | GP | 0.095 | 0.157 | 0.966 | G-Logistic | 0.112 | 0.403 | 0.918 | Ln-Normal | 0.186 | 0.7289 | 0.881 |

Triangular | 0.096 | 0.178 | 0.954 | Normal | 0.112 | 0.642 | 0.903 | Log10-Normal | 0.186 | 0.810 | 0.869 | |

GEV | 0.099 | 0.294 | 0.938 | Uniform | 0.119 | 0.666 | 0.892 | Exponential | 0.315 | 1.316 | 0.850 | |

Shahmirzad | Exponential | 0.097 | 0.006 | 0.965 | GEV | 0.142 | 0.011 | 0.928 | Normal | 0.167 | 0.104 | 0.894 |

Triangular | 0.126 | 0.009 | 0.951 | Uniform | 0.150 | 0.091 | 0.903 | GP | 0.254 | 0.111 | 0.879 | |

Meyami | GP | 0.105 | 0.001 | 0.973 | Normal | 0.130 | 0.005 | 0.931 | Log10-Normal | 0.210 | 0.017 | 0.897 |

GEV | 0.118 | 0.002 | 0.953 | Uniform | 0.130 | 0.007 | 0.923 | Exponential | 0.210 | 0.098 | 0.885 | |

G-Logistic | 0.126 | 0.004 | 0.942 | Ln-Normal | 0.147 | 0.013 | 0.906 | Empirical | 0.336 | 0.101 | 0.867 |

Table 5 shows the goodness fit test results of statistical distributions for *P*_{max24h} in the studied stations. According to Table 5, in 50% of the stations, the best distribution is related to the GP distribution. About 16.67% of stations follow Log normal, Exponential distribution, and Value Generalized Extreme distributions. Table 6 shows the goodness fit test outcomes of statistical distributions for the ratio of *P*_{max24h} to *P*_{year} in the studied stations. According to Table 6, in 50% of the stations, the best distribution is related to the GEV distribution. About 16.67% of stations follow normal, triangular, and GP distribution. In Tables 4–6, the best Distribution has been highlighted, and other statistical distributions at lower ranks have been shown.

Stations . | Distribution . | K–S . | χ^{2}
. | R^{2}
. | Distribution . | K–S . | χ^{2}
. | R^{2}
. | Distribution . | K–S . | χ^{2}
. | R^{2}
. |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Semnan | GP | 0.102 | 0.007 | 0.971 | Ln-Normal | 0.140 | 0.032 | 0.934 | Normal | 0.166 | 0.042 | 0.887 |

Triangular | 0.123 | 0.008 | 0.965 | Log10-Normal | 0.140 | 0.037 | 0.921 | G-Logistic | 0.178 | 0.055 | 0.872 | |

Uniform | 0.123 | 0.009 | 0.952 | GEV | 0.144 | 0.041 | 0.902 | Exponential | 0.402 | 0.177 | 0.854 | |

Shahrood | GP | 0.098 | 0.091 | 0.935 | Gumbel | 0.114 | 0.206 | 0.903 | Empirical | 0.120 | 0.548 | 0.874 |

GEV | 0.104 | 0.184 | 0.924 | Triangular | 0.117 | 0.247 | 0.892 | Shifted Exponential | 0.141 | 0.551 | 0.865 | |

Logistic | 0.110 | 0.197 | 0.910 | Gamma | 0.122 | 0.357 | 0.884 | G-Logistic | 0.159 | 0.642 | 0.843 | |

Dameghan | GEV | 0.074 | 0.049 | 0.943 | Ln-Normal | 0.098 | 0.068 | 0.901 | GP | 0.102 | 0.078 | 0.871 |

Triangular | 0.083 | 0.051 | 0.932 | Log10-Normal | 0.098 | 0.069 | 0.889 | Uniform | 0.107 | 0.079 | 0.863 | |

Normal | 0.092 | 0.053 | 0.914 | G-Logistic | 0.099 | 0.073 | 0.881 | Exponential | 0.362 | 0.091 | 0.852 | |

Garmsar | GP | 0.085 | 0.166 | 0.932 | Triangular | 0.088 | 0.390 | 0.891 | Ln-Normal | 0.123 | 0.677 | 0.851 |

GEV | 0.086 | 0.304 | 0.923 | G-Logistic | 0.093 | 0.424 | 0.887 | Log10-Normal | 0.123 | 0.721 | 0.842 | |

Normal | 0.087 | 0.331 | 0.910 | Uniform | 0.115 | 0.654 | 0.862 | Exponential | 0.387 | 0.852 | 0.827 | |

Shahmirzad | Exponential | 0.298 | 0.002 | - | Normal | 0.301 | 0.102 | - | Triangular | 0.421 | 0.274 | - |

G-Logistic | 0.299 | 0.103 | - | GEV | 0.335 | 0.197 | - | GP | 0.441 | 0.569 | - | |

Meyami | Ln-normal | 0.137 | 0.229 | 0.912 | Ln-Normal | 0.154 | 0.327 | 0.886 | 4 Parameter Beta | 0.222 | 0.382 | 0.852 |

GEV | 0.139 | 0.254 | 0.902 | Normal | 0.176 | 0.328 | 0.873 | Empirical | 0.231 | 0.420 | 0.842 | |

Triangular | 0.142 | 0.314 | 0.897 | Exponential | 0.191 | 0.330 | 0.861 | G-Logistic | 0.239 | 0.543 | 0.831 |

Stations . | Distribution . | K–S . | χ^{2}
. | R^{2}
. | Distribution . | K–S . | χ^{2}
. | R^{2}
. | Distribution . | K–S . | χ^{2}
. | R^{2}
. |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Semnan | GP | 0.102 | 0.007 | 0.971 | Ln-Normal | 0.140 | 0.032 | 0.934 | Normal | 0.166 | 0.042 | 0.887 |

Triangular | 0.123 | 0.008 | 0.965 | Log10-Normal | 0.140 | 0.037 | 0.921 | G-Logistic | 0.178 | 0.055 | 0.872 | |

Uniform | 0.123 | 0.009 | 0.952 | GEV | 0.144 | 0.041 | 0.902 | Exponential | 0.402 | 0.177 | 0.854 | |

Shahrood | GP | 0.098 | 0.091 | 0.935 | Gumbel | 0.114 | 0.206 | 0.903 | Empirical | 0.120 | 0.548 | 0.874 |

GEV | 0.104 | 0.184 | 0.924 | Triangular | 0.117 | 0.247 | 0.892 | Shifted Exponential | 0.141 | 0.551 | 0.865 | |

Logistic | 0.110 | 0.197 | 0.910 | Gamma | 0.122 | 0.357 | 0.884 | G-Logistic | 0.159 | 0.642 | 0.843 | |

Dameghan | GEV | 0.074 | 0.049 | 0.943 | Ln-Normal | 0.098 | 0.068 | 0.901 | GP | 0.102 | 0.078 | 0.871 |

Triangular | 0.083 | 0.051 | 0.932 | Log10-Normal | 0.098 | 0.069 | 0.889 | Uniform | 0.107 | 0.079 | 0.863 | |

Normal | 0.092 | 0.053 | 0.914 | G-Logistic | 0.099 | 0.073 | 0.881 | Exponential | 0.362 | 0.091 | 0.852 | |

Garmsar | GP | 0.085 | 0.166 | 0.932 | Triangular | 0.088 | 0.390 | 0.891 | Ln-Normal | 0.123 | 0.677 | 0.851 |

GEV | 0.086 | 0.304 | 0.923 | G-Logistic | 0.093 | 0.424 | 0.887 | Log10-Normal | 0.123 | 0.721 | 0.842 | |

Normal | 0.087 | 0.331 | 0.910 | Uniform | 0.115 | 0.654 | 0.862 | Exponential | 0.387 | 0.852 | 0.827 | |

Shahmirzad | Exponential | 0.298 | 0.002 | - | Normal | 0.301 | 0.102 | - | Triangular | 0.421 | 0.274 | - |

G-Logistic | 0.299 | 0.103 | - | GEV | 0.335 | 0.197 | - | GP | 0.441 | 0.569 | - | |

Meyami | Ln-normal | 0.137 | 0.229 | 0.912 | Ln-Normal | 0.154 | 0.327 | 0.886 | 4 Parameter Beta | 0.222 | 0.382 | 0.852 |

GEV | 0.139 | 0.254 | 0.902 | Normal | 0.176 | 0.328 | 0.873 | Empirical | 0.231 | 0.420 | 0.842 | |

Triangular | 0.142 | 0.314 | 0.897 | Exponential | 0.191 | 0.330 | 0.861 | G-Logistic | 0.239 | 0.543 | 0.831 |

Station . | Distribution . | K–S . | χ^{2}
. | R^{2}
. | Distribution . | K–S . | χ^{2}
. | R^{2}
. | Distribution . | K–S . | χ^{2}
. | R^{2}
. |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Semnan | GEV | 0.081 | 0.055 | 0.971 | Pearson III | 0.090 | 0.105 | 0.902 | Normal | 0.129 | 0.155 | 0.842 |

Ln-Normal | 0.082 | 0.064 | 0.962 | Shifted Gamma | 0.090 | 0.106 | 0.892 | GP | 0.130 | 0173 | 0.832 | |

Log10-Normal | 0.082 | 0.075 | 0.953 | Beta | 0.095 | 0.126 | 0.887 | Logistic | 0.132 | 0.250 | 0.821 | |

Log-Pearson III | 0.083 | 0.089 | 0.942 | Log-Logistic | 0.099 | 0.127 | 0.872 | Shifted Exponential | 0.198 | 0.256 | 0.813 | |

Gumbel | 0.084 | 0.093 | 0.931 | Triangular | 0.120 | 0.141 | 0.863 | Exponential | 0.432 | 0.257 | 0.802 | |

Gamma | 0.086 | 0.104 | 0.923 | Uniform | 0.125 | 0.146 | 0.856 | Empirical | 0.441 | 0.475 | 0.797 | |

Shahrood | Uniform | 0.098 | 0.108 | 0.972 | Triangular | 0.114 | 0.781 | 0.921 | Beta | 0.120 | 0.907 | 0892 |

GEV | 0.103 | 0.164 | 0.962 | Gumbel | 0.115 | 0.798 | 0.912 | Shifted Exponential | 0.140 | 0.957 | 0.887 | |

Pearson III | 0.104 | 0.225 | 0.953 | G-Logistic | 0.115 | 0.869 | 0.903 | Normal | 0.152 | 0.976 | 0.862 | |

Logistic | 0.112 | 0.486 | 0.942 | Gamma | 0.119 | 0.870 | 0.892 | GP | 0.157 | 0.978 | 0.852 | |

Dameghan | Triangular | 0.084 | 0.259 | 0.932 | Shifted Gamma | 0.097 | 1.663 | 0.872 | Logistic | 0.118 | 2.832 | 0.811 |

Uniform | 0.086 | 0.301 | 0.921 | Beta | 0.097 | 1.906 | 0.865 | G-Logistic | 0.118 | 2.940 | 0.809 | |

GEV | 0.093 | 0.607 | 0.913 | Ln-Normal | 0.097 | 2.292 | 0.855 | Log-Logistic | 0.119 | 2.969 | 0.798 | |

Log-Pearson III | 0.093 | 0.809 | 0.905 | Log10-Normal | 0.097 | 2.431 | 0.846 | Gumbel | 0.119 | 3.015 | 0.788 | |

GP | 0.094 | 0.909 | 0.894 | Normal | 0.098 | 2.541 | 0.832 | Shifted Exponential | 0.204 | 3.184 | 0.765 | |

Pearson III | 0.097 | 1.565 | 0.888 | Gamma | 0.101 | 2.557 | 0.824 | Exponential | 0.407 | 3.384 | 0.742 | |

Garmsar | GEV | 0.081 | 0.056 | 0.952 | Pearson III | 0.090 | 0.105 | 0.898 | Normal | 0.129 | 0.155 | 0.831 |

Ln-Normal | 0.082 | 0.064 | 0.942 | Shifted Gamma | 0.090 | 0.106 | 0.888 | GP | 0.130 | 0.173 | 0.822 | |

Log10-Normal | 0.082 | 0.075 | 0.932 | Beta | 0.095 | 0.126 | 0.873 | Logistic | 0.132 | 0.249 | 0.817 | |

Log-Pearson III | 0.083 | 0.089 | 0.929 | Log-Logistic | 0.099 | 0.127 | 0.865 | Shifted Exponential | 0.198 | 0.256 | 0.809 | |

Gumbel | 0.084 | 0.093 | 0.914 | Triangular | 0.120 | 0.141 | 0.855 | Exponential | 0.432 | 0.475 | 0.798 | |

Gamma | 0.086 | 0.103 | 0.903 | Uniform | 0.125 | 0.146 | 0.839 | Empirical | 0.446 | 0.517 | 0.778 | |

Shahmirzad | Exponential | 0.126 | 0.003 | 0.954 | G-Logistic | 0.147 | 0.367 | 0.892 | GP | 0.157 | 0.417 | 0.821 |

Normal | 0.130 | 0.218 | 0.943 | Logistic | 0.147 | 0.368 | 0.886 | Beta | 0.167 | 0.519 | 0.810 | |

Pearson III | 0.131 | 0.299 | 0.921 | Gumbel | 0.148 | 0.375 | 0.852 | Gamma | 0.167 | 0.583 | 0.802 | |

Triangular | 0.131 | 0.330 | 0.911 | Uniform | 0.156 | 0.409 | 0.843 | GEV | 0.167 | 0.591 | 0.792 | |

Meyami | GP | 0.140 | 0.036 | 0.942 | Beta | 0.190 | 0.211 | 0.882 | Uniform | 0.206 | 0.805 | 0.821 |

Shifted Exponential | 0.142 | 0.044 | 0.932 | Triangular | 0.190 | 0.234 | 0.872 | Log-Logistic | 0.222 | 0.829 | 0.814 | |

Log-Pearson III | 0.165 | 0.073 | 0.929 | GEV | 0.192 | 0.255 | 0.862 | Normal | 0.235 | 0.842 | 0.809 | |

Pearson III | 0.166 | 0.104 | 0.912 | Gumbel | 0.195 | 0.267 | 0.856 | Logistic | 0.255 | 0.867 | 0.792 | |

Shifted Gamma | 0.166 | 0.127 | 0.902 | Ln-Normal | 0.204 | 0.275 | 0.843 | Exponential | 0.392 | 0.918 | 0.789 | |

Gamma | 0.188 | 0.203 | 0.892 | Log10-Normal | 0.204 | 0.447 | 0.832 | Empirical | 0.395 | 0.955 | 0.772 |

Station . | Distribution . | K–S . | χ^{2}
. | R^{2}
. | Distribution . | K–S . | χ^{2}
. | R^{2}
. | Distribution . | K–S . | χ^{2}
. | R^{2}
. |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Semnan | GEV | 0.081 | 0.055 | 0.971 | Pearson III | 0.090 | 0.105 | 0.902 | Normal | 0.129 | 0.155 | 0.842 |

Ln-Normal | 0.082 | 0.064 | 0.962 | Shifted Gamma | 0.090 | 0.106 | 0.892 | GP | 0.130 | 0173 | 0.832 | |

Log10-Normal | 0.082 | 0.075 | 0.953 | Beta | 0.095 | 0.126 | 0.887 | Logistic | 0.132 | 0.250 | 0.821 | |

Log-Pearson III | 0.083 | 0.089 | 0.942 | Log-Logistic | 0.099 | 0.127 | 0.872 | Shifted Exponential | 0.198 | 0.256 | 0.813 | |

Gumbel | 0.084 | 0.093 | 0.931 | Triangular | 0.120 | 0.141 | 0.863 | Exponential | 0.432 | 0.257 | 0.802 | |

Gamma | 0.086 | 0.104 | 0.923 | Uniform | 0.125 | 0.146 | 0.856 | Empirical | 0.441 | 0.475 | 0.797 | |

Shahrood | Uniform | 0.098 | 0.108 | 0.972 | Triangular | 0.114 | 0.781 | 0.921 | Beta | 0.120 | 0.907 | 0892 |

GEV | 0.103 | 0.164 | 0.962 | Gumbel | 0.115 | 0.798 | 0.912 | Shifted Exponential | 0.140 | 0.957 | 0.887 | |

Pearson III | 0.104 | 0.225 | 0.953 | G-Logistic | 0.115 | 0.869 | 0.903 | Normal | 0.152 | 0.976 | 0.862 | |

Logistic | 0.112 | 0.486 | 0.942 | Gamma | 0.119 | 0.870 | 0.892 | GP | 0.157 | 0.978 | 0.852 | |

Dameghan | Triangular | 0.084 | 0.259 | 0.932 | Shifted Gamma | 0.097 | 1.663 | 0.872 | Logistic | 0.118 | 2.832 | 0.811 |

Uniform | 0.086 | 0.301 | 0.921 | Beta | 0.097 | 1.906 | 0.865 | G-Logistic | 0.118 | 2.940 | 0.809 | |

GEV | 0.093 | 0.607 | 0.913 | Ln-Normal | 0.097 | 2.292 | 0.855 | Log-Logistic | 0.119 | 2.969 | 0.798 | |

Log-Pearson III | 0.093 | 0.809 | 0.905 | Log10-Normal | 0.097 | 2.431 | 0.846 | Gumbel | 0.119 | 3.015 | 0.788 | |

GP | 0.094 | 0.909 | 0.894 | Normal | 0.098 | 2.541 | 0.832 | Shifted Exponential | 0.204 | 3.184 | 0.765 | |

Pearson III | 0.097 | 1.565 | 0.888 | Gamma | 0.101 | 2.557 | 0.824 | Exponential | 0.407 | 3.384 | 0.742 | |

Garmsar | GEV | 0.081 | 0.056 | 0.952 | Pearson III | 0.090 | 0.105 | 0.898 | Normal | 0.129 | 0.155 | 0.831 |

Ln-Normal | 0.082 | 0.064 | 0.942 | Shifted Gamma | 0.090 | 0.106 | 0.888 | GP | 0.130 | 0.173 | 0.822 | |

Log10-Normal | 0.082 | 0.075 | 0.932 | Beta | 0.095 | 0.126 | 0.873 | Logistic | 0.132 | 0.249 | 0.817 | |

Log-Pearson III | 0.083 | 0.089 | 0.929 | Log-Logistic | 0.099 | 0.127 | 0.865 | Shifted Exponential | 0.198 | 0.256 | 0.809 | |

Gumbel | 0.084 | 0.093 | 0.914 | Triangular | 0.120 | 0.141 | 0.855 | Exponential | 0.432 | 0.475 | 0.798 | |

Gamma | 0.086 | 0.103 | 0.903 | Uniform | 0.125 | 0.146 | 0.839 | Empirical | 0.446 | 0.517 | 0.778 | |

Shahmirzad | Exponential | 0.126 | 0.003 | 0.954 | G-Logistic | 0.147 | 0.367 | 0.892 | GP | 0.157 | 0.417 | 0.821 |

Normal | 0.130 | 0.218 | 0.943 | Logistic | 0.147 | 0.368 | 0.886 | Beta | 0.167 | 0.519 | 0.810 | |

Pearson III | 0.131 | 0.299 | 0.921 | Gumbel | 0.148 | 0.375 | 0.852 | Gamma | 0.167 | 0.583 | 0.802 | |

Triangular | 0.131 | 0.330 | 0.911 | Uniform | 0.156 | 0.409 | 0.843 | GEV | 0.167 | 0.591 | 0.792 | |

Meyami | GP | 0.140 | 0.036 | 0.942 | Beta | 0.190 | 0.211 | 0.882 | Uniform | 0.206 | 0.805 | 0.821 |

Shifted Exponential | 0.142 | 0.044 | 0.932 | Triangular | 0.190 | 0.234 | 0.872 | Log-Logistic | 0.222 | 0.829 | 0.814 | |

Log-Pearson III | 0.165 | 0.073 | 0.929 | GEV | 0.192 | 0.255 | 0.862 | Normal | 0.235 | 0.842 | 0.809 | |

Pearson III | 0.166 | 0.104 | 0.912 | Gumbel | 0.195 | 0.267 | 0.856 | Logistic | 0.255 | 0.867 | 0.792 | |

Shifted Gamma | 0.166 | 0.127 | 0.902 | Ln-Normal | 0.204 | 0.275 | 0.843 | Exponential | 0.392 | 0.918 | 0.789 | |

Gamma | 0.188 | 0.203 | 0.892 | Log10-Normal | 0.204 | 0.447 | 0.832 | Empirical | 0.395 | 0.955 | 0.772 |

*P*

_{year}in Semnan, Damghan, Garmsar, and Meyami stations. Gumbel distribution and Exponential distribution are the top distributions of Shahroud and Shahmirzad stations, respectively. As can be seen in Figure 3, the experimental and theoretical values are in good agreement.

*P*

_{max24h}in Semnan, Shahroud, and Garmsar stations, GP distribution was the superior distribution. GEV, Normal and Exponential distributions are the top statistical distributions of Damghan, Meyami, and Shahmirzad stations, respectively. As can be seen in Figure 4, the experimental and theoretical values are in good agreement.

*P*

_{max24h}to

*P*

_{year}in Semnan, Garmsar, and Shahmirzad stations, GEV distribution was superior. GP, normal, and triangular distributions are the top statistical distributions of Meyami, Shahroud, and Damghan stations, respectively. As can be seen in Figure 5, the experimental and theoretical values are in good agreement.

### Spatial distribution

To evaluate the spatial distribution methods, primarily, estimated data were calculated for each of the observed data using the cross-evaluation method.

*P*

_{year}in the study area in the return periods of 10, 25, 50, 100, and 200 years was performed and is shown in Figure 6. According to Figure 6, the maximum annual precipitation increases with approaching 53.35 east longitude and 35.77 north latitude and decreases as it moves away from it, because the highest altitude is related to the mentioned latitude and longitude (Shahmirzad station), and the lowest altitude is associated with the southeast of the basin. Also, with the increase of the return periods from 10 to 200 years,

*P*

_{year}increases from 420 to 1,040 mm per year.

*P*

_{max24h}estimated under different return periods and its spatial distribution. According to Figure 7, by moving toward the north (to Shahmirzad position), precipitation increases for a maximum of 24 h. Also, by moving toward the south and southeast, 24-h rainfall decreases in all the return periods studied. In addition, with an increasing return period from 10 to 200 years, the

*P*

_{max24h}increases from about 50 to 150 mm per day.

In other words, in all return periods of 10, 25, 50, 100 and 200, the amount of annual rainfall and 2-h rainfall decreases from the north and northeast of Semnan province to the southeast. The amount of error of each method is shown using evaluation criteria based on Table 7.

. | . | P_{year}. | P_{max24h}. | ||||
---|---|---|---|---|---|---|---|

Interpolation . | Tr . | R^{2}
. | RMSE(mm)
. | MAE(mm)
. | R^{2}
. | RMSE(mm)
. | MAE(mm)
. |

Kriging | 10 | 0.64 | 266.36 | 212.23 | 0.31 | 39.04 | 38.14 |

IDW | 10 | 0.35 | 268.34 | 258.19 | 0.32 | 39.24 | 38.35 |

RBF | 10 | 0.35 | 276.63 | 264.34 | 0.33 | 40.20 | 39.04 |

Kriging | 25 | 0.64 | 344.96 | 265.84 | 0.70 | 52.15 | 41.65 |

IDW | 25 | 0.62 | 354.23 | 272.62 | 0.57 | 52.63 | 41.97 |

RBF | 25 | 0.66 | 384.19 | 299.19 | 0.70 | 54.21 | 42.81 |

Kriging | 50 | 0.36 | 402.83 | 370.65 | 0.72 | 61.81 | 48.60 |

IDW | 50 | 0.39 | 408.08 | 373.86 | 0.57 | 62.63 | 49.15 |

RBF | 50 | 0.66 | 455.40 | 349.06 | 0.71 | 51.16 | 50.05 |

Kriging | 100 | 0.67 | 501.76 | 385.32 | 0.72 | 71.15 | 55.18 |

IDW | 100 | 0.59 | 509.00 | 388.27 | 0.57 | 72.36 | 55.99 |

RBF | 100 | 0.65 | 532.98 | 402.48 | 0.70 | 74.71 | 56.85 |

Kriging | 200 | 0.67 | 568.48 | 432.21 | 0.72 | 81.42 | 62.25 |

IDW | 200 | 0.59 | 577.59 | 435.83 | 0.57 | 83.06 | 63.37 |

RBF | 200 | 0.65 | 606.09 | 452.51 | 0.70 | 85.86 | 64.17 |

. | . | P_{year}. | P_{max24h}. | ||||
---|---|---|---|---|---|---|---|

Interpolation . | Tr . | R^{2}
. | RMSE(mm)
. | MAE(mm)
. | R^{2}
. | RMSE(mm)
. | MAE(mm)
. |

Kriging | 10 | 0.64 | 266.36 | 212.23 | 0.31 | 39.04 | 38.14 |

IDW | 10 | 0.35 | 268.34 | 258.19 | 0.32 | 39.24 | 38.35 |

RBF | 10 | 0.35 | 276.63 | 264.34 | 0.33 | 40.20 | 39.04 |

Kriging | 25 | 0.64 | 344.96 | 265.84 | 0.70 | 52.15 | 41.65 |

IDW | 25 | 0.62 | 354.23 | 272.62 | 0.57 | 52.63 | 41.97 |

RBF | 25 | 0.66 | 384.19 | 299.19 | 0.70 | 54.21 | 42.81 |

Kriging | 50 | 0.36 | 402.83 | 370.65 | 0.72 | 61.81 | 48.60 |

IDW | 50 | 0.39 | 408.08 | 373.86 | 0.57 | 62.63 | 49.15 |

RBF | 50 | 0.66 | 455.40 | 349.06 | 0.71 | 51.16 | 50.05 |

Kriging | 100 | 0.67 | 501.76 | 385.32 | 0.72 | 71.15 | 55.18 |

IDW | 100 | 0.59 | 509.00 | 388.27 | 0.57 | 72.36 | 55.99 |

RBF | 100 | 0.65 | 532.98 | 402.48 | 0.70 | 74.71 | 56.85 |

Kriging | 200 | 0.67 | 568.48 | 432.21 | 0.72 | 81.42 | 62.25 |

IDW | 200 | 0.59 | 577.59 | 435.83 | 0.57 | 83.06 | 63.37 |

RBF | 200 | 0.65 | 606.09 | 452.51 | 0.70 | 85.86 | 64.17 |

### Estimation of maximum 24-h rainfall using annual rainfall

*P*

_{year}and

*P*

_{max24h}. Figure 8 shows a graph of the

*P*

_{max24h}distribution relative to the

*P*

_{year}. According to Figure 8, the highest correlation coefficient between

*P*

_{year}and

*P*

_{max24h}is associated with the return period of 100 years. The lowest is related to the return period of 200 years. Therefore, the correlation between

*P*

_{year}and

*P*

_{max24h}in all return periods is high. Also, after a return period of 50 years, this correlation decreases slightly, which can be due to the limited

*P*

_{max24h}.

*P*

_{year}and

*P*

_{max24h}for different return periods at the studied stations. With an increasing return period in all stations,

*P*

_{year}and

*P*

_{max24h}have an upward trend. The trendline slope in

*P*

_{year}and

*P*

_{max24h}for Shahmirzad station is steeper than other stations. But this slope for Semnan station is almost zero. With the increase of the

*P*

_{year}return period and the

*P*

_{max24h}, there is no significant change. Shahmirzad's climate is semi-arid based on the De Martonne method, while the rest of the studied stations have dry climatic conditions. The reason for the difference in slope can be in this regard.

*P*

_{year}data and

*P*

_{max24h}for 10-, 25-, 50-, 100-, and 200-year return periods (Figure 10). In the Taylor diagram, the longitudinal distance from the origin of the coordinates represents the correlation coefficient, and the segmental lines represent the square root values of the mean squares of the error. As the circle segment increases, the value of this parameter increases. It can be said that each point of the Taylor diagram represents the standard deviation, the correlation coefficient, and the mean squares of the square error at the same time (Taylor 2001).

### Proposing a model for forecasting maximum 24-h rainfall

*P*

_{max24h}led to the general relationship resulted in Equation (23). This relationship is defined based on the parameters of

*z*and

*P*

_{year}and

*Tr*. In Equation (24), the coefficients of the parameters are expressed using the genetic method. Figure 11 shows the data predicted by GA in terms of measurement data. The horizontal axis shows the observed 24-h maximum precipitation data (in millimeters), and the vertical axis shows the simulated maximum 24-h precipitation data (in millimeters). The lower the data scatter around the best fit line, the greater the correlation and the less error. As can be seen, the correlation between the measured and simulated precipitation data is 0.98:where

*a, b, c, d, e, f, and g*are constant coefficients for Equation (23).

*z*is the height of stations above the sea level,

*Tr*is the return period,

*P*

_{year}is the anuual rainfall, and

*P*

_{max24h}is the maximum 24-h rainfall.

The Kavir plain is located in the south of the Semnan province and there is no city, and therefore, no synoptic station and almost all cities of this province are located in the northern half of the province. For instance, Semnan city, which is the seat of the Semnan province, and Garmsar city is closer to the southern desert of the province and have an arid and warm climate. However, Shahmirzad city has higher rainfall than other regions of the province due to locating on the Alborz mountain range heights. Shahmirzad is only 25 km away from Semnan city, but they are different climate conditions. In this study, stations were studied whose data were complete. Also, in this study, we wanted to examine only changes in latitude and longitude.

Despite the many advantages, the existing problems in fitting the models are among the disadvantages of Kriging methods (Teymourzade *et al.* 2019). Although the IDW method is quick and easy, it does not provide information on dependency or estimation of spatial uncertainty (Dingman 2002).

## CONCLUSION

The climatic conditions of Iran have a great variety that has been created due to changes in latitude and altitude. Semnan province has a variety of climatic conditions in different stations, and these conditions have prevailed on the stations due to the locations of the stations. Statistical analyses of *P*_{year} and *P*_{max24h} and the ratio of *P*_{max24h} to *P*_{year} were performed in six stations in Semnan, Shahroud, Damghan, Garmsar, Shahmirzad, and Meyami. Based on the *P*_{year} analysis, the Shahmirzad station has the highest annual average and the lowest average *P*_{year} in the Damghan station. Then, spatial interpolation using Kriging and IDW methods and 10-, 25-, 50-, 100-, and 200-year return periods were studied. In general, both Kriging and IDW methods had good performance in spatial distributions. For all return periods, the *P*_{year} and *P*_{max24h} decreases to the southeast of Semnan province. At the same time, the closer we get to the north of the region, especially to the Shahmirzad station, the *P*_{year} increases to a *P*_{max24h}.

There is a good relationship between *P*_{year} and *P*_{max24h}. The values of *R*^{2} coefficient in the 10-, 25-, 50-, 100-, and 200-year return periods were 0.988, 0.992, 0.988, 0.983 and 0.98, respectively. In the temporal analysis, the GP distribution with about 58% in the studied stations was the selected distribution in the analysis of *P*_{year} and *P*_{max24h}. Half of the regions followed the GEV distribution for the ratio of a *P*_{max24h} to *P*_{year}. Based on the results obtained in the present study, it is suggested that researchers can use statistical methods as an optimal approach for locating other meteorological stations. One of the limitations of this study is the small number of stations that have complete data. New methods can also be used to predict time with different return periods.

## FUNDING

This article is not funded by any scientific institution.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

## REFERENCES

*Hydrologic Analysis of Flash Floods in Sana'a, Yemen’ in Watershed Management 2010*. American Society of Civil Engineers, Reston, VA, pp. 1248–1259. http://ascelibrary.org/doi/10.1061/41143%28394%29112

*β*-IDW approach