Introduction

Rainfall is considered as one of the crucial hydrological phenomena, as it affects numerous activities. It has more importance in hydrological, meteorological, and climatological studies (Ye et al. 2018). Many applications in hydrology necessitate an accurate calculation of rainfall depth and return period (T) based on historical data. Prediction of floods in river basins, water balance research, water management research, rainwater harvesting, detention and storage tank designs, evapotranspiration estimates, irrigation, and flood management are some of the applications where rainfall is used as a key input (Rahman et al. 2013; Sabarish et al. 2017; Ostad-Ali-Askar et al. 2018; Langat et al. 2019; Kousar et al. 2020; Samantaray and Sahoo 2020; Moccia et al. 2021). Water resource planning and management necessitate complete and trustworthy hydrological data. To assess the water availability of a given area, a thorough database is required, and its lack might lead to incorrect planning and management. Long-term data can be used to make accurate water resource predictions. When the data length is small, the degree of uncertainty rises. The pattern of rainfall at any given location is affected due to the influences of local atmospheric, geologic, and morphologic conditions. For proper water resource planning and management, an accurate and precise understanding of rainfall distribution patterns throughout time at a given site is essential (Ostad-Ali-Askari et al. 2018). The assessment of available maximum daily rainfall will help in the efficient utilization of available water resources (Subudhi 2007). We can assess the expected rainfall at different probability levels/T by conducting rainfall probability or frequency analysis (Bhakar et al. 2008). This information is useful for the prevention of drought and flood severity, planning and design of reservoir layout, design of soil, and water conservation structures (Asim and Nath 2015; Derakhshannia et al. 2020; Zamir et al. 2021). Extreme rainfall events will have devastating consequences on people’s living (Singh et al. 2021). Maximum rainfall has a significant influence on the agriculture-based economy (Alam et al. 2018). Extreme rainfall may affect biodiversity, agriculture system, ecological balance, and livelihood patterns. The characteristics of intense rainfall occurrences must be extracted to identify risk variables that may be used to design long-term policies to save people and assets. Rainfall is a dominant factor for making planning or operational schedule of different agricultural and other activities of any given area (Tabish et al. 2015; Ostad-Ali-Askari et al. 2020; Saleh et al. 2021). The supply of irrigation water to the agriculture sector depends upon the available water source through rainfall, river, and surface and sub-surface water. Rainfed agriculture depends on rainfall to meet the crop water demand. Agriculture water management is one of the crucial practices due to limited, non-assured, and highly varied rainfall patterns. It is necessary to analyze the variability of rainfall patterns to tackle climate change issues that are responsible for natural hazards like drought and floods (Esberto 2018; Ahmad et al. 2019). Some geographical areas of a region can experience a dry season (drought), and some other geographical areas experience flash floods at the same time. The analysis of rainfall patterns will help in the management and effective utilization of water resources (Waghaye et al. 2015).

Interpretation of future probability of occurrence of hydrologic events from historical records is an important step of hydrology. Reliable and accurate rainfall prediction is required for a variety of purposes, including water resource planning, strategy enhancement, and upkeep activities. Maximum monthly, annual, daily, or consecutive daily rainfall prediction is very useful in drainage system design, reservoir operations, and design of hydraulic structures like barrages, dams, spillways, and bridges across the river flow (Bhatt et al. 1996; Tabish et al. 2015; Dabral et al. 2016; Ghosh et al. 2016; Ostad-Ali-Askari et al. 2017; Vanani et al. 2017; Namitha and Vinothkumar 2019; Nwaogazie and Sam 2019; Coronado-Hernández et al. 2020). Rainfall forecasting corresponding to different years of T would be possible using different PDFs (Tabish et al. 2015). Many attempts have been made for analyzing the frequency of occurrence of maximum rainfall magnitudes in different parts of India (Prakash and Rao 1986; Bhatt et al. 1996; Upadhyaya and Singh 1998; Singh 2001; Kumar and Bhardwaj 2015; Tabish et al. 2015; Singh et al. 2016; Kumar and Shanu, 2017; Sabarish et al. 2017; Namitha and Vinothkumar 2019; Salehi-Hafshejani et al. 2019; Talebmorad et al. 2020; Chaudhari and Sharma, 2021; Kaur et al. 2021; Sharma et al. 2021). Many PDFs have been established to determine the probability of occurrences of extreme events. But none of them has been a universally accepted PDF for frequency analysis of hydrologic events at any gauging site (Izinyon and Ajumuka 2013; Hassan et al. 2019). However, GEV, LP-III, P-III, L-NM, and NM PDFs are mostly suggested for frequency analysis (Topaloglu 2002; Ojha et al. 2008; Izinyon and Ajumuka 2013; Amin et al. 2016). The selection of the particular type of PDF for a specific site depends on the characteristics of the historical climate data (Izinyon and Ajumuka 2013; Amin et al. 2016; Pirnazar et al. 2018; Ostad-Ali-Askari et al. 2019; Talebmorad et al. 2021). The selection of a specific type of PDF for extreme rainfall estimation at different time scale is a challenging task (Li et al. 2015; Alahmadi 2019). Therefore, it becomes necessary to analyze the predictive performance of different PDFs for estimating maximum rainfall for the area under investigation. To the best of the author’s knowledge, the comparative study about the performance evaluation of different PDFs for maximum rainfall estimation has not been explored for the selected study area. The novelty of this study includes the finding that the different PDFs perform the best at different time scales for the estimation of maximum rainfall of the study area. Hence, it was revealed that a particular PDF is not suitable to estimate maximum rainfall magnitude at different time scales in a given region. By considering this issue, the following goals were set for this research: (a) to estimate maximum rainfall magnitudes at different durations of T using WPP formula; (b) to estimate maximum rainfall magnitudes at different durations of T using GEV, LP-III, P-III, L-NM, and NM PDFs; and (c) to select the best PDF for extrapolating the maximum rainfall at different time scale for longer durations of T in the study area.

Material and methods

Study area

Parbhani district is the district of Maharashtra state, India. The climate in this study area is semi-arid. The study area receives an average annual rainfall of 955 mm. Therefore, it falls under moderate to moderately high rainfall zone. The mean maximum temperature at the study area is about 44.6 °C while the mean minimum temperature at the study area is about 21.8 °C. The mean of daily relative humidity at the study area varies from 30 to 98% throughout the year. The maximum and minimum temperature at the Parbhani district occurs in May and December, respectively. Most of the population depends on agriculture to fulfill their daily needs. The location of the Parbhani district in the Indian state of Maharashtra is shown in Fig. 1.

Fig. 1
figure 1

Location of the Parbhani district in the Maharashtra state of India

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Data set

Daily rainfall data from the years 1971 to 2019 were collected from an Indian Meteorological Department–recognized observatory located at Vasantrao Naik Marathwada Agriculture University, Parbhani. These collected rainfall data have been analyzed for annual, maximum monthly, maximum and 1, 2, 3, 4, and 5 consecutive days of rainfall of the study area. In this study, the WPP formula has been used to find out the observed rainfall magnitudes at different time scales considering different durations of T. Different continuous PDFs like GEV, LP-III, P-III, L-NM, and NM were applied to assess the best-fitting PDF in the study area.

Methodology adopted

In this study, collected rainfall data of 49 years were used for the selection of the best PDF for estimating the maximum rainfall magnitude at different time scales. The observed maximum rainfall magnitude corresponding to 2, 5, 10, 25, and 50 years of T at annual, monthly, 1 day, and consecutive 2, 3, 4, and 5 days were estimated using the WPP formula. Different PDFs like GEV, LP-III, P-III, L-NM, and NM were used to estimate maximum rainfall magnitudes corresponding to T of 2, 5, 10, 25, and 50 years at the time scale for annual, monthly, 1 day, and consecutive 2, 3, 4, and 5 days. In this study, the chi-square test was used for testing the goodness of fit of different PDFs at different time scales corresponding to different durations of T. The best PDF at different time scales was selected based on the minimum sum of chi-square values. Finally, the selected PDF was employed for extrapolation of maximum monthly rainfall magnitudes estimation at the time scale for annual, monthly, 1 day, and consecutive 2, 3, 4, and 5 days for higher durations of T such as 100 and 200 years.

WPP formula

In this study, the WPP formula has been employed to perform the frequency analysis and find out the maximum rainfall magnitudes at different time scales. In order to apply this formula, firstly the annual rainfall, maximum monthly rainfall, and consecutive 1, 2, 3, 4, and 5 days of maximum rainfall were arranged in descending order of magnitude and then maximum rainfall magnitudes corresponding to different durations of T were estimated. The WPP formula is represented as (Sabarish et al. 2017):

$$P=\frac{m}{n+1}$$
(1)
$$T=\frac{1}{P}$$
(2)

where P is the probability of exceedance of given maximum rainfall, T is the return period/recurrence interval of that selected maximum rainfall, m is the order of a given maximum rainfall value, and n is the total number of maximum rainfall values (n is 49 in this study).

Gumbel extreme value PDF

The GEV PDF is commonly used for extreme value analysis of hydrologic and meteorological data, such as maximum floods and maximum rainfalls (Sabarish et al. 2017). Chow (1951) demonstrated the most commonly used PDF in hydrological investigations expressed as

$${X}_{T}=\overline{X }+K\sigma$$
(3)

where XT is the value of the X variate in a given random maximum rainfall series of T, \(\overline{X }\), \(\sigma\), and \(K\) are the mean, standard deviation, and frequency factor of the X variate, respectively. But Eq. (3) is only applicable when the data length is infinite (n tend to ∞) which is practically impossible. Equation (3) is slightly modified to consider finite data length as

$${X}_{T}=\overline{X }+K{\sigma }_{n-1}$$
(4)

where \({\sigma }_{n-1}\) is the standard deviation of finite data length which is expressed as

$${\sigma }_{n-1}=\sqrt{\frac{\sum {(X-\overline{X })}^{2}}{n-1}}$$
(5)

The frequency factor K is represented as

$$K=\frac{{Y}_{T}-\overline{{Y }_{n}}}{{S}_{n}}$$
(6)

where \(\overline{{Y }_{n}}\) and \({S}_{n}\) are reduced mean and standard deviation, respectively, whose values depend on data length n. YT is the reduced variate which depends on T as

$${Y}_{T}=-\left\{ln.\mathrm{ln}(\frac{T}{T-1})\right\}$$
(7)

In this study, the values of \(\overline{{Y }_{n}}\) = 0.5481 and \({S}_{n}\) = 1.1590 are chosen from tables given by Subramanya (2009) corresponding to data length n = 49.

Log-Pearson type III PDF

The LP-III distribution is utilized for maximum rainfall frequency analysis. In this method, the maximum rainfall (X variate) magnitude selected from each year is firstly transformed into logarithmic form Z, and then this log-transformed data is analyzed for probability analysis.

$$Z=\mathrm{log} x$$
(8)

The maximum rainfall X variate is transformed into Z data series using Eq. (8). Now, Eq. (3) can be expressed as

$${Z}_{T}=\overline{Z }+K{\sigma }_{z-1}$$
(9)

where K = f (T, coefficient of skewness (Cz)), and \(\overline{Z }\) and \({\sigma }_{z-1}\) are the mean and standard deviation of the transformed finite maximum rainfall data. In order to estimate ZT at a given T, the values of \(\overline{Z }\), \({\sigma }_{z-1}\), and Cz were determined using the following formulas:

$$\overline{Z }=\frac{\sum {Z}_{i}}{n}$$
(10)

where n is the rainfall data length (years),

$${\sigma }_{z-1}=\sqrt{\frac{\sum {(Z-\overline{Z })}^{2}}{n-1}}$$
(11)
$${C}_{z}=\frac{n\sum {(Z-\overline{Z })}^{3}}{(n-1)(n-2){({\sigma }_{z-1})}^{3}}$$
(12)

The value of frequency factor K was selected from the theoretical table given by Subramanya (2009) corresponding to Cz and T, in this study. Finally, the value of variate X corresponding to different durations of T is determined as

$${X}_{T}=antilog {Z}_{T}$$
(13)

Pearson type III PDF

This PDF is also used for frequency analysis of rainfall. The maximum rainfall magnitude XT with respect to T is estimated as

$${X}_{T}=\overline{X }+K{\sigma }_{n-1}$$
(14)

where K = f (T, Cx). In order to estimate XT at a given T, the values of \(\overline{X }\), \({\sigma }_{n-1}\), and Cx were determined using the following formulas:

$$\overline{X }=\frac{\sum {X}_{i}}{n}$$
(15)
$${\sigma }_{n-1}=\sqrt{\frac{\sum {(X-\overline{X })}^{2}}{n-1}}$$
(16)
$${C}_{x}=\frac{n\sum {(X-\overline{X })}^{3}}{(n-1)(n-2){({\sigma }_{n-1})}^{3}}$$
(17)

In this study, the values of frequency factor K were selected from the theoretical table given by Subramanya (2009) by considering Cz and T.

Log-normal PDF

When the coefficient of skewness Cz becomes zero, LP-III distribution transforms into L-NM distribution. Here, the logarithm of random variable X is normally distributed. In this study, K values were selected from the theoretical table given by Subramanya (2009) by considering Cz = 0 at different durations of T. Finally, the maximum rainfall magnitude (XT) was estimated as

$${X}_{T}=antilog {Z}_{T}$$
(18)

Normal PDF

Chow (1951) demonstrated that the NM distribution is also used in hydrological investigations. The maximum rainfall magnitude XT with respect to T is estimated as

$${X}_{T}=\overline{X }+K{\sigma }_{x}$$
(19)

Here, frequency factor K = Z, where, Z is the standard normal variable which can be estimated as

$$Z=d-\frac{2.515517+0.802853\mathrm{ d }+ 0.010328{d}^{2} }{1 + 1.432788\mathrm{ d }+ 0.189269 {d}^{2}+ 0.001308 {d}^{3}}$$
(20)

where d is the intermediate variable, it can be calculated as

$$d=\sqrt{\mathrm{ln}\frac{1}{{P}^{2}}(0 <P\le 0.5)}$$
(21)

where P (P = 1/T) is the exceedance probability. Pegu and Malik (2015) and Amin et al. (2016) used Eq. (20) and Eq. (21) for estimation of XT in the frequency factor method.

Goodness-of-fit test

Chi-square test

The chi-square test is a popular method for fitting empirical data to various PDFs (Haan 1994). It is mostly used for testing the goodness of fit of different PDFs. This test is utilized for the assessment of the goodness of fit between observed and fitted distribution hydrologic extremes. The chi-square value will be estimated corresponding to different PDFs for different durations of T values. The chi-square value (χ2) can be estimated as

$${\chi }^{2}=\sum \frac{{({X}_{o}-{X}_{E})}^{2}}{{X}_{E}}$$
(22)

where Xo is the observed rainfall and XE is rainfall estimated with different PDFs. A PDF is said to be the best-fitting when the sum of all χ2 values is least (Sabarish et al. 2017). Generally, the PDF with the least χ2 value is being considered the best-fitting PDF for a given data set. When χ2 is zero, it indicates observed rainfall (Xo) and estimated rainfall (XE) matches exactly. In this study, χ2 values were estimated for different PDFs corresponding to different durations of T values.

Results and discussion

In this study, 49 years (1971 to 2019) daily rainfall data were used for probability analysis. The variation of annual rainfall from the year 1971 to 2019 is shown in Fig. 2. Here, it was observed that annual rainfall varies from 1711 to 518 mm for the study area. Maximum monthly rainfall magnitudes corresponding to different 49 years were also evaluated as shown in Fig. 3. Here, it was found that maximum monthly rainfall varies from 845 to 150 mm from the year 1971 to 2019. The magnitudes of maximum rainfall of 1, 2, 3, 4, and 5 days were assessed from the year 1971 to 2019 as shown in Fig. 4. Here, it was observed that maximum 1-day rainfall varies from 243 to 47 mm over a 49-year duration. Similarly, 2, 3, 4, and 5 days of maximum rainfall varies from 421 to 58 mm, 462 to 67 mm, 480 to 69 mm, and 482 to 83 mm, respectively.

Fig. 2
figure 2

Annual rainfall variation from the year 1971 to 2019 for the study area

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Fig. 3
figure 3

Maximum monthly rainfall variation from the year 1971 to 2019 for the study area

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Fig. 4
figure 4

Variation of 1, 2, 3, 4, and 5 days maximum rainfall from the year 1971 to 2019 for the study area

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The maximum rainfall magnitudes at 1, 2, 3, 4, and 5 days corresponding to a duration of 2, 5, 10, 25, and 50 years of T were estimated using the WPP formula which is presented in Table 1. The variation of 1-, 2-, 3-, 4-, and 5-day maximum rainfall magnitudes with respect to T is also shown in Fig. 5. The maximum rainfall magnitudes for 1, 2, 3, 4, and 5 days corresponding to a 50-year duration of T were found to be 243, 421, 462, 480, and 482 mm, respectively, as per the WPP formula. Similarly, for the 25-year duration of T, 1-, 2-, 3-, 4-, and 5-day maximum rainfall values were found to be 235, 388, 421, 439, and 441 mm, respectively. In the case of the 10-year duration of T, 1-, 2-, 3-, 4-, and 5-day maximum rainfall values were observed to be 165, 223, 273, 278, and 293 mm, respectively. On the basis of the 5-year duration of T, 1-, 2-, 3-, 4-, and 5-day maximum rainfall values were observed to be 137, 167, 201, 208, and 257 mm, respectively. For the 2-year duration of T, 1-, 2-, 3-, 4-, and 5-day maximum rainfall values were estimated to be 90, 120, 123, 138, and 159 mm, respectively. Here, it was revealed that the difference in maximum rainfall magnitude in between the first 2 days (i.e., 1 and consecutive 2 days) is more than the last 2 days (i.e., consecutive 4 and 5 days) for a given duration of T. The maximum rainfall values of the 4- and 5-day time scales were found to be more close to each other. Here, it is found that the maximum rainfall magnitude significantly increases from the 2- to 50-year duration of T corresponding to 1, 2, 3, 4, and 5 days of maximum rainfall. The difference in maximum rainfall between the 2- to 5-year duration of T is found to be more for all considered 1, 2, 3, 4, and 5 consecutive days.

Table 1 Observed values of maximum rainfall (mm) for 1, 2, 3, 4, and 5 days time scale as per WPP formula
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Fig. 5
figure 5

Observed 1, 2, 3, 4, and 5 days of maximum rainfall at different durations of T as per WPP formula

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The difference in maximum rainfall between the 25- and 5-year (X25 − X5) duration of T was found to be 98, 221, 220, 231, and 184 mm for the 1-, 2-, 3-, 4-, and 5-day time scale, respectively. Similarly, the difference in maximum rainfall between the 50- and 25-year (X50 − X25) duration of T was observed to be 8, 33, 41, 41, and 41 mm for the 1-, 2-, 3-, 4-, and 5-day time scale, respectively. Here, it is observed that the difference in maximum rainfall (X25 − X5) between 25 and 5 years is more than the difference in maximum rainfall (X50 − X25) between 50 and 25 years. This result is similar to the finding of Sabarish et al. (2017). It has revealed that hydraulic structures designed with the 5-year duration of T and 25-year duration of T can have significant differences in their component dimensions. Due to the relatively small difference in maximum rainfall magnitude between the 25- and 50-year duration of T, the hydraulic component’s dimensions do not change significantly when the design is based on either 25- or 50-year duration of T. This finding is similar to that of Sabarish et al. (2017) who stated that hydraulic design based on either 50 or 100 years of T does not have a sizable difference in the hydraulic component dimensions.

In this study, observed annual rainfall magnitudes at different durations of T were estimated using the WPP formula as presented in Table 2 and also shown in Fig. 6. Annual rainfall magnitudes corresponding to the 2-, 5-, 10-, 25-, and 50-year duration of T were found to be 849, 1122, 1452, 1565, and 1711 mm, respectively. The observed rainfall curve between the 2- and 10-year duration of T is steeper than the rest of the curve as shown in Fig. 6. In order to estimate the predictive potential of different PDFs, five different PDFs were used as presented in Table 2. The best-fitting PDFs were selected based on the minimum sum of chi-square values. The estimated annual rainfall and chi-square values of different PDFs corresponding to different duration of T are presented in Table 2. The sum of chi-square values of GEV, LP-III, P-III, L-NM, and NM PDF was observed to be 11.91, 18.22, 18.59, 32.96, and 62.11 mm, respectively, for annual rainfall estimation. Hence, the GEV PDF has a minimum sum of chi-square values for the annual rainfall estimation of the study area. Here, it is found that GEV PDF performs better for annual rainfall estimation and can be used for extrapolation of annual rainfall values for the 100- and 200-year duration of T for the study area. The annual rainfall magnitude corresponding to the 100- and 200-year duration of T was observed to be 1943 and 2119 mm, respectively, for the study area which is estimated as per GEV PDF.

Table 2 Estimated annual rainfall and chi-square values of different PDFs at different durations of T
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Fig. 6
figure 6

Observed and estimated annual rainfall at different durations of T

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In this study, the observed maximum monthly rainfall magnitudes corresponding to the 2-, 5-, 10-, 25-, and 50-year duration of T were estimated using the WPP formula as shown in Fig. 7. The difference in the maximum monthly rainfall between the 50- and 25-, 25- to 10-, 10- to 5-, and 5- to 2-year duration of T was found to be 334, 40, 44, and 139 mm, respectively. Here, it was observed that the slope of the observed maximum monthly rainfall curve between the 5- and 25-year duration of T is flatter than the rest of the curve. From the 25-year duration of T onward, the amount of maximum monthly rainfall increases dramatically. The observed maximum monthly rainfall and chi-square values of different PDFs are presented in Table 3. The potential of accurate estimation of maximum monthly rainfall by different PDFs was assessed from the chi-square values. The PDF is said to be the best when the sum of the chi-square value for all considered duration of T is less. In this study, the sum of chi-square values for GEV, LP-III, P-III, L-NM, and NM distribution for maximum monthly rainfall estimation was found to be 54.22, 57.39, 58.48, 70.56, and 123.7 mm, respectively. Hence, GEV PDF estimated maximum monthly rainfall at different durations of T with a minimum sum of chi-square value and can be applied for extrapolation of maximum monthly rainfall magnitudes for extrapolation at higher durations of T such as 100 and 200 years. As per GEV PDF, the magnitude of maximum monthly rainfall for the 100- and 200-year duration of T for the study area was found to be 768 and 846 mm, respectively.

Fig. 7
figure 7

Observed and estimated maximum monthly rainfall at different durations of T

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Table 3 Estimated maximum monthly rainfall and chi-square values of different PDFs at different durations of T
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Different PDFs were applied for the estimation of 1, 2, 3, 4, and 5 days of maximum rainfall at different durations of T. The estimated 1, 2, 3, 4, and 5 days of maximum rainfall by using different PDFs and corresponding observed rainfall are shown in Figs. 8, 9, 10, 11, and 12, respectively, at different time scales. One-day maximum rainfall values estimated as per different PDFs and corresponding chi-square values for the 2-, 5-, 10-, 25-, and 50-year duration of T are presented in Table 4. The sum of chi-square values for GEV, LP-III, P-III, L-NM, and NM PDF for 1-day maximum rainfall estimation was found to be 3.32, 3.65, 5.57, 8.99, and 23.58 mm, respectively. Here, the GEV PDF performed the best because it has a minimum sum of chi-square value than other PDFs. However, GEV PDF has slightly over-estimated 1-day maximum rainfall values for the 2-, 5-, and 10-year duration of T and slightly under-estimated 1-day maximum rainfall values for the 25- and 50-year duration of T. After comparing the predictive potential of all PDFs, it was found that the GEV PDF performs the best for estimating 1-day maximum rainfall and can be applied for extrapolation of 1-day maximum rainfall for higher durations of T. As per GEV distribution, the magnitude of 1-day maximum rainfall at the 100- and 200-year duration of T was found to be 270 and 299 mm, respectively in the study area.

Fig. 8
figure 8

Observed and estimated 1-day maximum rainfall at different durations of T

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Fig. 9
figure 9

Observed and estimated 2-day maximum rainfall at different durations of T

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Fig. 10
figure 10

Observed and estimated 3-day maximum rainfall at different durations of T

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Fig. 11
figure 11

Observed and estimated 4 days maximum rainfall at different durations of T

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Fig. 12
figure 12

Observed and estimated 5-day maximum rainfall at different durations of T

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Table 4 Estimated 1-day maximum rainfall and chi-square values of different PDFs at different durations of T
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For the 2-day maximum rainfall estimation, the estimated rainfall and chi-square values observed at different PDFs are presented in Table 5. The sum of chi-square values for estimation of 2 days of maximum rainfall for GEV, LP-III, P-III, L-NM, and NM PDF was found to be 41.64, 37.34, 37.63, 93.25, and 120.9 mm, respectively. LP-III PDF estimated 2-day maximum rainfall values more accurately than others. Hence, the performance of the LP-III PDF was found to be the best followed by P-III, GEV, L-NM, and NM PDF, respectively. However, LP-III PDF slightly under-estimated 2-day maximum rainfall values for T of 2, 25, and 50 years and slightly over-estimated 2-day maximum rainfall values for T of 5 and 10 years. All the PDFs performed worst for estimating 2-day maximum rainfall magnitude at the 25-year duration of T. As per LP-III PDF, the magnitude of 2 days of maximum rainfall for the duration of T of 100 and 200 years was found to be 437 and 522 mm, respectively, in the study area.

Table 5 Estimated 2-day maximum rainfall and chi-square values of different PDFs at different durations of T
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The estimated 3-day maximum rainfall and chi-square values of different PDFs corresponding to T of 2, 5, 10, 25, and 50 years are presented in Table 6. The sum of chi-square values for estimation of 3-day maximum rainfall for GEV, LP-III, P-III, L-NM, and NM PDF was observed to be 32.19, 29.33, 31.11, 92.4, and 115.4 mm, respectively. The best-fitting PDF for 3-day maximum rainfall estimation was selected on the basis of the minimum sum of the chi-square value. Here, LP-III PDF performed the best followed by P-III, GEV, L-NM, and NM distribution, respectively, for 3-day maximum rainfall estimation. The chi-square values corresponding to different PDFs for a duration of T of 25 years were found to be relatively more than other durations of T. In this study, it was observed that LP-III can estimate 3-day maximum rainfall magnitudes more accurately and can be utilized to extrapolate the 3-day maximum rainfall values for longer period of T. As per LP-III PDF, the magnitude of 3-day maximum rainfall for durations of T of 100 and 200 years was found to be 503 and 608 mm, respectively, for the study area.

Table 6 Estimated 3-day maximum rainfall and chi-square values of different PDFs at different durations of T
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The estimated 4-day maximum rainfall and chi-square values of different PDFs at different durations of T are presented in Table 7. The sum of chi-square values for estimation of the 4-day maximum rainfall using GEV, LP-III, P-III, L-NM, and NM PDFs was observed to be 30.68, 26.95, 29.70, 79.39, and 109.9 mm, respectively. Here, LP-III PDF performed the best followed by P-III, GEV, L-NM, and NM PDF, respectively, for the 4-day maximum rainfall estimation. However, LP-III PDF slightly over-estimated the rainfall magnitudes for durations of T of 2 and 5 years and slightly under-estimated the rainfall magnitudes for durations of T of 10, 25 and 50 years. Here, LP-III PDF was employed to estimate the 4-day maximum rainfall magnitudes at durations of T of 100 and 200 years for extrapolation purposes. As per LP-III PDF, the magnitude of 4-day maximum rainfall for durations of T of 100 and 200 years was found to be 515 and 614 mm, respectively, in the study area.

Table 7 Estimated 4-day maximum rainfall and chi-square values of different PDFs at different durations of T
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Similarly, 5-day maximum rainfall and chi-square values for different PDFs are presented in Table 8. The sum of chi-square values of GEV, LP-III, P-III, L-NM, and NM distribution for estimation of 5-day maximum rainfall was observed to be 16.47, 18.40, 22.68, 50.41, and 78.5 mm, respectively. On the basis of the minimum sum of chi-square value, GEV PDF performed the best followed by LP-III, P-III, L-NM, and NM PDF, respectively, for the 5-day maximum rainfall estimation. Here, it is observed that GEV PDF is more capable to capture the observed 5-day maximum rainfall values for all durations of T which is considered, and hence, it can be applied to extrapolate the 5-day maximum rainfall values at the longer duration of T. As per GEV PDF, the magnitude of 5-day maximum rainfall for the 100- and 200-year duration of T was found to be 486 and 538 mm, respectively, in the study area.

Table 8 Estimated 5-day maximum rainfall and chi-square values of different PDFs at different durations of T
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Ferdows and Hossain (2005) applied L-NM, GEV, and LP-III PDFs for annual maximum runoff estimation of main rivers in Bangladesh and found that LP-III PDF gives better results than others. Izinyon and Ajumuka (2013) applied L-NM, LP-III, and GEV PDFs at three different flow gauging stations, namely, Serav, Garsol, and Mayokam located in Nigeria and observed that L-NM, L-NM, and LP-III performs the best at Serav, Garsol, and Mayokam stations, respectively. Asim and Nath (2015) applied L-NM and GEV PDFs for the annual maximum rainfall estimation of the Allahabad station of India and concluded that GEV PDF performed better than L-NM. Kumar and Bhardwaj (2015) employed L-NM, GEV, and LP-III PDFs for daily maximum rainfall estimation at Ludhiana station of Punjab and revealed that LP-III PDF is the best fit for the annual 1-day maximum rainfall estimation. Pegu and Malik (2015) predicted maximum monthly rainfalls at different durations of T for the Dhemaji region in Assam, India, and found that L-NM PDF performed the best for maximum monthly rainfall prediction. Sabarish et al. (2017) proposed that P-III PDF performed the best for 1-day maximum rainfall estimation, and LP-III PDF performed the best for the 2-, 3-, 4-, 5-day maximum rainfall estimation at Tiruchirappalli City in the Cauvery River basin, South India. Kaur et al. (2021) proposed that GEV PDF performed the best for the 1-, 2-, and 3-day maximum rainfall estimation for Roorkee, Uttarakhand, North India. Hence, best-fitting PDFs are different at different climatic conditions. Therefore, it becomes necessary to find out the best-fitting PDF for the study area which is under investigation. In this study, GEV PDF was found to be the best for annual, maximum monthly, and 1- and 5 day-maximum rainfall estimation while LP-III PDF was found to be the best for the 2-, 3-, and 4-day maximum rainfall estimation.

The main objectives of this study are to estimate maximum rainfall magnitudes at 2, 5, 10, 25, and 50 years of T at different time scales and to select the best PDF for extrapolation of maximum rainfall values for the 100- and 200-year duration of T at different time scales. The maximum rainfall magnitude of the higher duration of T is crucial for the design of important hydraulic structures to prevent the failure of structures in extreme conditions. The selection of duration of T and consecutive day maximum rainfall receives an important role in the design of drainage systems for any area (Dabral et al. 2016). When the design is based on rainfall magnitudes of minimum years’ duration of T, then it will be risky because the designed dimensions of the drainage system will not be able to drain off the entire drainage discharge of the drainage area. Hence, the maximum years of T duration should be taken into consideration to prevent waterlogging in the area. The selection of appropriate T duration and consecutive days of maximum rainfall is crucial in the design of an appropriate drainage system. The results obtained in this study surely assist in the design of an appropriate drainage system that will prevent flooding in the study area. The information regarding maximum rainfall magnitudes for different durations of T will surely assist in the design of different hydraulic structures across the river flows in the study area.

Conclusion

Analysis of maximum rainfall plays an important role in the field of hydrology, hydraulic structure design, and risk analysis. In this study, observed annual, maximum monthly, and 1, 2, 3, 4, and 5 consecutive days of maximum rainfall magnitudes corresponding to the 2-, 5-, 10-, 25-, and 50-year duration of T were estimated using the WPP formula. Different PDFs, namely, GEV, LP-III, P-III, L-NM, and NM, were employed for estimation of annual, maximum monthly, and 1, 2, 3, 4, and 5 days of maximum rainfall. The observed rainfall values were compared with rainfall values estimated by different PDFs. The best-fitting PDF was selected based on the minimum chi-square value. In this study, GEV PDF performed the best for annual, maximum monthly, and 1- and 5-day maximum rainfall estimation of the study area. On the other hand, LP-III PDF performed the best for the 2-, 3-, and 4-day maximum rainfall estimation of the study area. The selected best-fitting PDFs were used for extrapolation of maximum rainfall magnitudes for longer durations of T like 100 and 200 years as the design of important hydraulic structures (water storage dams, bridges, tanks, etc.) needs consideration of longer durations of T. As per best-fitting PDFs, the estimated annual, maximum monthly, and 1, 2, 3, 4, and 5 consecutive days of maximum rainfall magnitudes for the 100- and 200-year duration of T were found to be 1943, 768, 270, 437, 503, 515, and 486 mm and 2119, 846, 299, 522, 608, 614, and 538 mm, respectively. This study will surely assist design engineers and planners in the design of appropriate drainage systems and hydraulic structures (dams, nala bunds, ponds, bridges, hydropower plants) in the study area or similar climatic conditions. In order to operate and manage different hydraulic structures, maximum rainfall magnitudes corresponding to different durations of T as per the selected PDF must be considered. The results obtained in this study can be utilized for planning irrigation water management and agricultural activities. This study will be helpful in the prediction of flood and drought magnitudes and disaster management. Future studies should be focused to estimate the predictive performance of different PDFs at different time scales in different climatic conditions.