1 Introduction

With the development of global socioeconomy over the past decades, human-induced greenhouse gas emissions has changed the composition of the atmosphere and caused global warming, which has greatly affected the extreme precipitation and temperature events, especially for floods and droughts (Sarhadi and Soulis 2017; Diffenbaugh et al. 2017; Boukhelifa et al. 2018; Tong et al. 2020). According to the IPCC fifth assessment report, the frequency of heavy precipitation events has increased in most land areas (IPCC 2013; Wang and Tian 2010). The increasing frequency of extreme events brings severe challenges to regional flood control safety and engineering design (Song et al. 2018). Simultaneously, the occurrence of extreme rainfall events has caused considerable damage to human life and property, the most serious of which is urban waterlogging (Duan et al. 2014). However, these extreme events show high spatiotemporal heterogeneity (Du et al. 2014; Soltani et al. 2016; Ntegeka and Willems 2008; McAfee et al. 2013; Panthou et al. 2014). For instance, a significant increase of the annual maximum 1-day rainfall is found in the global scope (Min et al. 2011; Westra et al. 2013). Extreme precipitation and the average intensity of extreme precipitation are increasing in China (Wang and Tian 2010; Wu et al. 2018; Ge et al. 2019). Jung et al. (2011) studied the rainfall data from 183 meteorological stations in South Korea and found that the annual precipitation data showed a significant increasing trend and the extreme rainfall series had strong spatial variability. Investigating extreme events in India, Guhathakurta et al. (2011) noted that heavy rainfall occurred less frequently in most parts of central and northern India while more frequently in eastern and northeastern India, with extreme precipitation and flood risks in the country increasing dramatically. Understanding the temporal and spatial variation of extreme rainfall plays a vital role in formulating effective and appropriate water resources management policies, especially in mitigating and preventing flood disasters (Jung et al. 2017; Fan et al. 2018; Wu et al. 2020).

How to represent the extreme rainfall events has attracted much attention. Most previous studies take the annual maximum daily rainfall or the maximum threshold method (Peak-Over-Threshold (POT)) to study extreme rainfall (Ntegeka and Willems 2008; Song et al. 2018). Some scholars argued that the variety of sub-daily extreme precipitation may be greater than that of the daily extreme under anthropogenic forcing (Olsson et al. 2015; Hosseinzadehtalaei et al. 2019). However, due to data limitation, it is difficult to quantify the change of extreme precipitation in sub-daily precipitation, especially on a large-scale basin (Hosseinzadehtalaei et al. 2020). In this study, the annual maximum 1-day, 3-day, 7-day, and 15-day rainfall are selected as evaluation indicators to analyze the changes in extreme events. Since these indicators reflect the frequent impact of extreme events on society, which in turn are used to predict rainfall in different return periods (Min et al. 2011).

Statistical methods can be used to predict the rainfall in different return periods so as to provide a reference for flood prevention in the future. The most commonly used probability distribution models are Gumbel, P-III, generalized extreme value (GEV), and generalized Pareto (GP) (Fischer et al. 2012; Xia et al. 2012; Coronado-Hern A Ndez et al. 2020; Samuel et al. 2020). For instance, Samuel et al. (2020) reported that Kaduna’s monthly rainfall data are fitted to a generalized extreme value distribution. Mo et al. (2019) believed that the parameters of the GP and GEV models are variable in a climate change environment, and the fitting results of the GP model are better than the GEV model. Park et al. (2011) used Gumbel distribution and GEV distribution to study the variation of rainfall extreme values in South Korea. Chaudhuri and Sharma (2020) indicated that compared with Gumbel, GEV predicts the higher intensity of extreme rainfall with different return periods and duration in Delhi, India. The extreme rainfall distribution functions widely used in China in recent decades were Gamma and P-III type distribution functions (Hanson and Vogel 2008; Fischer et al. 2012; Xia et al. 2012). Wang et al. (2008) used Gamma distribution and K-S test to detect the variation of extreme precipitation in southern China. Xia et al. (2012) used GEV, Gamma, and GP models to analyze the extreme precipitation in the HRB and found that the annual maximum series can be better applied to GEV model, and the POT series can be better applied to GP model.

The North Anhui Province Plain (NAHPP) is located in middle reaches of the Huaihe River Basin (HRB), belonging to the north–south transitional zone in China. The HRB is a sensitive area to climate change (Ye and Li 2017) and a prone area to floods. In the twentieth century, the probability of flood in the HRB ranked the second in China (Duan et al. 2014). Its unique geographical location and frequent floods exert negative effects on crop growth in Anhui, which is a major agricultural province in China (Wei and Zhang 2010). Yang et al. (2016) and Ye and Li (2017) reported that the leading cause of the frequent floods in the HRB is the increase of extreme rainfall events or continuous heavy rainfall events. However, research on extreme rainfall in the HRB remains limited, and few scholars considered the variation and a probability distribution of extreme precipitation in the NAHPP (Xia et al. 2012; Yin et al. 2016; Ye and Li 2017). Located in the middle of the HRB, the NAHPP is the most vulnerable area being frequently hit by flood disasters induced by extreme rainfall. Taking the serious flood event in 2020 as an example, flood in the Huaihe River needs to be released to the NAHPP to alleviate the flood control pressure. As a result, flood in the NAHPP caused numerous economic losses, a large number of fields submerged and agricultural production reduced.

This paper aims to analyze the spatial and temporal variation of extreme precipitation in the NAHPP and study the statistical characteristics of extreme precipitation. The main objectives of this study are as follows: (1) to identify the oscillation period of rainfall at different time scales for the study area; (2) to investigate the spatial distribution and variation trend of different rainfall extremes over the study area; and (3) to estimate the return period of the rainfall extremes by using the optimal fitting distribution function. The findings would provide scientific support for flood control and disaster mitigation for the NAHPP.

2 Materials and methods

2.1 Study area and data sources

The NAHPP (Fig. 1) lies in the north of Anhui Province (114°55′ ~ 118°10′ E, 32°25′ ~ 34°35′ N), located in the middle reach of the HRB. It encompasses 6 cities and 27 counties (districts) of Bengbu, Huainan, Fuyang, Bozhou, Huaibei, and Suzhou. It covers about 3.9 × 104 km2, accounting for about 30% of the total area of Anhui Province. The terrain is flat, except for a few hills in the north. The altitude ranges from 20 to 40 m, and the natural gradient is 1/7500 ~ 1/10,000. The region belongs to the warm temperate zone and the semi-humid monsoon zone, with dry winters and dry springs, and the same period of rain and heat. It has the transitional nature of the temperate and subtropical climate, with frequent meteorological disasters. Rainfall in the flood season (June to September), mostly in the form of heavy rain, accounts for 60–70% of the annual rainfall (Chen et al. 2018).

Fig. 1
figure 1

Location of rainfall gauges and segment of the study area

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Rainfall data at 61 rainfall stations from 1976 to 2018 in the NAHPP were collected from the local water conservancy bureaus. As shown in Fig. 1, the rainfall station distribution in the NAHPP is relatively uniform. To analyze the spatial heterogeneity of rainfall distribution in the region, the study area was divided into three regions by using the K-means cluster analysis (KCA) method (Amiri and Mesgari 2016). Region I covers 17 rainfall stations with the least rainfall, Region II includes 26 stations, and Region III contains 18 stations with the heaviest rainfall (Fig. 1).

2.2 Time series analysis

In this study, the ensemble empirical mode decomposition (EEMD) method was used to analyze the multi-scale rainfall so as to determine the oscillation mode structure characteristics of rainfall at different time scales. The EEMD is one of the methods to extract signals of changing trends and can assess the complex change of non-stationary signals (Wu and Huang 2009). It is an improvement in the empirical mode decomposition (EMD) method by adding an appropriate white noise to the original data and calculating the average of the set after many times (Huang et al. 2009). In this paper, the EEMD calculation process involved 1000 sets of samples, and the added amplitude of white noise was 20% of standard deviation for the synthetic sequence.

The Mann–Kendall (MK) trend test is a nonparametric test method that has been widely used in the trend test of hydrological and meteorological data series (Wang et al. 2017; Merabti et al. 2018; Dinpashoh et al. 2019). In this paper, it was used to analyze the variation trend of rainfall extreme values at various rainfall stations in the NAHPP. For a given time series {xi}, where i = 1, 2,…, n, the calculation is as follows:

$$ S = \sum\limits_{k = 1}^{n - 1} {\sum\limits_{i = k + 1}^{n} {{\text{sign}}(x_{i} - x_{k} )} } $$
(1)
$$ {\text{sign}}(x_{i} - x_{k} ) = \left\{ \begin{gathered} + 1\quad {\text{ if (}}x_{i} - x_{k} {\text{)}}\,{\text{ > }}\,{\text{0}} \hfill \\ {\text{0}}\quad \,\,\,\,{\text{ if (}}x_{i} - x_{k} {\text{)}}\,{\text{ = }}\,{\text{0}} \hfill \\ - 1\quad {\text{ if (}}x_{i} - x_{k} {\text{)}}\,{\text{ < }}\,{\text{0}} \hfill \\ \end{gathered} \right. $$
(2)

S statistics variance calculation method:

$$ {\text{Var}}(S) = \frac{n(n - 1)(2n + 5)}{{18}} $$
(3)

The standard normal statistical variable Z is:

$$ Z = \frac{{S - {\text{sign}}(S)}}{{\sqrt {{\text{Var}}(S)} }} $$
(4)

The trend is statistically significant at \(\alpha = 0.05\) significance level when |Z|> 1.96. When Z > 1.96, the sequence showed a significant increase, and when Z < − 1.96, the sequence showed a significant decrease.

2.3 Distribution function

In this paper, three distribution functions (Gumbel, P-III, and GEV distribution) were used to simulate and fit the rainfall extreme value series. The applicability of the three distribution functions to different extreme precipitation in the NAHPP was studied, and the optimal fitting function used to calculate the rainfall extremes distribution in different return periods.

2.3.1 Gumbel

The Gumbel distribution function is:

$$ F(x) = P(X < x) = \exp \left\{ { - \left[ {\exp \left( { - \frac{x - \beta }{\alpha }} \right)} \right]} \right\} $$
(5)

The parameters are estimated by the classical moment method, and the results are as follows: (Rahman et al. 2013; Qian et al. 2018):

$$ \alpha = \frac{{\mu_{x} }}{{\mu_{y} }} = \frac{\sqrt 6 }{\pi }\sigma_{x} $$
(6)
$$ \beta = E(x) - \alpha E(y) = E(x) - c*\frac{\pi }{\sqrt 6 }\sigma_{x} $$
(7)

\(\alpha\)(scale parameter) and \(\beta\)(position parameter) are unknown parameters, determined according to the measured data; c is the Euler constant, with a value of 0.5772;\(\sigma_{x}\) is the standard deviation of sequence x; E(x) is the mean value of sequence x.

2.3.2 P-III

The P-III curve is an unsymmetrical unimodal, positive deflection curve, and its distribution function is:

$$ F(x) = \frac{1}{\alpha \Gamma (\gamma )}\int\limits_{0}^{ + \infty } {\left( {\frac{x - \beta }{\alpha }} \right)}^{\gamma - 1} \exp \left( { - \left( {\frac{x - \beta }{\alpha }} \right)} \right){\text{d}}x $$
(8)

The lower formula can be obtained by the standardized transformation of the upper formula, and then, the rainfall can be calculated according to the design frequency (Jin et al. 2019):

$$ x_{p} = \overline{x}(1 + C_{v}^{2} \times C_{s} /C_{v} \times {\text{Gammainv}}\:(1 - p,\,\alpha ,\,\beta )/2 - 2C_{v} /C_{s} ) $$
(9)

where \(\gamma\)(shape parameter) is an unknown parameter; \(\Gamma (\gamma )\) is a Gamma function.

2.3.3 GEV

The GEV distribution function is:

$$ F(x) = \exp \left\{ { - \left[ {1 - \gamma \left( {\frac{x - \beta }{\alpha }} \right)} \right]^{{\frac{1}{\gamma }}} } \right\} $$
(10)

According to the above calculation, the rainfall in different return periods can be compared with the results of the empirical return period. For the calculation of the empirical return period, please refer to the Cunnane formula (Cunnane 1978; Song et al. 2018):

$$ T = \frac{1}{{\frac{n}{N}\left( {1 - \frac{i - 0.4}{{n + 0.2}}} \right)}} $$
(11)

where n is the sample number of extreme sequences and N is the total number of years of extreme sequence, i = 1, 2… n.

2.4 Goodness-of-fit tests

The Kolmogorov–Smirnov (K-S) test was used to test the goodness of fit of the distribution function, which reflects the nonparametric test method of the degree of deviation between the hypothesis and the real population distribution (Shu et al. 2017). It is mainly to compare the theoretical distribution and empirical distribution to determine the maximum difference value D between them. The smaller D value means the better the fitting result. The trend is statistically significant at \(\alpha { = 0}{\text{.05}}\) significance level when \(D < D_{\alpha } = 0.208\). If more than one distribution function satisfies this condition, the distribution with the smallest D value will be taken as the optimal fitting distribution. Some researchers use root mean square error (RMSE) to judge the goodness of fit (Qian et al. 2018), but the principle is basically the same.

3 Results and discussions

3.1 Temporal and spatial variation of annual rainfall

The annual rainfall in the NAHPP is unevenly distributed and gradually decreases from south to north. Based on the MK test, the annual rainfall variation of each subregion was calculated and illustrated in Fig. 2. It indicates that the annual rainfall of the three subregions has no significant increasing trend, which is consistent with the results of Jin et al. (2014).

Fig. 2
figure 2

Temporal evolution of annual rainfall anomaly in region I (a), region II (b) and region III (c). The red lines are the 3-year moving average curve, and the blue bars are the annual rainfall anomaly

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To obtain the characteristics of the annual rainfall variations on different time scales, the EEMD was applied to the annual rainfall anomaly of each subregion. The three original precipitation anomaly series are broken down into four IMF components and a trend term RES (Fig. 3). Each IMF component reflects the fluctuation characteristics at different time scales from high to low frequency, and the trend term shows the trend of the original data over time. The four IMF components in Fig. 3 were analyzed to obtain the oscillation period and the variance contribution rate (Table 1). The contribution of the oscillation period of 2.2–2.3 years to the rainfall anomaly was the largest (more than 58%). On the interannual scale, precipitation in Regions I and II showed 3-year (IMF1) and 6-year (IMF2) climate variability, and precipitation in Region III showed 3-year (IMF1) and 4-year (IMF2) climate variability. On a chronological scale, the three subregions had climate variability of 21 years (IMF3 and IMF4). The results are consistent with the Morlet wavelet analysis of Wei et al. (2010). The interdecadal oscillation of summer precipitation in the study region was closely related to the interdecadal oscillation of the Pacific Ocean and the East Asian summer monsoon (Wei et al. 2010).

Fig. 3
figure 3

Decomposition of annual rainfall anomaly time series in three subregions using EEMD

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Table 1 Major cycles and variance contribution rates of each component of annual rainfall anomaly series in the NAHPP from 1978 to 2018
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3.2 Variation of rainfall extremes series

The 43-year extreme rainfall series (annual maximum 1-day, annual maximum 3-day, annual maximum 7-day, and annual maximum 15-day) at 61 rainfall stations in the NAHPP were calculated. The statistical characteristics of the maximum 1-day rainfall and the maximum 7-day rainfall in different stations and the statistical characteristics of the maximum 1-day rainfall and the maximum 7-day rainfall in different years are shown in Fig. 4. The number of rainfall stations is displayed in Table 2. The maximum daily rainfall of 50–100 mm in 32 stations is at rainstorm level, and the maximum daily rainfall of more than 100 mm in 29 stations is at rainstorm level. The average value of maximum 1-day rainfall and the average value of maximum 7-day rainfall fluctuated little at the different stations (Fig. 4a, b), and the difference between the average value and the maximum value is large. In general, the variation coefficient (CV) of the two extreme value series is between 0.3 and 0.5, but the CV value of the maximum 7-day rainfall at the same station is slightly higher than that of the maximum 1-day rainfall. It can also be found from the anomaly points that the maximum 7-day rainfall has many abnormal points. The degree of dispersion is large. The results show that the distribution of the extreme value of rainfall is not uniform at a different time at the same station. In different years, the extreme value of rainfall varies greatly (Fig. 4c, d), as does the mean value. Moreover, the maximum 1-day rainfall and the maximum 7-day rainfall CV easily appear at the extreme point. The spatial distribution is quite different at this extreme point.

Fig. 4
figure 4

Boxplot of rainfall extreme values at different stations a, b and in different years c, d

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Table 2 K-S test results of maximum 1-day rainfall at different rainfall stations
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In spatial distribution, the maximum 1-day rainfall CV = 0.06 and the maximum 7-day rainfall CV = 0.06 in the NAHPP. In temporal distribution, the maximum 1-day rainfall CV = 0.2 and the maximum 7-day rainfall CV = 0.3 in the NAHPP. According to the classification standard of CV (Wu et al. 2011), the rainfall extremes show weak variation (CV < 0.1) in spatial distribution, and the rainfall extreme value shows a medium variation (0.1 < CV < 1) in time distribution, which indicates that the temporal variation of rainfall extreme value in the NAHPP is higher than the spatial variation. The results are consistent with the variation range of the mean in Fig. 2. In all kinds of extreme hydrometeorology, spatial and temporal changes are very complex (Jung et al. 2017). The NAHPP is located in a transitional zone between the north and the south, and the influence of climate on rainfall is more complex.

In order to study the influence of geographical location on extreme rainfall (Fig. 5), the relationship between the extreme rainfall and latitude/longitude was analyzed by regression analysis. As displayed in Fig. 5a, b, the maximum 1-day rainfall correlated positively with longitude (R2 = 0.17), while the maximum 1-day rainfall was not correlated with latitude. As a result, in Fig. 5c, d, there was a significant positive correlation between the maximum 7-day rainfall and longitude (R2 = 0.19), and a very weak negative correlation between the maximum 7-day rainfall and latitude (R2 = 0.02). Thus, the extreme value of rainfall in the NAHPP increases with longitude and had little correlation with latitude. This conclusion is consistent with the shape characteristics of the boundary of the NAHPP (east–west long, north–south short). Longitude has great influence on the extreme value of rainfall. It is further proved that the spatial distribution of rainfall extremes in this area is not uniform.

Fig. 5
figure 5

Linear relationship between rainfall extreme value and longitude/latitude

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The MK trend test method was used to analyze the variation trend of rainfall extreme values in the NAHPP during 1976 and 2018, and the analysis results are shown in Fig. 6. The date of annual maximum daily rainfall of each station in the past 43 years was also calculated. For the whole NAHPP, the MK test indicated an insignificant increasing trend in different rainfall extremes series. Taking the annual maximum daily rainfall and annual maximum 3-day rainfall as examples, the changing trend of a single station was analyzed. Among the 61 rainfall stations, the annual maximum daily rainfall showed increasing trend in 52 stations and decreasing trend in 9 stations, all of which showed no significant (\(\alpha { = 0}{\text{.05}}\)) change (Fig. 6). Furthermore, the maximum daily rainfall mainly occurred between 1990 and 2010. The maximum 3-day rainfall showed an increasing trend in 54 stations, among which three stations showed a significant increasing trend (Fangdianzi, Dadian, and Huazhuangji), and seven stations showed no significant decreasing trend. Interestingly, Youji, Xuxialou, Gukouzha, and Yonggu showed no significant decreasing trends in four different rainfall extremes series (Fig. 6).

Fig. 6
figure 6

MK test results for each rainfall station. a Maximum 1-day rainfall; b Maximum 3-day rainfall; c Maximum 7-day rainfall; d Maximum 15-day rainfall;

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The rainfall extremes in the NAHPP showed no significant increasing trend. Although there is an increasing trend of precipitation in the NAHPP, the change is insignificant. The maximum 1-day rainfall at most stations showed nonsignificant increase trend, and a few stations showed nonsignificant decrease trend (Fig. 6). This is consistent with the results of Xia et al. (2012). The average annual precipitation increase in the HRB is 1.2 mm/10a, but the number of precipitation days decreases, and the radiation decrease is 0.8/10d (Wang et al. 2016). It should be noted that individual rainfall is increasing, which increases the risk of flooding. In addition, Zhang et al. (2014) have studied the HRB and suggested that the areas with the highest daily rainfall were along the middle and upper reaches of the HRB. The number of extreme precipitation days and precipitation intensity has an increasing trend (Ye and Li 2017). Moreover, the high-risk areas for rainstorms and floods in the HRB include Funan County in the NAHPP. The strong precipitation anomaly in the HRB may be related to the East Asian summer monsoon and the unique circulation pattern in East Asia (Wei and Zhang 2010). More importantly, there is a positive correlation between flooding and temperature in the HRB, with a positive coefficient of 0.88, and temperatures are expected to increase even faster in the future (Yang et al. 2012). Consequently, the flood disaster risk in the NAHPP cannot be ignored.

3.3 Probability distribution functions of rainfall extremes

The Gumbel, P-III, and GEV distribution functions were used to fit the four types of rainfall extremes series (maximum 1-day, maximum 3-day, maximum 7-day, and maximum 15-day rainfall), and the K-S test evaluated the goodness of fit of the three distribution functions. The K-S statistical value of maximum 1-day rainfall is presented in Table 2. The D value fitted by P-III distribution at Maji station (D = 0.214) did not pass the significance level test (\(\alpha = 0.05\)), but all the other stations did (\(\alpha = 0.05\)). The best fitted distribution functions of extreme rainfall are presented in Table 3. It can be found that the P-III distribution function accounts for the largest proportion of the optimal fitting distribution function of extreme rainfall, accounting for 52–66% (the number of rainfall stations is between 32 and 40). So, in this study, P-III distribution function can well fit the maximum 1-day, 3-day, 7-day, and 15-day precipitation in the NAHPP (Tables 2, 3). Hanson and Vogel (2008) obtained the same results for maximum 1-day precipitation data analysis. Especially in the maximum 15-day rainfall, the effect of P-III distribution fitting is more prominent.

Table 3 Proportion (station numbers) of optimal fitting distribution function of extreme rainfall
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In terms of annual maximum daily rainfall, 36 stations conformed to the P-III distribution, accounting for 59% of the total number of stations, 20 stations conformed to the GEV distribution, accounting for 33% of the total number of stations, and nine stations conformed to the Gumbel distribution, accounting for 15% of the total number of stations. (The optimal distribution is repeatedly counted.) Compared with the P-III and GEV distribution functions, the Gumbel distribution is not suitable for simulating the maximum 1-day rainfall in the NAHPP. Yang et al. (2010) and Khudri and Sadia (2013) have revealed the appearance of several opposite themes. Yang et al. (2010) analyzed the annual rainfall of 1-, 3-, 5-, and 7-day in the Pearl River Basin and found that GEV distribution function is one of the best fitting distribution functions. Khudri and Sadia (2013) found that the GEV and four parameters generalized Gamma distribution can well fit the annual maximum rainfall data of 22 weather stations in Bangladesh. The reason for this difference may be closely related to the local climate characteristics (Valenzuela and Garreaud 2019). Therefore, estimating the distribution of extreme rainfall in different regions is of great significance to accurately assess extreme rainfall.

3.4 Return period of rainfall extremes

Through three distribution functions, the rainfall of each station in different return periods (5a, 10a, 20a, 50a, and 100a) was calculated, and the optimal distribution function of the rainfall extremes value was obtained. The results were compared with the empirical return periods of samples (Fig. 7). It is clear that the fitting results of theoretical return periods and empirical return periods obtained by the three distribution functions are good for both individual stations and the NAHPP. Among them, the fitting results of the P-III and GEV distributions are very close to the extreme values of rainfall and are close to the empirical return period fitting results (Fig. 7). However, the Gumbel distribution correlated poorly, which is consistent with the K-S test results. The fitting results of three distribution functions of maximum 7-day rainfall are the closest. But when the return period is 10a, the fitting results of the three distribution functions and the empirical return period are unideal. When the return periods are 50a and 100a, the calculated extreme values of rainfall are slightly larger than those in the empirical return periods, which help to avoid the risk of flooding.

Fig. 7
figure 7

Comparison of the theoretical and empirical return periods of Bengbu a and NAHPP b

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According to the results of the K-S test and the fitting of the theoretical and empirical return periods, the rainfall in different return periods was estimated by using the better fitting P-III distribution. Then, the inverse distance weighting method in ArcGIS was used for spatial interpolation, and the spatial distribution of rainfall extremes in different return periods in the NAHPP was obtained (Fig. 8). As a result, in Fig. 8, the return periods of different extreme values are consistent. When the return period is 50a, the maximum value of the maximum 1-day rainfall mainly occurs in the northern NAHPP. Only some stations have relatively large rainfall (Xiaoxian and Heliu stations), and the difference between stations can reach 140 mm (Fig. 8a). It can be seen that the spatial distribution of rainfall in the NAHPP is uneven. The maximum value of the maximum 1-day rainfall mainly occurs in the southwestern NAHPP (Fig. 8c). The maximum values of the maximum 7-day rainfall (Fig. 8e) and the maximum 15-day rainfall mainly occur in the central NAHPP (Fig. 8g). The migration path of extreme rainfall is: north-southwest-central-central. The trend of rainfall extremes with a 100a return period is similar to that with a 50a return period.

Fig. 8
figure 8

Distribution of rainfall extremes in different return periods in the NAHPP.a maximum 1-day rainfall, return period 50a; b maximum 1-day rainfall, return period 100a; c maximum 3-day rainfall, return period 50a; d maximum 3-day rainfall, return period 100a; e maximum 7-day rainfall, return period 50a; f maximum 7-day rainfall, return period 100a; g maximum 15-day rainfall, return period 50a; h maximum 15-day rainfall, return period 100a

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The temporal and spatial variability of extreme rainfall is very complex. Although much work has been done on the coupling of rainfall dynamics and statistical description, the results obtained are not accurate (Bougadis and Adamowski 2006). Therefore, a scaling model was proposed to test the scale variability of extreme rainfall time series (Bougadis and Adamowski 2006). Nevertheless, there is no consensus on which distribution function to use in extreme rainfall. At present, the most widely used distribution functions are the GEV, Gumbel, P-III, Gamma, and Weibull distributions. Papalexiou et al. (2013) reported the applicability of several probability distribution models in global extreme daily rainfall and pointed out that the GEV distribution can be used to estimate the frequency of extreme daily rainfall. One of the focuses of this paper is to determine the probability distribution function suitable for the NAHPP, and our results indicate that the P-III distribution function has the best adaptability, followed by the GEV distribution function (Table 2 and Fig. 7). Using the P-III distribution function, the rainfall extreme value distribution under different return periods was calculated. The maximum 1-day rainfall in the 2-year return period is mostly less than 100 mm, and only two stations have rainfall of more than 100 mm. The maximum 1-day rainfall in the 5-year return period, 10-year return period, 20-year return period, and 50-year return period is basically 100–250 mm, and there are more stations with rainfall of over 250 mm in the maximum 1-day rainfall of 100-year return period, reaching the level of torrential rain.

Rainfall extremes are highly related to crop yields. Previous studies indicate that if groundwater level drops to 0.5 m below ground surface in three days after a rain storm, the rainstorm event will have little effect on crop yield. Otherwise, crop production will probably decrease to a certain extent (Wang and Ye 2008). If the drainage time exceeds three days, the crop yield will fall. Therefore, studying the distribution of rainfall extreme values can help improve farmland drainage in this area.

4 Conclusions

Based on observed rainfall data at 61 rainfall stations in the NAHPP from 1978 to 2018, temporal change trends and spatial distributions of the annual maximum 1-day, 3-day, 7-day, and 15-day rainfall in the NAHPP were analyzed by using the KCA, EEMD, MK, and K-S methods. Furthermore, the probability distribution of rainfall extremes in the NAHPP was investigated by using the Gumbel, P-III, and GEV models. This study can be concluded as follows:

  1. (1)

    The contribution of the oscillation period of 2.2–2.3 years to the rainfall anomaly was the largest (more than 58%), and rainfall shows a 2–3-year periodicity on the interannual scale and 21-year periodicity on the chronological scale.

  2. (2)

    The rainfall extremes in the NAHPP showed a weak variation in spatial distribution, and the rainfall extreme value showed a medium variation in time distribution. The spatial distribution of rainfall extremes in this area was not uniform, and the rainfall extremes in the NAHPP increased with longitude and had little correlation with latitude. The rainfall extremes showed nonsignificant increase trend over the whole NAHPP, but some stations showed nonsignificant decrease trend.

  3. (3)

    The D value fitted by P-III distribution at Maji station (D = 0.214) did not pass the significance level test (\(\alpha = 0.05\)), but all other stations did (\(\alpha = 0.05\)). The P-III distribution function can fit the extreme value of rainfall (the maximum 1-day rainfall: 59%), followed by the GEV distribution function (the maximum 1-day rainfall: 33%). It is recommended to use the P-III distribution function to fit rainfall extremes in the study area.

  4. (4)

    As the return period increased, the rainfall estimates using the three distribution functions were slightly larger than that in the empirical return period. The findings can provide scientific support on flood control standard design and flood risk mitigation.