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A regional frequency analysis of precipitation extremes in Mainland China


INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. (2017) Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/joc.5013

A regional frequency analysis of precipitation extremes in Mainland China with fuzzy c-means and L-moments approaches
Zhaoli Wang,a,b Zhaoyang Zeng,a Chengguang Lai,a,c* Xiaohong Chenc,d
a b

Wenxin Lin,a Xushu Wua and

School of Civil Engineering and Transportation, South China University of Technology, Guangzhou, China State Key Lab of Subtropical Building Science, South China University of Technology, Guangzhou, China c Center for Water Resources and Environment, Sun Yat-sen University, Guangzhou, China d Guangdong Engineering Technology Research Center of Water Security Regulation and Control for Southern China, Guangzhou, China

ABSTRACT: Owing to their essential influence on human society and natural environment exerted by inducing disasters, such as floods and droughts, further studies on precipitation extremes in China are needed. This study presents the regional frequency and spatial-temporal patterns of precipitation extremes in China based on a high-resolution (0.5? × 0.5? ) daily precipitation dataset from 1961 to 2013. With fuzzy c-means, L-moments methods and other scientific statistical tests, a regional frequency analysis (RFA) is conducted, aiming to further understand the regional and spatial distribution of precipitation extremes across China. The results show that: (1) the whole Mainland China can be divided into 50 homogeneous regions on the basis of the characteristics of mean annual precipitation and location indices. (2) For most of the regions, Generalized Extreme Value (GEV), Generalized Normal (GNO) and Pearson type III (PE3) distributions of precipitation extremes fit well, according to the results of goodness-of-fit (GOF) test. (3) For RX1DAY, GEV has the best-fit distribution in the east, northeast and southwest of China, whereas GNO distribution mostly fits the northern and parts of southwest and southeast; in addition, regions which fit PE3 and Generalized Logistic (GLO) distribute dispersedly across the country. (4) For RX5DAY, GEV mainly fits in the middle, southwestern and southern; GNO and PE3 apply best to the northeastern and northern, respectively. (5) Return periods of 20, 50 and 100 years for their best-fit distributions decrease gradually from southeastern China to northwestern China. Compared with the results of GEV distribution fitted to each grid, RFA may provide more accurate estimates of rainfall quantiles. Definitely, the study results will not only benefit further understanding of the unique and complex features of extreme precipitation in the whole Mainland China but also contribute to the nation-scale flood prevention, control and management in the backdrop of the changing climate.
KEY WORDS

precipitation extremes; regional frequency analysis; L-moments method; fuzzy c-means; Generalized Extreme Value distribution; Mainland China

Received 19 December 2015; Revised 12 December 2016; Accepted 8 January 2017

1. Introduction Under the circumstance of climate change, higher frequencies of extreme weather have a significant influence on human society (Dore, 2005; Alexander et al., 2006; Min et al., 2011). As declared by the Intergovernmental Panel on Climate Change (IPCC), there has been an observable increase in both the amount and intensity of global precipitation extremes (IPCC, 2007, 2012). However, enhanced precipitation extremes could induce more catastrophes such as floods, droughts and landslides which can directly threaten human welfare and security (Yang et al., 2010b; Hallegatte et al., 2013). Meanwhile, precipitation extremes have indirect effects on agriculture,

* Correspondence to: C. Lai, School of Civil Engineering and Transportation, South China University of Technology, Room 206, Jiaotong Building, Wushan Road, Tianhe District, Guangzhou 510641, China. E-mail: laichengguang@foxmail.com

food security and hydroelectric development (Piao et al., 2010). Besides, it is projected that precipitation extremes may turn the soil vulnerability to soil erosion (Scholz et al., 2008). All these could be the reasons why the spatial-temporal analysis of precipitation extremes is necessary and significant. Studies of precipitation extremes have received unprecedented attention from researchers worldwide in recent years (Min et al., 2011; Ghosh et al., 2012; Lee et al., 2012; Fu et al., 2013; Guo et al., 2013; Yang et al., 2013; Zakaria and Shabri, 2013). Regions are often delineated for specific purposes by analyzing the records of relevant rainfall patterns. Recently, regionalization and regional frequency analysis (RFA) in the L-moment framework have been intensively conducted (Smithers and Schulze, 2001; Fowler and Kilsby, 2003; Durrans and Kirby, 2004; Lin and Chen, 2006; Gaál et al., 2008; Satyanarayana and Srinivas, 2008; Yang et al., 2010a; Ngongondo et al. 2011; Dikbas et al. 2012; Bharath and Srinivas, 2014; Chen

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Figure 1. Distribution of the 11 basins and 3825 grid point locations across China. Numbers denote the river basins: (1) Songhua River Basin (SRB); (2) Liaohe River Basin (LB); (3) Haihe River Basin (HB); (4) Yellow River Basin (YB); (5) Huaihe River Basin (HRB); (6) Yangtze River Basin (YRB); (7) Minjiang River Basin (MRB); (8) Pearl River Basin (PRB); (9) Lantsang River Basin (LRB); (10) YarlungTsangpo River Basin (YTB); (11) Tarim River Basin (TRB).

et al., 2014; García-Marín et al., 2015; Nam et al., 2015). Smithers and Schulze (2001) applied RFA, which is based on the regional L-moment algorithm, to precipitation data in South Africa. Fowler and Kilsby (2003) utilized RFA for identifying the temporal changes of multi-day rainfall events in UK, while Gaál et al. (2008) compared different approaches for RFA in Slovakia. This method was also adopted in southern Malawi for more accurate identification of extreme precipitation in regions lack of station records, and the performance of RFA was satisfactory when validated for sites not included in the sample data (Ngongondo et al., 2011). Kysel? y et al. (2011) demonstrated that the region-of-influence (ROI) method could be more useful than at-site approach for modelling the probabilities of heavy precipitation. Dikbas et al. (2012) classified a rainfall series with the fuzzy c-means (FCM) cluster method. Nam et al. (2015) found that FCM-based regions are more appropriate for the precipitation in South Korea in term of the homogeneity of the identified regions. Bharath and Srinivas (2014) validated that the global fuzzy c-means (GFCM) cluster analysis is effective in reducing the number of regions to be delineated for RFA in India. The RFA applied to the mentioned study areas exhibits strong applicability and rationality, proving an effective tool for the analysis of regionalization and regional frequency. Nevertheless, there are few studies on the use of RFA in studying extreme rainfall in China. To date, RFA has been documented at basin scale only in the Yangtze River Basin (Chen et al., 2014) and the Pearl River Basin (PRB) (Yang et al., 2010a), as shown in Figure 1. As is known to all, China has suffered from enormous flood hazards over the
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past decades. Frequent precipitation-induced floods have been reported in many regions, such as the Yangtze River Basin in 1998 (Yin and Li, 2001; Zhang et al., 2007, 2008), the Yellow River Basin in 1993 (Yang et al., 2008) and the PRB in 1994 (Yang et al., 2010b; Lai et al., 2015; Wang et al., 2015). Precipitation regimes across different regions show that the spatial contrast and the frequency distribution patterns of extreme precipitation differ substantially from each other. Conventional practice (e.g. Huang et al., 2013) is criticized because the delineation of regions based on topography, climate and/or administrative boundaries does not guarantee their statistical homogeneity in rainfall, as extreme rainfall characteristics may vary considerably in a geographically contiguous region. A better understanding of Chinese precipitation extremes and spatiotemporal changes in the occurrences of floods and droughts necessitates the regionalization of extreme precipitation and research into the extreme frequency patterns across different regions in China. Nonetheless, regionalization and RFA of extreme rainfall across Mainland China have not been done yet. To overcome this shortage, this study aims to delineate the whole Mainland China into many homogeneous extreme precipitation regions and analyze the regional frequency patterns of extreme precipitation. These frequency patterns can not only characterize the extremes but also be typically utilized to estimate the probability of rare events, such as the 100-year return value, which are typically utilized in civil infrastructure design. The highlights and objectives of this study are to: (1) cluster hydrological homogeneous sites and identify
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Table 1. Definitions of extreme precipitation indices recommended by ETCCDI. Index CDD CWD R10mm R25mm R95T R99T RX1DAY RX5DAY PRCPTOT SDII WD Descriptive name Consecutive dry days Consecutive wet days Number of heavy precipitation days Number of very heavy precipitation days Very wet day precipitation Extremely wet day precipitation Maximum 1-day precipitation Maximum 5-day precipitation Annual total wet day precipitation Simple daily intensity index Wet days Definition Maximum number of consecutive days with RR < 1 mm Maximum number of consecutive days with RR ≧ 1 mm Annual count of days when RR ≧ 10 mm Annual count of days when RR≧25 mm Annual total precipitation when RR ≧ 95th percentile of 1961–2013 daily precipitation Annual total precipitation when RR ≧ 99th percentile of 1961–2013 daily precipitation Maximum 1-day precipitation Annual maximum 5-day precipitation Annual total precipitation in wet days (RR ≧ 1 mm) Annual total precipitation divided by the number of wet days in the year Annual count of days when RR ≧ 1 mm Unit Days Days Days Days mm mm mm mm mm mm day?1 Days

the homogeneous regions for precipitation extremes with FCM method; (2) determine the best probability distribution for each homogeneous region, and conduct RFA with the L-moments approaches; and (3) characterize the spatiotemporal patterns of extreme precipitation events in order to reveal the underlying impacts of climate variations. The study results will definitely benefit the further development and understanding of the unique and complex features of extreme precipitation in the whole Mainland China, and hence the nation-scale flood prevention, control and management in the backdrop of changing climate. The structure of this paper is as follow: Section 1 introduces the background introduction of precipitation extremes and RFA methods, Sections 2 and 3 gives a brief presentation of data and the methods adopted to analyze the precipitation extremes, Section 4 provides the results of regionalization and frequency analysis based on L-moments method, Section 5 presents a discussion of the results, and Section 6 is the conclusion.

2. Data A high-resolution (0.5? × 0.5? ) daily precipitation dataset recorded from 1961 to 2013 was obtained from China Meteorological Administration (CMA, http://cdc.nmic.cn/ home.do), which is officially in charge of monitoring, collecting, compiling and releasing high-quality meteorological data in China. This dataset was collected from 2474 rain gauge stations distributed across China based on the Thin Plate Spline (TPS) and GTOPO30 (Global 30 Arc-Second Elevation) data resampling. TPS interpolates the ratios of daily precipitation into the daily climatological magnitude at each location, which is thus considered to have the capacity to effectively eliminate the effects of topography on spatial interpolation. Moreover, as reported by the Assessment Report of China’s Ground Precipitation 0.5? × 0.5? Gridded Dataset released by the National Meteorological Information Centre (NMIC), each of the gridded box series is highly correlated with the station series
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with a small error. Therefore, this dataset can characterize precipitation variability in China with adequate accuracy on the whole (Wu et al., 2016). The gridded precipitation dataset represents the latest data that has not yet been widely used in published studies currently. The 3825-grid point locations and the distribution of the 11 basins of China are shown in Figure 1. In addition, the GIS maps from the resources and environment science data center of Chinese Academy of Sciences (see http://www.resdc.cn/ Default.aspx) were also utilized in the analysis of precipitation extremes in this study. The Commission for Climatology (CCl)/Climate Variability and Predictability (CLIVAR)/Joint WMO-IOC Technical Commission for Oceanography and Marine Meteorology (JCOMM) Expert Team on Climate Change Detection and Indices (ETCCDI) have recommended a total of 11 extreme precipitation indices (Table 1) that can characterize numerous aspects of precipitation extremes, including intensity, frequency and duration. In this study, two ETCCDI’s indices (ETIs) were selected, i.e. RX1DAY and RX5DAY, which are defined as monthly maximum 1-day precipitation and monthly maximum consecutive 5-day precipitation respectively (http://etccdi .pacificclimate.org/list_27_indices.shtml), so as to analyze the spatiotemporal variability of precipitation extremes in each basin. The reason for choosing these two indices is that they can not only characterize the extremes but also be typically utilized to represent the probability of rare events during the design of infrastructure and in other applications (Min et al., 2011; Wu et al., 2016). 3. Methodology

A RFA based on FCM and L-moment methods has been adopted to characterize the spatial pattern of precipitation extremes across Mainland China. 3.1. L-moments method

As a method to characterize linear functions of the dataset, L-moments method is utilized to describe probability
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distribution and estimate the quantiles of the distributions with L-skewness and L-kurtosis (Hosking and Wallis, 1993; Nathaniel, 1993). More details can be found in the study of Hosking and Wallis (1993, 1997). The theory has the advantage that it can be more robust in terms of sampling variability compared with other conventional approaches. The derivation is based on the order statistics obtained by sorting the sample {X 1, X 2, … , Xn} of n independent realizations of variable X in ascending order {X 1: n, X 2: n, … , Xn: n}; the subscript k: n denotes the kth smallest number in the sample of length n. L-moments k are defined as the expectations of linear combinations of these order statistics, k?1 ) ( ( ) k?1 1∑ E Xk?i∶k k = (1) (?1)i k i=0 i wherein, E denotes the expectation operator. L-moment ratios are the L-coefficients of variation (L-CV), L-skewness ( 3) and L-kurtosis ( 4); except for some special cases of small samples, their values shall be taken between ?1 and +1. The sample L-moment ratios are defined as t = l2 ∕l1 and tr = lr ∕l2 , r = 3,4, … (2)

China. The statistical methods utilized are introduced as follows. 3.2.1. Identification of homogenous regions through cluster analysis The FCM is adopted in this study to identify homogenous regions by clustering the grid points in China with similar precipitation characters. It is a method of clustering which allows one piece of data to belong to two or more clusters (Dunn, 1973; Bezdek, 1981; Wang et al., 2006; Basu and Srinivas, 2014; Bharath and Srinivas, 2014; Firat et al., 2014; Aydogdu and Firat, 2015). This method (developed by Dunn (1973) and improved by Bezdek (1981)) is frequently utilized in pattern recognition because this method can delineate the flexible regions, meaning that a site can be assigned to several regions with different membership. It can be stated that its results include more information on explaining hydrological processes better than the conventional methods, such as Hard K-Means and Wards’ method (Dikbas et al., 2012). It has been found that FCM is effective in reducing the number of regions to be delineated for RFA (Kulkarni and Kripalani, 1998; Shu and Burn, 2004; Dikbas et al., 2013; Bharath and Srinivas, 2014), which is based on the minimization of the following objective function: N C ∑ ∑ 2 um (4) Jm = ij ||xi ? cj ||, 1 ≤ m < ∞
i=1 j=1

wherein, lr is the unbiased rth L-moments, as an analogue to the traditional ratios. That is to say, t is the coefficient of variation (L-CV), t3 the L-skewness and t4 the L-kurtosis. The L-moment ratios will be utilized for homogeneity analysis in the RFA. Compared with conventional methods, L-moments methods have less biases in estimation, and their asymptotes are closer to the normal distribution in finite samples. In addition, the L-moments approach includes the characterization of probability distributions, the summary of observed data samples, the fitting of probability distributions to data, and the testing of the distributional form (Yang et al., 2010a; Bharath and Srinivas, 2014). 3.2. RFA based on L-moments method

wherein, m is any real number greater than 1, uij is the degree of membership of xi in the cluster j, xi is the ith d-dimensional measured data, cj is the d-dimension center of the cluster, and ||*|| is any norm to express the similarity between any measured data and the center. Furthermore, fuzzy partitioning was carried out through an iterative optimization of the objective function shown above, with the update of membership uij and the cluster centers cj according to:
N ∑

Provided that there are N sites in a region with sample size n1 , n2 , … , nN , respectively, and the L-moment ratios (L-CV, L-skewness and L-kurtosis) at site i are denoted by t(i) , t3 (i) and t4 (i) , respectively, the regional weighted average of L-moment ratios can be expressed by: t=
N ∑ i=1 N N N ∑ ∑ ∑ ( i) ni t(i) ∕ ni and tr = ni tr ∕ ni r = 3,4, … i=1 i=1 i=1

uij =

c ( ) 2 ∑ ||xi ?cj || m?1 ||xi ?ck || k=1

1

um · xi ij (5) um ij

, cj =

i=1 N ∑ i=1

(3) The RFA with L-moments method consists of five steps (Hosking and Wallis, 1993, 1997; Yang et al., 2010a): (1) identification of homogenous regions through cluster analysis; (2) data scanning with the discordancy measure Di ; (3) homogeneity testing with the heterogeneity measure H ; (4) distribution selection with the goodness-of-fit (GOF) measure Z ; and (5) regional estimation of precipitation quantiles with the L-moment approach. These five steps were followed to make a RFA for the whole Mainland
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This iteration will stop when there is a termination criterion between 0 and 1, whereas k represents the iteration steps. This procedure converges to a local minimum or a saddle point of Jm . The output from the cluster analysis is not the final results. The Xie and Beni Index (XB), a wide-used selection criteria of homogeneity division, was applied for determining of optimum number of clusters (Xie and Beni, 1991). Subjective adjustments can be conducted to improve the physical coherence of regions and reduce the heterogeneity of regions as measured through the heterogeneity measure. Several adjustments of regions may be recommended (Hosking and Wallis, 1997; Chen et al., 2014), such as: (1) moving a site or a few sites from one region to another; (2) deleting a site or a few sites from the
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dataset; (3) subdividing the region; (4) dividing the region by reassigning its sites to other regions; (5) combining the region with another or others; (6) combining two or more regions and redefining the groups; and (7) obtaining more data and redefining the groups. 3.2.2. Data scanning with the discordancy measure For ui = [t(i) , t3 (i) , t4 (i) ]T as the vector containing the t, t3 , and t4 values for site i, the superscript T denotes the transposition of a vector or matrix. The regional average is defined as (Hosking and Wallis, 1993):
N ∑

was used, as suggested by Hosking and Wallis (1993) with the L-kurtosis, where 4 DIST is the L-kurtosis of the fitted distribution to the data with the employment of the candidate distribution, and:
N sim

4 =

∑(m ) t4 ? t4 ∕Nsim

(11)

m=1

ui (6)

is the bias of t4 estimated through the simulation technique as before with t4 m being the sample L-kurtosis of the mth simulation and: [N ]}1∕2 { sim ∑ (m ) ?1 2 t4 ? t4 ? Nsim 4 (12) 4 = Nsim ? 1
m=1

N Then, the discordancy measure for site i is defined as (Hosking and Wallis, 1993): )T ) ( 1 ( Di = N ui ? u A?1 ui ? u (7) 3 wherein, A =
N ∑ ( i=1

u=

i=1

ui ? u

)(

)T ui ? u .

If Di value is large, the discordancy of site i with other sites is obvious. Hosking and Wallis (1997) found that there was no single fixed number which can be considered to be a ‘large’ Di value, and suggested some critical values for discordancy test which are dependent on the number of sites in the study region. 3.2.3. The homogeneity measure H L-moment based H -statistic (Hosking and Wallis, 1997): ( ) V ? V (8) H= V where V is the weighted standard deviation of at-site sample coefficients of L-variation; V , and V represent the mean and the standard deviation of V values computed from a large number of realizations of extreme rainfall, with the rainfall simulated from к distribution fitted using regional average L-moment ratios. A region is considered as ‘Acceptably Homogeneous (AH)’ if H < 1, ‘Possibly Heterogeneous (PH)’ if 1 ≤ H < 2, and ‘Definitely Heterogeneous (DH)’ if H ≥ 2. If extreme precipitation data are cross-correlated in a region, adjust H value to H adj (Castellarin et al., 2008). Hadj = H + 0.122 × (Nc – 1)
2 2

is the standard deviation of t4 . The fit is considered to be adequate if |Z DIST | is sufficiently close to zero, with the reasonable criterion being |Z DIST | ≤1.64 (at 90% confidence level). The GOF test descried by Equation (10) allows the keeping of distribution, while the selection of among the distributions kept is not allowed. As a result, if more than one candidate distributions are acceptable, the L-moment ratio diagram has been used to identify the distribution through the comparison of its closeness with the L-skewness and L-kurtosis combination in the L-moment ratio diagram (Hosking and Wallis, 1997; Yang et al., 2010a; Chen et al., 2014). 3.2.5. Regional estimation of precipitation quantiles

The computation of precipitation quantile of any recurrence interval is based on pooled regional information. In terms of regional frequency distribution, it is fitted with the regional average L-statistics determining a dimensionless quantile function known as regional growth curve. Following this, based on the index flood method, at-site precipitation quantile of desired recurrence interval is estimated with the assistance of the regional growth curve (Dalrymple, 1960; Hosking and Wallis, 1997; Satyanarayana and Srinivas, 2008). The term ‘index flood’ arose because early applications of the procedure were to flood data in hydrology, but the method can be used with any kind of data (Hosking and Wallis, 1997; Satyanarayana and Srinivas, 2008).

4. 4.1.

Results Regionalization of precipitation extremes

(9)

where indicates the average of squares of cross-correlations of concurrent extreme rainfall values for the sites in a cluster comprising of Nc sites. 3.2.4. Distribution selection with the goodness-fit measure For each candidate distribution, the GOF measure: Z DIST =
DIST 4 ? t4 + 4

4

(10)

While the climatic types and geographical characteristics across China are complicated, homogeneous regions should be identified so as to understand the precipitation patterns in Chinese mainland. In our study, a FCM Clustering method was used to identify homogeneous regions considering the latitude, longitude, elevation and mean annual precipitation. As a result, there are nearly 50 homogeneous regions relating to the precipitation extreme indices, namely RX1DAY and RX5DAY, based on above four physiographic and precipitation-related indices.
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(a)

(b)

RX1DAY

RX5DAY

Figure 2. Homogeneous extreme precipitation regions clustered by adjusting the fuzzy clusters. Removed sites are those that were eliminated during adjustments to heterogeneous clusters and are stippled.

(a)

(b)

RX1DAY

RX5DAY

Figure 3. Spatial distribution of best-fit distribution for 50 regions across China.

However, the output of the cluster analysis is not the final result. For the discordance test, it was done for 3825 grid points in order to find out the number of discordant sites (D < 3) and keep it in acceptable range. First, some adjustment towards these discordance sites shall be made to improve the homogeneity and physical coherence of regions and to decrease the heterogeneity of regions. Detailed description of these methods is demonstrated in Section 3.2.1. In terms of those failing to pass the discordance test, they shall be excluded from the dataset in our study. In this case, 96 and 107 grid points were excluded out of 3825 grid points for RX1DAY and RX5DAY, respectively (Figure 2). The final result of the adjustment is shown in Figure 3, in which China is categorized into 50 homogeneous regions for extreme precipitation. In order to further verify the rationality of the category, 2 to 100
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homogeneous regions were tested, respectively and the XB indexes were calculated (Figure 4). The XB value will be the minimum with the homogeneity region set to 4; however, more than 50% grid points fail to pass the discordance test. Further, above 90% of the points can pass the discordance test when China divided into 40–60 homogeneity regions. We finally determined 50 homogeneity regions after several adjustments with the consideration of the high homogeneity (>90%) as well as good visual effects. Then the homogeneity measure H 1 as defined in Section 3.2.2. was done for each region and those regions were detected to be statistically homogeneous when H1 < 2. The final result is shown in Table 2. It can be seen from Table 2 and Figure 2 that the numbers of AH and PH regions for RX1DAY are 45 and 5, respectively, with no DH region being found. Meanwhile,
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Figure 4. Variation of XB index with fuzzy c-means based on the number of clusters.

the numbers of AH and PH regions for RX5DAY are 38 and 12, respectively. Again, no DH region is detected for RX5DAY. In terms of RX1DAY and RX5DAY series, the distribution patterns of homogeneous regions are strongly similar. Large homogeneous regions, such as region 31, 16 and 37, are mainly distributed in the Northwest, Southwest and Southeast, respectively. By comparison, the Northeast and mid China have complex distribution patterns of homogeneous regions. 4.2. Selection of probability functions For each region delineated based on the FCM cluster analysis method, the temporal distribution of RX1DAY and RX5DAY, can be fitted by Generalized Extreme Value (GEV), Generalized Logistic (GLO), Generalized Normal (GNO), Generalized Pareto (GPA) and Pearson type III (PE3), detailed introduction in the Supporting Information (Section 1), distributions with L-moment based on regional GOF test (Hosking and Wallis, 1997). Spatial distribution of L-moment ratios (i.e. t, t_3, t_4 and t_5) for each regions across China using L-moments based RFA approach was presented in Figure S1 and Table S1. The GOF test was conducted to obtain the results in Table 3, which is considered acceptable at 90% confidence level if the absolute value of GOF statistic Z is less than or equal to 1.64. In case of more than one acceptable distribution, the distribution due to which the absolute value of the GOF test statistic is sufficiently close to zero was considered to be the best-fit distribution (Hosking and Wallis, 1997; Bharath and Srinivas, 2014; Chen et al., 2014). Under the condition that none of the distributions would pass the 90% confident test, however, one of them is close to 1.64, for which it would be accepted as the better-fit distribution. For those values far away from 1.64, Wakeby distribution instead could be used to fit them, as recommended by Hosking and Wallis (1997) and Bharath and Srinivas (2014). Wakeby is a kind of
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distribution, concerning which five indices are employed to describe it so that there is less error compared with other distribution in case of describing temporal pattern of precipitation extremes. In Table 3, the best-fit distribution of each homogeneous region corresponding to RX1DAY and RX5DAY has been demonstrated, which indicates that GEV, GNO and PE3 perform well in fitting extreme precipitation in each region. In addition, their spatial distributions are shown in Figure 3, which shows the spatial variety across China in terms of best-fit temporal distribution for precipitation extremes. As portrayed in Figure 3(a), GEV of RX1DAY has the best-fit distribution in the east, northeast and southwest of China, but most of the north areas, local areas in southwest and southeast present the best-fit of GNO distribution; additionally, regions fit PE3 best are mainly located in the northwest and south of China. As shown in Figure 3(b), GEV of RX5DAY fits the most areas, especially in the middle, southwest and south of China, followed by GNO and PE3 that mainly fit in the northeast and north respectively; in addition, other regions do not show obvious regular distribution. Wakeby distribution was used to fit the regions failing to pass the confidence test. Spatial distribution of parameters of best-fit distribution for each region based on RFA approach is also presented in Figure S2 and Table S2. The L-moment ratio plots for RX1DAY and RX5DAY in 50 homogeneous regions were utilized to describe the best-fit distribution through the comparison of its closeness to the L-skewness and L-kurtosis combination. As an example, the L-moment ratio diagram for RX1DAY and RX5DAY in Region 1 is shown in Figure 5. It can be observed that GNO performs best in fitting RX1DAY in this region, as proved by the GOF test. Then the GNO distribution was used to estimate the regional growth curve and associated frequency for region 1. Besides, the regional growth curves of RX1DAY and RX5DAY for each homogeneous region with 90% error bands were
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Table 2. Characteristics of 50 regions obtained by adjusting clusters resulting from the use of FCM method. Region TN PN 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 77 118 25 52 60 28 82 16 44 121 151 31 54 78 105 166 99 63 51 55 64 123 118 80 77 65 81 56 50 71 162 105 107 79 112 81 118 96 110 98 47 46 16 47 74 33 28 63 35 107 77 113 24 51 60 27 80 16 43 118 149 29 53 76 103 162 96 62 49 54 62 120 116 78 77 63 78 56 47 71 159 103 103 75 111 79 112 93 105 95 46 46 16 46 73 32 28 61 35 101 RX1DAY FN 0 5 1 1 0 1 2 0 1 3 2 2 1 2 2 4 3 1 2 1 2 3 2 2 0 2 3 0 3 0 3 2 4 4 1 2 6 3 5 3 1 0 0 1 1 1 0 2 0 6 H ?1.104 ?3.255 ?4.566 ?1.484 ?0.991 ?2.299 0.833 ?0.845 ?1.299 ?1.577 ?1.149 ?3.312 1.163 ?0.538 ?2.978 1.605 0.574 ?0.369 0.592 ?1.133 1.847 ?0.703 0.222 ?0.202 ?1.157 ?1.475 ?2.108 ?2.701 ?1.796 ?0.618 0.137 ?0.670 ?0.025 ?1.631 ?1.470 1.742 ?0.003 ?0.993 0.643 ?0.884 ?1.266 ?2.189 0.058 0.574 0.684 ?1.047 1.395 ?0.479 0.891 ?0.959 PN 76 117 25 52 60 28 80 16 41 115 147 30 53 76 102 160 95 62 49 52 64 119 118 77 75 65 81 54 49 68 157 100 103 75 107 79 112 93 108 96 47 45 16 44 72 32 28 61 34 103 RX5DAY FN 1 1 0 0 0 0 2 0 3 6 4 1 1 2 3 6 4 1 2 3 0 4 0 3 2 0 0 2 1 3 5 5 4 4 5 2 6 3 2 2 0 1 0 3 2 1 0 2 1 4 H 0.511 0.823 ?0.816 1.400 1.273 ?2.692 1.010 ?1.887 ?2.822 ?0.458 1.118 0.509 ?0.338 ?1.696 1.704 ?0.072 ?2.672 ?2.055 0.688 ?2.851 1.973 ?1.394 ?0.343 ?0.746 ?1.327 0.422 ?1.704 ?5.013 ?0.174 ?2.675 ?2.189 0.411 0.179 ?2.310 ?0.412 0.876 1.809 0.252 ?0.582 ?0.036 1.983 1.661 1.661 0.184 ?0.084 ?2.503 ?0.084 ?2.791 1.864 1.293

TN, the total number of points in each region; PN, the number of points pass the homogeneity measure; FN, the number of points fail to pass the homogeneity measure.

delineated in our study. The L-moment ratio plots of RX1DAY and RX5DAY for other regions were shown in Figures S3 and S5. As shown in Figure 6, taking region 1 for an example, the regional growth curve for RX1DAY in this region is evidently concave while for RX5DAY the growth curve is nearly a straight line. The other regional growth curves for RX1DAY and RX5DAY are shown in Figures S4 and S6.
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4.3. Spatial patterns of precipitation extremes for different return period For water resources management and disaster prediction, it is of significantly practical use to understand the spatial distribution of precipitation extremes. Therefore, we used the growth curve based on the best-fit distribution to estimate different return periods. In order to maintain the spatial associations of extreme precipitation, the sites
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Table 3. Goodness-of-fit test of regional distribution for each homogeneous region. Region (a) RX1DAY 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 (b) RX5DAY 1 2 3 4 5 6 7 8 9 10 11 GLO 8.112 6.537 3.176 7.197 6.909 4.120 6.260 6.915 7.029 8.165 10.623 1.572 5.279 11.142 9.555 6.361 8.781 5.771 5.530 2.135 6.409 21.139 2.236 7.531 5.790 4.443 5.675 6.234 10.040 5.079 7.041 9.335 13.190 6.594 8.769 8.951 11.423 4.176 10.055 6.846 4.886 8.085 4.334 4.798 4.496 5.402 6.794 8.539 9.303 11.655 2.694 9.507 0.977 4.459 7.056 3.189 8.890 6.433 4.991 4.778 11.373 GEV 2.911 0.633 ?0.443 1.927 1.142 ?0.013 1.154 3.709 1.847 1.203 2.402 ?1.069 0.838 4.327 3.709 1.723 3.450 1.570 0.287 ?0.754 1.671 9.892 ?3.786 0.447 ?1.503 ?0.705 ?0.560 ?0.172 4.067 ?1.025 ?1.803 2.297 3.590 ?0.207 1.141 1.500 2.116 ?2.861 1.958 ?1.435 1.077 1.843 0.939 0.259 ?0.277 1.052 1.198 3.928 3.534 5.244 ?3.351 1.622 ?2.256 ?1.993 1.069 ?0.207 2.205 3.028 ?0.417 ?1.028 1.339 GNO 0.985 ?1.455 ?0.855 1.183 0.125 ?0.489 ?0.627 3.169 1.153 ?0.688 0.612 ?2.053 ?0.513 3.210 1.501 ?1.896 1.055 ?0.142 ?0.496 ?2.514 0.099 9.903 ?5.463 ?0.163 ?1.769 ?1.919 ?1.469 ?0.554 3.530 ?1.899 ?3.190 1.005 3.552 ?0.930 ?0.300 1.025 1.454 ?3.556 0.972 ?1.609 ?0.309 1.524 0.724 ?0.768 ?1.789 0.503 1.588 2.177 3.476 3.122 ?3.997 0.094 ?2.747 ?1.748 0.188 ?0.877 1.314 2.692 ?0.636 ?3.128 0.531 PE3 ?2.629 ?5.401 ?2.008 ?0.692 ?2.207 ?1.814 ?4.003 1.908 ?0.640 ?4.471 ?3.201 ?3.896 ?3.144 0.571 ?2.627 ?8.235 ?3.316 ?3.304 ?2.419 ?5.634 ?2.907 7.914 ?8.797 ?2.164 ?3.377 ?4.441 ?3.725 ?2.147 1.813 ?4.087 ?6.525 ?1.914 1.800 ?3.029 ?3.520 ?0.868 ?0.996 ?5.659 ?1.696 ?3.283 ?2.915 0.035 ?0.137 ?2.930 ?4.709 ?0.955 0.999 ?1.094 2.384 ?0.939 ?5.869 ?3.292 ?3.944 ?2.630 ?1.991 ?2.351 ?0.991 1.675 ?1.856 ?7.080 ?2.232 GPA ?9.723 ?13.630 ?8.441 ?9.850 ?11.926 ?9.151 ?11.161 ?3.534 ?9.698 ?15.145 ?16.522 ?7.489 ?9.710 ?11.039 ?10.528 ?10.982 ?9.835 ?8.760 ?11.468 ?8.309 ?9.690 ?13.728 ?17.960 ?15.024 ?17.089 ?12.636 ?14.518 ?14.004 ?9.000 ?14.677 ?21.688 ?13.696 ?16.620 ?15.187 ?16.229 ?14.617 ?18.082 ?18.312 ?15.993 ?19.009 ?8.158 ?11.584 ?6.404 ?10.223 ?11.665 ?8.609 ?10.161 ?7.308 ?8.645 ?10.125 ?16.669 ?16.365 ?9.511 ?15.303 ?12.342 ?7.964 ?12.686 ?4.450 ?11.996 ?15.093 ?20.523 Best-fit GNO GEV GEV PE3 GNO GEV GNO PE3 PE3 GNO GNO GEV GEV PE3 GNO GEV GNO GNO GNO GEV GEV WAK WAK GNO GEV GEV GEV GNO PE3 GEV GEV GNO WAK GEV GNO GNO PE3 GEV GNO GEV GNO PE3 PE3 GEV GEV GNO PE3 PE3 WAK PE3 WAK GNO GLO GNO GNO GEV PE3 PE3 GEV GEV GNO

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Table 3. Continued Region 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 GLO 6.485 4.205 7.012 10.835 6.121 11.549 6.507 10.069 0.548 0.163 12.936 6.050 5.882 6.440 5.421 4.735 3.824 9.257 4.822 8.910 9.920 7.128 12.730 6.167 6.793 7.930 5.991 9.937 10.970 6.599 6.064 1.382 5.608 6.063 3.420 4.921 9.194 6.206 10.856 GEV 2.136 ?0.198 1.663 3.924 1.453 5.905 2.300 3.264 ?2.138 ?3.290 3.277 ?1.646 ?1.238 ?0.996 0.360 ?0.874 ?0.916 3.865 ?1.386 ?1.552 3.453 ?1.911 4.375 ?2.275 ?0.692 0.409 ?0.809 2.857 1.916 0.599 0.089 ?1.381 1.037 ?0.925 ?1.054 0.066 5.087 2.081 4.391 GNO 1.517 ?1.517 0.050 2.233 ?2.261 3.674 0.614 3.105 ?3.757 ?5.008 2.698 ?2.801 ?1.639 ?1.252 ?0.990 ?2.240 ?1.993 3.067 ?2.028 ?1.903 1.679 ?2.160 4.147 ?2.772 ?1.213 ?0.786 ?1.956 1.261 1.559 0.461 ?0.275 ?1.741 0.080 ?1.176 ?1.431 0.080 3.013 1.168 2.263 PE3 ?0.038 ?4.093 ?3.097 ?1.243 ?8.762 ?0.468 ?2.509 1.712 ?6.630 ?8.106 0.291 ?5.633 ?3.382 ?2.873 ?3.702 ?5.053 ?4.256 1.098 ?3.921 ?4.174 ?1.859 ?4.057 2.397 ?4.866 ?3.169 ?3.645 ?4.634 ?2.102 ?0.471 ?0.767 ?1.772 ?2.682 ?1.985 ?2.712 ?2.684 ?0.769 ?0.724 ?0.765 ?1.815 GPA ?7.586 ?10.641 ?11.030 ?12.145 ?11.386 ?7.917 ?8.023 ?11.191 ?9.147 ?12.025 ?17.581 ?18.905 ?16.587 ?16.872 ?11.503 ?13.911 ?11.866 ?8.217 ?15.045 ?23.881 ?11.746 ?21.149 ?13.407 ?20.497 ?16.924 ?16.516 ?16.171 ?13.489 ?17.457 ?12.143 ?12.820 ?7.506 ?9.453 ?15.856 ?10.818 ?10.123 ?5.325 ?7.427 ?11.097 Best-fit PE3 GNO GNO PE3 GEV PE3 GNO PE3 GLO GLO PE3 GEV GEV GEV GEV GEV GEV PE3 GEV GEV GNO GEV WAK GEV GEV GEV GEV GNO PE3 GNO GEV GEV GNO GEV GEV GEV PE3 PE3 PE3

which were detected discordant are not excluded from the dataset while at the same time we could also use the best-fit distribution for 3825 grid points to obtain their 20-, 50-, 100-year return value. Generally, Figure 7 shows that nearby points close to the boundary between two adjacent regions do not show unreasonably different quantiles. It can be observed from Figure 7(a) that the peak values of RX1DAY for the return periods appear in the south-eastern coastal (around 110? –120? E; 20? –30? N), specifically in the Minjiang River Basin, lower PRB and lower Yangtze River Basin. Meanwhile, it can be seen that there are also a few high-value sites appearing in southern Yarlung Tsangpo River Basin. In general, the 20-, 50-, 100-year return values have shown similar spatial patterns: gradual decrease from southeast to northwest. As for RX5DAY, both the south-eastern coastal and lower YRB have shown higher tendency of return values but they have become less in inland of China. Furthermore, the very low RX5DAY quantile values are located in the northwestern China.
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5. Discussions RFA based on L-moments method has been used in this paper to figure out the spatial patterns of precipitation extremes in China. In RFA, 50 homogeneous regions were clustered through a FCM method. As for the homogeneity measure H , it was used to stipple the sites failing to pass the homogeneity test. From Figure 2, it could be found that extreme rainfall characteristics may vary considerably in a geographically contiguous region (e.g. region 6 and region 7; region 42 and region 44). In addition, FCM method was found to be effective in delineation of regions and reducing the number of regions to be delineated for RFA for the reason that it can break through the limitations of topography, climate and/or administrative boundaries. This is in agreement with the conclusions made by Nam et al. (2015), Basu and Srinivas (2014) and Bharath and Srinivas (2014). The temporal distribution of precipitation extremes for each region can be fitted by GEV, GLO, GNO, GPA,
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(a)

(b)

RX1DAY
Figure 5. L-moment ratio plot for region 1.

RX5DAY

(a)

(b)

RX1DAY

RX5DAY

Figure 6. Estimated regional growth curves of Region 1 with 90% error bands.

PE3 and Wakeby distributions with the employment of the L-moment method. The results of GOF test demonstrate that GEV, GNO and PE3 perform better in fitting the distributions. Compared with the conclusion drawn by Chen et al. (2014), the Yangtze River Basin (YRB) was divided into around 10 homogeneous regions in our study instead of 6. Based on the result of our study, the precipitation with 20 return period years has a gradual increase from the upper to the lower YRB, the result of which is similar to that in the previous study. That means there have been higher risks of floods in the middle and lower YRB. Besides, the GOF test results indicated that PE3, GEV and GNO performed well in fitting regional distributions in YRB, which is similar as that in our study. In our study, GEV performed best in the most regions of YRB with four and six main regions fitting best in RX1DAY and RX5DAY respectively, followed by GNO with four and two best-fit regions in RX1DAY and RX5DAY respectively, but only one region had PE3/Wakeby distribution in RX1DAY/RX5DAY. As for the PRB, Yang et al. (2010a) found out that in case the whole basin is categorized into
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six regions, the spatial variations of precipitation in different return periods (Return period = 50–100 years) would increase from the upstream to downstream at the regional scale, which has been proved by the result of our study. For most of the areas of PRB, our results show that the GNO, GEV and PE3 perform better in GOF test while the GNO, GLO, GEV and PE3 distributions fit well with the study conducted by Yang et al. (2010a). However, it is worth noting that there is little difference between them. The possible cause is that there is some difference between the number of sites and homogeneous regions, while at the same time the present study used the updated longer observational data series from 1961 to 2013. In short, both comparisons above indicate the RFA which has been used is reasonable and creditable. Moreover, the return levels are calculated by fitting a GEV distribution to the RX1DAY and RX5DAY series at each grid, also known as at-site frequency analysis (ASFA). The GOF is estimated through the Kolmogorov–Smirnov (KS) test, on which basis, if the test fails, an empirical distribution rather than a GEV
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(a)

(b)

RX1DAY

RX5DAY

Figure 7. Spatial distributions of RX1DAY and RX5DAY with 20-, 50-, 100-year return period using L-moments based on RFA approach.

is fit. The fitted GEV parameters (see Figure S7) were used to calculate the return levels for the series at each grid (Figure 8). However, some errors can be expected in terms of the application to large geographical areas, such as entire Chinese mainland, where spatial variation
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in distribution parameters is not smooth. Compared with the results of GEV distribution (Figure 8), the result of the spatial pattern of RFA (Figure 7) seems to be smoother and more correlated to natural features, with the similar tendency exhibited. Moreover, we performed the Monte Carlo
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(a)

(b)

RX1DAY

RX5DAY

Figure 8. Spatial distribution of RX1DAY and RX5DAY for return periods of 20, 50 and 100 years using ASFA approach.

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Figure 9. Spatial distribution of the RMSE for return period 100 year using L-moments based RFA approach. The grids of no data were eliminated during the adjustments to heterogeneous clusters.

Figure 10. Spatial distribution of the RMSE for return period of 100 year using ASFA approach.

simulation to generate 1000 samples for RX1DAY and RX5DAY and then sampled 100 times randomly (sequential sampling 100 samples from the total 1000 samples with replacement) to fit 100 distribution curves in each grid. Whereafter, we used the 100 values respectively corresponding to 20-, 50- and 100-year return values to fit the normal distribution and then calculated the RMSE at 90% confidence interval based on RFA and ASFA approach, respectively. Taking the return period of 100 years for example, Figures 9 and 10 present that the RMSE values of the two approaches gradually increase from northwest to southeast in general. However, the vast majority of RMSE values for RFA are apparently smaller than that for ASFA in both RX1DAY and RX5DAY. Similar conclusions (see Figures S8 and S9) can also be obtained in 20 and 50 return period years, suggesting that RFA may provide more accurate estimates of extreme rainfall quantiles. This is consistent with the conclusions made by Nam et al. (2015) and Bharath and Srinivas (2014).
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Spatial patterns of RX1DAY and RX5DAY for return periods of 20, 50 and 100 years indicate that the return period values would gradually decrease from southeast to northwest in China. Specifically for RX1DAY, for a return period of 20 years along the southeast coastline of the Chinese Mainland and on the Hainan Island, the amount of RX1DAY would exceed 135 mm (Figure 7(a)), thus suggesting a high risk of flooding in these regions. As expected, the RX5DAY distribution for return periods (Figure 7(b)) is very similar to the RX1DAY distribution. The main reason for the phenomenon is that the south-eastern area of China is near the Pacific, as a result of which it is easier for water vapor to transfer and is more likely to be effected by the East Asian summer monsoon (EASM, more detail definition can be found in http://ljp .gcess.cn/dct/page/1), the South Asian summer monsoon (SASM) and El Ni?o Southern Oscillation (ENSO) (Xiao et al., 2014; Wu et al., 2016). In contrast, northwest inland far away from the ocean is subject to the influence of a
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weakened Asian summer monsoon, for which the amount of water vapor transmission would thus greatly reduce and accordingly rainfall intensity would become small, implying a low risk of flooding in these regions. In summary, RFA is an advanced, reasonable and effective method at regional or national scale. However, the distribution and trend of extreme precipitation in China is complex for the multiple climatic feature and various geographical characters, due to which more detailed analysis is required. This paper is based on a general view upon the whole country. Even if there are some conclusions concerning basins and catchments, the study for each catchment is not detailed enough and more factors that would affect the conclusions should be taken into consideration. Therefore, there are a lot more work to be done for extreme precipitation analysis in China. 6. Conclusions RFA of precipitation extremes with FCM and L-moments approaches has been conducted in Chinese mainland based on a high-resolution (0.5? × 0.5? ) daily precipitation dataset from 1961 to 2013. Conclusions have been made as follows: 1. The entire Chinese mainland is delineated into 50 homogeneous regions through FCM method based on extreme rainfall and location indices (including latitude, longitude, elevation and mean annual precipitation). 2. As a result of GOF test and L-moment ratio plots analysis for each homogeneous region, GEV, GNO and PE3 distributions fit well for most of the regions in China regarding to precipitation extremes. For RX1DAY, GEV has the best-fit distribution in the east, northeast and southwest of China, whereas GNO distribution mostly fits the northern and local regions of southwest and southeast; in addition, regions fitting PE3 and GLO distribute dispersedly across the country. For RX5DAY, GEV fits most areas, especially in the middle, southwest and south of China; GNO and PE3 function best in the northeast and north of China, respectively. The Wakeby distribution is conducted to fit the regions failing to pass the confidence test. 3. Each grid point precipitation quantile is estimated with the regional growth curve based on the best-fit distribution for each homogeneous region. From the spatial distribution of quantile estimation, the 20-, 50-, 100-year return values would gradually decrease from the southeastern to the northwestern part of China, thereby suggesting a high risk of flooding along the southeast coastline of the Chinese Mainland and on the Hainan Island. 4. Compared with the result of GEV distribution fitted to each site, the spatial patterns of the return value for RFA are smoother and more correlated to natural features. RFA may provide more accurate estimates of rainfall quantiles than at-site frequency analysis (ASFA).
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Acknowledgements The research is financially supported by the National Natural Science Foundation of China (Grant Nos 51579105, 51210013, 51479216, 91547202), Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase), the Water Resource Science and Technology Innovation Program of Guangdong Province (2016-25, 2015-17). Supporting information The following supporting information is available as part of the online article: Figure S1. Spatial distribution of L-moment ratios (t, t_3, t_4 and t_5) for each region across China using L-moments based on RFA. Figure S2. Spatial distribution of the parameters (xi, alpha and k) of best-fit distribution for each region across China using L-moments based on RFA. Figure S3. L-moment ratio plot of RX1DAY for each region. Figure S4. Estimated regional growth curves of RX1DAY for each region. Figure S5. L-moment ratio plot of RX5DAY for each region. Figure S6. Estimated regional growth curves of RX5DAY for each region. Figure S7. Spatial distribution of the parameters (xi, alpha and k) using ASFA approach. Figure S8. Spatial distribution of the RMSE for return periods of 20, 50 and 100 year using L-moments based RFA approach. Figure S9. Spatial distribution of the RMSE for the return periods of 20, 50 and 100 year using ASFA approach. Table S1. L-moment ratios (t, t_3, t_4 and t_5) for each region across China using L-moments based on RFA approach. Table S2. Three parameters (location, scale and shape) of best-fit frequency distribution for each homogeneous region.

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