1 Introduction

Observations reveal that, as the climate warms, the hydrological cycle involves not only an average increase in rainfall, but also changes of the intensity, frequency, and duration (IFD) of extreme rainfall over many regions (IPCC, 2013; Yu et al., 2014; Liang et al., 2015; Fan and Chen, 2016). In the next few decades, extreme rainfall would be more intense and frequent (Sun et al., 2007; Lau et al., 2013; Ma and Zhou, 2015). With a large population that is heavily dependent on agriculture, China is much more vulnerable to extreme rainfall, which has caused tremendous losses in societal, economic, and ecological aspects (Guo et al., 2014; Ke and Guan, 2014; Li M. G. et al., 2016). Therefore, future changes of the extreme precipitation IFD in China are of crucial importance to the water management, agricultural activities, civil infrastructure design, and other activities focused on adaptation to climate change (Cheng and AghaKouchak, 2014; Fan et al., 2015).

To measure historical trends and possible projected changes of extreme rainfall events, various indices were proposed to depict the IFD characteristics of such events (Chen et al., 2011; You et al., 2011; van der Schrier et al., 2013; Jiang et al., 2015; Wu et al., 2016). One of the most popular sets of indices is the 27 Climate Extremes Indices (http://www.climdex.org/indices.html) (Karl et al., 1999; Alexander et al., 2006). In the present work, we focus on statistical modeling of rainfall extremes, specifically, of the characteristics of extreme wet spells. Wet spells are defined as a period of persistent rainy days with daily precipitation exceeding a certain threshold (She et al., 2013; Li X. et al., 2016).

Previous studies have suggested that the wet spell climatological distribution in China and their temporal variations in recent decades show a marked regional variability (Xie, 2002; Su et al., 2005; Zhai et al., 2005; Bai et al., 2007; Zhang et al., 2011; Qian et al., 2014). The mean intensities of wet spells decrease from Southeast to Northwest China. Meanwhile, the mean wet spell durations increase from north to south, with the longest spells over eastern Tibetan Plateau and Southeast China. The wet spell numbers present a slight increase over most of western China, but significant decrease over North, central, and Southwest China (Zhai et al., 2005; Bai et al., 2007; Zhao et al., 2009; Ma and Zhou, 2015). Zhang et al. (2011) measured the peak length of series of consecutive wet days, and showed that although the annual mean length is roughly constant, a slight increase in length since the 1990s can be detected in both summer and winter. Notably, the complex geography and regionally varying climatology of China make the selection of thresholds and definition of wet spells not uniform, and varied with the study area in previous investigations (Bai et al., 2007; Zhao et al., 2009; Qian et al., 2014). Compared to the low thresholds used in previous studies, a relatively high percentile threshold is chosen in this study to capture the IFD probabilistic characteristics of extreme rainfall spells during summer.

The extreme value theory (EVT) methods are commonly employed to characterize distribution of extreme meteorological events (Coles, 2001; Li et al., 2005; Sugahara et al., 2009; Acero et al., 2011; Mondal and Mujumdar, 2015). Two types of extreme distributions, obtained by using different sampling methods, have been used extensively. One is the generalized extreme value distribution (GEV), which models block maxima (BM) of the variable (e.g., the annual maximum in daily precipitation). The second is the generalized Pareto distribution (GPD), which models all exceedances above a predefined threshold, which is known as peaks-over-threshold (POT) approach. As considerably more data points are used than is the case with BM, POT generally yields more reliable results (Coles, 2001).

In statistical studies of extreme precipitation, it is convenient to model both severity (intensity) and frequency, based on the point process (PP) approach (Katz, 2013). In other words, the number of occurrences across a threshold is fitted as a Poisson distribution and the corresponding exceedance magnitudes (i.e., intensity) as a GPD (Coles, 2001; Ding et al., 2008; Wan et al., 2013; Cheng and AghaKouchak, 2014). However, such methods cannot provide a complete description of extreme precipitation IFD, particularly for wet spells. Furrer et al. (2010) extended the PP method to additionally model the IFD of hot spells. Specifically, the intensity of hot spell is modeled by a conditional GPD, the frequency by the Poisson distribution, and the duration by the geometric distribution, respectively. However, there have been few studies of statistical modeling of extreme rainfall spells in China before. In this work, the above method is introduced to characterize the wet spell IFD and their projected changes over China in the late 21st century.

In recent years, the global climate models (GCMs), from the Coupled Model Intercomparison Project Phase 5 (CMIP5), have increasingly been used in analysis of the climate change impact on hydrology (Taylor et al., 2012). However, most GCMs are known to exhibit systematic biases for precipitation over China. Generally, extreme precipitation is underestimated in eastern China in summer, but overestimated in western China (Ou et al., 2013; Zhou B. T. et al., 2014; Jiang et al., 2015), meaning that bias correction is necessary. Therefore, many bias correction techniques are proposed recently, such as, linear scaling, the delta-change approach (DEL), variance scaling, local intensity scaling, power transformation, and quantile–quantile mapping (QM) (Teutschbein and Seibert, 2012; Fang et al., 2015). An adjusted bias correction procedure, based on the QM and DEL, is proposed and applied to the CMIP5 projected wet spell changes in China in this study.

The remainder of this paper is organized as follows. Section 2 introduces the data and methodology. Section 3 describes the statistical modeling of IFD of observed extreme wet spells during 1961–2005, and Section 4 analyzes the statistical attributes of projected wet spell IFD changes in the late 21st century (2071–2100). Finally, the results are summarized and discussed in Section 5.

2 Data and methodology 2.1 Data

Daily observed precipitation, from 756 meteorological stations throughout China, covering 1951–2012, is obtained from the National Meteorological Information Center (NMIC), which is a division under the China Meteorological Administration. This dataset has been quality-controlled by the NMIC, and widely used for recent climate change impact analysis (Sun and Ao, 2013; Jiang et al., 2015). Observed precipitation in summer (i.e. June, July, and August; JJA) from 1961 to 2005 are used in this study, as data in the 1950s are sparse, and the primary set of CMIP5 historical simulations ends in 2005. Stations with missing values from 1961 to 2005 are also rejected, which leaves a total of 553 stations, as shown in Fig. 1, mainly over East China, but sparse in West China, especially the Qinghai–Tibetan Plateau.

Daily precipitation simulations were extracted from 27 GCMs (Table 1) of CMIP5 datasets (Taylor et al., 2012), which can be downloaded at the data portal http://pcmdi9.llnl.gov/. The historical runs cover the same period 1961–2005 as the observations, while future projections are under the two emission scenarios: RCP4.5 and RCP8.5 for 2071–2100 (Moss et al., 2010). One realization, for each GCM, is selected for the climate change analysis, due to the large number of experiments included in the CMIP5 framework. For comparison with observations, all daily GCM precipitation data are spatially interpolated to the 553 observed stations based on the bivariate interpolation in R software (R Core Team, 2016), which applied the bilinear or bicubic spline interpolation function (see more details in Akima, 1978).

Similar to Furrer et al. (2010), we define an extreme wet spell as persistent rainy days when rainfall is higher than a high percentile-based threshold (i.e., the 95th percentile of JJA daily precipitation from 1971 to 2000), which enables a comparison of changes between different regions (Tolika and Maheras, 2005). As shown in Fig. 1 (shading), the thresholds of wet spells in China broadly increase from northwest to southeast. Furthermore, three indices are calculated to quantify the extreme rainfall spell characteristics: 1) the intensity of wet spell, defined as the maximum daily precipitation, which benefits addressing temporal dependence within each wet spell (Mondal and Mujumdar, 2015); 2) the duration of wet spell, as the length of period with amounts exceeding the threshold; and 3) the frequency of wet spell, which corresponds to the number of occurrences of wet spells per summer.

2.2 GCM bias correction

It is important to first remove systematic biases from the GCM outputs (Teutschbein and Seibert, 2012). The quantile–quantile scaling (QQS) correction method, which is based on QM and DEL, is therefore proposed, to obtain more accurate future projections for the present study. Unlike other bias correction methods, QQS does not adjust the GCM simulations directly, but rather combines observations with GCM mean change signals in each precipitation quantile interval. In another word, future precipitation is derived by scaling the observed data (1961–2005) according to the GCM quantile interval mean change signals of the GCM projected rainfalls (2071–2100) relative to the historical runs (1971–2000) under a given climate change scenario.

As GCMs simulate too many low-intensity rainfall events compared with observations (Teutschbein and Seibert, 2012), a GCM-specific precipitation threshold (P_{mod, th}) at each station is extracted by comparing the observed and simulated differences of the empirical cumulative distribution functions (ECDFs) for the historical period (1961–2005), and this threshold is used to adjust the wet day frequencies by setting the historical (P_{mod, his}) and future projection (RCP4.5 and RCP8.5) daily precipitation (P_{mod, fut}) rates to 0 mm day^–1 where they are less than P_{mod, th}:

$P_{\rm mod, \,his}^{*1} = \left\{ {\begin{array}{*{20}{l}}{0, }&{{\rm if}\;{P_{\rm mod, \,his}} < {P_{\rm mod,\, th}}}\\{{P_{\rm mod, \,his}}, }&{\rm otherwise}\end{array}} \right., $

(1)

$P_{\rm mod, \,fut}^{*1} = \left\{ {\begin{array}{*{20}{l}}{0, }&{{\rm if}\;{P_{\rm mod, \,fut}} < {P_{\rm mod, \,th}}}\\{{P_{\rm mod, \,fut}}, }&{\rm otherwise}\end{array}} \right.. $

(2)

Then, projections of future daily precipitation $P_{\rm mod, \,fut}^{*}$ can be obtained by multiplying the observed precipitation during 1961–2005 by the GCM quantile mean scaling factors (QMS), in each precipitation quantile interval:

$\begin{array}{l}\!\!\!P_{\rm mod, \,fut}^* = {P_{\rm obs}}\left({{P_{\rm obs}} \in (x_{\rm obs}^{{\alpha _{k - 1}}}, x_{\rm obs}^{{\alpha _k}}\;)} \right) \\\!\cdot\left\{\! {\frac{{{\mu _m}\left[ {P_{\rm mod, \,fut}^{*1}\left({P_{\rm mod, \,fut}^{*1}\! \in \!(x_{\rm mod, \,fut}^{{\alpha _{k - 1}}}, x_{\rm mod, \,fut}^{{\alpha _k}})} \right)} \right]}}{{{\mu _m}\left[ {P_{\rm mod, \,his}^{*1}\left({P_{\rm mod, \,his}^{*1} \!\in \!(x_{\rm mod, \,his}^{{\alpha _{k - 1}}}, x_{\rm mod, \,his}^{{\alpha _k}})} \right)} \right]}}}\! \right\}, \end{array}$

(3)

where ${\mu _m}\left[ \cdot \right]$ represents the mean value operator, and $x_{\rm obs}^{{\alpha _k}}$ , $x_{\rm mod, \,his}^{{\alpha _k}}$ , and $x_{\rm mod, \,fut}^{{\alpha _k}}$ are the precipitation quantile boundaries for the observed historical, GCM historical, and GCM future periods, respectively, corresponding to the probability ${{\alpha _k}}$ , which can be calculated from the ECDFs (Themeßl et al., 2012). The series of $\left\{ {{\alpha _k}} \right\}$ takes values from 0.15 to 0.95 in steps of 0.2, for k = 1, 2, …, 5, while ${{\alpha _0}}$ and ${{\alpha _6}}$ indicate the minimum and maximum probability (i.e., 0 and 1), respectively.

The QMS [the right-most factor in Eq. (3)] calculates GCM-simulated future changes as the ratio of mean precipitation between the future and historical periods within sequential quantile intervals, and uses this ratio to adjust the observed data. Thus, QMS alters the magnitudes of wet spell intensity and frequency directly, and affects duration indirectly. The QQS correction method is employed to individual GCM at each station, to remove systematic biases and calculate new daily precipitation future projections $P_{\rm mod,\, fut}^{*}$ for RCP4.5 and RCP8.5.

2.3 Statistical model

The statistical wet spell model (WSM) was first proposed by Furrer et al. (2010) to fit hot spells. In this study, we use WSM to model the IFD characteristics of extreme rainfall spells (Mondal and Mujumdar, 2015). Following Furrer et al. (2010), for a sufficiently high threshold u, the wet spell intensities can be modeled with a GPD, whose cumulative distribution function is

$\begin{aligned}F\left({x;\xi, {\sigma _u}} \right) = 1 - {\left[ {1 + \xi \left({\frac{{x - u}}{{{\sigma _u}}}} \right)} \right]^{ - 1/\xi }}, x > u, \\ \quad 1 + \xi \left({\frac{{x - u}}{{{\sigma _u}}}} \right) > 0, \end{aligned}$

(4)

where $\xi \;{\rm and}\; {\sigma _u}\left({ > 0} \right)$ are the shape and scale parameters. The parameter $\xi > 0\left({ = 0, < 0} \right)$ implies that wet spell intensity has a heavy (unbounded light, bounded finite) tailed GPD distribution.

The number of wet spells per summer (i.e., frequency) usually follows the Poisson distribution, for which its probability distribution function (PDF) is given by

$P\left({Y = k} \right) = \frac{{{\lambda ^k}e - \lambda }}{{k!}},\left({\lambda > 0, k = 0, 1, 2, \cdots } \right).$

(5)

Wet spells occur at an average rate of $\lambda $ per year, with a variance of $\lambda $ .

The aforementioned distributions of the wet spell intensity and frequency are jointly called the Poisson–GP method. The WSM is completed by using the simplest plausible model: the geometric distribution, given its memory less property, to describe the wet spell durations. The geometric distribution function is as follows:

$P\left({Z = k} \right) = {\left({1 - \theta } \right)^{k - 1}}\theta ,\left({\theta > 0, k = 1, 2, \cdots } \right),$

(6)

with the expectation of $1/\theta $ and variance of $\left({1 - \theta } \right)\!/{\theta ^2}$ .

In term of the maximum likelihood method, we can calculate the WSM parameter estimations (i.e., those in Eqs. (4), (5), and (6)), and their standard errors by further using the observed information matrix and delta method. Substituted into Eqs. (4), (5), and (6), the mean characteristics and corresponding probabilities can also be obtained. Furthermore, the K-S (for GPD) and Chi-square (for Poisson and geometric distributions) tests are used to evaluate the goodness of fit of WSM (for details, see Coles, 2001; She et al., 2013).

3 Wet spells in observational records

To demonstrate the application of the WSM and to measure its performance, three observation stations in eastern China (i.e., Beijing, Wuhan, and Guangzhou; see Fig. 1) are used. Taken together, the stations are representative of major cities in northern, central, and southern China (Wang et al., 2015). In recent years, with increasing population and accelerating urbanization, in areas where urban surface rainwater drainage infrastructure has been inadequately maintained, heavy rainfall has led to increasing events of urban flooding, involving major damage to property and loss of life in these cities. For example, the heaviest rain in six decades fell on Beijing, the capital of China, during a 20-h period on July 21–22, 2012, resulting in unprecedented losses, with 79 people killed, at least 8200 houses destroyed, and a total economic cost of more than $1.86 billion (Wang et al., 2013; Wen et al., 2015).

Using the 95th percentile of precipitation during 1971–2000 gives thresholds of 44.8, 56.6, and 47.3 mm day^–1 for Beijing, Wuhan, and Guangzhou, respectively (Table 2). These are used to extract wet spells from the observations. Then, the IFD of summer extreme rainfall spells from these representative stations are fitted to the WSM (i.e., the GPD, Poisson, and geometric distributions). The maximum likelihood estimates of the WSM parameters and their standard errors are listed in Table 2. The results (Fig. 2) show that the WSM realistically models summer extreme wet spells at the three stations, and all of them passed the K-S and Chi-square significance tests. The estimated scale parameters ( $\sigma $ ) for GPD are 19.58, 32.96, and 30.26 for Beijing, Wuhan, and Guangzhou, respectively. Compared with previous modeling of hot spells (Furrer et al., 2010; Wang et al., 2015), wet spell intensities show greater variability and complexity. Figures 2a–c suggest that the distribution of wet spell intensities can be well fitted by GPD. The mean wet spell intensities, calculated from Eq. (4), are 67.5, 97.8, and 79.4 mm day^–1 for Beijing, Wuhan, and Guangzhou, respectively (Table 3).

The estimated Poisson parameters (λ), which represent the mean wet spell frequencies, are 2.13, 1.93, and 3.07 events yr^–1, for Beijing, Wuhan, and Guangzhou, respectively (Tables 2 and 3). The number of observed wet spells per summer and the estimated Poisson PDFs (obtained by using Eq. (5); Figs. 2d–f), show that, although there are some discrepancies between the two distributions, the observations justify the use of a Poisson distribution in the WSM. The lower panels of Fig. 2 also suggest a good agreement between the observed and fitted distributions of wet spell durations for the three sites. Note that wet spell duration estimates provided here are limited in their precision by the daily resolution of the precipitation data. Their mean wet spell durations are 1.104, 1.149, and 1.087 days, respectively.

Following the same approach, the WSM is used to model extreme rainfall spells at all 553 gauge stations in China, and most of which passed the K-S and Chi-square significance tests (figure omitted). Figure 3 provides the spatial patterns of mean wet spell intensity (MWSI), frequency (MWSF), and duration (MWSD), estimated by the WSM using the observed precipitation during 1961–2005, and related probabilistic measures, namely, the 100-yr return level (RL100), the probability of a frequency of at least 5 events per year (PrF5), and the probability of a wet spell duration of at least 2 days (PrD2). The MWSI distribution (Fig. 3a) is similar to the threshold pattern (Fig. 1), but the magnitudes are much larger. Thresholds vary from 1.20 to 98.85 mm day^–1, while MWSI values lie in the range 3.2–160.3 mm day^–1. Both spatial patterns resemble the climatological precipitation distribution, in which magnitudes decrease from southeast to northwest, with the lowest values over western China, while the highest over the Yangtze–Huai River basin and the southeast coast of China.

As shown in Figs. 3c, e, wet spells with higher MWSF (3.3–4.1 events per summer) and longer MWSD (1.16–1.32 days) are found in Southwest China. This may be related to the lower wet spell thresholds and mountainous terrain of this region. We also find higher MWSF but relatively shorter MWSD over Northeast China. Overall, the MWSF over China is around 1–4 events per summer, and the MWSD is around 1.0–1.3 days.

The wet spell intensity return level is an important quantity in decision-making with respect to planning and water resource management (She et al., 2013). The RL100 distribution (obtained from Eq. (4); Fig. 3b) is consistent with MWSI, with values decreasing from southeast to northwest, spanning a range of 16.6–740.7 mm day^–1. The PrF5 distribution matches that of the MWSF, with largest values in Southwest China, where the probability of five or more wet spells in one summer is 25%–40% (Fig. 3d). Finally, the PrD2 distribution is also similar to that of the MWSD with highest values in Southwest China (Fig. 3f)

4 CMIP5 future projection

Compared with the previous generation of climate models (CMIP3), the CMIP5 GCMs, with a higher resolution, provide more robust predictions of climate extremes (Taylor et al., 2012). Due to complex interactions between climate phenomena, including the East Asian summer monsoon, the precipitation simulation over China remains challenging, and most GCMs still exhibit systematic biases in the region (Jiang et al., 2015). We now use the WSM to evaluate the GCMs and select those with the best representation of extreme rainfall spells. Following this, the QQS bias correction method is applied to the selected models by mapping the GCM quantile interval mean change signals (between the historical and projection experiments) onto observed daily precipitation. This produces corrected GCM future daily precipitation projections that are used to discuss possible future changes of wet spells in Subsection 4.3.

4.1 GCM model evaluation and selection

The WSM is applied to the historical experiment data (1961–2005) from the 27 GCMs for which accumulated daily precipitation are available for both the historical and projection experiments. Four indices, namely the JJA mean precipitation (MP), MWSI, MWSF, and MWSD, estimated separately from the GPD, Poisson, and geometric distributions, are calculated for evaluating the individual GCM performance of mean summer daily precipitation and wet spell IFD characteristics.

Figure 4 provides the spatial correlation coefficients between the 27 GCMs and observations during the historical period of 1961–2005, for the four estimated indices. Here, t-test is used to test correlations for significance, and yields a critical correlation coefficient, at the 5% level, of 0.083. It is found that most of the pattern correlations between GCMs and observations are significantly positive. For MP, the majority of models (22 out of 27) show spatial correlation coefficients with observations of 0.5–0.8. This indicates that most of the GCMs simulate reasonably well the spatial distribution of MP. Similarly, MWSI is well simulated by the majority of GCMs, with correlation coefficients exceeding 0.5 for 18 models. It is notable that better simulation of MP does not necessarily mean better reproduction of MWSI; for example, inmcm4 shows a correlation coefficient of 0.78 for MP, but only 0.28 for MWSI. The lowest correlation coefficient among these two indices is 0.14, for MWSI in CanESM2.

For MWSF, 14 of 27 GCMs show pattern correlation coefficients above 0.5. However, MIROC-ESM and MIROC-ESM-CHEM display negligible correlations with the observations. This suggests that these GCMs have less skill in simulating wet spell frequency than in simulating MP and MWSI. Correlations for MWSD are generally lower, with only 5 out of 27 correlation coefficients exceeding 0.5. The worst-performing GCMs (CanESM2, CNRM-CM5, and IPSL-CM5B-LR) show correlations of around 0.3.

In summary, all GCMs show better simulation in MP and wet spell intensity, and worse performance in the frequency and duration of extreme rainfall spells. Based on the correlation coefficients shown in Fig. 4, we select the five GCMs with the best overall performance: ACCESS1.3, BCC-CSM1.1-m, CCSM4, HadGEM2-ES, and MPI-ESM-MR. Note that, there are many other possible ways to measure the performance of climate models, but we use one relative criterion here, in line with the objectives of this study.

4.2 Scaling future daily rainfall

The QQS bias correction procedure is now applied to the five selected GCMs, to obtain corrected GCM projections of future daily precipitation, by scaling a climate change signal, that is, QMS in Eq. (3) to the observed daily precipitation. Figure 5 illustrates the spatial distribution of the multi-model ensemble (MME) QMS change (i.e., QMS minus 1; QMSC) of the five best GCMs in six quantile segment intervals. It is notable that wetter future values are simulated over the Qinghai–Tibetan Plateau (28°–40°N, 75°–95°E) in all quantile intervals. This is caused in part by model overestimation at sparse observation stations (Jiang et al., 2015). Generally, MME precipitation changes are most positive (wetter) for higher quantiles, and negative (drier) for lower quantiles. This means that most of China is projected to see more extreme rainfall and less low-intensity precipitation in the late 21st century.

In RCP4.5 (Figs. 5a–f), except for positive values over the Qinghai–Tibetan Plateau and in southeastern coastal areas, the QMSC in the lowest quantile interval of [0, 0.15) is negative over most of China, and especially in the upper–middle Yangtze River (YZB) and Yellow River basin (YRB), and in Northwest Xinjiang. In contrast, the QMSC in the interval [0.15, 0.35) is weakly positive over most of eastern China. Moving to higher quantile intervals, areas showing decreasing precipitation become smaller, and only small negative areas over Northwest China remain in the interval [0.95, 1]. The QMSC in RCP8.5 (Figs. 5g–l) presents a similar distribution to those in RCP4.5, but with stronger positive and negative values.

Although the five GCMs show similar spatial patterns over most of China, there are differences in the QMSC in some areas in the different quantile segment intervals. Figure 6 depicts the QMSC of 5 GCMs and their MME in the interval [0.95, 1], which is of particular importance to the analysis that follows. In RCP4.5 (Figs. 6a–f), the QMSC over the middle–upper YZB and YRB is weakly negative for BCC-CSM1.1-m, but positive for the other four models. South and Northeast China show similar features. Overall, since the positive values are larger than the negative values, the MME QMSC shows positive change ratios over most of China. Compared with RCP4.5, QMSC in RCP8.5 (Figs. 6g–l) are similar, but much wetter. For instance, QMSC for BCC-CSM1.1m is shifted from negative in RCP4.5 to positive in RCP8.5 over central regions of YZB and YRB, and the same occurs for CCSM4 in South China.

4.3 Projected changes in wet spells in the late 21st century

As above, three representative stations are used to illustrate the future projections of extreme rainfall spell characteristics estimated by the QQS-corrected WSM. Note that the future projections of precipitation are essentially observations scaled by the QMS, and we use the same thresholds as for the observations to define extreme wet spells. Figure 7 shows the estimated IFD PDFs for the MME of five selected GCMs, at the three stations in RCP4.5 and RCP8.5. The corresponding estimations of MWSI, MWSF, and MWSD at the three sites are given in Table 3. Observed IFD PDFs and their mean values, for comparison, are also shown in Fig. 7 and Table 3.

Generally, in the late 21st century, both wet spell intensity and frequency of the three stations show a remarkable increase (one exception being in frequency at Wuhan), higher in RCP8.5 but lower in RCP4.5 (Figs. 7a–f and Table 3). As the GPD scale parameter is larger in both RCP4.5 and RCP8.5, the PDFs of wet spell intensity are much flatter, with more substantial tails, than the observed PDFs, meaning that it is more likely to have stronger wet spells in the future scenarios. The changes in MWSI confirm this interpretation, as values at the three sites are increased over historical observations by 4.2–12.9 mm day^–1 in RCP4.5, and by 9.1–14.2 mm day^–1 in RCP8.5, with the largest increase at Beijing (Fig. 7a and Table 3). The PDFs for wet spell frequency at Beijing and Guangzhou in both RCP4.5 and RCP8.5 show a substantial positive shift. The estimated MWSFs at Beijing (Guangzhou) are increased by 0.84 (0.84) events yr^–1 in RCP8.5 compared with RCP4.5, and by 0.91 (1.15) events yr^–1 compared with the historical observations. The PDFs at Wuhan show a smaller positive shift, and MWSF is increased by only around 0.22 events yr^–1. In contrast, wet spell durations change almost imperceptibly in future scenarios, and MWSDs show very small changes (from –0.007 to 0.045 days) from the observations. There are two possible reasons for this result: the restriction of the precipitation data to daily resolution, as mentioned above, and the relatively high threshold definition used for wet spells. The QQS method is able to capture wet spell intensity and frequency changes, but the high thresholds mean that wet spell durations tend not to change greatly.

Table 4 lists the corresponding probabilistic measures (RL100, PrF5, and PrD2) of wet spells at the three stations for the future scenarios and the observed period. RL100 shows increases at all three sites, which is consistent with the changes in the GPD PDFs and MWSI. Compared to the observed period, the increases in RL100 are 46.1 (51.7) mm day^–1 at Beijing and 47.3 (72.6) mm day^–1 at Guangzhou in RCP4.5 (RCP8.5). At Wuhan, RL100 shows a smaller change of only 26.9 (19.0) mm day^–1 in RCP4.5 (RCP8.5). This is partly due to the much higher threshold and MWSI than at the other two stations. PrF5 at the three sites behaves similarly, with probabilities increased by more than 10% at Beijing and 15% at Guangzhou, but only around 2% at Wuhan, under the two emissions. Similarly to MWSD, PrD2 at the three stations shows small positive changes (1%–3.6%) at Wuhan and Guangzhou, and a negative or near-zero change at Beijing.

Next, we present MME-averaged projections of future extreme rainfall spell characteristics in RCP4.5 and RCP8.5 (figures omitted), as estimated by the WSM, at all 553 stations, along with changes (Fig. 8) from historical observations. Wet spell intensity and frequency both show substantial increases. MWSI ranges from 3.3 to 170.6 mm day^–1, with a highest increase over historical observations of 15.6 mm day^–1 in RCP4.5 (Fig. 8a), and from 3.3 to 180.7 mm day^–1, with a highest increase of 22.3 mm day^–1 in RCP8.5 (Fig. 8b). It is notable that increases are seen almost everywhere, with the exception of small areas around the upper YZB and Northwest Xinjiang. The greatest increase areas are over the middle–lower YZB and YRB. MWSF ranges from 1.1 to 5.5 events yr^–1, with a greatest increase over historical observations of 1.6 events yr^–1 in RCP4.5 (Fig. 8c), and from 1.1 to 6.4 events yr^–1 in RCP8.5, with a greatest increase of 2.8 events yr^–1 (Fig. 8d). There are the biggest increases in southeastern and northeastern China, especially in RCP8.5, while differences over Northwest Xinjiang and the lower YZB are near-zero or negative. Increases in MWSD are weak, by comparison, and vary among different areas (Figs. 8e, f). The largest increase in MWSD is 0.08 (0.12) days in RCP4.5 (RCP8.5). Decreases in MWSD, although weak, are seen over the upper, middle–lower YZB and YRB, in contrast to increases in intensity and frequency seen in the same areas (Figs. 8e, f). This suggests that future extreme rainfall spells will be more intense and more frequent, but shorter in duration, over YZB and YRB.

Further to the changes in mean wet spell characteristics, Fig. 9 shows the projected changes to extreme rainfall spell probability distributions. In RCP4.5 (RCP8.5), RL100 is increased by an average of 22.6 (37.8) mm day^–1, and by a maximum of 139.8 (205.1) mm day^–1, respectively (Figs. 9a, b). The pattern of increases in RL100 is largely consistent with the increases in MWSI. The largest increases are over the middle YZB, lower YRB, and South China. Similarly, PrF5 shows increases consistent with the changes in MWSF. PrF5 values range from 0.6% to 65.1% in RCP4.5, with a maximum increase over historical observations of 30%, and from 0.6% to 76.0% in RCP8.5, with a maximum increase of 46.6% (Figs. 9c, d). Consistent with the small changes in MWSD, PrD2 shows relatively little future changes (Figs. 9e, f). Changes in PrD2 range from –1.9% to +7.5% in RCP4.5, and from –2.4% to +8.2% in RCP8.5. Since the probability is determined by MWSD, the sign of the change in PrD2 presents a similar distribution to the sign of the change in MWSD (Figs. 8e, f).

5 Conclusions and discussion

We used WSM (Furrer et al., 2010) to model and project extreme wet spells over China in 1961–2005 and in 2071–2100. A wet spell is defined as the consecutive rainy days where daily precipitation exceeds a relatively high percentile-based threshold (the 95th percentile of daily precipitation), which makes it possible to compare changes among different regions. Three indices were introduced to characterize the wet spells, namely, the maximum rainfall (intensity), the persistent days (duration), and the occurrences per summer (frequency). Furthermore, wet spell intensity is fitted with a GPD, frequency with a Poisson distribution, and duration with a geometric distribution, respectively. The results show that the WSM is capable of realistically modeling extreme rainfall spells during 1961–2005, as inferred from comparisons with observations at 553 stations throughout China.

The difficulty encountered in simulating the climate system suggests that most GCMs exhibit systematic biases in precipitation over China (Jiang et al., 2015). To more reliably project future changes in extreme rainfall spells in the late 21st century, two optimization measures were used on outputs from 27 GCMs. Firstly, the five best GCMs were selected according to their ability to reproduce major features of observed historical precipitation distributions. This ability was objectively assessed by measuring spatial correlations between model output and observations in terms of MP and by using the three indices mentioned previously (MWSI, MWSF, and MWSD). Most of the GCMs performed relatively well in simulating the patterns of MP and MWSI, and less well in MWSF and MWSD. The five GCMs selected were ACCESS1.3, BCC-CSM1.1-m, CCSM4, HadGEM2-ES, and MPI-ESM-MR.

Next, a bias correction procedure (QQS), which combines quantile–quantile mapping and the delta-change approach (Teutschbein and Seibert, 2012; Fang et al., 2015), was applied to the five leading GCMs. Unlike most other bias correction procedures, QQS does not adjust the GCM output directly, but instead captures GCM projections of future changes by mapping the GCM mean change signals for different quantile intervals [measured between the historical (1971–2000) and future (2071–2100) experiments] onto observed daily precipitation patterns. In this way, corrected GCM projections of daily precipitation in two RCP scenarios were obtained.

The WSM was used to calculate IFD and probability distribution projections of extreme wet spells, using the same threshold definition as for observations, from the MME of QQS-corrected GCM daily precipitation. The results showed that in the late 21st century, the wet spell intensity and frequency will have substantial increases, with the largest increases in intensity over the mid–lower YZB and YRB, while the highest increases in frequency over southeastern and northeastern China. The RCP8.5 scenario was found to produce much larger increases than RCP4.5. The probabilistic projections indicate that RL100 will be improved by over 10.9% (20.1%), while PrF5 by 5.6% (8.2%) over most stations of China in RCP4.5 (RCP8.5). These results mean that China will experience more severe and more frequent extreme rainfall in the late 21st century, which are consistent with the previous studies (e.g., Chen, 2013; Zhou T. J. et al., 2014; Chen and Sun, 2015). In contrast, changes in wet spell duration were found to be small and regionally varying.

Under the assumption of stationary EVT, the WSM was employed to study future changes based on direct QQS scaling of rainfall. The atmospheric circulations that drive the changes in wet spell characteristics have not been explicitly modeled. In reality, changes in global average temperature, local temperature, wind speed, humidity, and large-scale circulation also contribute to changes in wet spell IFD. It is not obvious how these mechanisms can be handled by EVT. To address this problem, non-stationary EVTs are introduced and widely used in recent years.Cheng and AghaKouchak (2014) investigated changes in precipitation IFD, with respect to their possible impact on infrastructure design, based on non-stationary GEV using Bayesian inference. Wan et al. (2013) used the non-stationary GPD to study patterns of the extreme monthly rainfall over China and their association with the ENSO. However, due to the complexity of China's climatology, non-stationary extensions of EVT remain less commonly used to study extreme rainfall spells and their relation to circulation changes, which suggests an interesting direction for future work.

In this study, we have proposed QQS for capturing the GCM change signal between historical and future periods. The choice of bias correction scheme has a significant impact on the climate change projection (Teutschbein and Seibert, 2012). Additional GCM bias correction methods should be tested in order to clarify the sensitivity of projected changes in wet spells to this procedure. In addition, most bias corrections are assumed to be stationary, that is, the relations between observations and model output are the same in the historical and future periods. Non-stationary model correction methods, developed recently (Kallache et al., 2011), are worthy of further attention. Furthermore, the 553 station rainfall dataset is used in this paper, and we may not obtain higher resolution climate changes. The gridded observations in China of CN05, derived from more than 2400 stations, are widely used and will make an attractive alternative (Xu et al., 2009; Wu and Gao, 2013). The related work and the comparison will be done in the future study.