Frequency analysis of precipitation extremes under a changing climate: a case study in Heihe River basin, China

The stationary assumption for the traditional frequency analysis of precipitation extremes has been challenged due to natural climate variability or human intervention. To overcome this challenge, this paper, taking Heihe River basin as the case study, performed the frequency analysis by developing a nonstationary GEV model for those seasonal maximum daily precipitation (SMP) time series with nonstationary characteristics by employing the GEV conditional density estimation network. In addition, the confidence intervals (CIs) of estimated return levels were also investigated by using the residual bootstrap technique. Results showed that, 7 of 12 SMP series were nonstationary. The parameters in the nonstationary model were specified as functions of time varying or correlated climate indices varying covariates. The frequency analysis showed that the return levels varied linearly or nonlinearly with covariates. Precipitation extremes with the same magnitude in the study area were found to be occurring more frequently in the future. The CIs of such return levels increased with time passing, especially those from the more complex GEV11 model, embedding a nonlinear increasing trend in model scale parameters. It implied that the increase of model complexity is likely to result in the increase of uncertainty in estimates. This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (CC BY 4.0), which permits copying, adaptation and redistribution, provided the original work is properly cited (http://creativecommons.org/licenses/by/4.0/). doi: 10.2166/wcc.2020.170 om http://iwaponline.com/jwcc/article-pdf/12/3/772/893800/jwc0120772.pdf er 2021 Qingyun Tian Zhanling Li (corresponding author) Xueli Sun School of Water Resources and Environment, China University of Geosciences, Beijing 100083, China and MOE Key Laboratory of Groundwater Circulation and Environmental Evolution, China University of Geosciences (Beijing), Beijing 100083, China E-mail: zhanling.li@cugb.edu.cn

Studies on precipitation extremes have caused widespread concern due to their considerable impacts on agriculture,

STUDY AREA AND DATASETS
Heihe River basin ( Figure 1) is the second largest inland basin in northwest China with an area of 142,900 km 2 . It is located in the climatic intersection between the Westerlies and the East Asian summer monsoon (Qin et al. ), which belongs to a typical continental arid climate. The basin has an important strategic position in northwest China. The middle reach of the basin, on the ancient 'Silk Road' and the current Asia-Europe Continental Bridge, is one of the top ten commodity grain bases in China, with a long history of agriculture; the Ejina Oasis in the lower reach is an important ecological security barrier. However, water resources in the basin are scarce. Understating the precipitation extremes in the basin is of great importance not only for the local water allocation but also for the ecological restoration.
The basin is divided into three reaches, the upper, middle, and lower reaches, which correspond to mountainous areas, oasis areas, and arid Gobi Desert areas, respectively. The upper reach of the basin, ranging from Qilian Mountain to Yingluoxia gorge, is the area where most of the runoff is generated. The annual average temperature is less than 2 C and the annual precipitation is 200-500 mm, but reaches 700 mm in some mountain areas. The middle reach, ranging from Yingluo gorge to Zhengyi gorge, is the farmland oasis where most

METHODS AND MATERIALS
Trend and stationary tests indicates how strong the trend is and whether it is increasing or decreasing. More details about the test can be found in studies of Libiseller & Grimvall () and Gao et al.
The stationarity test is carried out using the augmented Dickey-Fuller test (ADF), which is an augmented version of the original Dickey-Fuller test, to verify the stationarity of a larger and more complicated set of time series (Said & Dickey ). The null hypothesis in the ADF test is nonstationary, that is, a unit root is present in the time series. The statistic Dickey-Fuller used in the test is a negative number.
The more negative it is, the stronger the rejection of the null hypothesis that there is a unit root at some level of con- Taking time t as the covariate as an example, the cumulative density function (cdf) of a random variable Y drawn from a nonstationary GEV distribution is given by: where μ(t), α(t), and κ(t) are the location, scale, and shape parameters, respectively. The location parameter specifies the center of the distribution, the scale parameter gives an indication of the size of deviations around the location, and the shape parameter governs the tail behavior of GEV distribution. These parameters are the functions of the covariate t.
A flowchart of applied GEV-CDN model is shown in activation function m (·) to the inner product between the covariates and the input-hidden layer weights w (1) ji plus the bias b (1) j : The m (·) is taken to be a sigmoidal function, e.g., the hyperbolic tangent function tanh (·) for the nonlinear GEV-CDN network. The identity function is adopted if the GEV-CDN mapping is to be strictly linear. The value of the k th output from the network o k is then given by the hidden-output layer weights w (2) kj and hidden-output layer biases b (2) k . In the present study, the value of κ* is set to be 0.5 according to the approach of generalized maximum like- where L is the likelihood function, P is the number of model parameters, and N represents the sample size. The model that minimizes AICc is deemed to give the best trade-off between maximizing model fit and minimizing model complexity and is selected for the final use.

Return level estimation
Constant parameters correspond to a stationary GEV model. From Equation (1), the probabilistic quantile q p can be obtained from the following equation: where p is the non-exceedance probability (0 < p < 1), and the annual maximum (or minimum) q p corresponds to the where μ(t), α(t) and κ(t) are time-dependent GEV parameters. 2. transforming residuals from the fitted model to be identically distributed:

RESULTS AND DISCUSSION
Trend and stationary tests

Stationary and nonstationary GEV modeling
GEV-CDN is then used for stationary and nonstationary extreme value analysis for all the 12 SMP series. The performance of four candidate GEV-CDN models (GEV0, GEV1, GEV2, and GEV11) are tested by using AICc. A smaller AICc value indicates a better fitting.
The AICc values are summarized in Table 2  Besides the variable of time t, the climate indices EASMI and WPI are also taken into account as the covariate in the nonstationary GEV models. The correlations between SMP Note: * and ** represent the significance level of 0.1 and 0.05.

CONCLUSIONS
In this study, the frequency analysis of seasonal maximum 1day precipitation (SMP) in the Heihe River basin is conducted by using the flexible, efficient, and robust GEV- The best nonstationary GEV model is used to estimate the 2-year, 20-year and 100-year return levels of precipitation extremes. As a result, precipitation extremes increase in frequency over time linearly or nonlinearly, and this increase will continue in the future, which means that the occurrence of precipitation extremes in the study area will increase in the future. This study developed nonstationary GEV models for fitting SMP with nonstationary characteristics in the study area by introducing time t or climate indices of WPI and EASMI as covariates. Cannon () and Vasiliades et al.
() considered that a single covariate may be not enough to explain the changes of precipitation. Thus, more covariates together with their interactions need to be considered in developing nonstationary models in future studies. In addition, using just one index of maximum daily precipitation to indicate the precipitation extremes in this study also has a certain limitation since precipitation extremes can be characterized in many different ways, not only the intensity of heavy precipitation, but also its duration and frequency, the accumulated amount of precipitation for