椭圆型Copulas函数在西安站干旱特征分析中的应用

本文研究了干旱发生的联合概率、条件概率和重现期等干旱特征.以陕西省西安站月降水为例,应用Meta-Gaussian Copula和Student t Copula构造了干旱历时、干旱烈度和烈度峰值的联合概率分布,并进行了多变量分布拟合优质评价及拟合检验,在此基础上计算了联合分布的重现期以及2变量和3变量情形下的条件概率与条件重现期.研究表明,Meta-Gaussian Copula可以描述干旱历时、干旱烈度和烈度峰值三者的联合分布.由于多元联合分布可以考虑到多个变量之间的不同组合,能够求得不同干旱历时、干旱烈度或烈度峰值下的条件概率和条件重现期,因而能够更加全面客观地反映干旱的特征.     

Archimedean族Copulas函数在多变量干旱特征分析中的应用

以甘肃省陇西站月降水资料为例,应用9种3维Archimedean Copulas函数构造了干旱历时、干旱烈度和烈度峰值的联合概率分布,并进行了多变量的拟合优度评价,利用优选出的3维非对称型M12 Copula函数,计算联合分布的重现期以及不同组合下的条件概率与条件重现期。结果表明,M12Copula函数可以描述干旱历时、干旱烈度和烈度峰值间的联合分布。由于Copula函数能够用来构建不同边缘分布的联合分布,可以获得变量间不同组合下的重现期,因而能够更全面客观地反映干旱的特征,是描述干旱多变量分布的一种有效途径。     

非参数方法在干旱频率分析中的应用

运用非参数核密度估计方法研究干旱发生的联合概率、条件概率和重现期等干旱特征,以宁夏盐池站的月降水为例,应用单变量核概率密度函数估计干旱历时D的边缘分布,进行参数方法和非参数方法的拟合效果比较。在此基础上,采用双变量核概率密度函数估计方法构建了历时D与烈度S、历时D与峰值P的两变量联合概率分布,并计算了联合分布的重现期、条件概率与条件重现期。结果表明:与参数方法相比,非参数核密度估计方法能够描述干旱历时D、烈度S和峰值P两两之间的联合分布,是研究干旱频率的另一种新途径。     

基于区域农业用水量的干旱重现期计算方法

为研究实际水利条件下农业干旱的发生规律,简化农业干旱事件的评估方法,提出基于区域农业用水量的干旱重现期计算方法。通过构建农业用水量距平百分率干旱指标W_A,在基于降雨量距平百分率干旱指标P_A识别干旱事件的基础上,提取W_A干旱指标下的干旱历时和干旱烈度特征变量,并根据以P_A为干旱指标的干旱烈度频率分布曲线F_S（x）和干旱历时频率分布曲线F_D（x）,运用Copula的简化方法计算基于W_A的干旱事件重现期T,最后结合基于P_A的干旱事件重现期T₀,回归分析出T与T₀间关系的计算公式。选取干旱灾害影响严重的亳州市为实证区域开展应用研究,计算得到1975-2007年各场干旱事件的T₀和T以及T₀与T的经验关系式。结果表明:T比T₀更合理地反映区域农业实际受旱状况,重现期T₀和T间存在高度的相关关系,采用T的回归方程可简化计算考虑区域实际抗旱能力下的干旱事件重现期,在区域防旱减灾实践中具有推广应用价值。     

基于 Copula 函数的宜昌水文干旱特征分析

基于宜昌站1951～2014年的实测月径流资料,选用标准化径流指数(SSI),运用游程理论识别干旱,应用Copula函数构建干旱特征变量间的多维联合概率分布,进而对宜昌站的干旱特征进行分析。结果表明:(1)宜昌在1950～1980年代,干旱次数呈现交替变化,自1990年代以来,特别是进入21世纪后,宜昌干旱事件增多、持续时间增大、干旱烈度和峰值增高,干旱情势有加重的趋势;(2)Copula函数可很好地描述宜昌地区干旱特征变量间的联合概率分布,多变量的联合重现期和同现重现期可分别作为实际单变量重现期区间估计的下限和上限,用以评估宜昌地区不同干旱变量值所代表的干旱事件发生的频率;(3)宜昌站近60年出现两次严重的干旱事件,一次发生于1978年9月～1979年7月,该事件的干旱历时和干旱烈度均达到了历史极值,这两个变量的联合重现期约为32a,同现重现期约为110a;该事件的干旱历时、干旱烈度和烈度峰值三个变量的联合重现期为9a,同现重现期约为115a。另一次干旱事件发生于2006年6月～12月,其烈度峰值达到了历史极值,其重现期接近90a;该次事件的干旱历时、干旱烈度和烈度峰值三者的联合重现期只有13a左右,同现重现期则超过了231a。     

基于Copula函数的组合变量联合概率分布研究及应用

基于Copula函数原理,利用武江流域实测水文资料,以广义GDP为洪峰洪量边缘分布,构建了流域组合变量Copula概率分布模型,分析了洪峰与洪量、洪量与洪水历时、洪峰与洪水历时的联合概率分布,绘制各种变量组合下的联合分布图及重现期等值线图,并比较了同重现期条件下,洪水单变量设计值与多维联合设计值的区别。结果表明:广义GDP分布能很好的描述洪峰、洪量边缘分布,而基于广义GDP分布和指数分布构建的两变量Copula联合概率分布模型不限定变量的边缘分布,对各种类型的水文变量联合分布拟合效果较好;能全面反映洪水各特征属性不同等级下的联合发生频率,对同一频率下联合分布推求的洪水设计值比单变量设计值偏于安全。基于Copula函数的组合变量概率分布模型描述洪峰流量、洪量、洪水历时等特征的联合分布,较为全面地反映组合特征的洪水发生的概率和重现期,进一步反映洪水风险。     

基于Copulas函数的二维干旱变量联合分布

通过构建干旱变量的联合分布揭示干旱演变规律,可作为干旱分析的重要手段。基于8种单参数族的Copulas函数进行新疆乌鲁木齐和石河子气象站二维干旱变量的联合分布。经拟合优度评价:Frank Copula对干旱历时和干旱烈度、干旱历时和烈度峰值的拟合度最好;Clayton Copula对于干旱烈度和烈度峰值的拟合效果最好。二维变量联合超越概率值随单变量值的减小而增大;单变量的重现期介于二维变量联合重现期与同现重现期之间。表明Copulas函数能够描述二维干旱特征变量的联合分布。     

基于暴雨衰减特性的上海市长历时综合暴雨公式

编制适用于不同历时的综合暴雨公式是协调城市管网排水与区域防洪治涝的重要基础。选用上海市代表雨量站徐家汇站65 a实测雨量资料,建立不同重现期暴雨强度与历时关系,解析暴雨衰减规律,编制单一重现期暴雨公式,结合雨力公式推求适用不同重现期的长历时综合暴雨公式,并推导出暴雨重现期公式。结果表明:不同重现期1~24 h历时暴雨强度均以0.74的衰减指数衰减,据此推求的长历时综合暴雨公式可计算1~24 h任意历时、2~100 a任意重现期的设计暴雨,且平均相对和平均绝对均方根误差分别为1.9%和0.009 mm/min,符合规范要求;暴雨重现期公式可估算1~24 h历时内任意场次暴雨的重现期,高效地服务于城市洪涝防治决策。成果已纳入上海市治涝地方标准,对其他城市具有参考价值。     

水文双变量重现期分析及在干旱中应用

单变量水文统计中一些广为接受的概念在多变量环境下尚缺乏深入分析,也易被误解,如N年内重现期大于等于T的多变量事件发生的次数与N/T的关系。实践中,多变量联合重现期与其边缘分布变量重现期的一些经验关系被发现并通过了案例验证分析,但缺乏解释和推导。基于GH Copula推导了双变量联合重现期与边缘分布变量重现期的关系以及双变量事件发生次数与其重现期、变量相关程度间的定量关系。以昆明56年的逐月SPI(Standardized Precipitation Index)和SRI(Standardized Runoff Index)识别了干旱事件,采用GH Copula构建了干旱历时和烈度的联合分布函数,验证了双变量联合重现期与边缘分布变量重现期的关系以及多变量事件发生次数与其重现期的定量关系。表明不宜以“and”第1重现期是否接近于比该干旱事件的旱情更重的干旱发生的平均时间间隔来说明干旱特征值重现期分析的合理性。变量的相关性不强时,需谨慎采用边缘分布变量重现期的较大值近似代替“and”事件的第1重现期。     

区域供水系统干旱历时特性综合分析

根据区域供水系统中供水水库(群)的蓄水状态及其运行调度策略在满足区域供水目标方面的重要作用,提出了以水库(群)的时段初始蓄水量与时段内来水量之和,以及时段内供水系统的目标需水量作为主要控制参数,以两者的差值正、负来表达供水系统处于非干旱状态和干旱状态的区域干旱鉴别方法。为了多角度地分析干旱和非干旱历时分布、转移概率以及干旱重现期等干旱历时特性以较全面地掌握区域干旱事件特征,针对一具体区域供水系统,运用模拟统计法和概率解析法两种技术手段来研究区域干旱历时特性,同时针对两种方法的计算结果差异,指出了各种方法的特征以及产生差异的原因。     

Severity–duration–frequency curves in the mitigation of drought impact: an agricultural case study

Severity–duration–frequency (SDF) curves are very useful in the analysis of drought phenomena. Station-level information obtained from SDF curves can be interpolated to obtain severity maps for fixed return period, in order to jointly analyse the spatial variability of drought characteristics (e.g. severity, duration and frequency). This approach is limited because the severity is usually quantified through indices that use hydrological and meteorological data, depending on the type of requirements. Therefore, drought indices can only reflect hydrological conditions, but are unable to quantify economic losses associated with droughts. In other words, SDF curves do not allow effective quantification of the impact expected with a certain return period. This paper proposes the methodology drought economic risk assessment (DERA) as an approach that emphasizes the importance of the relationship between a generic drought index (which quantifies water deficit) and the economic impact of the failure to meet water demand. Using integrated SDF curves, this relationship enables drought severity and corresponding impacts to be mapped. This procedure was applied to agricultural droughts (sunflower crop) in Umbria Region (central Italy). The agricultural drought impact variable was identified by sunflower yield (Y); the economic impact variable by net benefit depletion (EL); and the drought index by Relative Severity Index (RSI), which is quantifiable by a soil–water balance model. The relationships Y = g(RSI) and EL = f(Y) were specifically determined. Using DERA, it was possible to derive curves for SDF, impact–duration–frequency, and economic losses–duration–frequency (ELDF), which were then used to map severity, impact and economic losses for the assigned return period and duration. From the ELDF curves, further information was obtained by mapping critical drought durations for the assigned return period and economic loss threshold. The case study supports the potential of the proposed approach, both in the planning and real-time management of drought effects.     

A comparative frequency analysis of three standardized drought indices in the Poyang Lake basin,China

Regional drought frequency analysis was carried out in the Poyang Lake basin (PLB) from 1960–2014 based on three standardized drought indices: the standardized precipitation index (SPI), the standardized precipitation evapotranspiration index (SPEI) and the standardized Palmer drought index (SPDI). Drought events and characteristics were extracted. A Gumbel–Hougaard (GH) copula was selected to construct the bivariate probability distribution of drought duration and severity, and the joint return periods (T _a ) were calculated. Results showed that there were 50 (50 and 40) drought events in the past 55 years based on the SPI (SPEI and SPDI), and 9 (8 and 10) of them were severe with T _a more than 10 years, occurred in the 1960s, the 1970s and the 2000s. Overall, the three drought indices could detect the onset of droughts and performed similarly with regard to drought identification. However, for the SPDI, moisture scarcity was less frequent, but it showed more severe droughts with substantially higher severity and longer duration droughts. The conditional return period (Ts|d) was calculated for the spring drought in 2011, and it was 66a and 54a, respectively, based on the SPI and SPDI, which was consistent with the record. Overall, the SPI, only considering the precipitation, can as effectively as the SPEI and SPDI identify the drought process over the PLB under the present changing climate. However, drought is affected by climate and land-cover changes; thus, it is necessary to integrate the results of drought frequency analysis based on different drought indices to improve the drought risk management.     

Spatial hydrological drought characteristics in Karkheh River basin,southwest Iran using copulas

Investigation on drought characteristics such as severity, duration, and frequency is crucial for water resources planning and management in a river basin. While the methodology for multivariate drought frequency analysis is well established by applying the copulas, the estimation on the associated parameters by various parameter estimation methods and the effects on the obtained results have not yet been investigated. This research aims at conducting a comparative analysis between the maximum likelihood parametric and non-parametric method of the Kendall \(\tau \) estimation method for copulas parameter estimation. The methods were employed to study joint severity–duration probability and recurrence intervals in Karkheh River basin (southwest Iran) which is facing severe water-deficit problems. Daily streamflow data at three hydrological gauging stations (Tang Sazbon, Huleilan and Polchehr) near the Karkheh dam were used to draw flow duration curves (FDC) of these three stations. The \(Q_{75}\) index extracted from the FDC were set as threshold level to abstract drought characteristics such as drought duration and severity on the basis of the run theory. Drought duration and severity were separately modeled using the univariate probabilistic distributions and gamma–GEV, LN2–exponential, and LN2–gamma were selected as the best paired drought severity–duration inputs for copulas according to the Akaike Information Criteria (AIC), Kolmogorov–Smirnov and chi-square tests. Archimedean Clayton, Frank, and extreme value Gumbel copulas were employed to construct joint cumulative distribution functions (JCDF) of droughts for each station. Frank copula at Tang Sazbon and Gumbel at Huleilan and Polchehr stations were identified as the best copulas based on the performance evaluation criteria including AIC, BIC, log-likelihood and root mean square error (RMSE) values. Based on the RMSE values, nonparametric Kendall-\(\tau \) is preferred to the parametric maximum likelihood estimation method. The results showed greater drought return periods by the parametric ML method in comparison to the nonparametric Kendall \(\tau \) estimation method. The results also showed that stations located in tributaries (Huleilan and Polchehr) have close return periods, while the station along the main river (Tang Sazbon) has the smaller return periods for the drought events with identical drought duration and severity.     

Drought characterization using a new copula-based trivariate approach

Meteorological drought is a natural climatic phenomenon that occurs over various time scales and may cause significant economic, environmental and social damages. Three drought characteristics, namely duration, average severity and peak intensity, are important variables in water resources planning and decision making. This study presents a new method for construction of three-dimensional copulas to describe the joint distribution function of meteorological drought characteristics. Using the inference function for margins, the parameters for six types of copulas were tested to select the best-fitted copulas. According to the values of the log-likelihood function, Galambos, Frank and Clayton were the selected copula models to describe the dependence structure for pairs of duration–severity, severity–peak and duration–peak, respectively. Trivariate cumulative probability, conditional probability and drought return period were also investigated based on the derived copula-based joint distributions. The proposed model was evaluated over the observed data of a Qazvin synoptic station, and the results were compared with the empirical probabilities. For measuring the model accuracy, R ², root mean square error (RMSE) and the Nash–Sutcliffe efficiency (NSE) criteria were used. Results indicated that R ², RMSE and NSE were equal to 0.91, 0.098 and 0.668, respectively, which demonstrate sufficient accuracy of the proposed model. Drought probabilistic characteristics can provide useful information for water resource planning and management.     

变化环境下东江流域水文干旱特征及缺水响应

在干旱事件不确定性和枯期径流变异性的双重影响下,水文干旱特征时序非一致性问题为其联合分布模拟带来困难。基于东江干流测站日径流过程数据,采用游程理论提取水文干旱事件,并结合干旱特征均值变化、时序一致性分析及边缘分布模拟,以确定干旱事件融合及剔除评判标准的合理取值。基于Rosenblatt变换Cramer-von Mises检验统计量拟合方法,构建水文干旱特征两变量联合分布Copula模型,并根据同频法设计两变量组合值。通过对比枯期径流变点分隔子序列干旱特征,分析变化环境下东江流域水文干旱特征及缺水响应。结果表明:水文干旱事件融合和剔除的评判标准值分别取0.1和0.3比较合理。干旱特征两变量之间具有较高的正相关性,但不同时间系列不同变量之间的联合分布及边缘分布最优模型并不一致。流域水库尤其是新丰江水库的径流调节作用,对于缓解东江中下游水文干旱效果明显,超阈联合重现期为2年的设计干旱持续时间、总缺水量和最大日缺水量分别减少了63%~71%、71%~84%和30%~47%,但如果要满足东江河道内最小管理流量目标,其依然分别达到了12~18 d、6 114万~9 030万m3和715.0万~929.0万m3。     

Preparing stream flow drought severity–duration–frequency curves using threshold level method

Drought is a natural phenomenon which occurs in different climate regimes. In the present study, hydrological drought of the Roud Zard basin has been investigated based on run theory. Daily runoff data of Mashin hydrometery station during 1970 to 2012 was assessed using 70 % of mean daily runoff as threshold level. Results showed that the maximum drought duration of 309 days occurred in 1998 and 1999 and max drought deficit of 117.217 million cubic meters per second in 1983 with 275 days duration. Time series of the annual maxima values of duration and volume deficit showed similar trend of increase and decreasing. Burr statistical distribution, as the most suitable one fitted to the drought duration data, forecasted 339 days duration for drought event with 20 years return period and generalized extreme value forecasted 37.9 million cubic meters of deficit volume for this return period. Severity-duration-frequency (SDF) curves were prepared, classifying drought durations to four intervals and fitting statistical distribution to each. Resulted SDF curves showed that, in each period, increase of duration resulted in increased value of the volume deficit with a non-linear trend, though predicted drought volume with 20 years return period was 2.1 million cubic meters for 1 to 10 days duration, 6.9 for 11 to 30 days, 34.5 for 31 to 120 days, and 79.1 for more than 120 days duration drought event. Drought deficit volume increasing rate was also different in each class of duration interval. Drought SDF curves derived in this study can be used to quantify water deficit for natural stream and reservoir. SDFs could also be extended to allow for drought regional frequency analysis to be used in ungauged sites.