Bivariate drought frequency analysis using the copula method

Droughts are major natural hazards with significant environmental and economic impacts. In this study, two-dimensional copulas were applied to the analysis of the meteorological drought characteristics of the Sharafkhaneh gauge station, located in the northwest of Iran. Two major drought characteristics, duration and severity, as defined by the standardized precipitation index, were abstracted from observed drought events. Since drought duration and severity exhibited a significant correlation and since they were modeled using different distributions, copulas were used to construct the joint distribution function of the drought characteristics. The parameter of copulas was estimated using the method of the Inference Function for Margins. Several copulas were tested in order to determine the best data fit. According to the error analysis and the tail dependence coefficient, the Galambos copula provided the best fit for the observed drought data. Some bivariate probabilistic properties of droughts, based on the derived copula-based joint distribution, were also investigated. These probabilistic properties can provide useful information for water resource planning and management.     

Probabilistic assessment of flood risks using trivariate copulas

In this paper, a copula-based methodology is presented for probabilistic assessment of flood risks and investigated the performance of trivariate copulas in modeling dependence structure of flood properties. The flood is a multi-attribute natural hazard and is characterized by mutually correlated flood properties peak flow, volume, and duration of flood hydrograph. For assessing flood risk, many studies have used bivariate analysis, but a more effective assessment can be possible considering all three mutually correlated flood properties simultaneously. This study adopts trivariate copulas for multivariate analysis of flood risks, and applied to a case study of flood flows of Delaware River basin at Port Jervis, NY, USA. On evaluation of various probability distributions for representation of flood variables, it is found that the flood peak flow and volumes can be best represented by Fréchet distribution, whereas flood duration by log-normal distribution. The joint distribution is modeled using four trivariate copulas, namely, three fully nested form of Archimedean copulas: Clayton, Gumbel–Hougaard, Frank copulas; and one elliptical copula: Student’s t copula. Based on distance-based performance measures, graphical tests, and tail-dependence measures, it is found that the Student’s t copula best representing the trivariate dependence structure of flood properties as compared to the other copulas. Similar results are found for bivariate copula modeling of flood variables pairs, where Student’s t copula performed better than the other copulas. The obtained copula-based joint distributions are used for multivariate analysis of flood risks, in terms of primary and secondary return periods. The resultant trivariate return periods are compared with univariate and bivariate return periods, and addressed the necessity of multivariate flood risk analysis. The study concludes that the trivariate copula-based methodology is a viable choice for effective risk assessment of floods.     

Analysis of drought condition and risk in Peninsular Malaysia using Standardised Precipitation Index

Contrary to the belief that Peninsular Malaysia experiences wet condition throughout the year, prolonged dry condition has lately become a recurrent phenomenon in this region. As a result, country's agricultural sector and water resources have been under severe constraints from this situation. To get a clearer picture of the dry condition in Peninsular Malaysia, the Standardised Precipitation Index, based on the data of monthly rainfall from 50 stations, is derived. Spatial analysis is used to illustrate the percentage of occurrences of dry and very dry events. To evaluate the potential risk due to the dry conditions, we modelled the joint distribution of severity and duration of dry condition by means of bivariate copula. Several copula models were tested, and the model, which best represents the relationship between severity and duration, is determined using Akaike information criterion. Based on the results, the return period for the drought severity, based on the longest duration of drought at each station, can be estimated. This enables the drought risk to be calculated, thus planning on the measures to minimise the impact of a prolonged drought to the societies, which can be done by the relevant authorities.     

基于Copula函数的北京强降水频率及危险性分析

客观分析强降水事件的发生频率及其致灾因子危险性,能为局地洪涝灾害的防灾、减灾规划及灾害预警提供科学依据。探讨了基于二元Copula函数的强降水致灾变量联合分布及其在强降水危险性分析中的应用。利用北京地区2005-2014年逐时降水资料提取强降水事件案例,通过建立能反映两个主要致灾因素--降水持续时间和过程降水量依存关系的二元联合分布模型,计算了北京地区强降水事件条件重现期,并以此为基础开展危险性分析。研究表明,北京地区强降水事件的持续时间多小于24 h,且主要服从广义极值和对数正态分布,而过程降水量则更适用于广义极值分布;通过Gumbel Copula函数能较好刻画过程降水量与持续时间的相互依存关系。北京地区短时强降水重现期受持续时间影响明显,仅基于降水量的重现期估算会低估其致灾危险性,利用基于Copula函数的条件重现期能更合理描述不同强降水情景致灾因子的危险性特征及其空间差异性特征。北京地区持续时间小于12 h、过程降水量在50 mm以上的强降水事件多呈东北-西南走向,而持续时间在6 h以内的50 mm以上强降水则在北京城区及东北部地区更加频繁。     

Parameter estimation of copula functions using an optimization-based method

Application of the copulas can be useful for the accurate multivariate frequency analysis of hydrological phenomena. There are many copula functions and some methods were proposed for estimating the copula parameters. Since the copula functions are mathematically complicated, estimating of the copula parameter is an effortful work. In the present study, an optimization-based method (OBM) is proposed to obtain the parameters of copulas. The usefulness of the proposed method is illustrated on drought events. For this purpose, three commonly used copulas of Archimedean family, namely, Clayton, Frank, and Gumbel copulas are used to construct the joint probability distribution of drought characteristics of 60 gauging sites located in East-Azarbaijan province, Iran. The performance of OBM was compared with two conventional methods, namely, method of moments and inference function for margins. The results illustrate the supremacy of the OBM to estimate the copula parameters compared to the other considered methods.     

Probabilistic characterization of hydrological droughts

It is proposed to use the joint distribution of probabilities of minimum water discharges and the river runoff depth to characterize the severity and danger of durable low-flow periods. The copula theory is used to construct this distribution that enables to take account of differences in the types of one-dimensional probability distributions of initial variables and of the correlation between them. The studies were carried out for the vegetation periods using the data of the runoff depth measurements in Georghiu-Dej on the Don River in 1895–1980 and in Kirov on the Vyatka River in 1878–1980. The selection of copulas was based on comparing empirical and theoretical joint distributions of low-flow characteristics. It is demonstrated that the best results for the approximation of the joint probability distribution of low-flow characteristics are obtained from the Frank copula. Plotted are the dependences of low-flow frequency on the specified values of corresponding minimum water discharges and the river runoff depth during vegetation periods. Estimated are the probability and frequency of some most severe lowflow periods observed on the Don and Vyatka rivers.     

A copula-based multivariate analysis of Canadian RCM projected changes to flood characteristics for northeastern Canada

In the present work, climate change impacts on three spring (March–June) flood characteristics, i.e. peak, volume and duration, for 21 northeast Canadian basins are evaluated, based on Canadian regional climate model (CRCM) simulations. Conventional univariate frequency analysis for each flood characteristic and copula based bivariate frequency analysis for mutually correlated pairs of flood characteristics (i.e. peak–volume, peak–duration and volume–duration) are carried out. While univariate analysis is focused on return levels of selected return periods (5-, 20- and 50-year), the bivariate analysis is focused on the joint occurrence probabilities P1 and P2 of the three pairs of flood characteristics, where P1 is the probability of any one characteristic in a pair exceeding its threshold and P2 is the probability of both characteristics in a pair exceeding their respective thresholds at the same time. The performance of CRCM is assessed by comparing ERA40 (the European Centre for Medium-Range Weather Forecasts 40-year reanalysis) driven CRCM simulated flood statistics and univariate and bivariate frequency analysis results for the current 1970–1999 period with those observed at selected 16 gauging stations for the same time period. The Generalized Extreme Value distribution is selected as the marginal distribution for flood characteristics and the Clayton copula for developing bivariate distribution functions. The CRCM performs well in simulating mean, standard deviation, and 5-, 20- and 50-year return levels of flood characteristics. The joint occurrence probabilities are also simulated well by the CRCM. A five-member ensemble of the CRCM simulated streamflow for the current (1970–1999) and future (2041–2070) periods, driven by five different members of a Canadian Global Climate Model ensemble, are used in the assessment of projected changes, where future simulations correspond to A2 scenario. The results of projected changes, in general, indicate increases in the marginal values, i.e. return levels of flood characteristics, and the joint occurrence probabilities P1 and P2. It is found that the future marginal values of flood characteristics and P1 and P2 values corresponding to longer return periods will be affected more by anthropogenic climate change than those corresponding to shorter return periods but the former ones are subjected to higher uncertainties.     

Frequency analysis of short‐duration rainfalls

Abstract

An investigation was carried out to determine the appropriateness of a probability distribution and fitting technique commonly used in Canada for rainfall frequency analysis.

The extreme value type 1 (EV1) distribution was assessed for three long‐term Canadian stations: Victoria, St Thomas and Québec. The EV1 distribution appears to provide a reasonable fit for durations varying from 5 min to 6 h, but is not clearly superior to another two‐parameter distribution, the lognormal.

The fitting technique, known as modified moments or regression, was assessed by comparing it with three other fitting techniques: moments, maximum likelihood and adjusted maximum likelihood. This comparison was carried out using Monte Carlo simulation techniques over the parameter space deemed to be representative of short duration rainfall data for Canada. In terms of estimating rainfall amounts of specified return period, the modified moments technique was the poorest with regard to both bias and efficiency. In general, the maximum likelihood estimates were the most efficient and were relatively unbiased.     

植被对云南气候要素影响的敏感性试验

利用MM5模式对云南两个具有代表性的强降水过程进行高分辨率模拟,通过下垫面植被的敏感性试验,考察云南气候要素(降水、温度、湿度、风等)对下垫面植被状况的敏感性,从而达到了解自然环境及人类活动在云南天气、气候及气候变化中的作用与影响,以期提高对未来天气、气候变化、环境变化及其对人类社会发展影响的预测和评估能力。试验结果表明,下垫面植被状况的改变对云南气候要素值的影响非常明显,这种影响一般在近地面至700 hPa之间;下垫面植被覆盖率的锐减,使云南降水范围、降水量、空气湿度急剧减小,温度、风力、蒸发能力、干旱指数迅速增大,加剧了云南的干旱化和沙漠化,最终将导致云南的天气气候和环境生态系统偏离本来的平衡状态和演变过程。但下垫面植被状况的改变对降水中心、冷(暖)中心、干(湿)中心位置影响不大。     

Role of intra-seasonal oscillations on monsoon floods and droughts over India

Aim of this diagnostic study is to investigate the impact of intra-seasonal oscillations in terms of number, duration and intensity on rainfall during June through September, 1979–2006. Analysis of wavelet spectra for winds at 850 hPa field for monsoon period reveals number and duration of oscillations, which exercise profound influence on monsoon rainfall. Results indicate that four to six oscillations appear in normal rainfall or flood cases, while two or three oscillations are identified in the years of drought episodes. Though total duration of above oscillations is varied from 25 to 85 days, the duration is short (20 to 35 days) obviously in the years of less number of oscillations and also the number of oscillations are directly related to the monsoon rainfall. The coefficient of correlation between them is 0.56, which is significant at 1% level. To examine the strength of intra-seasonal oscillations in terms of different indices on seasonal rainfall is investigated. The Madden and Julian Oscillation Index shows an inverse relationship with rainfall, where as a direct relationship is noticed between Monsoon Shear Index and rainfall for the study period. Both results are significant at 5% level. To consolidate the above statistical relationships, seasonal circulation changes in the contrasting years of monsoon rainfall have been examined; present study reveals that anomaly negative outgoing longwave radiation is noticed over most of Arabian Sea, Indian sub-continent and the Bay of Bengal during June through September in flood year (1988). But opposite convective activity is true in drought year (2002). Similarly the spatial U-850 hPa field distribution showed much stronger monsoon winds in 1988, while zonal circulation was very weak in 2002. Such differences are observed in the anomaly zonal wind field at 200 hPa also. Over the monsoon region U-850 hPa field is almost a mirror image of U-200 hPa distribution of wind field. Finally annual cycles of U-850 and U-200 hPa fields reflect striking difference at 200 hPa level during the summer monsoon period in flood and drought years.     

压能、湿焓场与暴雨落区的诊断分析

通过分析压能场与湿焓场的分布特征，讨论了它们对暴雨发生可能产生的影响，这种影响实质上是由于动力学与热力学的相互作用所造成的，在本文分析的暴雨仆例中，暴雨过程都与等压能线密集带和等湿焓线密集带有一定联系，在暴雨发生前24h一般有强的正湿焓平流，暴雨主要落区一般出现在平流最强的地方，因此，分析压能场与湿焓场可以获取一些新的信息，这对提高暴雨的预报能力可以起到一定的作用。     

Severity, duration and frequency of drought in SE England from 1697 to 2011

Severe droughts have affected much of Europe over the last 40 years. A limitation to current understanding of droughts is based around drought characteristics (e.g. frequency, severity and duration) as there are limited long series (>100 years) with well documented severe droughts. This is further complicated with future climate projections, and the potential implications that these will have on drought characteristics. This paper presents reconstructed drought series from 1697, 1726 and 1767 to 2011 for three sites in southeast England. Precipitation and temperature series are reconstructed to generate long drought series using the self-calibrated Palmer Drought Severity Index, enabling determination of drought characteristics. The reconstructions identify multiple drought-rich periods, 1730–1760 and 1890-present, with an increasing tendency towards more severe droughts during the latter period. Prolonged rainfall deficiencies are found to be the primary cause of severe droughts, with rising temperatures increasing soil moisture deficit, therefore intensifying drought conditions. Cycles at the 6–10 year period identify a sub-decadal to decadal signal during drought-rich periods. Analysis of the spatial variability of droughts finds that whilst severe events are predominantly regionally coherent, there are notable variations in severity and duration between sites, which are attributed to localised rainfall variability. This study extends the temporal range of previous drought studies and places recent drought events in a longer context improving upon existing ‘benchmark’ drought analyses in southeast England; with far-reaching implications for local, national and continental scale reduction of drought vulnerability and risk.     

基于马尔科夫链转移概率极限分布的降水过程持续性研究

各地不同气候条件所具有的天气状态转移概率的极限分布实际上代表了各地天气气候的持续性和转折性特征,同时也表明了它的可预报期限。从马尔科夫链理论出发,初步研究了中国160个代表测站逐日天气状态演变过程的极限分布。结果表明,转移概率的极限分布不但其空间分布有明显差异,而且不同季节的极限分布也有明显差异。例如,有的地区仅有2 d持续期,有的可达4—6 d或更长,充分反映出不同地区因其影响的天气系统差异所造成的逐日天气气候的持续性和转折性特征的差异。其研究意义在于,由此可作为天气气候分型区划的一种理论依据。统计分析结果表明,就全中国平均而言,夏季持续期最短,持续期由北向南、由西向东呈增加趋势,且春季平均降水持续期为5.1 d,夏季平均为5.0 d,秋季平均为6.5d,冬季平均为6.2 d。可见夏半年比冬半年的降水持续期短,这可能是因为春夏季的天气系统比较复杂且中小尺度天气系统较多的缘故。这从另一侧面再次证明,各地逐日降水天气状态演变过程具有一定的天气气候状态自然转折的持续性即自然天气周期的气候状况,从而为短期天气预报提供了气候背景。     

珠海市年雨量和年最大日雨量多年一遇的极值计算

用皮尔逊-III型分布分别对珠海市年雨量和年最大日雨量两种变量进行拟合,进而计算了它们在不同重现期下的极值。结果表明:珠海市50年一遇的估算年雨量和年最大日雨量分别是2 908 mm和565 mm,100年一遇的分别是3 042 mm和648 mm。用澳门和珠海的雨量历史资料验证发现:澳门在104年间出现的最大年雨量是3 041.4 mm,与珠海百年一遇估算雨量吻合;珠海平均7.2年出现一次2 500 mm以上年雨量,而该雨量的估算重现期是7.5年,亦十分接近;年最大日雨量的估算重现期则比实际重现期小些。可见拟合良好,对其不同重现期下的极值估计结果可信。     

Analysis of mid-twentieth century rainfall trends and variability over southwestern Uganda

A methodology has been applied to investigate the spatial variability and trends existent in a mid-twentieth century climatic time series (for the period 1943–1977) recorded by 58 climatic stations in the Albert–Victoria water management area in Uganda. Data were subjected to quality checks before further processing. In the present work, temporal trends were analyzed using Mann–Kendall and linear regression methods. Heterogeneity of monthly rainfall was investigated using the precipitation concentration index (PCI). Results revealed that 53 % of stations have positive trends where 25 % are statistically significant and 45 % of stations have negative trends with 23 % being statistically significant. Very strong trends at 99 % significance level were revealed at 12 stations. Positive trends in January, February, and November at 40 stations were observed. The highest rainfall was recorded in April, while January, June, and July had the lowest rainfall. Spatial analysis results showed that stations close to Lake Victoria recorded high amounts of rainfall. Average annual coefficient of variability was 19 %, signifying low variability. Rainfall distribution is bimodal with maximums experienced in March–April–May and September–October–November seasons of the year. Analysis also revealed that PCI values showed a moderate to seasonal rainfall distribution. Spectral analysis of the time components reveals the existence of a major period around 3, 6, and 10 years. The 6- and 10-year period is a characteristic of September–October–November, March–April–May, and annual time series.     

Falling Lake Victoria water levels: Is climate a contributing factor?

Recently, and perhaps most threatening, Lake Victoria water level has been receding at an alarming rate. A recent study suggested the possibility of the expanded hydroelectric power station in Uganda. However, since the lake receives 80% of its refill through direct rainfall and only 20% from the basin discharge, climatic contributions cannot be ignored, since the 80% water is directly dependant on it. It is therefore necessary to investigate climatic contribution to the declining Lake Victoria water level observed over a long period, i.e., 30 years. This contribution uses 30 years period anomalies for rainfall, river discharge and lake level changes of stations within Lake Victoria basin to analyse linear and cyclic trends of climate indicators in relation to Lake levels. Linear trend analysis using the Student’s t test indicate a decreasing pattern in rainfall anomalies, with the slope being statistically similar to those of water levels at both Kisumu, Maziba and Jinja stations for the same period of time (1976–1999), thus showing a strong correlation. On the other hand, cyclic trend analysis using Discrete Fourier Transform (DFT) shows cyclic period of water level to coincide with those of droughts and rainfall. The strong relationship between climatic indicators of drought and rainfall on one-hand and lake levels on the other hand signifies the need to incorporate climate information in predicting, monitoring and managing lake level changes.     

Detecting rainfall trends in twentieth century (1871–2006) over Orissa State, India

Climate change has affected the temperature and rainfall characteristics worldwide. However, the changes are not equal for all regions and have localized intensity and must be quantified locally to manage the natural resources. Orissa is an eastern state in India where agricultural activities mainly depends on the rainfall and thus face problems due to changing patterns of rainfall due to changing climate. In the present study, attempts were made to study temporal variation in monthly, seasonal and annual rainfall over the state during the period from 1871 to 2006. Long term changes in rainfall characteristics were determined by both parametric and non-parametric tests. The analysis revealed a long term insignificant decline trend of annual as well as monsoon rainfall, where as increasing trend in post-monsoon season over the state of Orissa. Rainfall during winter and summer seasons showed an increasing trend. Statistically monsoon rainfall can be considered as very dependable as the coefficient of variation is 14.2%. However, there is decreasing monthly rainfall trend in June, July and September, where as increasing trend in August. This trend is more predominant in last 10?year. Based on departure from mean, rainfall analysis also showed an increased number of dry years compared to wet years after 1950. This changing rainfall trend during monsoon months is major concern for the rain-fed agriculture. More over, this will affect hydro power generation and reservoir operation in the region.     

利用二维极值分布模拟我国几个代表站的强降水概率特征

基于中国6个代表站5-9月的逐日降水资料,利用二维Gumbel-Logistic分布,研究了中国不同区域的过程降水量和日最大强降水雨量的联合概率特征。结果表明,各代表性台站的过程雨量和强降水雨量的联合分布均符合二维Gumbel分布。强降水雨量与过程降雨量联合分布所描述的极端事件是更小的小概率事件。相同强降水雨量条件下,过程雨量越大,重现期越长当强降水雨量增大时,同一过程雨量的重现期也延长。在同级强降水雨量出现的条件下,各地过程降雨量往往是愈往南方其条件概率愈大,而其出现的过程雨量也随之增大。这为研究强降水极端状况的全方位特征做出了新的试验．也曼加客观地揭示了极端气候事件的多方面概率特征．     

北京市强降雨分区及重现期研究

利用北京观象台1841—2008年年降水量资料及近50年北京市20个气象站和82个雨量站资料分析了北京地区降水量的时间和空间变化规律,对北京地区的降雨进行分区研究,并结合观象台站逐分钟降雨资料应用广义偏态分布（GPD）方法分析了北京地区不同历时降雨量重现期。结果表明：近168年来,北京地区有两个多雨时段和两个少雨时段,目前北京处于20世纪90年代至今的少雨时段内。70至80年代,北京地区强降雨主要为全市区域性降雨,90年代之后北京的短历时强降雨呈现出局地性的特征,降水分布不均,强降水中心大致成东北一西南向带状分布。根据北京市降雨EOF分析,将北京市划分为4个降雨分区,分别是山后区、城市中心区、东北部山区和东南部平原区。其中城区代表站观象台站多个历时不同重现期降雨量分析结果经过与现行排水规范对比表明,重现期模拟结果可靠。     

Probable maximum precipitation for a 2-day duration over the Indian Peninsula

Summary For the purpose of providing information to hydrologists for designing costly and large hydraulic structures, estimates of probable maximum precipitation (PMP) for a 2-day duration for stations in the Indian Peninsula lying between 8° N to 20° N were calculated using the Hershfield statistical techniqe. Maximum annual 2-day rainfall data of 80 years from 1901 for 131 stations in the region were used. In order to obtain values of PMP, an enveloping frequency factor (k _m ) curve based on the actual rainfall data of stations in the region was developed. The enveloping curve was then utilised to estimate 2-day PMP values of all the 131 stations. Based on these PMP estimates, a generalised chart showing the spatial distribution of 2-day PMP was prepared. It was found that 2-day PMP estimates over the Indian peninsula varied from 40 to 95 cm and the average ratio of 2-day PMP to the highest observed 2-day rainfall was found to be 1.76.With 2 Figures