Joint Return Periods Research Articles

Trivariate Frequency Analyses of Peak Discharge, Hydrograph Volume and Suspended Sediment Concentration Data Using Copulas

Copula functions are often used for multivariate frequency analyses, but discharge and suspended sediment concentrations have not yet been modelled together with the use of 3-dimensional copula functions. One hydrological station from Slovenia and five stations from USA with watershed areas from 920 km2 to 24,996 km2 were used for trivariate frequency analyses of peak discharges, hydrograph volumes and suspended sediment concentrations. Different parametric marginal distributions were applied and parameters were estimated with the method of L-moments. Maximum pseudo-likelihood method was used for copula parameters estimation. With the use of statistical and graphical tests we selected the most appropriate copula model. Symmetric and asymmetric versions of Archimedean copulas were applied according to the dependence characteristics of the individual stations. We selected Gumbel-Hougaard copula as the most appropriate model for all discussed stations. Primary joint return periods OR and secondary Kendall’s return periods were calculated and comparison between selected copula functions was made. We can conclude that copula functions are useful mathematical tool, which can also be used for modelling variables that are presented in this paper.

Evaluation of trends and multivariate frequency analysis of droughts in three meteorological subdivisions of western India

ABSTRACTThis article presents evaluation of trends and multivariate frequency analysis of droughts in three meteorological subdivisions of western India, namely, western Rajasthan, Saurashtra and Kutch and Marathwada regions. These regions are frequently affected by droughts and there is an urgent need for effective planning and management of droughts. Meteorological drought is modelled using Standardized Precipitation Index (SPI) at a time scale of 6 months over 110 years during 1896–2005. Trends in SPI time series are investigated by using nonparametric Mann–Kendall trend test for different time windows: entire study period of 1896–2005 and then splitting into three time windows of 1896–1931, 1932–1966 and 1967–2005. For total study period, the long‐term trend in SPI time series is found to be in an upward direction for the three regions. However, statistically significant downward trend is observed during 1932–1966 for western Rajasthan region, and Saurashtra and Kutch region in the month of June, indicating increase in number of drought occurrences during this period. Further, drought is a multivariate natural calamity characterizing severity, duration and peak; hence, probabilistic assessment of drought characteristics is investigated using copula method. The joint distribution of drought properties is modelled using three fully nested forms of Archimedean copulas: Clayton, Gumbel–Hougaard and Frank and one elliptical class of Student's t copula. On performing various statistical tests as well as upper tail dependence test it is found that Student's t copula better represents trivariate drought properties when compared with other copula families. The joint distribution obtained from the copula is utilized for computation of conditional probabilities and joint return periods. The importance of trivariate frequency analysis of drought is elucidated over univariate and bivariate frequency analysis. Overall, the results of the study could provide valuable insight towards regional drought risk management under changing climate.

Estimating storm surge intensity with Poisson bivariate maximum entropy distributions based on copulas

This paper introduces four kinds of novel bivariate maximum entropy distributions based on bivariate normal copula, Gumbel–Hougaard copula, Clayton copula and Frank copula. These joint distributions consist of two marginal univariate maximum entropy distributions. Four types of Poisson bivariate compound maximum entropy distributions are developed, based on the occurrence frequency of typhoons, on these novel bivariate maximum entropy distributions and on bivariate compound extreme value theory. Groups of disaster-induced typhoon processes since 1949–2001 in Qingdao area are selected, and the joint distribution of extreme water level and corresponding significant wave height in the same typhoon processes are established using the above Poisson bivariate compound maximum entropy distributions. The results show that all these four distributions are good enough to fit the original data. A novel grade of disaster-induced typhoon surges intensity is established based on the joint return period of extreme water level and corresponding significant wave height, and the disaster-induced typhoons in Qingdao verify this grade criterion.

Joint occurrence period of wind speed and wave height based on both service term and risk probability

Return periods calculated for different environmental conditions are key parameters for ocean platform design. Many codes for offshore structure design give no consideration about the correlativity among multi-loads and over-estimate design values. This frequently leads to not only higher investment but also distortion of structural reliability analysis. The definition of design return period in existing codes and industry criteria in China are summarized. Then joint return periods of different ocean environmental parameters are determined from the view of service term and danger risk. Based on a bivariate equivalent maximum entropy distribution, joint design parameters are estimated for the concomitant wave height and wind speed at a site in the Bohai Sea. The calculated results show that even if the return period of each environmental factor, such as wave height or wind speed, is small, their combinations can lead to larger joint return periods. Proper design criteria for joint return period associated with concomitant environmental conditions will reduce structural size and lead to lower investment of ocean platforms for the exploitation of marginal oil field.

Bivariate Rainfall and Runoff Analysis Using Entropy and Copula Theories

Multivariate hydrologic frequency analysis has been widely studied using: (1) commonly known joint distributions or copula functions with the assumption of univariate variables being independently identically distributed (I.I.D.) random variables; or (2) directly applying the entropy theory-based framework. However, for the I.I.D. univariate random variable assumption, the univariate variable may be considered as independently distributed, but it may not be identically distributed; and secondly, the commonly applied Pearson’s coefficient of correlation (g) is not able to capture the nonlinear dependence structure that usually exists. Thus, this study attempts to combine the copula theory with the entropy theory for bivariate rainfall and runoff analysis. The entropy theory is applied to derive the univariate rainfall and runoff distributions. It permits the incorporation of given or known information, codified in the form of constraints and results in a universal solution of univariate probability distributions. The copula theory is applied to determine the joint rainfall-runoff distribution. Application of the copula theory results in: (i) the detection of the nonlinear dependence between the correlated random variables-rainfall and runoff, and (ii) capturing the tail dependence for risk analysis through joint return period and conditional return period of rainfall and runoff. The methodology is validated using annual daily maximum rainfall and the corresponding daily runoff (discharge) data collected from watersheds near Riesel, Texas (small agricultural experimental watersheds) and Cuyahoga River watershed, Ohio.

The Return Period Analysis of Natural Disasters with Statistical Modeling of Bivariate Joint Probability Distribution

New features of natural disasters have been observed over the last several years. The factors that influence the disasters' formation mechanisms, regularity of occurrence and main characteristics have been revealed to be more complicated and diverse in nature than previously thought. As the uncertainty involved increases, the variables need to be examined further. This article discusses the importance and the shortage of multivariate analysis of natural disasters and presents a method to estimate the joint probability of the return periods and perform a risk analysis. Severe dust storms from 1990 to 2008 in Inner Mongolia were used as a case study to test this new methodology, as they are normal and recurring climatic phenomena on Earth. Based on the 79 investigated events and according to the dust storm definition with bivariate, the joint probability distribution of severe dust storms was established using the observed data of maximum wind speed and duration. The joint return periods of severe dust storms were calculated, and the relevant risk was analyzed according to the joint probability. The copula function is able to simulate severe dust storm disasters accurately. The joint return periods generated are closer to those observed in reality than the univariate return periods and thus have more value in severe dust storm disaster mitigation, strategy making, program design, and improvement of risk management. This research may prove useful in risk-based decision making. The exploration of multivariate analysis methods can also lay the foundation for further applications in natural disaster risk analysis.

Evaluation of dam overtopping risk based on univariate and bivariate flood frequency analyses

There is a growing tendency to assess safety of dams by mathematical and statistical methods in hydrosystem engineering. This research presents the application of risk and uncertainty analysis to dam overtopping based on univariate and bivariate flood frequency analyses by applying Gumbel logistic distribution. The bivariate frequency analyses produced six inflow hydrographs with a joint return period of 100 years. Afterward, the overtopping risk of the Doroudzan Dam was evaluated for all six inflow hydrographs by considering quantile of flood peak discharge, initial depth of water in the reservoir, and discharge coefficient of spillway as uncertain variables and using two uncertainty analysis methods; Monte Carlo simulation and Latin hypercube sampling. Finally, the results of both univariate and bivariate frequency analyses were compared to show the significance of bivariate analysis on dam overtopping.

호우사상의 크기순서에 대한 고찰

본 연구에서는 이변량 빈도해석을 이용하여 호우사상에 재현기간을 부여할 때 나타나는 특성을 고찰하였다. 호우사상은 서로 상반된 성격의 구성요소들을 가지고 있어 다른 호우사상과 규모를 비교하는 것이 쉽지 않다. 이는 호우사상의 분석결과를 수문학적 설계에서 활용하는데 어려움을 주는 요인이 되고 있다. 이변량 빈도해석은 이러한 문제를 해결하는데 도움이 될 수 있으나 대상변량 및 결합재현기간의 선택에 따라 그 해석결과가 크게 달라질 수 있으므로 주의가 필요하다. 본 연구에서 이변량 빈도해석 결과에 따라 실제 관측된 호우사상을 크기순서대로 나열할 때 나타나는 특성을 살펴보았다. 그 결과 지속기간이나 조건부재현기간을 적용하는 것은 호우사상의 재현기간 규모를 판단하는데 왜곡된 결과를 줄 수 있는 것으로 검토되었다. 평균강우강도만이 강조되어 크기순위가 결정되기 때문이다. 바람직한 적용방법은 대상변량으로 총강우량과 평균강우강도를 선택하고 결합재현기간을 적용하는 것으로 나타났다. In this study, the return periods of storm event is investigated using bivariate frequency analysis. Ranking storm events is known to be very ambiguous due to contradictory characteristics of storm events components. This make it difficult to use storm event analysis in the hydrologic design. Bivariate frequency analysis of storm events is helpful to deal with ranking storm events, the caution is needed to select bivariate variables and choice appropriate joint return periods. Storm duration or conditional joint return periods adversely affect the decision of magnitude order for storm events. Only mean rainfall intensity is major factor in this case. It is useful for a appropriate ranking of storm events to select total depth and mean intensity of storm events and to use AND joint return periods.

독립호우사상의 확률론적 해석 : 2. 호우사상의 재현기간

본 연구에서는 이변량 극치분포를 이용하여 연최대치 호우사상을 평가하였다. 이를 위해 특정 재현기간을 가지는 호우사상의 강우량을 비동시결합 재현기간, 동시결합 재현기간 그리고 구간조건부 결합재현기간의 세 가지를 이용하여 산정하였다. 이때, 결합재현기간별 호우사상의 값의 크기가 서로 다르게 산정되는 이유를 이변량 분포의 확률특성을 보여주는 사분면을 이용하여 설명하였다. 호우지속기간 24시간인 경우에 동시결합재현기간을 이용하여 산정한 확률강우량은 전통적인 방법으로 얻어진 강우지속기간 24시간의 확률강우량과 유사하게 나타났다. 이러한 결과는 전통적인 강우빈도해석의 제약사항을 극복하는데 도움이 될 것으로 보여진다. 이변량 빈도해석으로 얻어진 확률호우사상은 저류시설물의 계획시 통계적으로 보다 유용하면서도 간단한 설계 호우사상을 제공할 수 있을 것으로 보여진다. In this study, annual maximum storm events are evaluated by applying the bivariate extremal distribution. Rainfall quantiles of probabilistic storm event are calculated using OR case joint return period, AND case joint return period and interval conditional joint return period. The difference between each of three joint return periods was explained by the quadrant which shows probability calculation concept in the bivariate frequency analysis. Rainfall quantiles under AND case joint return periods are similar to rainfall depths in the univariate frequency analysis. The probabilistic storm events overcome the primary limitation of conventional univariate frequency analysis. The application of these storm event analysis provides a simple, statistically efficient means of characterizing frequency of extreme storm event.

Copula-based evaluation of high-and low-flows frequency of the Lake Poyang Basin and the risk assessment

在以气温上升为主要特征的全球气候变化导致区域水循环加剧、极端气候水文事件频发的背景下, 对极端水文事件, 特别是水文极值遭遇频率的科学认识, 对区域防洪抗旱具有重要意义.基于此, 本文引入当前多变量分析中较常用的Copula函数.分析鄱阳湖流域主要支流赣江与抚河、乐安河与昌江的洪水、枯水流量联合概率特征, 并对引起该流域水文极值频率变化的原因及影响进行有益探讨.结果表明:(1)洪水、枯水的联合重现期小于其设计重现期, 而其间现重现期则大于设计重现期, 洪水、枯水联合重现期与设计重现期的差值小于同现氧现期与设计重现期的差值;(2)在设计重现期相同的情况下, 外洲站和李家渡站洪水联合重现期大于枯水联合蕈现期, 而洪水同现重现期小于枯水同现重现期, 并且重现期小于lO年洪水同时遭遇的机率比较大, 重现期大于lO年洪水遭遇的频率迅速的减小.虎山站和李家渡站的洪水和枯水的联合重现期和同现重现期基本一致, 遭受洪水和枯水的频率基本一致;(3)在设计重现期相同的情况下, 外洲站与李家渡站洪水联合重现期略大于虎山站与渡峰坑站, 洪水同现重现期小于虎山站与渡峰坑站.外洲站与李家渡站枯水联合重现期小于虎山站与渡峰坑站, 枯水的同现重现期大于虎山站与渡峰坑站。;The currently global climate changes characterized by increasing temperature have the potential to alter the hydrological cycle in regional and global scale.The intensifying meteor-hydrological extremes should be due to altered hydrological cycle.Sound understanding and good knowledge of probability behaviors of meteor-hydrological extremes are the first step into effective management of resources and enhancement of human mitigalion to natural hazards.In this case, the multivariate analysis method and the Copula function are used in this study with aim to investigate joint probability behaviors of high flow and low flow of the major tributaries of the Lake Poyang, the largest freshwater wetland in China.The results indicate that:(1) The joint return periods (JP) of high and low flows are shorter than the designed ones.However, the joint return periods given X>x and Y>y (JPS) are much longer than the designed ones.(2) Given the same designed return periods, the JPs of high flow at the Waizhou and Lijiadu stations are larger than those of low flow events.Besides, the JPSs of high flow are usually shorter than those of low flow, implying higher probability of high flow events at the Waizhou and Lijiadu stations.The probability of concurrent occurrence of high flow with return periods of < 10 years is larger than that of > 10 years.The JPs and JPSs of high flow and low flow events at the Hushan and Lijiadu stations are approximately similar, implying the similar probability of concurrent occurrence of high and low flow e-vents.(3) Given the same designed return periods, the JPs (JPSs) of high flow events at the Waizhou and Lijiadu stations are larger (smaller) than those at the Hushan and Dufengkeng stations.The JPs (JPSs) of low flow events at the Waizhou and Lijiadu stations are smaller (larger) than those at the Hushan and Dufengkeng stations.

Modelling bivariate rainfall distribution and generating bivariate correlated rainfall data in neighbouring meteorological subdivisions using copula

AbstractThe rainfall patterns of neighbouring meteorological subdivisions of India are similar because of similar climatological and geographical characteristics. Analysing the rainfall pattern separately for these meteorological subdivisions may not always capture the correlation and tail dependence. Furthermore, generating the multivariate rainfall data separately may not preserve the correlation. In this study, copula method is used to derive the bivariate distribution of monsoon rainfall in neighbouring meteorological subdivisions. Different Archimedean copulas are used for this purpose and the best copula is selected based on nonparametric test and tail dependence coefficient. The fitted copula is then applied to derive the bivariate distribution, joint return period and conditional distribution. Bivariate rainfall data is generated with the fitted copula and it is observed with the increase of sample size, the generated data is able to capture the correlation as well as tail dependence. The methodology is demonstrated with the case study of two neighbouring meteorological subdivisions of North‐East India: Assam and Meghalaya meteorological subdivision and Nagaland, Manipur, Mizoram and Tripura meteorological subdivision. Copyright © 2010 John Wiley &amp; Sons, Ltd.

Design Flood Volume of the Three Gorges Dam Project

With an increasing tendency of the flood disaster frequency and intensity in China, risk analysis of design flood volume for the Three Gorges Dam Project (TGP) should be of paramount importance for about one-half of the population and gross domestic product in China. How to predict the floods of Yangtze River with high precision is always a difficult problem, for the vast drainage basin and numerous tributaries. After a brief introduction of extreme value distribution and compound extreme value distribution, the TGP design flood volume predicted by the Chinese Hydraulic Design Code (CHDC) recommended annual maximum data sampling based Pearson type III extrapolation method is compared with the stochastic simulation technique based on the important sampling procedure (ISP) and the multivariate compound extreme value distribution (MCEVD), respectively. Comparison results show that the 100-year joint return period of the 3-day flood volume predicted by ISP and MCEVD is greater than 1,000-year return volume pr...

Assessment of overtopping reliability and benefits of a flood detention dam

There is a growing tendency to assess safety levels of existing dams and to design new dams using probabilistic approaches according to project characteristics and site-specific conditions. This study is a probabilistic assessment of the overtopping reliability of a dam, which will be designed for flood detention purpose, and will compute the benefits that can be gained as a result of the implementation of this dam. In a case study, a bivariate flood frequency analysis was carried out using a five-parameter bivariate gamma distribution. A family of joint return period curves relating the runoff peak discharges to the runoff volumes at the dam site was derived. A number of hydrographs were also obtained under a joint return period of 100 years to observe the variation of overtopping tendency. The maximum reservoir elevation and overtopping reliability were determined by performing a probabilistic reservoir routing based on Monte Carlo simulations.

Probabilistic characterization of drought properties through copulas

This paper shows an application of copulas to the probabilistic analysis of drought characteristics. Drought occurrences are analyzed by the Standardized Precipitation Index (SPI) computed on the mean areal precipitation, aggregated at 6months, observed in Sicily between 1921 and 2003. Assuming a drought period as a consecutive number of intervals where SPI values are less than −1, several characteristics are determined, namely: drought length, mean and minimum SPI values, and drought mean areal extent.Results of a preliminary analysis based on Kendall’s correlation and upper tail dependence coefficient, computed on observed and resampled data, show significant dependence properties between almost all the considered pairs. The four-dimensional joint distribution required to correctly model the stochastic structure of variables is determined by resorting to copula approach. This allows flexibility in choosing suitable marginals and dependence structure, and in simplifying the inference procedure as well. Drought return periods are then computed as mean interarrival time, taking into account two drought characteristics at a time by means of the corresponding bivariate marginals of the fitted four-dimensional distribution.Application of the proposed methodology to Sicilian precipitation series shows a good correspondence between empirical and theoretical joint return periods, thus indicating that copulas are adequate to jointly model drought characteristics and to compute exceedance probabilities of drought events.

Discussion of “Bivariate Flood Frequency Analysis Using the Copula Method” by L. Zhang and V. P. Singh

The authors have applied the theory of Copulas to flood fr quency analysis, modeling the statistical dependence betwe each pair of the following variables: peak, volume, and duratio of a flood episode, and computing some joint return periods interest in hydrologic design and survey. The present commenta would like to make some points clear. The authors have provided appropriate references to statisti of Copula, but missed any reference to the areas of hydrology an water resources. Further, they state that “The copula method h been developed by Sklar 1959 , Genest and McKay 1986 , Ge est and Rivest 1993 , and Nelsen 1999 , but has yet to recei much attention of the environmental and water resources comm nity, its importance notwithstanding.” A confutation of this stat ment is reported below. In recent years, papers have been published describing sever research teams who approached the development and applicatio of Copulas to hydrologic modeling. De Michele and Salvado 2003 and Salvadori and De Michele 2004a modeled the d pendence between the storm duration and storm intensity of rainfall event. Salvadori 2004 and Salvadori and De Miche 2004b discussed the concept of return period of a bivaria event in terms of Copulas, and also introduced the so-called se ondary return period. Favre et al. 2004 studied the combinatio of peak flows at different sites, and the joint modeling of pe flows and volumes. In addition, they used some mixtures of di tributions as marginals, exploiting the possibility offered by cop las to derive the appropriate joint probability distributions. D Michele et al. 2005 modeled the association between the pe and volume of a flood event to calculate the flood hydrographs f a given return period. Recently, Salvadori and De Michele 200 proposed a trivariate frequency analysis for storm intensity an duration of wet and dry runs, through coupling variables wi positive and negative dependency. In addition, they introduc the link of Copulas with renewal processes, providing a nov statistical approach to the calculation of the Antecedent Moistu Condition. Note that some of the above references deal with floo frequency analysis as approached by the authors. The authors were aware of the studies mentioned above, for least two reasons: 1 participation in conferences where the paper were presented e.g., DEMOSTAFI 2004 in Quebec Ci

A comparison of two bivariate extreme value distributions

There are two distinct bivariate extreme value distributions constructed from Gumbel marginals, namely Gumbel mixed (GM) model and Gumbel logistic (GL) model. These two models have completely different structures and their dependence ranges are different. The product-moment correlation coefficient for the former is ρ∈[0,2/3] and the latter is ρ∈[0,1]. It is natural to ask which one is more appropriate for representing the joint probabilistic behavior of two correlated Gumbel-distributed variables. This study compares these two models by numerical experiments. The comparison is based on that: (i) if the two distribution models are identical, then the joint probability and the joint return period computed by the GM model should be the same as those by the GL model; and (ii) if a selected distribution is the true distribution from which sample data are drawn, then the probabilities computed by the theoretical model should provide a good fit to empirical ones. Comparison results indicate that in the range of correlation coefficient ρ∈[0,2/3], both models provide identical joint probabilities and joint return periods, and both indicate a good fit to empirical probabilities; while for ρ∈(2/3,1), only the Gumbel logistic model can be used.

Bivariate frequency analysis: discussion of some useful concepts in hydrological application

AbstractBasic concepts such as conditional probability distributions, conditional return periods, and joint return periods are important to understand and to interpret multivariate hydrological events such as floods and storms. However, these concepts are not well documented in the open literature. This paper assembles and clarifies these concepts, and illustrates their practical utility. Relationships between joint return periods and univariate return periods are also derived. These concepts and relationships are demonstrated by applying a bivariate extreme value distribution to represent the joint distribution of flood peak and volume from an actual basin. Copyright © 2002 John Wiley &amp; Sons, Ltd.

CJ89-52. Numerical analysis of sheet metal deep-drawing process : By Deng Zhi (Beijing University of Sci. and Tech., Beijing, China) and Chen Hezheng (Beijing Inst. of Aero. and Astro., Beijing, China). Chinese J. Mech. Engng25, No. 4 (Dec. 1989), pp. 33–40

In the context of global climate change, drought has become an important factor that affects the maize yield in China. To analyse the impact of drought on maize yield loss in Northeast China in current and future climate scenarios, the Composite Meteorological Drought Index (CI) is introduced to reconstruct the following drought indicators: drought accumulative days (DAD), drought accumulative intensity (DAI), and consecutive drought days (CDD). These three drought indicators are used to describe the three-dimensional characteristics of drought in this study. Sen’s slope method and three-dimensional copula functions are adopted to analyse the variety of drought features, and Ensemble Empirical Mode Decomposition (EEMD) is used to analyse the variations in maize yield. A temporal assessment of the standardized yield residuals series (SYRS) of maize from 1961 to 2014 is conducted. A panel regression model is applied to demonstrate the drought impact on maize yield at various growth stages under the RCP4.5 scenario. The results show that the drought risk level for midwest Jilin Province, western Liaoning, and eastern Heilongjiang increase with global warming in the current scenario. The shorter three-dimensional joint return periods, 44–80 yr, were mainly located in western Jilin Province, Liaodong Peninsula, and northwestern Liaoning. Eastern Heilongjiang has a slightly longer joint return period of 80–100 yr. The SYRS shows a strong statistical correlation with drought indicator variations; drought-prone regions exhibit strong positive correlations. In comparison, excess precipitation regions show strong negative correlations with drought indicators in most growth stages. Drought indicators have a relatively strong association with SYRS at the milky–mature maize growth stage, and the occurrence of drought during this period primarily determines the maize yield changes in the future. Maize yield changes are −2.04%, −2.65% and −1.57% for Liaoning, Jilin, and Heilongjiang Province under the RCP4.5 scenario. These results can be used as a tool for early warning of drought risk to maize, and will accelerate the progress of drought disaster risk management.

EVALUATION OF CONCURRENT CHARACTERISTICS OF RAINFALL AND STORM SURGE BY A STOCHASTIC TYPHOON MODEL

A stochastic typhoon model with three properties of central pressure, moving direction and moving velocity is constructed by using B-spline functions and multi-variate autoregressive models. This stochastic typhoon model generates hourly data of typhoon location, central pressure, moving direction and velocity. Predictors against the rainfall intensity and the storm surge at Osaka are selected by using a stepwise regression analysis. About 2820 typhoons for 600 years are generated, and rainfall intensities and storm surges associated with the simulated typhoons are generated at Osaka. The concurrent characteristics of rainfall and storm surge; such as the frequency distribution of time lags, the spatial distributions of typhoon locations, when the peak rainfall intensities and the maximum storm surges occurred. At the same time the joint return periods of heavy rainfall intensities and large storm surges, are evaluated by using observed and simulated data.