Multivariate Hydrological Frequency Analysis Research Articles

Nonstationary Multivariate Hydrological Frequency Analysis in the Upper Zhanghe River Basin, China

Design annual runoff is a classical issue in traditional univariate hydrological frequency analysis (HFA). We developed a multivariate HFA approach for designs for a study region covering the confluence of two streams. HFA relies on the assumption that probability distribution is consistent in the past, present, and future; however, it has been asserted as incorrect under an uncertain and changing environment. A change-point was detected in our study and adopted to divide runoff into two periods, with no significant trends in all subseries. The post-change design annual runoff witnessed dramatic mean value decline by about half at four frequencies (50%, 75%, 90% and 95%), which were selected in the bivariate analysis. Probability distribution models were constructed with univariate p-III distributions through Clayton, Frank, and Gumbel copulas and independence. Frank copula showed a generally better match with observations than others. The traditional approach, adding up the same-frequency results from both tributaries independently, was disproved by the systematically smaller design values in independence model than copulas and the 40% asynchronous encounter probability. The 25.6% worst synchronous dry-dry encounter situation may be a concern for water resource managers. Consequently, multivariate HFA should prevail as design approach in terms of water resources security.

Multivariate missing data in hydrology – Review and applications

Abstract Water resources planning and management require complete data sets of a number of hydrological variables, such as flood peaks and volumes. However, hydrologists are often faced with the problem of missing data (MD) in hydrological databases. Several methods are used to deal with the imputation of MD. During the last decade, multivariate approaches have gained popularity in the field of hydrology, especially in hydrological frequency analysis (HFA). However, treating the MD remains neglected in the multivariate HFA literature whereas the focus has been mainly on the modeling component. For a complete analysis and in order to optimize the use of data, MD should also be treated in the multivariate setting prior to modeling and inference. Imputation of MD in the multivariate hydrological framework can have direct implications on the quality of the estimation. Indeed, the dependence between the series represents important additional information that can be included in the imputation process. The objective of the present paper is to highlight the importance of treating MD in multivariate hydrological frequency analysis by reviewing and applying multivariate imputation methods and by comparing univariate and multivariate imputation methods. An application is carried out for multiple flood attributes on three sites in order to evaluate the performance of the different methods based on the leave-one-out procedure. The results indicate that, the performance of imputation methods can be improved by adopting the multivariate setting, compared to mean substitution and interpolation methods, especially when using the copula-based approach.

Spatial Pattern Characterization and Multivariate Hydrological Frequency Analysis of Extreme Precipitation in the Pearl River Basin, China

Consecutive extreme rainfall events, especially those having unfavourable spatio-temporal patterns, always trigger large floods. This paper aims to examine, through the multivariate hydrological frequency analysis, the probability of the synchronous occurrence of rainfall extremes in the Pearl River basin. The copula method together with the stationarity and independence tests, which are crucial to the valid use of statistical methods in regional frequency analyses, were applied in the study. The obtained results indicate that: (1) major precipitation events of the annual maximum 1-, 3-, 5- and 7-day rainfall recorded at 42 stations are the flat looking series and variables are independent, (2) the marginal distribution of all extreme rainfall variables in four homogeneous hydrologic regions fits the log-normal probability distribution and most of their joint distribution fits the Gumbel-Hougaard distribution, (3) on that basis the contour maps of the joint distribution of annual maximum 1-, 3-, 5- and 7-day rainfall between different regions are drawn and the probability of the synchronous occurrence of the extreme rainfalls in different regions are estimated. These findings have great practical value for the regional water resources and flood risk management and are important in exploration of the spatial patterns of rainfall extremes in the Pearl River basin in order to reveal the underlying linkages between precipitation and floods from a broader geographical perspective.

Depth-based multivariate descriptive statistics with hydrological applications

[1] Hydrological events are often described through various characteristics which are generally correlated. To be realistic, these characteristics are required to be considered jointly. In multivariate hydrological frequency analysis, the focus has been made on modeling multivariate samples using copulas. However, prior to this step, data should be visualized and analyzed in a descriptive manner. This preliminary step is essential for all of the remaining analysis. It allows us to obtain information concerning the location, scale, skewness, and kurtosis of the sample as well as outlier detection. These features are useful to exclude some unusual data, to make different comparisons, and to guide the selection of the appropriate model. In the present paper we introduce methods measuring these features, which are mainly based on the notion of depth function. The application of these techniques is illustrated on two real-world streamflow data sets from Canada. In the Ashuapmushuan case study, there are no outliers and the bivariate data are likely to be elliptically symmetric and heavy-tailed. The Magpie case study contains a number of outliers, which are identified to be real observed data. These observations cannot be removed and should be accommodated by considering robust methods for further analysis. The presented depth-based techniques can be adapted to a variety of hydrological variables.

Reducing uncertainty in estimates of frequency distribution parameters using composite likelihood approach and copula‐based bivariate distributions

Conventional multivariate hydrological frequency analysis utilizes only the concurrent parts of data sets, leaving a lot of nonconcurrent data unutilized. Simultaneous inclusion of such nonconcurrent data can significantly reduce uncertainty in hydrologic design estimates. The methodology proposed in this paper allows varied length multivariate data to be combined and analyzed in an integrated framework through a “Composite Likelihood Approach.” The method employs copula‐based multivariate distributions in order to provide necessary flexibility of admitting arbitrary marginals. The paper presents the theoretical basis of the approach and highlights its advantages through two applications. A significant reduction in uncertainty in design flood quantiles of a relatively shorter flood series is achieved by utilizing an associated downstream flood data. The advantage of the methodology is further demonstrated by establishing significant information gain for six different combinations of Gaussian and non‐Gaussian marginals. The proposed approach marks a paradigm shift in hydrologic design procedures, particularly for partially gauged basins, wherein a higher precision in hydrologic designs is achieved by leveraging associated information that has hitherto remained unutilized. It is opined that the approach will enable offsetting the impact of dwindling hydrological observation networks around the world by enhancing information that is derivable from existing networks.

Trivariate copulas for characterisation of droughts

Australian agriculture is at serious risk from drought, and water resource infrastructure and management can mitigate the effects. The consequences of droughts depend on their intensity, duration and severity. These variables are correlated and the dependence structure is here described by copulas. Copulas are multivariate uniform distributions which allow for the dependence structure to be modelled independently of the marginal distributions. Trivariate Gaussian and Gumbel copulas are fitted to the data from a rainfall district in NSW. We assess the goodness of fit of the data to the different forms using several criteria. The data are best described by a Gumbel copula and three parameter Weibull marginal distributions. References P. Embrechets, F. Lindskog and A. J. McNeil. Modelling dependence with copulas and applications to risk management. In: Rachev S. T., editor. Handbook of heavy tailed distributions in finance. Amsterdam, North--Holland: Elsevier, 2003. A-C Favre, S. E. Adlouni, L. Perreault, N. Thiemonge and B. Bobee. Multivariate hydrological frequency analysis using copulas. Water Resources Research , 2004, 40 , W01101. doi:10.1029/2003WR002456 C. Genest and J. MacKay. The joy of copulas: bivariate distributions with uniform marginals. The American Statistician , 1986, 40 (4), 280--283. C. Genest and A-C Favre. Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrologic Engineering , 2006, 12 (4), 347--368. doi:10.1061/(ASCE)1084-0699(2007)12:4(347) S. Grimaldi and F. Serinaldi. Asymmetric copula in multivariate flood frequency analysis. Advances in Water Resources , 2006, 29 , 1155--1167. doi:10.1016/j.advwatres.2005.09.005 H. Joe. Multivariate models and dependence concepts . New York: Chapman and Hall, 1997. T. B. McKee, N. J. Doesken and J. Kliest. The relationship of drought frequency and duration to time scales. In Proceeding of the 8th Conference on Applied Climatology , 17--22 January 1993, Anaheim, CA, American Meteorological Society: Boston, MA, USA, 179--184. R Development Core Team. R: A Language and Environment for Statistical Computing . R Foundation for Statistical Computing. 2007. http://www.R-project.org J. T. Shiau. Fitting Drought Duration and Severity with Two-Dimensional Copulas. Water Resources Management , 2006, 20 , 795--815. doi:10.1007/s11269-005-9008-9 A. Sklar. Fonctions de repartitions a $n$ dimensions et leurs marges. Publications de l'Institut de Statistique de l'Universite de Paris , 1959, 8 , 229--231. The Australian. El Nino bad news for drought-stricken east . The Australian, 14th September. Accessed 19th September. http://www.theaustralian.news.com.au/printpage/0,5942,2041027 L. Zhang and V. P. Singh. Bivariate Flood Frequency Analysis using Copula Method. Journal of Hydrologic Engineering , 2006, 11 (2), 150--164. doi:10.1061/(ASCE)1084-0699(2006)11:2(150)

Multivariate hydrological frequency analysis using copulas

This article presents the modeling of multivariate extreme values using copulas. Our approach allows us to model the dependence structure independently of the marginal distributions, which is not possible with standard classical methods. The methodology has been applied on two different problems in hydrology. The first application is concerned with the combined risk in the framework of frequency analysis. Four copulas have been tested on peak flows from the watershed of Peribonka in Québec, Canada. The second application relates to the joint modeling of peak flows and volumes. Three copulas have been applied to the watershed of the Rimouski River in Québec, Canada. This approach using copulas is promising since it allows us to take into account a wide range of correlation which can happen in hydrology.