On the distribution of flood volume in partial duration series analysis of flood phenomena

A methodology based on the theory of stochastic processes is applied to the analysis of floods. The approach will be based on some results of the theory of extreme values over a threshold. In this paper, we focus on the estimation of the distribution of the flood volume in partial duration series analysis of flood phenomena, by using a bivariate exponential distribution of discharge exceedances and durations over a base level.     

A new seasonal design flood method based on bivariate joint distribution of flood magnitude and date of occurrence

Abstract

Seasonal design floods which consider information on seasonal variation are very important for reservoir operation and management. The seasonal design flood method currently used in China is based on seasonal maximum (SM) samples and assumes that the seasonal design frequency is equal to the annual design frequency. Since the return period associated with annual maximum floods is taken as the standard in China, the current seasonal design flood cannot satisfy flood prevention standards. A new seasonal design flood method, which considers dates of flood occurrence and magnitudes of the peaks (runoff), was proposed and established based on copula function. The mixed von Mises distribution was selected as marginal distribution of flood occurrence dates. The Pearson Type III and exponential distributions were selected as the marginal distribution of flood magnitude for annual maximum flood series and peak-over-threshold samples, respectively. The proposed method was applied at the Geheyan Reservoir, China, and then compared with the currently used seasonal design flood methods. The case study results show that the proposed method can satisfy the flood prevention standard, and provide more information about the flood occurrence probabilities in each sub-season. The results of economic analysis show that the proposed design flood method can enhance the floodwater utilization rate and give economic benefits without lowering the annual flood protection standard.

Citation Chen, L., Guo, S. L., Yan, B. W., Liu, P. & Fang, B. (2010) A new seasonal design flood method based on bivariate joint distribution of flood magnitude and date of occurrence. Hydrol. Sci. J. 55(8), 1264–1280.     

Derivation of a curve number and kinematic-wave based flood frequency distribution

Abstract

The physically-based flood frequency models use readily available rainfall data and catchment characteristics to derive the flood frequency distribution. In the present study, a new physically-based flood frequency distribution has been developed. This model uses bivariate exponential distribution for rainfall intensity and duration, and the Soil Conservation Service-Curve Number (SCS-CN) method for deriving the probability density function (pdf) of effective rainfall. The effective rainfall-runoff model is based on kinematic-wave theory. The results of application of this derived model to three Indian basins indicate that the model is a useful alternative for estimating flood flow quantiles at ungauged sites.     

Preliminary flood frequency estimates for Lebanon

Abstract

This paper describes a first attempt at developing a regional flood estimation methodology for Lebanon. The analyses are based on instantaneous flood peak data for the whole country, and cover the period from the start of observations in the 1930s to the start of the civil war in the mid-1970s. Three main flood-generating zones are identified, and regional flood growth curves are derived for each zone using the Generalized Extreme Value distribution fitted by probability-weighted moments. Typical parameter values are presented, together with regression coefficients for estimating the mean annual flood. Based on this work, several recommendations are made on the future data collection and analysis requirements to develop a national flood estimation methodology for Lebanon.     

A bivariate gamma distribution for use in multivariate flood frequency analysis

A gamma distribution is one of the most frequently selected distribution types for hydrological frequency analysis. The bivariate gamma distribution with gamma marginals may be useful for analysing multivariate hydrological events. This study investigates the applicability of a bivariate gamma model with five parameters for describing the joint probability behavior of multivariate flood events. The parameters are proposed to be estimated from the marginal distributions by the method of moments. The joint distribution, the conditional distribution, and the associated return periods are derived from marginals. The usefulness of the model is demonstrated by representing the joint probabilistic behaviour between correlated flood peak and flood volume and between correlated flood volume and flood duration in the Madawask River basin in the province of Quebec, Canada. Copyright © 2001 John Wiley & Sons, Ltd.     

Non-linear growth of cumulative flood losses with time

Statistical data over the past 30 years show that the cumulative sum of losses caused by floods S(t) has been increasing with time approximately as t^1·3, i.e. faster than the linear growth expected for a stationary process. (Losses are evaluated by the number of homeless caused by floods, since these data are the most systematically reported.) At the same time, the factors determining flood losses (the rate of floods and single loss distribution) appear to be stationary over the period of observation. An explanation of this paradox is suggested based on a heavy-tail distribution function of losses, i.e. a distribution function with infinite expectation value. The proposed stochastic model predicts a faster than linear growth of the cumulative losses until some limiting time, which corresponds to the recurrence period of the maximal possible single loss. Similar pseudo-non-stationary effects can be observed for other types of catastrophes and hydrological characteristics with heavy-tail distributions © 1998 John Wiley & Sons, Ltd.     

A cross entropy method for flood frequency analysis

The cross-entropy method with fractile constraints has been developed to estimate a random variable when the data are a set of independent observations of the variable. The method can claim several advantages over existing methods. It uses a reference distribution like the prior distribution in Bayesian analysis and likewise generates a posterior distribution.The method is of interest, in particular, because it satisfies two fundamental requirements for selfconsistency in the analysis of a probabilistic system based on data: a principle of invariance and a principle of data monotonicity.The method is applied to flood analysis. Robustness of the minimum cross-entropy method is compared with other methods: the methods of moments and the maximum likehood.     

Use of a derived distribution approach for flood prediction in poorly gauged basins: A case study in Italy

A derived distribution approach is developed for flood prediction in poorly gauged basins. This couples information on the expected storm scaling, condensed into Depth Duration Frequency curves, with soil abstractions modeled using Soil Conservation Service Curve Number method and hydrological response through Nash’s Instantaneous Unit Hydrograph. A simplified framework is given to evaluate critical duration for flood design. Antecedent moisture condition distribution is included. The method is tested on 16 poorly gauged Mediterranean watersheds in Tyrrhenian Liguria, North Western Italy, belonging to a homogeneous hydrological regions. The derived flood distribution is compared to the regional one, currently adopted for flood design. The evaluation of Curve Number is critical for peak flood evaluation and needs to be carefully carried out. This can be done including local Annual Flood Series data in the estimation of the derived distribution, so gathering the greatest available information. However, Curve Number influence decreases for the highest return periods. When considerable return periods are required for flood design and few years of data are available, the derived distribution provides more accurate estimates than the approach based on single site distribution fitting. A strategy based on data availability for application of the approach is then given. The proposed methodology contributes to the ongoing discussion concerning PUB (Prediction in Ungauged Basins) decade of the IAHS association and can be used by researchers and practitioners for those sites where no flood data, or only a few, are available, provided precipitation data and land use information are at hand.     

Impulse response of a linear diffusion analogy model as a flood frequency probability density function

Abstract

The impulse response of a linear convective-diffusion analogy (LD) model used for flow routing in open channels is proposed as a probability distribution for flood frequency analysis. The flood frequency model has two parameters, which are derived using the methods of moments and maximum likelihood. Also derived are errors in quantiles for these parameter estimation methods. The distribution shows that the two methods are equivalent in terms of producing mean values—the important property in case of unknown true distribution function. The flood frequency model is tested using annual peak discharges for the gauging sections of 39 Polish rivers where the average value of the ratio of the coefficient of skewness to the coefficient of variation equals about 2.52, a value closer to the ratio of the LD model than to the gamma or the lognormal model. The likelihood ratio indicates the preference of the LD over the lognormal for 27 out of 39 cases. It is found that the proposed flood frequency model represents flood frequency characteristics well (measured by the moment ratio) when the LD flood routing model is likely to be the best of all linear flow routing models.     

Bayesian flood frequency analysis in the light of model and parameter uncertainties

The specific objective of the paper is to propose a new flood frequency analysis method considering uncertainty of both probability distribution selection (model uncertainty) and uncertainty of parameter estimation (parameter uncertainty). Based on Bayesian theory sampling distribution of quantiles or design floods coupling these two kinds of uncertainties is derived, not only point estimator but also confidence interval of the quantiles can be provided. Markov Chain Monte Carlo is adopted in order to overcome difficulties to compute the integrals in estimating the sampling distribution. As an example, the proposed method is applied for flood frequency analysis at a gauge in Huai River, China. It has been shown that the approach considering only model uncertainty or parameter uncertainty could not fully account for uncertainties in quantile estimations, instead, method coupling these two uncertainties should be employed. Furthermore, the proposed Bayesian-based method provides not only various quantile estimators, but also quantitative assessment on uncertainties of flood frequency analysis.     

Cautionary note on multicomponent flood distributions for annual maxima

Multicomponent probability distributions such as the two‐component Gumbel distribution are sometimes applied to annual flood maxima when individual floods are seen as belonging to different classes, depending on physical processes or time of year. However, hydrological inconsistencies may arise if only nonclassified annual maxima are available to estimate the component distribution parameters. In particular, an unconstrained best fit to annual flood maxima may yield some component distributions with a high probability of simulating floods with negative discharge. In such situations, multicomponent distributions cannot be justified as an improved approximation to a local physical reality of mixed flood types, even though a good data fit is achieved. This effect usefully illustrates that a good match to data is no guarantee against degeneracy of hydrological models. Copyright © 2016 John Wiley & Sons, Ltd.     

基于二次重现期的多变量洪水风险评估

由于洪水是一种具有多个特征属性的随机事件,频率分析成为洪水风险评估的一种有效手段,多变量重现期与设计值的定义与计算则是洪水频率分析中的重点和难点.本文通过构造洪水历时、洪峰与洪量的联合分布,介绍了一种新的多变量重现期定义——二次重现期,并探讨了"或"重现期、"且"重现期和二次重现期对安全与危险域识别的差异性,以及在洪水风险管理与工程设计中的合理性与可靠性.传统的"或"和"且"多变量重现期对安全与危险域的识别存在局限性,利用Kendall函数定义的二次重现期则提供了更加合理的安全与风险域识别,避免了对安全事件与危险事件的错误判定,更有利于指导洪水风险的管理.在给定的二次重现期条件下,依据出现概率最大原则推算的历时、洪峰与洪量设计值组合可以满足工程设计以较低成本承受较大风险的追求,相比于单变量设计值,考虑了洪水多个属性联合特征的多变量设计值提供了更加全面和可靠的参考信息.     

A Bayesian approach for estimating extreme flood probabilities with upper-bounded distribution functions

Some recent research on fluvial processes suggests the idea that some hydrological variables, such as flood flows, are upper-bounded. However, most probability distributions that are currently employed in flood frequency analysis are unbounded to the right. This paper describes an exploratory study on the joint use of an upper-bounded probability distribution and non-systematic flood information, within a Bayesian framework. Accordingly, the current PMF maximum discharge appears as a reference value and a reasonable estimate of the upper-bound for maximum flows, despite the fact that PMF determination is not unequivocal and depends strongly on the available data. In the Bayesian context, the uncertainty on the PMF can be included into the analysis by considering an appropriate prior distribution for the maximum flows. In the sequence, systematic flood records, historical floods, and paleofloods can be included into a compound likelihood function which is then used to update the prior information on the upper-bound. By combining a prior distribution describing the uncertainties of PMF estimates along with various sources of flood data into a unified Bayesian approach, the expectation is to obtain improved estimates of the upper-bound. The application example was conducted with flood data from the American river basin, near the Folsom reservoir, in California, USA. The results show that it is possible to put together concepts that appear to be incompatible: the deterministic estimate of PMF, taken as a theoretical limit for floods, and the frequency analysis of maximum flows, with the inclusion of non-systematic data. As compared to conventional analysis, the combination of these two concepts within the logical context of Bayesian theory, contributes an advance towards more reliable estimates of extreme floods.     

太湖流域设计暴雨时空分布对太湖洪水位影响分析

太湖是太湖流域最大的调蓄水体,合理地推求太湖流域设计暴雨,对于太湖设计洪水位确定非常重要.针对近年来太湖流域变化环境造成的暴雨特性及产汇流机制的变异,采用水文水动力学模型模拟分析了现状条件下太湖流域设计暴雨控制时段及时空分布对太湖洪水位影响.结果表明,以30、60、90日为控制时段的设计雨量与太湖最高洪水位关联密切,控制时段低于30日的暴雨时程分配对太湖最高洪水位基本没有影响.当设计暴雨中心位于太湖上游区域时,模拟的太湖洪水位具有明显升高的趋势,表明太湖洪水位对上游暴雨更为敏感.分析了1999、2016、2020年暴雨为典型的设计暴雨场景,结果表明,暴雨时程分配对太湖洪水位影响显著,主雨峰位于暴雨后期的设计暴雨可以造成更高的太湖洪水位.从太湖防洪安全考虑,采用30、60、90日为控制时段,暴雨中心位于上游,且雨峰位于暴雨过程后期的设计暴雨推求太湖洪水位是合适的.建议将2016、2020年暴雨过程列入太湖设计暴雨计算的备选典型,并作进一步分析论证.     

Applicability of Wakeby distribution in flood frequency analysis: a case study for eastern Australia

Parametric method of flood frequency analysis (FFA) involves fitting of a probability distribution to the observed flood data at the site of interest. When record length at a given site is relatively longer and flood data exhibits skewness, a distribution having more than three parameters is often used in FFA such as log‐Pearson type 3 distribution. This paper examines the suitability of a five‐parameter Wakeby distribution for the annual maximum flood data in eastern Australia. We adopt a Monte Carlo simulation technique to select an appropriate plotting position formula and to derive a probability plot correlation coefficient (PPCC) test statistic for Wakeby distribution. The Weibull plotting position formula has been found to be the most appropriate for the Wakeby distribution. Regression equations for the PPCC tests statistics associated with the Wakeby distribution for different levels of significance have been derived. Furthermore, a power study to estimate the rejection rate associated with the derived PPCC test statistics has been undertaken. Finally, an application using annual maximum flood series data from 91 catchments in eastern Australia has been presented. Results show that the developed regression equations can be used with a high degree of confidence to test whether the Wakeby distribution fits the annual maximum flood series data at a given station. The methodology developed in this paper can be adapted to other probability distributions and to other study areas. Copyright © 2014 John Wiley & Sons, Ltd.     

Models for extremes using the extended three-parameter Burr XII system with application to flood frequency analysis / Modèles d’extrêmes utilisant le système Burr XII étendu à trois paramètres et application à l’analyse fréquentielle des crues

Abstract

Abstract A new theoretically-based distribution in frequency analysis is proposed. The extended three-parameter Burr XII distribution includes the generalized Pareto distribution, which is used to model the exceedences over threshold; log-logistic distribution, which is also advocated in flood frequency analysis; and Weibull distribution, which is a part of the generalized extreme value distribution used to model annual maxima as special cases. The extended Burr distribution is flexible to approximate extreme value distribution. Note that both the generalized Pareto and generalized extreme value distributions are limiting results in modelling the exceedences over threshold and block extremes, respectively. From a modelling perspective, generalization might be necessary in order to obtain a better fit. The extended three-parameter Burr XII distribution is therefore a meaningful candidate distribution in the frequency analysis. Maximum likelihood estimation for this distribution is investigated in the paper. The use of the extended three-parameter Burr XII distribution is demonstrated using data from China.     

Non-identical models for seasonal flood frequency analysis

Abstract

The aim is to build a seasonal flood frequency analysis model and estimate seasonal design floods. The importance of seasonal flood frequency analysis and the advantages of considering seasonal design floods in the derivation of reservoir planning and operating rules are discussed, recognising that seasonal flood frequency models have been in use for over 30 years. A set of non-identical models with non-constant parameters is proposed and developed to describe flows that reflect seasonal flood variation. The peak-over-threshold (POT) sampling method was used, as it is considered to provide significantly more information on flood seasonality than annual maximum (AM) sampling and has better performance in flood seasonality estimation. The number of exceedences is assumed to follow the Poisson distribution (Po), while the peak exceedences are described by the exponential (Ex) and generalized Pareto (GP) distributions and a combination of both, resulting in three models, viz. Po-Ex, Po-GP and Po-Ex/GP. Their performances are analysed and compared. The Geheyan and the Baiyunshan reservoirs were chosen for the case study. The application and statistical experiment results show that each model has its merits and that the Po-Ex/GP model performs best. Use of the Po-Ex/GP model is recommended in seasonal flood frequency analysis for the purpose of deriving reservoir operation rules.     

Asymmetric copula in multivariate flood frequency analysis

The univariate flood frequency analysis is widely used in hydrological studies. Often only flood peak or flood volume is statistically analyzed. For a more complete analysis the three main characteristics of a flood event i.e. peak, volume and duration are required. To fully understand these variables and their relationships, a multivariate statistical approach is necessary. The main aim of this paper is to define the trivariate probability density and cumulative distribution functions. When the joint distribution is known, it is possible to define the bivariate distribution of volume and duration conditioned on the peak discharge. Consequently volume–duration pairs, statistically linked to peak values, become available. The authors build trivariate joint distribution of flood event variables using the fully nested or asymmetric Archimedean copula functions. They describe properties of this copula class and perform extensive simulations to highlight differences with the well-known symmetric Archimedean copulas. They apply asymmetric distributions to observed flood data and compare the results those obtained using distributions built with symmetric copula and the standard Gumbel Logistic model.