Impulse response of the kinematic diffusion model as a probability distribution of hydrologic samples with zero values

It is hypothesized that the unit impulse response of a linearized kinematic diffusion (KD) model is a probability distribution suitable for frequency analysis of hydrologic samples with zero values. Such samples may include data on monthly precipitation in dry seasons, annual low flow, and annual maximum peak discharge observed in arid and semiarid regions. The hypothesized probability distribution has two parameters, which are estimated using the methods of moments (MOM) and maximum likelihood (MLM). Also estimated are errors in quantiles for MOM and MLM. The distribution shows an equivalency of MOM and MLM with respect to the mean value—an important property for ML-estimation in the case of the unknown true distribution function. The hypothesized KD distribution is tested on 44 discharge data series and compared with the Muskingum-like (M-like) probability distribution function. A comparison of empirical distribution with KD and M-like distributions shows that MOM better reproduces the upper tail of the distribution, while MLM is more robust for higher sample values and more conditioned on the value of the probability of the zero value event. The KD-model is suitable for frequency analysis of short samples with zero values and it is more universal than the M-like model as its modal value cannot be only equaled to zero value but also to any positive value.     

Three‐parameter discontinuous distributions for hydrological samples with zero values

A consistent approach to the frequency analysis of hydrologic data in arid and semiarid regions, i.e. the data series containing several zero values (e.g. monthly precipitation in dry seasons, annual peak flow discharges, etc.), requires using discontinuous probability distribution functions. Such an approach has received relatively limited attention. Along the lines of physically based models, the extensions of the Muskingum‐based models to three parameter forms are considered. Using 44 peak flow series from the USGS data bank, the fitting ability of four three‐parameter models was investigated: (1) the Dirac delta combined with Gamma distribution; (2) the Dirac delta combined with two‐parameter generalized Pareto distribution; (3) the Dirac delta combined with two‐parameter Weibull (DWe) distribution; (4) the kinematic diffusion with one additional parameter that controls the probability of the zero event (KD3). The goodness of fit of the models was assessed and compared both by evaluation of discrepancies between the results of both estimation methods (i.e. the method of moments (MOM) and the maximum likelihood method (MLM)) and using the log of likelihood function as a criterion. In most cases, the DWe distribution with MLM‐estimated parameters showed the best fit of all the three‐parameter models. Copyright © 2005 John Wiley & Sons, Ltd.     

The impact of seasonal flood peak dependence on annual maxima design quantiles

If the maximum annual peak flow series are a mixture of summer and winter flows, a seasonal approach to flood frequency analysis is necessary. While considering seasonal maxima as mutually independent events, the annual maxima distribution is defined as the product of seasonal distributions. However, if the independency assumption does not hold, a bivariate approach with dependent margins should be applied, i.e. the copula approach. The impact of dependency on design quantiles is investigated here in the context of the Fréchet-Hoeffding inequality defining copula bounds and the definition of dependency. The results of the two approaches are compared using six catchments in the San River basin, where in four cases the dependency of seasonal maxima has been identified as positive significant and no strong dominance of any one season is observed. The product model leads to higher estimates of design quantiles than do models where the dependency is taken into account and, therefore, is safe.

EDITOR R. Woods ASSOCIATE EDITOR A. Fiori     

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.     

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 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.     

Multi-decadal variations in annual maximum peak flows in semi-arid and temperate regions of Chile

Abstract

In determining the possible influence of climate change, it is important to understand the temporal and spatial variability in streamflow response for diverse climate zones. Thus, the aim of this study was to determine the presence of changes in annual maximum peak flow for two climate zones in Chile over the past few decades. A general analysis, a flood frequency analysis and a trend analysis were used to study such changes between 1975 and 2008 for a semi-arid (29°S–32°S) and a temperate (36°S–38°S) climatic zone. The historic annual maxima, minima and mean flows, as well as decadal mean peak flow, were compared over the period of record. The Gumbel distribution was selected to compare the 30-year flood values of two ±15-year intervals, which showed that streamflow decreased by an average of 19.5% in the semi-arid stations and increased by an average of 22.6% in the temperate stations. The Mann-Kendall test was used to investigate the temporal changes in streamflows, with negative trends being observed in 87% of the stations analysed in the semi-arid zone, and positive trends in 57% of those analysed in the temperate zone. These differences in streamflow response between climate zones could be related to recent documented increases in altitude of the zero-degree isotherm in the Andes Mountains of Chile, since most of the significant positive and negative changes were detected in first-order rivers located closer to this mountain range.

Editor D. Koutsoyiannis; Associate editor H. Lins

Citation Pizarro, R., Vera, M., Valdés, R., Helwig, B., and Olivares, C., 2013. Multi-decadal variations in annual maximum peak flows in semi-arid and temperate regions of Chile. Hydrological Sciences Journal, 59 (2), 300–311.     

Probability prediction of peak break‐up water level through vine copulas

Prediction of the peak break‐up water level, which is the maximum instantaneous stage during ice break‐up, is desirable to allow effective ice flood mitigation, but traditional hydrologic flood routing techniques are not efficient in addressing the large uncertainties caused by numerous factors driving the peak break‐up water level. This research provides a probability prediction framework based on vine copulas. The predictor variables of the peak break‐up water level are first chosen, the pair copula structure is then constructed by using vine copulas, the conditional density distribution function is derived to perform a probability prediction, and the peak break‐up water level value can then be estimated from the conditional density distribution function given the conditional probability and fixed values of the predictor variables. This approach is exemplified using data from 1957 to 2005 for the Toudaoguai and Sanhuhekou stations, which are located in the Inner Mongolia Reach of the Yellow River, and the calibration and validation periods are divided at 1986. The mean curve of the peak break‐up water level estimated from the conditional distribution function can capture the tendency of the observed series at both the Toudaoguai and Sanhuhekou stations, and more than 90% of the observed values fall within the 90% prediction uncertainty bands, which are approximately twice the standard deviation of the observed series. The probability prediction results for the validation period are consistent with those for the calibration period when the nonstationarity of the marginal distributions for the Sanhuhekou station are considered. Compared with multiple linear regression results, the uncertainty bands from the conditional distribution function are much narrower; moreover, the conditional distribution function is more capable of addressing the nonstationarity of predictor variables, and the conclusions are confirmed by jackknife analysis. Scenario predictions for cases in which the peak break‐up water level is likely to be higher than the bankfull water level can also be conducted based on the conditional distribution function, with good performance for the two stations.     

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.     

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.     

Hydrological model for urban catchments – analytical development using copulas and numerical solution

Abstract

A model based on analytical development and numerical solution is presented for estimating the cumulative distribution function (cdf) of the runoff volume and peak discharge rate of urban floods using the joint probability density function (pdf) of rainfall volume and duration together with information about the catchment's physical characteristics. The joint pdf of rainfall event volume and duration is derived using the theory of copulas. Four families of Archimedean copulas are tested in order to select the most appropriate to reproduce the dependence structure of those variables. Frequency distributions of runoff event volume and peak discharge rate are obtained following the derived probability distribution theory, using the functional relationship given by the rainfall–runoff process. The model is tested in two urban catchments located in the cities of Chillán and Santiago, Chile. The results are compared with the outcomes of continuous simulation in the Storm Water Management Model (SWMM) and with those from another analytical model that assumes storm event duration and volume to be statistically independent exponentially distributed variables.

Citation Zegpi, M. & Fernández, B. (2010) Hydrological model for urban catchments – analytical development using copulas and numerical solution. Hydrol. Sci. J. 55(7), 1123–1136.     

Relationship between the yearly maxima of peak and daily discharge for some basins in Tuscany

ABSTRACT

The paper deals with the estimation of the probability distribution of the yearly maximum of the peak discharge Q by means of the distribution of the maximum of the daily discharge q and the distribution of the ratio R = Q/q. The study was carried out for some catchments in Tuscany, analysing the dependence of the parameters of R on the geomorphic catchment parameters. The values of the discharge Q, relevant to an assigned return period, obtained by the methods given herein agree rather well (the error is about 20%) with those directly obtained from the observed values of Q.     

最大峰值加速度与有效峰值加速度的大小比例关系及影响因素探讨

分别以最大峰值加速度(以下简称PGA)和有效峰值加速度(以下简称EPA)为参数,对金沙江流域上12个工程场点进行了地震危险性分析,得到了各个场点在不同的年超越概率下的基岩PGA和EPA值。通过对PGA、EPA值比较分析认为:PGA与EPA值的大小比例关系主要受年超越概率大小的影响,当年超越概率较大时,表现为PGA>EPA;当年超越概率较小时,PGA与EPA的比例关系还与场点周围的潜源分布形式及潜源的震级上限的大小有关,不同的年超越概率、不同的潜源分布形式和震级上限,可使PGA>EPA,也可使PGA相似文献   

Random forests as a tool to understand the snow depth distribution and its evolution in mountain areas

The small scale distribution of the snowpack in mountain areas is highly heterogeneous, and is mainly controlled by the interactions between the atmosphere and local topography. However, the influence of different terrain features in controlling variations in the snow distribution depends on the characteristics of the study area. As this leads to uncertainties in high spatial resolution snowpack simulations, a deeper understanding of the role of terrain features on the small scale distribution of snow depth is required. This study applied random forest algorithms to investigate the temporal evolution of snow depth in complex alpine terrain using as predictors various topographical variables and in situ snow depth observations at a single location. The high spatial resolution (1 m x 1 m) snow depth distribution database used in training and evaluating the random forests was derived from terrestrial laser scanner (TLS) devices at three study sites, in the French Alps (2 sites) and the Spanish Pyrenees (1 site). The results show the major importance of two topographic variables, the topographic position index and the maximum upwind slope parameter. For these variables the search distances and directions depended on the characteristics of each site and the TLS acquisition date, but are consistent across sites and are tightly related to main wind directions. The weight of the different topographic variables on explaining snow distribution evolves while major snow accumulation events still take place and minor changes are observed after reaching the annual snow accumulation peak. Random forests have demonstrated good performance when predicting snow distribution for the sites included in the training set with R² values ranging from 0.82 to 0.94 and mean absolute errors always below 0.4 m. Oppositely, this algorithm failed when used to predict snow distribution for sites not included in the training set, with mean absolute errors above 0.8 m.     

Hybrid method to update the design flood shape for spillway operation in a dam: case study of the Huites Dam,Mexico

To design and review the operation of spillways, it is necessary to estimate design hydrographs, considering their peak flow, shape and volume. A hybrid method is proposed that combines the shape of the design hydrograph obtained with the UNAM Institute of Engineering Method (UNAMIIM) with the peak flow and volume calculated from a bivariate method. This hybrid method is applied to historical data of the Huites Dam, Sinaloa, Mexico. The goal is to estimate return periods for the maximum discharge flows (that account for the damage caused downstream) and the maximum levels reached in the dam (measure of the hydrological dam safety) corresponding to a given spillway and its management policy. Therefore, to validate the method, the results obtained by the flood routing of the 50-year hydrograph are compared with those obtained by the flood routing of the three largest historical floods. Both maximum flow and elevation were in the range of values observed within 37.5–75 years corresponding to the length of the historical record.     

On seasonal approach to flood frequency modelling. Part II: flood frequency analysis of Polish rivers

The annual peak flow series of Polish rivers are mixtures of summer and winter flows. As Part II of a sequence of two papers, practical aspects of applicability of seasonal approach to flood frequency analysis (FFA) of Polish rivers are discussed. Taking A Two‐Component Extreme Value (TCEV1) model as an example it was shown in the first part that regardless of estimation method, the seasonal approach can give profit in terms of upper quantile estimation accuracy that rises with the return period of the quantile and is the greatest for no seasonal variation. In this part, an assessment of annual maxima (AM) versus seasonal maxima (SM) approach to FFA was carried out with respect to seasonal and annual peak flow series of 38 Polish gauging stations. First, the assumption of mutual independence of the seasonal maxima has been tested. The smoothness of SM and AM empirical probability distribution functions was analysed and compared. The TCEV1 model with seasonally estimated parameters was found to be not appropriate for most Polish data as it considerably underrates the skewness of AM distributions and upper quantile values as well. Consequently, the discrepancies between the SM and AM estimates of TCEV1 are observed. Taking SM and TCEV1 distribution, the dominating season in AM series was confronted with predominant season for extreme floods. The key argument for presumptive superiority of SM approach that SM samples are more statistically homogeneous than AM samples has not been confirmed by the data. An analysis of fitness to SM and AM of Polish datasets made for seven distributions pointed to Pearson (3) distribution as the best for AM and Summer Maxima, whereas it was impossible to select a single best model for winter samples. In the multi‐model approach to FFA, the tree functions, i.e., Pe(3), CD3 and LN3, should be involved for both SM and AM. As the case study, Warsaw gauge on the Vistula River was selected. While most of AM elements are here from winter season, the prevailing majority of extreme annual floods are the summer maxima. The upper quantile estimates got by means of classical annual and two‐season methods happen to be fairly close; what's more they are nearly equal to the quantiles calculated just for the season of dominating extreme floods. Copyright © 2011 John Wiley & Sons, Ltd.     

Discussion on the influence of truncation of ground motion residual distribution on probabilistic seismic hazard assessment

Recent studies on assessment of a very low annual probability of exceeding (APE) ground motions, 10-4 or less, have highlighted the importance of the upper bound of ground motions when very low probability results are acquired. The truncation level adopted in probabilistic seismic hazard analysis (PSHA) should be determined by an aleatory uncertainty model (i.e., distribution model) of ground motions and the possible maximum and minimum ground motion values of a specific earthquake. However, at the present ...     

Assessing possible changes in flood frequency due to climate change in mid‐sized watersheds in New York State,USA

The frequency of flooding is often presumed to increase with climate change because of projected increases in rainfall intensities. In this paper, using 50‐plus years of historical discharge and meteorological data from three watersheds in different physiographic regions of New York State, USA, we find that annual maximum stream discharges are associated with 20% or less of the annual maximum rainfall events. Instead of rainfall events, approximately 20% of annual maximum stream discharges are associated with annual maximum snowmelt events while 60% of annual maximum discharges are associated with moderate rainfall amounts and very wet soil conditions. To explore the potential for changes in future flood risk, we employed a compound frequency distribution that assumes annual maximum discharges can be modelled by combining the cumulative distribution functions of discharges resulting from annual maximum rainfall, annual maximum snowmelt, and occurrences of moderate rain on wet soils. Basing on a compound frequency distribution comprised of univariate general extreme value (GEV) and gamma distributions, we found that a hypothetical 20% increase in the magnitude of rainfall‐related stream discharge results in little change in 96th percentile annual maximum discharge. For the 99th percentile discharge, two waterbodies in our study had a 10% or less increase in annual maximum discharge when annual maximum rainfall‐related discharges increased 20% while the third waterbody had a 16% increase in annual maximum discharges. Additionally, in some cases, annual maximum discharges could be offset by a reduction in the discharge resulting from annual maximum snowmelt events. While only intended as a heuristic tool to explore the interaction among different flood‐causing mechanisms, use of a compound flood frequency distribution suggests a case can be made that not all waterbodies in humid, cold regions will see extensive changes in flooding due to increased rainfall intensities. Copyright © 2011 John Wiley & Sons, Ltd.     

Extreme value theory based on the r largest annual events: a robust approach

In Smith (1986, J. Hydrol. 86, 27–43), a family of statistical distributions and estimators for extreme values based on a fixed number r > = 1 of the largest annual events are presented. The method of estimation was numerical maximum likelihood. In this paper, we consider the robust estimation of parameters in such families of distributions. The estimation technique, which is based on optimal B-robust estimates, will assign weights to each observation and give estimates of the parameters based on the data which are well modeled by the distribution. Thus, observations which are not consistent with the proposed distribution can be identified and the validity of the model can be assessed. The method is illustrated on Venice sea level data.