On the use of partial probability weighted moments in the analysis of hydrological extremes

The use of partial probability weighted moments (PPWM) for estimating hydrological extremes is compared to that of probability weighted moments (PWM). Firstly, estimates from at‐site data are considered. Two Monte Carlo analyses, conducted using continuous and empirical parent distributions (of peak discharge and daily rainfall annual maxima) and applying four different distributions (Gumbel, Fréchet, GEV and generalized Pareto), show that the estimates obtained from PPWMs are better than those obtained from PWMs if the parent distribution is unknown, as happens in practice. Secondly, the use of partial L‐moments (obtained from PPWMs) as diagnostic tools is considered. The theoretical partial L‐diagrams are compared with the experimental data. Five different distributions (exponential, Pareto, Gumbel, GEV and generalized Pareto) and 297 samples of peak discharge annual maxima are considered. Finally, the use of PPWMs with regional data is investigated. Three different kinds of regional analyses are considered. The first kind is the regression of quantile estimates on basin area. The study is conducted applying the GEV distribution to peak discharge annual maxima. The regressions obtained with PPWMs are slightly better than those obtained with PWMs. The second kind of regional analysis is the parametric one, of which four different models are considered. The congruence between local and regional estimates is examined, using peak discharge annual maxima. The congruence degree is sometimes higher for PPWMs, sometimes for PWMs. The third kind of regional analysis uses the index flood method. The study, conducted applying the GEV distribution to synthetic data from a lognormal joint distribution, shows that better estimates are obtained sometimes from PPWMs, sometimes from PWMs. All the results seem to indicate that using PPWMs can constitute a valid tool, provided that the influence of ouliers, of course higher with censored samples, is kept under control. Copyright © 2007 John Wiley & Sons, Ltd.     

Partial L-moments for the analysis of censored flood samples / Utilisation des L-moments partiels pour l’analyse d’échantillons tronqués de crues

Abstract

Abstract A parameter estimation method is proposed for fitting the generalized extreme value (GEV) distribution to censored flood samples. Partial L-moments (PL-moments), which are variants of L-moments and analogous to ?partial probability weighted moments?, are defined for the analysis of such flood samples. Expressions are derived to calculate PL-moments directly from uncensored annual floods, and to fit the parameters of the GEV distribution using PL-moments. Results of Monte Carlo simulation study show that sampling properties of PL-moments, with censoring flood samples of up to 30% are similar to those of simple L-moments, and also that both PL-moment and LH-moments (higher-order L-moments) have similar sampling properties. Finally, simple L-moments, LH-moments, and PL-moments are used to fit the GEV distribution to 75 annual maximum flow series of Nepalese and Irish catchments, and it is found that, in some situations, both LH- and PL-moments can produce a better fit to the larger flow values than simple L-moments.     

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.     

Occurrence of phthalate diesters in rivers of the Paris district (France)

The occurrence and fate of six phthalate esters were investigated in the River Seine and two tributaries in the Paris area, at six sites from upstream to downstream, throughout different seasons, with special attention given to a flood episode. At all sites, except the River Marne, whatever the period, diethylhexyl phthalate (DEHP) displayed the highest concentration (0·323–0·779 µg l^?1 as mean values), followed by di‐n‐butyl phthalate (DnBP; 0·211–0·526 µg l^?1 as mean values), which is consistent with the production pattern. High phthalate concentrations appeared to be related to population density and industrial distribution. The strongest values were located downstream from wastewater treatment plant effluent inputs. Seasonal variations were closely related to river flow and air and water temperature. The annual evolution of phthalates was linked to the hydrological conditions. During the flood episode at Paris, the DEHP concentration displayed a sharp increase (260 to 1123 ng l^?1) occurring 9 days before the flood peak. The simultaneous fluctuations of NH₄, typical of domestic origin, and of DEHP suggests a similar origin from combined sewer overflows for both compounds. Other elevated values corresponded to the highest precipitation amounts, and demonstrated the contribution of non‐point‐source runoff and wastewater overflow to DEHP inputs. The DEHP concentration relationship with river flow displayed a clockwise hysteresis loop, corresponding to a ‘first‐flush’ effect in which the finite supplies are swept along in the early stage of the flood. Annual fluxes for 2004 at Paris were 2257 kg of DEHP and 1613 kg of DnBP. The flood period seemed to have a minor impact on annual phthalate transport (29·5% and 41% of the total for DEHP and for DnBP, respectively), which is contrary to annual suspended solids transport (69%). Copyright © 2007 John Wiley & Sons, Ltd.     

A theoretically based Wakeby distribution for annual flood series

Abstract

The exact distribution of the ratio of any magnitude to the sum of all magnitudes in an annual flood series satisfying the usual distribution-free assumptions of independence and identical distribution, and the additional parametric assumption of exponential tail behaviour with truncation, is shown to be a beta distribution of the first kind. A two-parameter linear transformation of the beta distribution completes the derivation and yields a Wakeby distribution which has the number of members in a series as a given parameter. The Wakeby distribution is developed to illustrate how, in principle, some perceived deficiencies in current flood frequency analysis may be met: more complex parametric assumptions should lead to distributions of wider application. In particular, the distribution has a secure theoretical basis and is hydrologically more realistic because it bounds the variate and requires the definition of a temporally finite annual series. Analytical expressions are obtained for estimating the two distribution parameters; the quantite standard error and a plotting rule. An example is given of the application of the distribution to the design flood problem and an annual flood series is modelled. It is further suggested that a suitable design value for the largest flood to be withstood by a protection work is a statistic of the largest flood occurring during its lifetime. For the derived Wakeby distribution this criterion specifies risk and probability of non-exceedance of the design flood once a lifetime is selected.     

Process controls on the statistical flood moments ‐ a data based analysis

In this paper, the controls of different indicators on the statistical moments (i.e. mean annual flood (MAF), coefficient of variation (CV) and skewness (CS)) of the maximum annual flood records of 459 Austrian catchments are analysed. The process controls are analysed in terms of the correlation of the flood moments within five hydrologically homogeneous regions to two different types of indicators. Indicators of the first type are static catchment attributes, which are associated with long‐term observations such as mean annual precipitation, the base flow index, and the percentage of catchment area covered by a geological unit or soil type. Indicators of the second type are dynamic catchment attributes that are associated with the event scale. Indicators of this type used in the study are event runoff coefficients and antecedent rainfall. The results indicate that MAF and CV are strongly correlated with indicators characterising the hydro‐climatic conditions of the catchments, such as mean annual precipitation, long‐term evaporation and the base flow index. For the catchments analysed, the flood moments are not significantly correlated with static catchment attributes representing runoff generation, such as geology, soil types, land use and the SCS curve number. Indicators of runoff generation that do have significant predictive power for flood moments are dynamic catchment attributes such as the mean event runoff coefficients and mean antecedent rainfall. The correlation analysis indicates that flood runoff is, on average, more strongly controlled by the catchment moisture state than by event rainfall. Copyright © 2008 John Wiley & Sons, Ltd.     

Scaling properties of Canadian flood flows

This study investigated the spatial scaling properties of Canadian flood flows, namely, annual maximum mean 1‐, 5‐ and 7‐day flows using both the product moments (PMs) and probability weighted moments (PWMs). Both approaches demonstrate that flood flows in climatic regions 1 (Pacific), 2 (South British Columbia mountains), 3 (Yukon and northern British Columbia), 6 (Northeastern forest), 7 (Great Lakes and St. Lawrence rivers), 8 (Atlantic), and 10 (Arctic tundra) exhibit simple scaling with scaling exponent θ/H close to 0·90, while flood flows in regions 4 (Prairie provinces), 5 (Northwestern forest), and 9 (Mackenzie) does not with scaling exponent θ/H close to 0·50. The plots of coefficient of variations of flood flows versus drainage area indicate that Cv remains almost constant in regions 1, 2, 3, 6, 7, 8, and 10, while it decreases as drainage area increases in regions 4, 5, and 9. These results demonstrate that the index flood method is applicable in climatic regions 1, 2, 3, 6, 7, 8, and 10, while it is not in climatic regions 4, 5, and 9. The physical backgroud of the simple scaling of flood flows in most Canadian climatic regions is that snowmelt or rain‐on‐snow runoff is a dominant flood‐generating mechanism across the country. Copyright © 2008 John Wiley & Sons, Ltd.     

Understanding changes in annual runoff following land use changes: a systematic data‐based approach

A simple modelling framework for assessing the response of ungauged catchments to land use change in South‐Western Australia is presented. The framework uses knowledge of transpiration losses from native vegetation and pasture and then partitions the ‘excess’ water (resulting from reduced transpiration after land use change) between runoff and deep storage. The simple partitioning is achieved by using soft information (satellite imagery, previous mapping and field assessment) to delimit the spread of the permanently saturated area close to the stream. Runoff is then assumed to increase in proportion to the saturated area, with the residual difference going to deep storage. The model parameters to describe the annual water yield are obtained a priori and no calibration to streamflow is required. We tested the model using gauged records over 25 years from paired catchment experiments in South‐Western Australia. Very good estimates of runoff were obtained from high rainfall (>1100 mm yr⁻¹) catchments (R² > 0·9) and for low rainfall (<900 mm yr⁻¹) catchments after clearing (R² = 0·96) but results were poorer (R² = 0·55) for an uncleared low rainfall catchment, although overall balances were reasonable. In the drier uncleared catchments, the within‐year distributions of rainfall may exert a substantial influence on runoff response that is not completely captured by the presented model. Copyright © 2005 John Wiley & Sons, Ltd.     

Probability Estimates of Extreme Temperature Events: Stochastic Modelling Approach vs. Extreme Value Distributions

The paper deals with the probability estimates of temperature extremes (annual temperature maxima and heat waves) in the Czech Republic. Two statistical methods of probability estimations are compared; one based on the stochastic modelling of time series of the daily maximum temperature (TMAX) using the first-order autoregressive (AR(1)) model, the other consisting in fitting the extreme value distribution to the sample of annual temperature peaks.The AR(1) model is able to reproduce the main characteristics of heat waves, though the estimated probabilities should be treated as upper limits because of deficiencies in simulating the temperature variability inherent to the AR(1) model. Theoretical extreme value distributions do not yield good results when applied to maximum annual lengths of heat waves and periods of tropical days (TMAX 30°C), but it is the best method for estimating the probability and recurrence time of annual one-day temperature extremes. However, there are some difficulties in the application: the use of the two-parameter Gumbel distribution and the three-parameter generalized extreme value (GEV) distribution may lead to different results, particularly for long return periods. The resulting values also depend on the chosen procedure of parameter estimation. Based on our findings, the shape parameter testing for the GEV distribution and the L moments technique for parameter estimation may be recommended.The application of the appropriate statistical tools indicates that the heat wave and particularly the long period of consecutive tropical days in 1994 were probably a more rare event than the record-breaking temperatures in July 1983 exceeding 40°C. An improvement of the probability estimate of the 1994 heat wave may be expected from a more sophisticated model of the temperature series.     

Unbiased estimation of probability weighted moments and partial probability weighted moments from systematic and historical flood information and their application to estimating the GEV distribution

Unbiased estimators of probability weighted moments (PWM) and partial probability weighted moments (PPWM) from systematic and historical flood information are derived. Applications are made to estimating parameters and quantiles of the generalized extreme value (GEV) distribution. The effect of lower bound censoring, which might be deliberately introduced in practice, is also considered.     

Reassessment of flood frequency using historical information for the River Ouse at York,UK (1200–2000)

Abstract

The reassessment of flood risk at York, UK, is pertinent in light of major flooding in November 2000, and heightened concerns of a perceived increase in flooding nationally. Systematic flood level readings from 1877 and a wealth of documentary records dating back as far as 1263 AD give the City of York a long and rich history of flood records. This extended flood record provides an opportunity to reassess estimates of flood frequency over a time scale not normally possible within flood frequency analysis. This paper re-evaluates flood frequency at York, considering the strengths and weaknesses in estimates resulting from four contrasting methods of analysis and their corresponding data: (a) single-site analysis of gauged annual maxima; (b) pooled analysis of multi-site gauged annual maxima; (c) combined analysis of systematic annual maxima augmented with historical peaks, and (d) analysis of only the very largest peaks using a Generalized Pareto Distribution. Use of the historical information was found to yield risk estimates which were lower and considered to be more credible than those achieved using gauged records alone.

Citation Macdonald, N. & Black, A. R. (2010) Reassessment of flood frequency using historical information for the River Ouse at York, UK (1200–2000). Hydrol. Sci. J. 55(7), 1152–1162.     

Using regional regression within index flood procedures and an empirical Bayesian estimator

Studies have illustrated the performance of at-site and regional flood quantile estimators. For realistic generalized extreme value (GEV) distributions and short records, a simple index-flood quantile estimator performs better than two-parameter (2P) GEV quantile estimators with probability weighted moment (PWM) estimation using a regional shape parameter and at-site mean and L-coefficient of variation (L-CV), and full three-parameter at-site GEV/PWM quantile estimators. However, as regional heterogeneity or record lengths increase, the 2P-estimator quickly dominates. This paper generalizes the index flood procedure by employing regression with physiographic information to refine a normalized T-year flood estimator. A linear empirical Bayes estimator uses the normalized quantile regression estimator to define a prior distribution which is employed with the normalized 2P-quantile estimator. Monte Carlo simulations indicate that this empirical Bayes estimator does essentially as well as or better than the simpler normalized quantile regression estimator at sites with short records, and performs as well as or better than the 2P-estimator at sites with longer records or smaller L-CV.     

Flood frequency analysis of southern Africa: II. Identification of regional distributions

Abstract

Regional frequency analysis of annual maximum flood data comprising 407 stations from 11 countries of southern Africa is presented. Forty-one homogeneous regions are identified. The L-moments of the observed data indicate that the possible underlying frequency distributions are Pearson type 3 (P3), lognormal 3-parameter (LN3), General Pareto (GPA) or General Extreme Value (GEV). Simulation experiments for the selection of the most suitable flood frequency procedure indicate that Pearson type 3/Probability Weighted Moments (P3/PWM) and log-Pearson type 3/Method of Moments (LP3/MOM) are suitable procedures for the region.     

Assessment of river flow with significant lateral inflow through reverse routing modeling

The discharge hydrograph estimation in rivers based on reverse routing modeling and using only water level data at two gauged sections is here extended to the most general case of significant lateral flow contribution, without needing to deploy rainfall–runoff procedures. The proposed methodology solves the Saint‐Venant equations in diffusive form also involving the lateral contribution using a “head‐driven” modeling approach where lateral inflow is assumed to be function of the water level at the tributary junction. The procedure allows to assess the discharge hydrograph at ends of a selected river reach with significant lateral inflow, starting from the stage recorded there and without needing rainfall data. Specifically, the MAST 1D hydraulic model is applied to solve the diffusive wave equation using the observed stage hydrograph at the upstream section as upstream boundary condition. The other required data are (a) the observed stage hydrograph at the downstream section, as benchmark for the parameter calibration, and (b) the bathymetry of the river reach, from the upstream section to a short distance after the downstream gauged section. The method is validated with different flood events observed in two river reaches with a significant intermediate basin, where reliable rating curves were available, selected along the Tiber River, in central Italy, and the Alzette River, in Luxembourg. Very good performance indices are found for the computed discharge hydrographs at both the channel ends and along the tributaries. The mean Nash‐Sutcliffe value (NS_q) at the channel ends of two rivers is found equal to 0.99 and 0.86 for the upstream and downstream sites, respectively. The procedure is also validated on a longer stretch of the Tiber River including three tributaries for which appreciable results are obtained in terms of NS_q for the computed discharge hydrographs at both the channel ends for three investigated flood events.     

Appraisal of regional and index flood quantile estimators

Our results illustrate the performance of at-site and regional GEV/PWM flood quantile estimators in regions with different coefficients of variation, degrees of regional heterogeneity, record lengths, and number of sites. Analytic approximations of bias and variance are employed. For realistic GEV distributions and short records, the index-flood quantile estimator performs better than a 2-parameter GEV/PWM quantile estimator with a regional shape parameter, or a 3-parameter at-site GEV/PWM quantile estimator, in both humid and especially in arid regions, as long as the degree of regional heterogeneity is moderate. As regional heterogeneity or record lengths increases, 2-parameter estimators quickly dominate. Flood frequency models that assign probabilities larger than 2% to negative flows are unrealistic; experiments employing such distributions provide questionable results. This appraisal generally demonstrates the value of regionalizing estimators of the shape of a flood distribution, and sometimes the coefficient of variation.     

Regionalization of floods in New Brunswick (Canada)

This study uses the method of peaks over threshold (P.O.T.) to estimate the flood flow quantiles for a number of hydrometric stations in the province of New Brunswick, Canada. The peak values exceeding the base level (threshold), or `exceedances', are fitted by a generalized Pareto distribution. It is known that under the assumption of Poisson process arrival for flood exceedances, the P.O.T. model leads to a generalized extreme value distribution (GEV) for yearly maximum discharge values. The P.O.T. model can then be applied to calculate the quantiles X <sub>T</sub> corresponding to different return periods T, in years. A regionalization of floods in New Brunswick, which consists of dividing the province into `homogeneous regions', is performed using the method of the `region of influence'. The 100-year flood is subsequently estimated using a regionally estimated value of the shape parameter of the generalized Pareto distribution and a regression of the 100-year flood on the drainage area. The jackknife sampling method is then used to contrast the regional results with the values estimated at site. The variability of these results is presented in box-plot form. Received: June 1, 1997     

Regionalization of floods in New Brunswick (Canada)

This study uses the method of peaks over threshold (P.O.T.) to estimate the flood flow quantiles for a number of hydrometric stations in the province of New Brunswick, Canada. The peak values exceeding the base level (threshold), or `exceedances', are fitted by a generalized Pareto distribution. It is known that under the assumption of Poisson process arrival for flood exceedances, the P.O.T. model leads to a generalized extreme value distribution (GEV) for yearly maximum discharge values. The P.O.T. model can then be applied to calculate the quantiles X <sub>T</sub> corresponding to different return periods T, in years. A regionalization of floods in New Brunswick, which consists of dividing the province into `homogeneous regions', is performed using the method of the `region of influence'. The 100-year flood is subsequently estimated using a regionally estimated value of the shape parameter of the generalized Pareto distribution and a regression of the 100-year flood on the drainage area. The jackknife sampling method is then used to contrast the regional results with the values estimated at site. The variability of these results is presented in box-plot form. Received: June 1, 1997     

A soft hydrological monitoring approach for comparing runoff on a network of small poorly gauged catchments

Catchments in many parts of the world are either ungauged or poorly gauged, and the dominant processes governing their streamflow response are still poorly understood. The analysis of runoff coefficients provides essential insight into catchment response, particularly if both range of catchments and a range of events are compared. This paper investigates how well the hydrological runoff of 11 small, poorly gauged catchments with ephemeral streams (0·1‐0·6 km²) can be compared using estimated runoff with the associated uncertainty. Data of rainfall and water depth at a catchment's outlet were recorded using automatic logging equipment during 2008‐2009. The hydrological regime is intermittent and the annual precipitation ranged between 569 and 727 mm. Discharge was estimated using Manning's equation and channel cross‐section measurements. Innovative work has been performed under controlled experimental conditions to estimate Manning's coefficient values for the different cover types observed in studied streams: non‐aquatic vegetations (giant reed, bramble and thistle), grass and coarse granular deposits. The results show that estimates derived using roughness coefficients differ from those previously established for larger streams with aquatic vegetation. Catchment runoff was compared at both the event and the annual scale. The results indicate significant variability between the catchment's responses. This variability allows for classification in spite of all the uncertainty associated with runoff estimation. This study highlights the potential of using a network of poorly gauged catch ments. From almost no catchment understanding the proposed methodology allows to compare poorly gauged catchments and highlights similarity/dissimilarity between catchment responses. Copyright © 2011 John Wiley & Sons, Ltd.     

Appraisal of regional and index flood quantile estimators

Our results illustrate the performance of at-site and regional GEV/PWM flood quantile estimators in regions with different coefficients of variation, degrees of regional heterogeneity, record lengths, and number of sites. Analytic approximations of bias and variance are employed. For realistic GEV distributions and short records, the index-flood quantile estimator performs better than a 2-parameter GEV/PWM quantile estimator with a regional shape parameter, or a 3-parameter at-site GEV/PWM quantile estimator, in both humid and especially in arid regions, as long as the degree of regional heterogeneity is moderate. As regional heterogeneity or record lengths increases, 2-parameter estimators quickly dominate. Flood frequency models that assign probabilities larger than 2% to negative flows are unrealistic; experiments employing such distributions provide questionable results. This appraisal generally demonstrates the value of regionalizing estimators of the shape of a flood distribution, and sometimes the coefficient of variation.