Moments of truncated continuous univariate distributions

Truncated moment expressions (TMEs), defined as moment equations for truncated or incomplete distributions, are derived for several continuous univariate distributions commonly applied to hydrologic problems, including normal, lognormal, Pearson type III, log Pearson type III, and extreme value (Weibull and Gumbel) distributions. Solutions for gamma, tanks-in-series, and exponential distributions result as special cases. For most of the distributions considered here, closed form TMEs are presented for Nth order moments for the general case of double truncation (both upper and lower bounds). For the normal and Gumbel distributions, TMEs are presented only for moments of order N={0,1,2,3} and {0,1}, respectively. The derived TMEs are used to evaluate the effect of truncation on measured moments. The relative error between the first four truncated and complete moments is calculated as a function of both upper and lower truncation point.     

Establishing acceptance regions for L-moments based goodness-of-fit tests for the Pearson type III distribution

Goodness-of-fit tests based on the L-moment-ratio diagram for selection of appropriate distributions for hydrological variables have had many applications in recent years. For such applications, sample-size-dependent acceptance regions need to be established in order to take into account the uncertainties induced by sample L-skewness and L-kurtosis. Acceptance regions of two-parameter distributions such as the normal and Gumbel distributions have been developed. However, many hydrological variables are better characterized by three-parameter distributions such as the Pearson type III and generalized extreme value distributions. Establishing acceptance regions for these three-parameter distributions is more complicated since their L-moment-ratio diagrams plot as curves, instead of unique points for two-parameter distributions. Through stochastic simulation we established sample-size-dependent 95% acceptance regions for the Pearson type III distribution. The proposed approach involves two key elements—the conditional distribution of population L-skewness given a sample L-skewness and the conditional distribution of sample L-kurtosis given a sample L-skewness. The established 95% acceptance regions of the Pearson type III distribution were further validated through two types of validity check, and were found to be applicable for goodness-of-fit tests for random samples of any sample size between 20 and 300 and coefficient of skewness not exceeding 3.0.     

Frequency analyses of annual extreme rainfall series from 5 min to 24 h

The parameter estimation methods of (1) moments, (2) maximum‐likelihood, (3) probability‐weighted moments (PWM) and (4) self‐determined PWM are applied to the probability distributions of Gumbel, general extreme values, three‐parameter log‐normal (LN3), Pearson‐3 and log‐Pearson‐3. The special method of computing parameters so as to make the sample skewness coefficient zero is also applied to LN3, and hence, altogether 21 candidate distributions resulted. The parameters of these distributions are computed first by original sample series of 14 successive‐duration annual extreme rainfalls recorded at a rain‐gauging station. Next, the parameters are scaled by first‐degree semi‐log or log‐log polynomial regressions versus rainfall durations from 5 to 1440 min (24 h). Those distributions satisfying the divergence criterion for frequency curves are selected as potential distributions, whose better‐fit ones are determined by a conjunctive evaluation of three goodness‐of‐fit tests. Frequency tables, frequency curves and intensity–duration–frequency curves are the outcome. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical investigation of apparent multifractality of samples from processes subordinated to truncated fBm

We investigate numerically apparent multi‐fractal behavior of samples from synthetically generated processes subordinated to truncated fractional Brownian motion (tfBm) on finite domains. We are motivated by the recognition that many earth and environmental (including hydrological) variables appear to be self‐affine (monofractal) or multifractal with Gaussian or heavy‐tailed distributions. The literature considers self‐affine and multifractal types of scaling to be fundamentally different, the first arising from additive and the second from multiplicative random fields or processes. It has been demonstrated theoretically by one of us that square or absolute increments of samples from Gaussian/Lévy processes subordinated to tfBm exhibit apparent/spurious multifractality at intermediate ranges of separation lags, with breakdown in power‐law scaling at small and large lags as is commonly exhibited by real data. A preliminary numerical demonstration of apparent multifractality by the same author was limited to Gaussian fields having nearest neighbor autocorrelations and led to rather noisy results. Here, we adopt a new generation scheme that allows us to investigate apparent multifractal behaviors of samples taken from a broad range of processes including Gaussian with and without symmetric Lévy and log‐normal (as well as potentially other) subordinators. Our results shed new light on the nature of apparent multifractality, which has wide implications vis‐a‐vis the scaling of many hydrological as well as other earth and environmental variables. Copyright © 2011 John Wiley & Sons, Ltd.     

Probability distribution type of Canadian annual minimum streamflow / Type de distribution de probabilité du débit minimum annuel au Canada

Abstract

The method of L-moment ratio diagrams along with the averaged weighted distance (AWD) is applied to identify a probability distribution of annual minimum streamflow, namely annual minimum daily streamflow in 11 climatic regions of Canada. Across the entire country, the Pearson type III probability distribution is an acceptable distribution for describing annual minimum streamflow with the 3-parameter lognormal and log Pearson type III distributions as potential candidates. Some minor differences in the probability distribution type among different climatic regions are also observed, which may be taken into account in the selection of the distribution type of annual minimum streamflow.     

Flood frequency analysis of Turkey using L‐moments method

The index flood procedure coupled with the L‐moments method is applied to the annual flood peaks data taken at all stream‐gauging stations in Turkey having at least 15‐year‐long records. First, screening of the data is done based on the discordancy measure (D_i) in terms of the L‐moments. Homogeneity of the total geographical area of Turkey is tested using the L‐moments based heterogeneity measure, H, computed on 500 simulations generated using the four parameter Kappa distribution. The L‐moments analysis of the recorded annual flood peaks data at 543 gauged sites indicates that Turkey as a whole is hydrologically heterogeneous, and 45 of 543 gauged sites are discordant which are discarded from further analyses. The catchment areas of these 543 sites vary from 9·9 to 75121 km² and their mean annual peak floods vary from 1·72 to 3739·5 m³ s^?1. The probability distributions used in the analyses, whose parameters are computed by the L‐moments method are the general extreme values (GEV), generalized logistic (GLO), generalized normal (GNO), Pearson type III (PE3), generalized Pareto (GPA), and five‐parameter Wakeby (WAK). Based on the L‐moment ratio diagrams and the |Z_dist|‐statistic criteria, the GEV distribution is identified as the robust distribution for the study area (498 gauged sites). Hence, for estimation of flood magnitudes of various return periods in Turkey, a regional flood frequency relationship is developed using the GEV distribution. Next, the quantiles computed at all of 543 gauged sites by the GEV and the Wakeby distributions are compared with the observed values of the same probability based on two criteria, mean absolute relative error and determination coefficient. Results of these comparisons indicate that both distributions of GEV and Wakeby, whose parameters are computed by the L‐moments method, are adequate in predicting quantile estimates. Copyright © 2011 John Wiley & Sons, Ltd.     

Stochastic simulation of bivariate gamma distribution: a frequency-factor based approach

A frequency-factor based approach for stochastic simulation of bivariate gamma distribution is proposed. The approach involves generation of bivariate normal samples with a correlation coefficient consistent with the correlation coefficient of the corresponding bivariate gamma samples. Then the bivariate normal samples are transformed to bivariate gamma samples using the well-known general equation of hydrological frequency analysis. We demonstrate that the proposed bivariate gamma simulation approach is capable of generating random sample pairs which not only have the desired marginal densities of component random variables but also their correlation coefficient. Scatter plots of simulated bivariate sample pairs also exhibit appropriate linear patterns (dependence structure) that are commonly observed in environmental and hydrological applications. Caution should also be exercised when specifying combinations of coefficients of skewness and the correlation coefficient for bivariate gamma simulation.     

Application of L‐moments and Bayesian inference for low‐flow regionalization in Sefidroud basin,Iran

Reliable estimation of low flows at ungauged catchments is one of the major challenges in water‐resources planning and management. This study aims at providing at‐site and ungauged sites low‐flow frequency analysis using regionalization approach. A two‐stage delineating homogeneous region is proposed in this study. Clustering sites with similar low‐flow L‐moment ratios is initially conducted, and L‐moment‐based discordancy and heterogeneity measures are then used to detect unusual sites. Based on the goodness‐of‐fit test statistic, the best‐fit regional model is identified in each hydrologically homogeneous region. The relationship between mean annual 7‐day minimum flow and hydro‐geomorphic characteristics is also constructed in each homogeneous region associated with the derived regional model for estimating various low‐flow quantiles at ungauged sites. Uncertainty analysis of model parameters and low‐flow estimations is carried out using the Bayesian inference. Applied in Sefidroud basin located in northwestern Iran, two hydrologically homogeneous regions are identified, i.e. the east and west regions. The best‐fit regional model for the east and west regions are generalized logistic and Pearson type III distributions, respectively. The results show that the proposed approach provides reasonably good accuracy for at‐site as well as ungauged‐site frequency analysis. Besides, interval estimations for model parameters and low flows provide uncertainty information, and the results indicate that Bayesian confidence intervals are significantly reduced when comparing with the outcomes of conventional t‐distribution method. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Climate‐informed low‐flow frequency analysis using nonstationary modelling

Stationarity is often assumed for frequency analysis of low flows in water resources management and planning. However, many studies have shown that flow characteristics, particularly the frequency spectrum of extreme hydrologic events, were modified by climate change and human activities. Thus, the conventional frequency analysis that fails to consider the nonstationary characteristics may lead to costly design. The analysis presented in this paper was based on the more than 100 years of daily flow data from the Yichang gauging station 44 km downstream of the Three Gorges Dam. The Mann–Kendall trend test under the scaling hypothesis showed that the annual low flows had a significant monotonic trend, whereas an abrupt change point was identified in 1936 by the Pettitt test. The climate‐informed low‐flow frequency analysis and the divided and combined method were employed to account for the impacts from related climate variables and nonstationarities in annual low flows. Without prior knowledge of the probability density function for the gauging station, six distribution functions including the generalized extreme values (GEV), Pearson Type III, Gumbel, Gamma, Lognormal and Weibull distributions have been tested to find the best fit, in which the local likelihood method is used to estimate the parameters. Analyses show that GEV had the best fit for the observed low flows. This study has also shown that the climate‐informed low‐flow frequency analysis is able to exploit the link between climate indices and low flows, which would account for the dynamic feature for reservoir management and provide more accurate and reliable designs for infrastructure and water supply. Copyright © 2014 John Wiley & Sons, Ltd.     

A discrete Bayesian network to investigate suspended sediment concentrations in an Alpine proglacial zone

In this study, a Bayesian Network (BN) is used to model the suspended sediment concentrations (SSC) in the catchments of the glaciers Noir and Blanc in the Ecrins National Park, France, and at the distal end of the proglacial zone into which both torrents drain. Relationships between air temperature, glacier discharge and SSC are represented as random variables; thereby taking the natural next step from proposed modified rating curve methods which increasingly approximate random variable approaches. Hydrological relationships are propagated through the network via conditional probability distributions computed from 980 field records obtained at three monitoring sites during July 2005. Rainfall affected data are removed from the modelling process. A two‐sample Kolmogorov–Smirnov goodness‐of‐fit (two‐sample KS) test (n = 5) shows good agreement between the probability distributions of SSC predicted by the BN, and those recorded in the field at the outflow of the proglacial zone over an air temperature range of 5–25 °C. The BN performs poorly for air temperatures between 25 and 30 °C and this is attributed to limited field records covering this temperature range. Discussion of the significant limitations surrounding the widespread application of BNs in hydrological modelling are offered with a focus on data volume and temporal limitations. Copyright © 2008 John Wiley & Sons, Ltd.     

A comparison of three methods for downscaling daily precipitation in the Punjab region

Many downscaling techniques have been developed in the past few years for projection of station‐scale hydrological variables from large‐scale atmospheric variables simulated by general circulation models (GCMs) to assess the hydrological impacts of climate change. This article compares the performances of three downscaling methods, viz. conditional random field (CRF), K‐nearest neighbour (KNN) and support vector machine (SVM) methods in downscaling precipitation in the Punjab region of India, belonging to the monsoon regime. The CRF model is a recently developed method for downscaling hydrological variables in a probabilistic framework, while the SVM model is a popular machine learning tool useful in terms of its ability to generalize and capture nonlinear relationships between predictors and predictand. The KNN model is an analogue‐type method that queries days similar to a given feature vector from the training data and classifies future days by random sampling from a weighted set of K closest training examples. The models are applied for downscaling monsoon (June to September) daily precipitation at six locations in Punjab. Model performances with respect to reproduction of various statistics such as dry and wet spell length distributions, daily rainfall distribution, and intersite correlations are examined. It is found that the CRF and KNN models perform slightly better than the SVM model in reproducing most daily rainfall statistics. These models are then used to project future precipitation at the six locations. Output from the Canadian global climate model (CGCM3) GCM for three scenarios, viz. A1B, A2, and B1 is used for projection of future precipitation. The projections show a change in probability density functions of daily rainfall amount and changes in the wet and dry spell distributions of daily precipitation. Copyright © 2011 John Wiley & Sons, Ltd.     

Artificial intelligence for identifying hydrologically homogeneous regions: A state‐of‐the‐art regional flood frequency analysis

Due to the severity related to extreme flood events, recent efforts have focused on the development of reliable methods for design flood estimation. Historical streamflow series correspond to the most reliable information source for such estimation; however, they have temporal and spatial limitations that may be minimized by means of regional flood frequency analysis (RFFA). Several studies have emphasized that the identification of hydrologically homogeneous regions is the most important and challenging step in an RFFA. This study aims to identify state‐of‐the‐art clustering techniques (e.g., K ‐means, partition around medoids, fuzzy C‐means, K ‐harmonic means, and genetic K ‐means) with potential to form hydrologically homogeneous regions for flood regionalization in Southern Brazil. The applicability of some probability density function, such as generalized extreme value, generalized logistic, generalized normal, and Pearson type 3, was evaluated based on the regions formed. Among all the 15 possible combinations of the aforementioned clustering techniques and the Euclidian, Mahalanobis, and Manhattan distance measures, the five best were selected. Several watersheds' physiographic and climatological attributes were chosen to derive multiple regression equations for all the combinations. The accuracy of the equations was quantified with respect to adjusted coefficient of determination, root mean square error, and Nash–Sutcliffe coefficient, whereas, a cross‐validation procedure was applied to check their reliability. It was concluded that reliable results were obtained when using robust clustering techniques based on fuzzy logic (e.g., K ‐harmonic means), which have not been commonly used in RFFA. Furthermore, the probability density functions were capable of representing the regional annual maximum streamflows. Drainage area, main river length, and mean altitude of the watershed were the most recurrent attributes for modelling of mean annual maximum streamflow. Finally, an integration of all the five best combinations stands out as a robust, reliable, and simple tool for estimation of design floods.     

Finding the most appropriate precipitation probability distribution for stochastic weather generation and hydrological modelling in Nordic watersheds

Six precipitation probability distributions (exponential, Gamma, Weibull, skewed normal, mixed exponential and hybrid exponential/Pareto distributions) are evaluated on their ability to reproduce the statistics of the original observed time series. Each probability distribution is also indirectly assessed by looking at its ability to reproduce key hydrological variables after being used as inputs to a lumped hydrological model. Data from 24 weather stations and two watersheds (Chute‐du‐Diable and Yamaska watersheds) in the province of Quebec (Canada) were used for this assessment. Various indices or statistics, such as the mean, variance, frequency distribution and extreme values are used to quantify the performance in simulating the precipitation and discharge. Performance in reproducing key statistics of the precipitation time series is well correlated to the number of parameters of the distribution function, and the three‐parameter precipitation models outperform the other models, with the mixed exponential distribution being the best at simulating daily precipitation. The advantage of using more complex precipitation distributions is not as clear‐cut when the simulated time series are used to drive a hydrological model. Although the advantage of using functions with more parameters is not nearly as obvious, the mixed exponential distribution appears nonetheless as the best candidate for hydrological modelling. The implications of choosing a distribution function with respect to hydrological modelling and climate change impact studies are also discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

Two kinds of moment ratio diagrams and their applications in hydrology

We refocus attention on moment ratio diagrams and their uses in hydrology with four major objectives: (1) to summarize the information available in the literature about possible uses of the traditional moment ratio diagram introduced by Karl Pearson, which uses the coefficient of skewness and of kurtosis to compare the shapes of various distributions commonly used in hydrology; (2) to complete this traditional MRD by integrating into it the regions occupied by the log-Pearson Type III and generalized gamma distributions which are more and more used in hydrology; (3) to present another MRD which uses ratios of moments of orders –1 (harmonic mean), quasi zero (geometric mean) and 1 (arithmetic mean); (4) to stress the need to consider the different MRD's (along with the more recently introduced L-moment ratio diagrams) as complementary tools for choosing between distributions fitted to hydrologic data. Finally, using Monte Carlo simulation we compare the two types of diagrams as tools to identify and discriminate between different distributions.     

Influence of irregular topography and random soil properties on coherency loss of spatial seismic ground motions

Coherency functions are used to describe the spatial variation of seismic ground motions at multiple supports of long span structures. Many coherency function models have been proposed based on theoretical derivation or measured spatial ground motion time histories at dense seismographic arrays. Most of them are suitable for modelling spatial ground motions on flat‐lying alluvial sites. It has been found that these coherency functions are not appropriate for modelling spatial variations of ground motions at sites with irregular topography (Struct. Saf. 1991; 10 (1):1–13). This paper investigates the influence of layered irregular sites and random soil properties on coherency functions of spatial ground motions on ground surface. Ground motion time histories at different locations on ground surface of the irregular site are generated based on the combined spectral representation method and one‐dimensional wave propagation theory. Random soil properties, including shear modulus, density and damping ratio of each layer, are assumed to follow normal distributions, and are modelled by the independent one‐dimensional random fields in the vertical direction. Monte‐Carlo simulations are employed to model the effect of random variations of soil properties on the simulated surface ground motion time histories. The coherency function is estimated from the simulated ground motion time histories. Numerical examples are presented to illustrate the proposed method. Numerical results show that coherency function directly relates to the spectral ratio of two local sites, and the influence of randomly varying soil properties at a canyon site on coherency functions of spatial surface ground motions cannot be neglected. Copyright © 2010 John Wiley & Sons, Ltd.     

Bivariate frequency analysis of nonstationary low‐flow series based on the time‐varying copula

Many studies have analysed the nonstationarity in single hydrological variables due to changing environments. Yet, few researches have been done to investigate how the dependence structure between different individual hydrological variables is affected by changing environments. To investigate how the reservoirs have altered the dependence structure between river flows at different locations on the Hanjiang River, a time‐varying copula model, which takes the nonstationarity in the marginal distribution and/or the time variation in dependence structure between different hydrological series into consideration, is presented in this paper to perform a bivariate frequency analysis for the low‐flow series from two neighbouring hydrological gauges. The time‐varying moments model with either time or reservoir index as explanatory variables is applied to build the time‐varying marginal distributions of the two low‐flow series. It's found that both marginal distributions are nonstationary, and the reservoir index yields better performance than the time index in describing the nonstationarities in the marginal distributions. Then, the copula with the dependence parameter expressed as a function of either time or reservoir index is applied to model the variable dependence between the two low‐flow series. The copula with reservoir index as the explanatory variable of the dependence parameter has a better fitting performance than the copula with the constant or the time‐trend dependence parameter. Finally, the effect of the time variation in the joint distribution on three different types of joint return periods (i.e. AND, OR and Kendall) of low flows at two neighbouring hydrological gauges is presented. Copyright © 2014 John Wiley & Sons, Ltd.     

Types of probability distributions in the evaluation of extreme floods

The development of an optimal scheme for evaluation of maximal water discharges is discussed, including adequate probability distribution laws, an effective procedure for their approximation based on observational data, and reliable goodness-of-fit tests for analytical and empirical distributions. One-dimensional probability distribution laws are systematized. Promising distributions were identified, including generalized distribution of extreme values, lognormal distribution, Pearson type V power distribution, and GPD, for evaluating maximal discharges. The available methods for approximating analytical curves, including the up-to-date method of L-moments are considered. Parameter estimation algorithm based on L-moment method for Pearson type III distribution is considered. Pearson type III distribution, lognormal distribution, GEV, and GPD are compared in the approximation of maximal water discharges in rivers of Austria, Siberia, Far East, and the Hawaiian Islands.     

Development of a modelling methodology for the investigation of riparian hydrological processes

The riparian zone is in intimate contact with the river and, as such, is a critical zone for understanding hydrological problems. This paper presents a general modelling methodology for the assessment of riparian hydrological processes. It is applicable to a wide range of riparian spaces and incorporates current expertise in numerical methods. A core part of the modelling methodology is the random walk particle method (RWPM). We develop an RWPM as part of the ESTEL2D subsurface flow model, test it against analytical solutions and apply it to the simulation of parcels of water as they move through the riparian zone. The modelling methodology provides a new opportunity to assess fundamental hydrological process issues such as the proportioning of pre‐event and event water storm runoff, and reversals of flow in floodplains. Copyright © 2005 John Wiley & Sons, Ltd.     

Evaluation and hydrologic validation of TMPA satellite precipitation product downstream of the Pearl River Basin,China

With high spatio‐temporal resolution and wide coverage, satellite‐based precipitation products can potentially fill the deficiencies of traditional in situ gauge precipitation observations and provide an alternative data source for ungauged areas. However, due to the relatively poor accuracy and high uncertainty of satellite‐based precipitation products, it remains necessary to assess the quality and applicability of the products for each investigated area. This study evaluated the accuracy and error of the latest Tropical Rainfall Measuring Mission Multi‐satellites Precipitation Analysis 3B42‐V7 satellite‐based precipitation product and validated the applicability of the product for the Beijiang and Dongjiang River Basins, downstream of the Pearl River Basin in China. The study first evaluated the accuracy, error, and bias of the 3B42‐V7 product during 1998–2006 at daily and monthly scale via comparison with in situ observations. The study further validated the applicability of the product via hydrologic simulation using the variable infiltration capacity hydrological model for three hydrological stations in the Beijiang River Basin, considering two scenarios: a streamflow simulation with gauge‐calibrated parameters (Scenario I) and a simulation after recalibration with the 3B42‐V7 product (Scenario II). The results revealed that (a) the 3B42‐V7 product produced acceptable accuracy both at the daily scale and high accuracy at the monthly scale while generally tending to overestimate precipitation; (b) the product clearly overestimated the frequency of no rainfall events at the grid cell scale and light rainfall (<1 mm/day) events at the region scale and also overestimated the amount of heavy rain (25–50 mm/day) and hard rain (≥50 mm/day) events; (c) under Scenario I, the 3B42‐V7 product performed poorly at three stations with gauge‐calibrated parameters; under Scenario II, the recalibrated model provided significantly improved performance of streamflow simulation with the 3B42‐V7 product; (d) the variable infiltration capacity model has the ability to reveal the hydrological characteristics of the karst landform in the Beijiang Basin when using the 3B42‐V7 product.