Weighted estimate of extreme quantile: an application to the estimation of high flood return periods

Parametric models are commonly used in frequency analysis of extreme hydrological events. To estimate extreme quantiles associated to high return periods, these models are not always appropriate. Therefore, estimators based on extreme value theory (EVT) are proposed in the literature. The Weissman estimator is one of the popular EVT-based semi-parametric estimators of extreme quantiles. In the present paper we propose a new family of EVT-based semi-parametric estimators of extreme quantiles. To built this new family of estimators, the basic idea consists in assigning the weights to the k observations being used. Numerical experiments on simulated data are performed and a case study is presented. Results show that the proposed estimators are smooth, stable, less sensitive, and less biased than Weissman estimator.     

A distribution function based bandwidth selection method for kernel quantile estimation

The key problem in nonparametric frequency analysis of flood and droughts is the estimation of the bandwidth parameter which defines the degree of smoothing. Most of the proposed bandwidth estimators have been based on the density function rather than the cumulative distribution function or the quantile that are the primary interest in frequency analysis. We propose a new bandwidth estimator derived from properties of quantile estimators. The estimator builds on work by Altman and Léger (1995). The estimator is compared to the well-known method of least squares cross-validation (LSCV) using synthetic data generated from various parametric distributions used in hydrologic frequency analysis. Simulations suggest that our estimator performs at least as well as, and in many cases better than, the method of LSCV. In particular, the use of the proposed plug-in estimator reduces bias in the estimation as compared to LSCV. When applied to data sets containing observations with identical values, typically the result of rounding or truncation, the LSCV and most other techniques generally underestimates the bandwidth. The proposed technique performs very well in such situations.     

The flood probability distribution tail: how heavy is it?

This paper empirically investigates the asymptotic behaviour of the flood probability distribution and more precisely the possible occurrence of heavy tail distributions, generally predicted by multiplicative cascades. Since heavy tails considerably increase the frequency of extremes, they have many practical and societal consequences. A French database of 173 daily discharge time series is analyzed. These series correspond to various climatic and hydrological conditions, drainage areas ranging from 10 to 10⁵ km², and are from 22 to 95 years long. The peaks-over-threshold method has been used with a set of semi-parametric estimators (Hill and Generalized Hill estimators), and parametric estimators (maximum likelihood and L-moments). We discuss the respective interest of the estimators and compare their respective estimates of the shape parameter of the probability distribution of the peaks. We emphasize the influence of the selected number of the highest observations that are used in the estimation procedure and in this respect the particular interest of the semi-parametric estimators. Nevertheless, the various estimators agree on the prevalence of heavy tails and we point out some links between their presence and hydrological and climatic conditions.     

A theoretical study on terrestrial gravimetric data refinement by Earth gravity models

The idea of this paper is to present estimators for combining terrestrial gravity data with Earth gravity models and produce a high‐quality source of the Earth's gravity field data through all wavelengths. To do so, integral and point‐wise estimators are mathematically developed, based on the spectral combination theory, in such a way that they combine terrestrial data with one and/or two Earth gravity models. The integral estimators are developed so that they become biased or unbiased to a priori information. For testing the quality of the estimators, their global mean square errors are generated using an Earth gravity model08 model and one of the recent products of the gravity field and steady‐state ocean circulation explorer mission. Numerical results show that the integral estimators have smaller global root mean square errors than the point‐wise ones but they are not efficient practically. The integral estimator of the biased type is the most suited due to its smallest global root mean square error comparing to the rest of the estimators. Due largely to the omission errors of Earth gravity models the point‐wise estimators are not sensitive to the Earth gravity model commission error; therefore, the use of high‐degree Earth gravity models is very influential for reduction of their root mean square errors. Also it is shown that the use of the ocean circulation explorer Earth gravity model does not significantly reduce the root mean square errors of the presented estimators in the presence of Earth gravity model08. All estimators are applied in the region of Fennoscandia and a cap size of 2° for numerical integration and a maximum degree of 2500 for generation of band‐limited kernels are found suitable for the integral estimators.     

Expected probability weighted moment estimator for censored flood data

Two well-known methods for estimating statistical distributions in hydrology are the Method of Moments (MOMs) and the method of probability weighted moments (PWM). This paper is concerned with the case where a part of the sample is censored. One situation where this might occur is when systematic data (e.g. from gauges) are combined with historical data, since the latter are often only reported if they exceed a high threshold. For this problem, three previously derived estimators are the “B17B” estimator, which is a direct modification of MOM to allow for partial censoring; the “partial PWM estimator”, which similarly modifies PWM; and the “expected moments algorithm” estimator, which improves on B17B by replacing a sample adjustment of the censored-data moments with a population adjustment. The present paper proposes a similar modification to the PWM estimator, resulting in the “expected probability weighted moments (EPWM)” estimator. Simulation comparisons of these four estimators and also the maximum likelihood estimator show that the EPWM method is at least competitive with the other four and in many cases the best of the five estimators.     

Land cover effects on runoff patterns in eastern Piedmont (USA) watersheds

Physiography and land cover determine the hydrologic response of watersheds to climatic events. However, vast differences in climate regimes and variation of landscape attributes among watersheds (including size) have prevented the establishment of general relationships between land cover and runoff patterns across broad scales. This paper addresses these difficulties by using power spectral analysis to characterize area‐normalized runoff patterns and then compare these patterns with landscape features among watersheds within the same physiographic region. We assembled long‐term precipitation and runoff data for 87 watersheds (first to seventh order) within the eastern Piedmont (USA) that contained a wide variety of land cover types, collected environmental data for each watershed, and compared the datasets using a variety of statistical measures. The effect of land cover on runoff patterns was confirmed. Urban‐dominated watersheds were flashier and had less hydrologic memory compared with forest‐dominated watersheds, whereas watersheds with high wetland coverage had greater hydrologic memory. We also detected a 10–15% urban threshold above which urban coverage became the dominant control on runoff patterns. When spectral properties of runoff were compared across stream orders, a threshold after the third order was detected at which watershed processes became dominant over precipitation regime in determining runoff patterns. Finally, we present a matrix that characterizes the hydrologic signatures of rivers based on precipitation versus landscape effects and low‐frequency versus high‐frequency events. The concepts and methods presented can be generally applied to all river systems to characterize multiscale patterns of watershed runoff. Copyright © 2013 John Wiley & Sons, Ltd.     

Effect of time on dynamic properties of cohesive soils using improved transfer function estimators

Improved estimators of the transfer function (H2, H3, H4), in addition to the conventional estimator (H1), were used to evaluate the dynamic soil properties and to study the effect of confinement duration on damping and shear modulus of soils. In this study, two types of cohesive soils, a kaolinite and a bentonite, were tested using a resonant column apparatus under random torsional excitation conditions. Root meam square strain levels in the range of 10⁻³–10⁻² and confining pressures in the range of 34.47–150 kPa were considered. The confinement duration ranged from 500 to 20 000 min. The results of this study indicate that as time increases, the difference in damping values obtained by the various estimators of the transfer function decreases, whereas the shear moduli are not influenced by the variation in confinement duration.     

Multigaussian kriging for point-support estimation: incorporating constraints on the sum of the kriging weights

In the geostatistical analysis of regionalized data, the practitioner may not be interested in mapping the unsampled values of the variable that has been monitored, but in assessing the risk that these values exceed or fall short of a regulatory threshold. This kind of concern is part of the more general problem of estimating a transfer function of the variable under study. In this paper, we focus on the multigaussian model, for which the regionalized variable can be represented (up to a nonlinear transformation) by a Gaussian random field. Two cases are analyzed, depending on whether the mean of this Gaussian field is considered known or not, which lead to the simple and ordinary multigaussian kriging estimators respectively. Although both of these estimators are theoretically unbiased, the latter may be preferred to the former for practical applications since it is robust to a misspecification of the mean value over the domain of interest and also to local fluctuations around this mean value. An advantage of multigaussian kriging over other nonlinear geostatistical methods such as indicator and disjunctive kriging is that it makes use of the multivariate distribution of the available data and does not produce order relation violations. The use of expansions into Hermite polynomials provides three additional results: first, an expression of the multigaussian kriging estimators in terms of series that can be calculated without numerical integration; second, an expression of the associated estimation variances; third, the derivation of a disjunctive-type estimator that minimizes the variance of the error when the mean is unknown.     

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.     

Trend analysis in Turkish precipitation data

This study aims to determine trends in the long‐term annual mean and monthly total precipitation series using non‐parametric methods (i.e. the Mann–Kendall and Sen's T tests). The change per unit time in a time series having a linear trend was estimated by applying a simple non‐parametric procedure, namely Sen's estimator of slope. Serial correlation structure in the data was accounted for determining the significance level of the results of the Mann–Kendall test. The data network used in this study, which is assumed to reflect regional hydroclimatic conditions, consists of 96 precipitation stations across Turkey. Monthly totals and annual means of the monthly totals are formed for each individual station, spanning from 1929 to 1993. In this case, a total of 13 precipitation variables at each station are subjected to trend detection analysis. In addition, regional average precipitation series are established for the same analysis purpose. The application of a trend detection framework resulted in the identification of some significant trends, especially in January, February, and September precipitations and in the annual means. A noticeable decrease in the annual mean precipitation was observed mostly in western and southern Turkey, as well as along the coasts of the Black Sea. Regional average series also displayed trends similar to those for individual stations. Copyright © 2005 John Wiley & Sons, Ltd.     

Accuracy assessment of optimum interpolation in multichannel precipitation analysis

The accuracy of an optimum interpolation technique in filling missing values in multichannel (or multisite) hydrologic series containing time-coincident data gaps is examined. The applied methodology is based on the maximum entropy method (MEM) of spectral estimation or multivariate autoregressive modeling and heavily depends upon the properties of multichannel prediction error filter (PEF). Six precipitation time series spatially located within a hydrologic basin are used and time-coincident artificial gaps are created in all six series. The performance of the technique is assessed by comparing the filled-in series to the observed and by employing spectral analysis. The results reveal the usefulness of the method in multichannel hydrologic analysis.     

Accuracy assessment of optimum interpolation in multichannel precipitation analysis

The accuracy of an optimum interpolation technique in filling missing values in multichannel (or multisite) hydrologic series containing time-coincident data gaps is examined. The applied methodology is based on the maximum entropy method (MEM) of spectral estimation or multivariate autoregressive modeling and heavily depends upon the properties of multichannel prediction error filter (PEF). Six precipitation time series spatially located within a hydrologic basin are used and time-coincident artificial gaps are created in all six series. The performance of the technique is assessed by comparing the filled-in series to the observed and by employing spectral analysis. The results reveal the usefulness of the method in multichannel hydrologic analysis.     

Rainfall estimation using raingages and radar — A Bayesian approach: 2. An application

In Seo and Smith (this issue), a set of estimators was built in a Bayesian framework to estimate rainfall depth at an ungaged location using raingage measurements and radar rainfall data. The estimators are equivalent to lognormal co-kriging (simple co-kriging in the Gaussian domain) with uncertain mean and variance of gage rainfall. In this paper, the estimators are evaluated via cross-validation using hourly radar rainfall data and simulated hourly raingage data. Generation of raingage data is based on sample statistics of actual raingage measurements and radar rainfall data. The estimators are compared with lognormal co-kriging and nonparametric estimators. The Bayesian estimators are shown to provide some improvement over lognormal co-kriging under the criteria of mean error, root mean square error, and standardized mean square error. It is shown that, if the prior could be assessed more accurately, the margin of improvement in predicting estimation variance could be larger. In updating the uncertain mean and variance of gage rainfall, inclusion of radar rainfall data is seen to provide little improvement over using raingage data only.     

Statistical corrections of spatially interpolated missing precipitation data estimates

Spatial interpolation methods used for estimation of missing precipitation data generally under and overestimate the high and low extremes, respectively. This is a major limitation that plagues all spatial interpolation methods as observations from different sites are used in local or global variants of these methods for estimation of missing data. This study proposes bias‐correction methods similar to those used in climate change studies for correcting missing precipitation estimates provided by an optimal spatial interpolation method. The methods are applied to post‐interpolation estimates using quantile mapping, a variant of equi‐distant quantile matching and a new optimal single best estimator (SBE) scheme. The SBE is developed using a mixed‐integer nonlinear programming formulation. K‐fold cross validation of estimation and correction methods is carried out using 15 rain gauges in a temperate climatic region of the U.S. Exhaustive evaluation of bias‐corrected estimates is carried out using several statistical, error, performance and skill score measures. The differences among the bias‐correction methods, the effectiveness of the methods and their limitations are examined. The bias‐correction method based on a variant of equi‐distant quantile matching is recommended. Post‐interpolation bias corrections have preserved the site‐specific summary statistics with minor changes in the magnitudes of error and performance measures. The changes were found to be statistically insignificant based on parametric and nonparametric hypothesis tests. The correction methods provided improved skill scores with minimal changes in magnitudes of several extreme precipitation indices. The bias corrections of estimated data also brought site‐specific serial autocorrelations at different lags and transition states (dry‐to‐dry, dry‐to‐wet, wet‐to‐wet and wet‐to‐dry) close to those from the observed series. Bias corrections of missing data estimates provide better serially complete precipitation time series useful for climate change and variability studies in comparison to uncorrected filled data series. Copyright © 2013 John Wiley & Sons, Ltd.     

Distributed modelling of future changes in hydrological processes of Spencer Creek watershed

The hydrologic impact of climate change has been largely assessed using mostly conceptual hydrologic models. This study investigates the use of distributed hydrologic model for the assessment of the climate change impact for the Spencer Creek watershed in Southern Ontario (Canada). A coupled MIKE SHE/MIKE 11 hydrologic model is developed to represent the complex hydrologic conditions in the Spencer Creek watershed, and later to simulate climate change impact using Canadian global climate model (CGCM 3·1) simulations. Owing to the coarse resolution of GCM data (daily GCM outputs), statistical downscaling techniques are used to generate higher resolution data (daily precipitation and temperature series). The modelling results show that the coupled model captured the snow storage well and also provided good simulation of evapotranspiration (ET) and groundwater recharge. The simulated streamflows are consistent with the observed flows at different sites within the catchment. Using a conservative climate change scenario, the downscaled GCM scenarios predicted an approximately 14–17% increase in the annual mean precipitation and 2–3 °C increase in annual mean maximum and minimum temperatures for the 2050s (i.e., 2046–2065). When the downscaled GCM scenarios were used in the coupled model, the model predicted a 1–5% annual decrease in snow storage for 2050s, approximately 1–10% increase in annual ET, and a 0·5–6% decrease in the annual groundwater recharge. These results are consistent with the downscaled temperature results. For future streamflows, the coupled model indicated an approximately 10–25% increase in annual streamflows for all sites, which is consistent with the predicted changes in precipitation. Overall, it is shown that distributed hydrologic modelling can provide useful information not only about future changes in streamflow but also changes in other key hydrologic processes such as snow storage, ET, and groundwater recharge, which can be particularly important depending on the climatic region of concern. The study results indicate that the coupled MIKE SHE/MIKE 11 hydrologic model could be a particularly useful tool for understanding the integrated effect of climate change in complex catchment scale hydrology. Copyright © 2010 John Wiley & Sons, Ltd.     

地震波动强度变化的数学建模分析与仿真

传统利用灰色关联分析方法对地震波动强度变化进行数学建模分析与仿真时,对地震波动强度变化的数列进行仿真分析时,忽略了地震波动强度的时间属性对结果的影响,导致分析结果准确性较低。本论述提出新的地震波动强度变化数学建模分析与仿真方法,通过地震波动强度序列的经验分布确定门限自回归模型的门限值,依据该门限值、AIC最小准则以及最小残差平方等方法获取地震波动强度序列的门限自回归模型,分析自回归模型的极限环和振荡的属性特点,得到地震波动强度变化的初步数值模拟结果。本论述构建了基于均生函数的地震波动强度序列的数学模型,通过均生函数数学建模方法拟合地震波动强度时间序列,依据时间序列基于双评分准则选取拟合周期,实现地震波动强度的数值仿真。实验结果表明,所提方法对地震波动强度变化模型具有较高的准确性和稳定性。     

The effects of dam construction and precipitation variability on hydrologic alteration in the Lancang River Basin of southwest China

This paper presents an evaluation of the spatio-temporal patterns of hydrologic alteration induced by dam construction and precipitation variability in the Lancang River Basin of southwest China from 1957 to 2000. Analyses were conducted using the linear regression method, the Mann–Kendall test, and the Range of Variability Approach. The results indicate that there was considerable variation in the average monthly precipitation between the pre- and post-dam periods in the Lancang River Basin. Second, the magnitude of monthly runoff was strongly related to precipitation, which showed an up-down annual variation, and was significantly altered by dam construction and precipitation variability. In the modified series (hydrologic series with the precipitation impacts removed), runoff deviations between the pre- and post-dam periods became larger. Third, the extreme runoff cycles were influenced by dam construction and precipitation variability downstream from the dam, and the monthly maximum runoff increased from the pre-dam to post-dam period at all hydrologic stations. Fourth, the degree of hydrologic alteration (DHA) indicates that the precipitation variability not only affected the hydrologic regime of unregulated river reach but also modified the negative impacts of dam construction, which could provide a modest mitigation of the hydrologic alterations induced by dam construction, possibly decreasing the level of DHA. Last, the overall degree of hydrologic alteration in the observed series reached 25.2, 25.3, and 29.1 % for the upstream, midstream, and downstream areas, respectively. These results show that the hydrologic regimes of the Lancang River during the 1957–2000 period were affected by damming and precipitation variability, but the hydrologic alteration was relatively low in the upstream areas of the river without a dam.     

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.