Probabilistic modelling of flood events using the entropy copula

The estimation of flood frequency is vital for the flood control strategies and hydraulic structure design. Generating synthetic flood events according to statistical properties of observations is one of plausible methods to analyze the flood frequency. Due to the statistical dependence among the flood event variables (i.e. the flood peak, volume and duration), a multidimensional joint probability estimation is required. Recently, the copula method is widely used for multivariable dependent structure construction, however, the copula family should be chosen before application and the choice process is sometimes rather subjective. The entropy copula, a new copula family, employed in this research proposed a way to avoid the relatively subjective process by combining the theories of copula and entropy. The analysis shows the effectiveness of the entropy copula for probabilistic modelling the flood events of two hydrological gauges, and a comparison of accuracy with the popular copulas was made. The Gibbs sampling technique was applied for trivariate flood events simulation in order to mitigate the calculation difficulties of extending to three dimension directly. The simulation results indicate that the entropy copula is a simple and effective copula family for trivariate flood simulation.     

Spatio-temporal analysis and derivation of copula-based intensity–area–frequency curves for droughts in western Rajasthan (India)

This study presents spatio-temporal analysis of droughts in one of the most drought prone region in India–western Rajasthan and develops drought intensity-area-frequency curves for the region. The meteorological drought conditions are analyzed using 6-month standardized precipitation index (SPI-6) estimated at spatial resolution of 0.5° × 0.5°. Spatio-temporal analysis of SPI-6 indicates increase in frequency of droughts at the central part of the region. The non-parametric Mann–Kendall test for seasonal trend analysis showed increase in number of grids under drought during the study period. Further, bivariate frequency analysis of drought characteristics—intensity and areal extent is carried out using copula methods. For modeling joint dependence between drought variables, three copula families namely Gumbel-Hougaard, Frank and Plackett copulas are evaluated. Based on goodness-of-fit as well as upper tail dependence tests, it is found that the Gumbel-Hougaard copula best represents the drought properties. The copula-based joint distribution is used to compute conditional return periods and drought intensity–area–frequency (I–A–F) curves. The I–A–F curves could be helpful in risk evaluation of droughts in the region.     

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.     

Bivariate flood frequency analysis using the copula function: a case study of the Litija station on the Sava River

As an alternative to the commonly used univariate flood frequency analysis, copula frequency analysis can be used. In this study, 58 flood events at the Litija gauging station on the Sava River in Slovenia were analysed, selected based on annual maximum discharge values. Corresponding hydrograph volumes and durations were considered. Different bivariate copulas from three families were applied and compared using different statistical, graphical and upper tail dependence tests. The parameters of the copulas were estimated using the method of moments with the inversion of Kendall's tau. The Gumbel–Hougaard copula was selected as the most appropriate for the pair of peak discharge and hydrograph volume (Q‐V). The same copula was also selected for the pair hydrograph volume and duration (V‐D), and the Student‐t copula was selected for the pair of peak discharge and hydrograph duration (Q‐D). The differences among most of the applied copulas were not significant. Different primary, secondary and conditional return periods were calculated and compared, and some relationships among them were obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

Probabilistic earthquake hazard analysis for Cairo,Egypt

Cairo is the capital of Egypt and the largest city in the Arab world and Africa, and the sixteenth largest metropolitan area in the world. It was founded in the tenth century (969 ad) and is 1046 years old. It has long been a center of the region’s political and cultural life. Therefore, the earthquake risk assessment for Cairo has a great importance. The present work aims to analysis the earthquake hazard of Cairo as a key input’s element for the risk assessment. The regional seismotectonics setting shows that Cairo could be affected by both far- and near-field seismic sources. The seismic hazard of Cairo has been estimated using the probabilistic seismic hazard approach. The logic tree frame work was used during the calculations. Epistemic uncertainties were considered into account by using alternative seismotectonics models and alternative ground motion prediction equations. Seismic hazard values have been estimated within a grid of 0.1°?×?0.1 ° spacing for all of Cairo’s districts at different spectral periods and four return periods (224, 615, 1230, and 4745 years). Moreover, the uniform hazard spectra have been calculated at the same return periods. The pattern of the contour maps show that the highest values of the peak ground acceleration is concentrated in the eastern zone’s districts (e.g., El Nozha) and the lowest values at the northern and western zone’s districts (e.g., El Sharabiya and El Khalifa).     

Asymmetric copula in multivariate flood frequency analysis

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

Comparison of conventional and Monte Carlo simulation-based probabilistic seismic hazard analyses for Shiraz city,southern Iran

Estimation of ground-motion amplitudes of different hazard levels is of paramount importance in planning of urban development of any metropolis. Such estimation can be computed through a probabilistic seismic hazard analysis (PSHA). This paper concentrates on the PSHA of an area located in Shiraz city, southern Iran. The area includes whole of Shiraz city (i.e., one of the largest and most populous cities of Iran) and its outskirts. Conventional and Monte Carlo simulation-based approaches are utilized to perform the PSHA of the studied area. Two areal seismic source models are delineated, and thence seismicity parameters of all zones associated with their corresponding uncertainties are computed. Uncertainties in ground-motion prediction are accounted for via three ground-motion prediction equations (GMPEs) within the logic tree framework. These GMPEs are applied to estimate bedrock ground shaking (V_s30?=?760 m/s) for several return periods (i.e., 75, 475, 975, and 2475 years). In general, the results of the two abovementioned PSHA approaches show relatively similar results. However, the Monte Carlo simulation-based approach overpredicts bedrock spectral accelerations at periods of 0.4–2.5 s compared to the conventional PSHA approach for return periods of 475, 975, and 2475 years.     

Copula models for frequency analysis what can be learned from a Bayesian perspective?

Large spring floods in the Québec region exhibit correlated peakflow, duration and volume. Consequently, traditional univariate hydrological frequency analyses must be complemented by multivariate probabilistic assessment to provide a meaningful design flood level as requested in hydrological engineering (based on return period evaluation of a single quantity of interest). In this paper we study 47 years of a peak/volume dataset for the Romaine River with a parametric copula model. The margins are modeled with a normal or gamma distribution and the dependence is depicted through a parametric family of copulas (Arch 12 or Arch 14). Parameter joint inference and model selection are performed under the Bayesian paradigm. This approach enlightens specific features of interest for hydrological engineering: (i) cross correlation between margin parameters are stronger than expected , (ii) marginal distributions cannot be forgotten in the model selection process and (iii) special attention must be addressed to model validation as far as extreme values are of concern.     

Risk assessment of extreme air pollution based on partial duration series: IDF approach

The occurrences of extreme pollution events have serious effects on human health, environmental ecosystems, and the national economy. To gain a better understanding of this issue, risk assessments on the behavior of these events must be effectively designed to anticipate the likelihood of their occurrence. In this study, we propose using the intensity–duration–frequency (IDF) technique to describe the relationship of pollution intensity (i) to its duration (d) and return period (T). As a case study, we used data from the city of Klang, Malaysia. The construction of IDF curves involves a process of determining a partial duration series of an extreme pollution event. Based on PDS data, a generalized Pareto distribution (GPD) is used to represent its probabilistic behaviors. The estimated return period and IDF curves for pollution intensities corresponding to various return periods are determined based on the fitted GPD model. The results reveal that pollution intensities in Klang tend to increase with increases in the length of time between return periods. Although the IDF curves show different magnitudes for different return periods, all the curves show similar increasing trends. In fact, longer return periods are associated with higher estimates of pollution intensity. Based on the study results, we can conclude that the IDF approach provides a good basis for decision-makers to evaluate the expected risk of future extreme pollution events.     

Joint probability of precipitation and reservoir storage for drought estimation in the headwater basin of the Huaihe River,China

Reservoir storage plays an important role in water supply during the dry season when precipitation is insufficient. In a watershed where the streams are controlled by reservoirs, drought occurrences depend on not only precipitation variations but also reservoir regulation. In this study, the joint dependence structure of the reservoir storage and its relevant variables of precipitation and/or upstream outflow were analyzed for two cascade reservoirs in a headwater basin of the Huaihe River, China. Correlation analysis indicates that the reservoir storage in October (the end of the wet season) depends highly on the regional precipitation at time scales of several months, e.g., 7 months for the upstream and 9 months for the downstream. Additionally, the downstream storage is correlated with outflow from the upstream reservoir at the 5-month timescale significantly. For estimation of the joint probability of pairs of the storage and its relevant variables, univariate marginal distributions and bivariate copula were appropriately selected in terms of statistical tests. The bivariate return period of \(T(X < x,Y < y)\) and \(T(X \le x,Y \ge y)\) and the conditional probability of \(P(Y \ge y|X \le x)\) were estimated by using the selected Clayton copula. The results from contour lines of the bivariate return period demonstrate that the probability of drought occurrences affected by both reservoir storage and precipitation/outflow is smaller than that by either of the variables. Meanwhile, the concurrent drought probability between precipitation and reservoir storage in the upstream is higher than that in the downstream. The estimated conditional probability offers useful information on how much the regular storage could be remained under some specified drought levels of precipitation/upstream outflow. Therefore, the results are helpful for improving the operation strategies of the cascade reservoirs for the adaptive management of drought under different climate variations.     

Assessment of hydrological droughts for the Yellow River,China, using copulas

Droughts are one of the normal and recurrent climatic phenomena on Earth. However, recurring prolonged droughts have caused far‐reaching and diverse impacts because of water deficits. This study aims to investigate the hydrological droughts of the Yellow River in northern China. Since drought duration and drought severity exhibit significant correlation, a bivariate distribution is used to model the drought duration and severity jointly. However, drought duration and drought severity are often modelled by different distributions; the commonly used bivariate distributions cannot be applied. In this study, a copula is employed to construct the bivariate drought distribution. The copula is a function that links the univariate marginal distributions to form the bivariate distribution. The bivariate return periods are also established to explore the drought characteristics of the historically noticeable droughts. The results show that the return period of the drought that occurred in late 1920s to early 1930s is 105 years. The significant 1997 dry‐up phenomenon that occurred in the downstream Yellow River (resulting from the 1997–1998 drought) only has a return period of 4·4 years and is probably induced by two successive droughts and deteriorated by other factors, such as human activities. Copyright © 2007 John Wiley & Sons, Ltd.     

Seismic hazard assessment for Iceland in terms of macroseismic intensity using a site approach

We present the results of probabilistic seismic hazard assessment for Iceland in the framework of the EU project UPStrat-MAFA using the so-called site approach implemented in the SASHA computational code. This approach estimates seismic hazard in terms of macroseismic intensity by basically relying on local information about documented effects of past seismic events in the framework of a formally coherent and complete treatment of intensity data. In the case of Iceland, due to the lack of observed intensities for past earthquakes, local seismic histories were built using indirect macroseismic estimates deduced from epicentral information through an empirical attenuation relationship in probabilistic form. Seismic hazard was computed for four exceedance probabilities for an exposure time of 50 years, equivalent to average return periods of 50, 200, 475 and 975 years. For some localities, further return periods were examined and deaggregation analysis was performed. Results appear significantly different from previous seismic hazard maps, though just a semi-qualitative comparison is possible because of the different shaking measure considered (peak ground acceleration versus intensity), and the different computational methodology and input data used in these studies.     

The influence of meteorological variation on the upwelling system off eastern Hainan during summer 2007–2008

The influence of meteorological variation, i.e., typhoon and precipitation events, on the coastal upwelling off the eastern Hainan Island was studied based on observations taken during two upwelling seasons. The observations were made in August 2007 and July 2008, respectively. We found that, in principle, similar structure of sea surface temperature and bottom temperature prevailed in both observational periods, providing evidence that upwelling events occur frequently during the summer monsoon along the eastern Hainan shelf. Based on a simple momentum balance theory, we studied the balances between momentum fluxes, wind stress, and bottom stress. The results showed that the Burger number is S ≈ 1, indicating that the cross-shelf momentum flux divergence was balanced by the wind stress and the onshore return flow occurred in the interior of the water column. Hence, a conceptual model of the upwelling structure was built for further understanding of upwelling events. In addition, it was also observed that variations in the strength of upwelling are controlled by storm events, i.e., strong northerly winds change the structure of the thermocline on the shelf significantly. The strong mixing caused by wind reduces the strength of the thermocline, in particular in coastal seas. Based on our conceptual model, a frontal zone between mixed coastal water and offshore water develops which destabilizing the water column and hence decreases the upwelling strength. Freshwaters from the two main rivers in the Wenchang Bay are confined to the coastal area less than 20–30 m deep, as confirmed by our water mass analysis. Freshwater discharge stabilized the water column, inhibiting the upwelling as shown by the potential energy calculation. Consequently, estuarine water only inhibits the upwelling in the near coastal area. Therefore, it can be concluded that estuarine water does not have a significant impact on upwelling strength on the shelf.     

Advancing fracture fragility assessment of pre-Northridge welded column splices

A refined probabilistic assessment of seismic demands and fracture capacity of welded column splice (WCS) connections in welded steel moment resisting frames (WSMRFs) is presented. Seismic demand assessment is performed through cloud-based nonlinear time history analysis (NLTHA) for two case-study structures, i.e., a 4- and a 20- story WSMRFs. Results from NLTHA are used to derive fracture fragility of WCS connections. To this aim, the study investigates (1) optimal ground-motion intensity measures for conditioning probabilistic seismic demand models in terms of global (i.e., maximum inter-story drift ratio) and local (i.e., peak tensile stress in the flange of WCSs) engineering demand parameters of WSMRFs; (2) the effect of ground-motion vertical components on the longitudinal flange stress of WCS connections and their resulting fracture fragility; and (3) the effect of WCS capacity uncertainties on the fracture fragility estimates of those connections. For the latter case, an advanced finite element fracture mechanics-based approach proposed by the authors is employed to capture aleatory and epistemic uncertainties affecting fracture capacities. The focus is on pre-Northridge WCS connections featuring partial joint penetration and brittle materials, making them highly vulnerable to seismic fracture. Fracture fragility results for the case-study structures are compared and discussed, highlighting the importance of the considered issues on fragility estimates, particularly in the case of high-rise structures. Findings from the study contribute shedding some light on the influence of seismic demand and capacity uncertainties on the assessment of fracture fragility of WCS connections. These findings can guide similar performance-based assessment exercises for WSMRFs to inform, for instance, the planning and design of retrofitting strategies for those vulnerable connections.     

Probabilistic seismic hazard analysis in Nepal

The seismic ground motion hazard for Nepal has been estimated using a probabilistic approach. A catalogue of earthquakes has been compiled for Nepal and the surrounding region （latitude 26% N and 31.7% N and longitude 79° E and 90° E） from 1255 to 2011. The distribution of catalogued earthquakes, together with available geological and tectonic information were used to delineate twenty-three seismic source seismic source information and probabilistic earthquake hazard prediction relationship, peak ground accelerations （PGAs） have zones in Nepal and the surrounding region. By using the parameters in conjunction with a selected ground motion been calculated at bedrock level with 63%, 10%, and 2% probability of exceedance in 50 years. The estimated PGA values are in the range of 0.07-0.16 g, 0.21 0.62 g, and 0.38-1.1 g for 63%, 10%, and 2% probability of exceedance in 50 years, respectively. The resulting ground motion maps show different characteristics of PGA distribution, i.e., high hazard in the far-western and eastern sections, and low hazard in southern Nepal. The quantified PGA values at bedrock level provide information for microzonation studies in different parts of the country.     

Application of bivariate frequency analysis to the derivation of rainfall–frequency curves

Bivariate distributions have been recently employed in hydrologic frequency analysis to analyze the joint probabilistic characteristics of multivariate storm events. This study aims to derive practical solutions of application for the bivariate distribution to estimate design rainfalls corresponding to the desired return periods. Using the Gumbel mixed model, this study constructed rainfall–frequency curves at sample stations in Korea which provide joint relationships between amount, duration, and frequency of storm events. Based on comparisons and analyses of the rainfall–frequency curves derived from univariate and bivariate storm frequency analyses, this study found that conditional frequency analysis provides more appropriate estimates of design rainfalls as it more accurately represents the natural relationship between storm properties than the conventional univariate storm frequency analysis.     

Frequency analysis of droughts using the Plackett copula and parameter estimation by genetic algorithm

This study aims to model the joint probability distribution of drought duration, severity and inter-arrival time using a trivariate Plackett copula. The drought duration and inter-arrival time each follow the Weibull distribution and the drought severity follows the gamma distribution. Parameters of these univariate distributions are estimated using the method of moments (MOM), maximum likelihood method (MLM), probability weighted moments (PWM), and a genetic algorithm (GA); whereas parameters of the bivariate and trivariate Plackett copulas are estimated using the log-pseudolikelihood function method (LPLF) and GA. Streamflow data from three gaging stations, Zhuangtou, Taian and Tianyang, located in the Wei River basin, China, are employed to test the trivariate Plackett copula. The results show that the Plackett copula is capable of yielding bivariate and trivariate probability distributions of correlated drought variables.     

An analog period method for gap-filling of latent heat flux measurements

Practically all records of eddy-covariance flux measurements are affected by gaps, caused by several reasons. In this work, we propose analog period (AP) methods for gap-filling, and test them for filling gaps of latent heat flux at five AmeriFlux sites. The essence of the methods is to look for periods in the record that bear a strong resemblance, in the variable to be filled, to the periods immediately before and after the gap. Similarity between periods is gauged by the coefficient of determination, and the search for similar periods and their ranking is straightforward. The methods are developed in a univariate version (that uses only the latent heat flux data series itself) and several multivariate ones, that incorporate sensible heat flux, ground heat flux and net radiation data. For each set of independent variables used for gap-filling, the methods are tested against an existing gap-filling procedure with similar data requirements. Thus, the univariate version is tested against the mean diurnal variation method, and the multivariate versions are tested against corresponding simple and multiple linear regression techniques that use energy-budget data, and in one case the evaporative fraction as well. In our tests, the proposed univariate version performs better than the mean diurnal variation method, and the multivariate versions perform better than simple/multiple linear regression methods. An often used available computer package, REddyProc, was also tested as a basis of comparison. In general, the proposed methods (in univariate and multivariate versions) and simple/multiple linear regressions performed better than REddyProc. For the datasets analysed, gap filling via the evaporative fraction method performed poorly.     

Can we tell more than we can know? The limits of bivariate drought analyses in the United States

The joint occurrence of extreme hydroclimatic events, such as simultaneous precipitation deficit and high temperature, results in the so-called compound events, and has a serious impact on risk assessment and mitigation strategies. Multivariate frequency analysis (MFA) allows a probabilistic quantitative assessment of this risk under uncertainty. Analyzing precipitation and temperature records in the contiguous United States (CONUS), and focusing on the assessment of the degree of rarity of the 2014 California drought, we highlight some critical aspects of MFA that are often overlooked and should be carefully taken into account for a correct interpretation of the results. In particular, we show that an informative exploratory data analysis (EDA) devised to check the basic hypotheses of MFA, a suitable assessment of the sampling uncertainty, and a better understanding of probabilistic concepts can help to avoid misinterpretation of univariate and multivariate return periods, and incoherent conclusions concerning the risk of compound extreme hydroclimatic events. Empirical results show that the dependence between precipitation deficit and temperature across the CONUS can be positive, negative or not significant and does not exhibit significant changes in the last three decades. Focusing on the 2014 California drought as a compound event and based on the data used, the probability of occurrence strongly depends on the selected variables and how they are combined, and is affected by large uncertainty, thus preventing definite conclusions about the actual degree of rarity of this event.     

Application of copulas for derivation of drought severity–duration–frequency curves

This study presents copula‐based multivariate probabilistic approach to model severity–duration–frequency (S‐D‐F) relationship of drought events in western Rajasthan, India. Drought occurrences are analysed using standardized precipitation index computed on monthly mean areal precipitation, aggregated at a time scale of 6 months. After testing with a series of probability density functions, the drought variable severity is found to be better represented with log‐normal distribution, whereas duration is well fitted with exponential distribution. Four different classes of bivariate copulas – Archimedean, extreme value, Plackett, and elliptical families are evaluated for modelling joint distribution of drought characteristics. It is observed that the extreme value copula – Gumbel–Hougaard copula – performed better as compared with other classes of copulas, based on results of various statistical tests and upper tail dependence coefficient. The joint distribution obtained from best performing copula is then employed to determine conditional return period and to derive drought severity‐duration‐frequency (S‐D‐F) curves for the study region. The results of the study suggests that the copula method can be used effectively to derive the drought S‐D‐F curves, which can be helpful in planning and adopting suitable drought mitigation strategies in drought‐prone areas. Copyright © 2011 John Wiley & Sons, Ltd.