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Advances in Multivariate Analysis of Environmental Phenomena: Celebrating the 15th Anniversary of Copulas in Hydrology

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A special issue of Water (ISSN 2073-4441). This special issue belongs to the section "Hydrology".

Deadline for manuscript submissions: closed (30 September 2019) | Viewed by 49538

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Special Issue Editors

Dr. Gianfausto Salvadori

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Guest Editor

Dipartimento di Matematica e Fisica, Università del Salento, Provinciale Lecce-Arnesano, P.O.Box 193, Lecce I-73100, Italy
Interests: Multivariate Analysis and Modeling of Environmental Phenomena

Prof. Dr. Carlo De Michele

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Guest Editor

Department Civil and Environmental Engineering, Politecnico di Milano, 20133 Milano, Italy
Interests: snow hydrology; extremes in hydrologic variables
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Dear Colleagues,

In 2003, a seminal paper introduced the notion of Copula in hydrology: The target was to provide a new statistical tool to conveniently deal with the modeling of multivariate environmental phenomena. Since then, thousands of works have used Copulas to approach awkward and tricky problems involving the (joint) random behavior of non-independent variables, coming up with new models and techniques of an unprecedented reach and scope. Several are the areas of hydrological sciences that have taken advantage of the power of Copulas: Among others, the study of floods, droughts, rainfall, and sea storms can now benefit from an increased capacity of (statistically) explaining the complex interactions of a number of variables. In turn, the predictive ability of many models has greatly improved, and the assessment of environmental risk has made substantial progresses. In addition, new findings concerning the quantification of hydrologic uncertainty have been obtained, and the assessment of basin similarities and regionalization techniques has received a great impulse. Moreover, thanks to innovative techniques of multivariate design, new cost-benefit strategies have been developed, resulting in less expensive structures as compared, e.g., to those planned via a traditional univariate approach.

However, the multivariate analysis via Copulas is yet in its infancy: several are the questions that are still open, and further startling discoveries are expected in the near future. The aim of this Special Issue is to present the most recent advances concerning the multivariate analysis of environmental phenomena (and especially the hydrological ones), with an eye to applications, and to provide a critical discussion of several important related issues.

Dr. Gianfausto Salvadori
Prof. Carlo De Michele
Guest Editors

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Keywords

Copula
Multivariate Analysis
Multivariate Design
Risk Assessment
Hydrology
Flood
Drought
Rainfall
Sea Storm

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Published Papers (11 papers)

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18 pages, 2967 KiB

Open AccessArticle

Effect of Multicollinearity on the Bivariate Frequency Analysis of Annual Maximum Rainfall Events

by Chulsang Yoo and Eunsaem Cho

Water 2019, 11(5), 905; https://doi.org/10.3390/w11050905 - 29 Apr 2019

Cited by 13 | Viewed by 3929

Abstract

A rainfall event, simplified by a rectangular pulse, is defined by three components: the rainfall duration, the total rainfall depth, and mean rainfall intensity. However, as the mean rainfall intensity can be calculated by the total rainfall depth divided by the rainfall duration, any two components can fully define the rainfall event (i.e., one component must be redundant). The frequency analysis of a rainfall event also considers just two components selected rather arbitrarily out of these three components. However, this study argues that the two components should be selected properly or the result of frequency analysis can be significantly biased. This study fully discusses this selection problem with the annual maximum rainfall events from Seoul, Korea. In fact, this issue is closely related with the multicollinearity in the multivariate regression analysis, which indicates that as interdependency among variables grows the variance of the regression coefficient also increases to result in the low quality of resulting estimate. The findings of this study are summarized as follows: (1) The results of frequency analysis are totally different according to the selected two variables out of three. (2) Among three results, the result considering the total rainfall depth and the mean rainfall intensity is found to be the most reasonable. (3) This result is fully supported by the multicollinearity issue among the correlated variables. The rainfall duration should be excluded in the frequency analysis of a rainfall event as its variance inflation factor is very high. Full article

(This article belongs to the Special Issue Advances in Multivariate Analysis of Environmental Phenomena: Celebrating the 15th Anniversary of Copulas in Hydrology)

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20 pages, 3428 KiB

Open AccessArticle

Synthetic Hydrographs Generation Downstream of a River Junction Using a Copula Approach for Hydrological Risk Assessment in Large Dams

by Jose Angel Aranda and Rafael García-Bartual

Water 2018, 10(11), 1570; https://doi.org/10.3390/w10111570 - 2 Nov 2018

Cited by 8 | Viewed by 4956 | Correction

Abstract

Peak flows values (Q) and hydrograph volumes (V) are obtained from a selected family of historical flood events (period 1957–2017), for two neighboring mountain catchments located in the Ebro river basin, Spain: rivers Ésera and Isábena. Barasona dam is located downstream of the river junction. The peaks over threshold (POT) method is used for a univariate frequency analysis performed for both variables, Q and V, comparing several suitable distribution functions. Extreme value copulas families have been applied to model the bivariate distribution (Q, V) for each of the rivers. Several goodness-of-fit tests were used to assess the applicability of the selected copulas. A similar copula approach was carried out to model the dependence between peak flows of both rivers. Based on the above-mentioned statistical analysis, a Monte Carlo simulation of synthetic design flood hydrographs (DFH) downstream of the river junction is performed. A gamma-type theoretical pattern is assumed for partial hydrographs. The resulting synthetic hydrographs at the Barasona reservoir are finally obtained accounting for flow peak time lag, also described in statistical terms. A 50,000 hydrographs ensemble was generated, preserving statistical properties of marginal distributions as well as statistical dependence between variables. The proposed method provides an efficient and practical modeling framework for the hydrological risk assessment of the dam, improving the basis for the optimal management of such infrastructure. Full article

(This article belongs to the Special Issue Advances in Multivariate Analysis of Environmental Phenomena: Celebrating the 15th Anniversary of Copulas in Hydrology)

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19 pages, 3124 KiB

Open AccessArticle

A Copula-Based Bayesian Network for Modeling Compound Flood Hazard from Riverine and Coastal Interactions at the Catchment Scale: An Application to the Houston Ship Channel, Texas

by Anaïs Couasnon, Antonia Sebastian and Oswaldo Morales-Nápoles

Water 2018, 10(9), 1190; https://doi.org/10.3390/w10091190 - 4 Sep 2018

Cited by 73 | Viewed by 7764

Abstract

Traditional flood hazard analyses often rely on univariate probability distributions; however, in many coastal catchments, flooding is the result of complex hydrodynamic interactions between multiple drivers. For example, synoptic meteorological conditions can produce considerable rainfall-runoff, while also generating wind-driven elevated sea-levels. When these drivers interact in space and time, they can exacerbate flood impacts, a phenomenon known as compound flooding. In this paper, we build a Bayesian Network based on Gaussian copulas to generate the equivalent of 500 years of daily stochastic boundary conditions for a coastal watershed in Southeast Texas. In doing so, we overcome many of the limitations of conventional univariate approaches and are able to probabilistically represent compound floods caused by riverine and coastal interactions. We model the resulting water levels using a one-dimensional (1D) steady-state hydraulic model and find that flood stages in the catchment are strongly affected by backwater effects from tributary inflows and downstream water levels. By comparing our results against a bathtub modeling approach, we show that simplifying the multivariate dependence between flood drivers can lead to an underestimation of flood impacts, highlighting that accounting for multivariate dependence is critical for the accurate representation of flood risk in coastal catchments prone to compound events. Full article

(This article belongs to the Special Issue Advances in Multivariate Analysis of Environmental Phenomena: Celebrating the 15th Anniversary of Copulas in Hydrology)

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14 pages, 428 KiB

Open AccessArticle

Confidence Regions for Multivariate Quantiles

by Maximilian Coblenz, Rainer Dyckerhoff and Oliver Grothe

Water 2018, 10(8), 996; https://doi.org/10.3390/w10080996 - 27 Jul 2018

Cited by 3 | Viewed by 3487

Abstract

Multivariate quantiles are of increasing importance in applications of hydrology. This calls for reliable methods to evaluate the precision of the estimated quantile sets. Therefore, we focus on two recently developed approaches to estimate confidence regions for level sets and extend them to provide confidence regions for multivariate quantiles based on copulas. In a simulation study, we check coverage probabilities of the employed approaches. In particular, we focus on small sample sizes. One approach shows reasonable coverage probabilities and the second one obtains mixed results. Not only the bounded copula domain but also the additional estimation of the quantile level pose some problems. A small sample application gives further insight into the employed techniques. Full article

(This article belongs to the Special Issue Advances in Multivariate Analysis of Environmental Phenomena: Celebrating the 15th Anniversary of Copulas in Hydrology)

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17 pages, 2528 KiB

Open AccessArticle

Application of Copula Functions for Rainfall Interception Modelling

by Nejc Bezak, Katarina Zabret and Mojca Šraj

Water 2018, 10(8), 995; https://doi.org/10.3390/w10080995 - 27 Jul 2018

Cited by 13 | Viewed by 3802

Abstract

Rainfall interception is an important process of the water cycle that can have significant influence on surface runoff and groundwater storage. Since rainfall interception measurements are rare and time consuming, rainfall interception estimation can be made indirectly using different meteorological variables. Experimental data of rainfall interception for birch and pine trees was measured at an experimental plot located in an urban area of Ljubljana, Slovenia in this study. A copula model was applied to predict the rainfall interception using meteorological variables, namely air temperature and vapour pressure deficit data. The copula model performance was compared to some other models such as decision trees, multiple linear regressions, and exponential functions. Using random sampling, we found that the copula model where Khoudraji-Liebscher copula functions were used yielded slightly smaller root mean square error (RMSE) and mean absolute error (MAE) values than other tested methods (i.e., RMSE and MAE results for birch trees were 24.2% and 18.2%, respectively and RMSE and MAE results for pine trees were 25.0% and 19.6%, respectively). The results demonstrate that the copula-based proposed method and other tested models could be used for the prediction of rainfall interception at the considered plot and in the wider surroundings. Furthermore, these models could also be applied for the prediction of rainfall interception for these two tree species in other locations under similar vegetation and meteorological conditions. Full article

(This article belongs to the Special Issue Advances in Multivariate Analysis of Environmental Phenomena: Celebrating the 15th Anniversary of Copulas in Hydrology)

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25 pages, 2993 KiB

Open AccessArticle

Spatio-Temporal Synthesis of Continuous Precipitation Series Using Vine Copulas

by Ana Claudia Callau Poduje and Uwe Haberlandt

Water 2018, 10(7), 862; https://doi.org/10.3390/w10070862 - 28 Jun 2018

Cited by 8 | Viewed by 4239

Abstract

Long and continuous series of precipitation in a high temporal resolution are required for several purposes, namely, urban hydrological applications, design of flash flood control structures, etc. As data of the temporally required resolution is often available for short period, it is advantageous to develop a precipitation model to allow for the generation of long synthetic series. A stochastic model is applied for this purpose, involving an alternating renewal process (ARP) describing a system consisting of spells that can take two possible states: wet or dry. Stochastic generation of rainfall time series using ARP models is straight forward for single site simulation. The aim of this work is to present an extension of the model to spatio-temporal simulations. The proposed methodology combines an occurrence model to define in which locations rainfall events occur simultaneously with a multivariate copula to generate synthetic events. Rainfall series registered in different regions of Germany are used to develop and test the methodology. Results are compared with an existing method in which long independent time series of rainfall events are transformed to spatially dependent ones by permutation of their order. The proposed model shows to perform as a satisfactory extension of the ARP model for multiple sites simulations. Full article

(This article belongs to the Special Issue Advances in Multivariate Analysis of Environmental Phenomena: Celebrating the 15th Anniversary of Copulas in Hydrology)

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23 pages, 14101 KiB

Open AccessArticle

A Cautionary Note on the Reproduction of Dependencies through Linear Stochastic Models with Non-Gaussian White Noise

by Ioannis Tsoukalas, Simon Michael Papalexiou, Andreas Efstratiadis and Christos Makropoulos

Water 2018, 10(6), 771; https://doi.org/10.3390/w10060771 - 12 Jun 2018

Cited by 18 | Viewed by 4646

Abstract

Since the prime days of stochastic hydrology back in 1960s, autoregressive (AR) and moving average (MA) models (as well as their extensions) have been widely used to simulate hydrometeorological processes. Initially, AR(1) or Markovian models with Gaussian noise prevailed due to their conceptual and mathematical simplicity. However, the ubiquitous skewed behavior of most hydrometeorological processes, particularly at fine time scales, necessitated the generation of synthetic time series to also reproduce higher-order moments. In this respect, the former schemes were enhanced to preserve skewness through the use of non-Gaussian white noise— a modification attributed to Thomas and Fiering (TF). Although preserving higher-order moments to approximate a distribution is a limited and potentially risky solution, the TF approach has become a common choice in operational practice. In this study, almost half a century after its introduction, we reveal an important flaw that spans over all popular linear stochastic models that employ non-Gaussian white noise. Focusing on the Markovian case, we prove mathematically that this generating scheme provides bounded dependence patterns, which are both unrealistic and inconsistent with the observed data. This so-called “envelope behavior” is amplified as the skewness and correlation increases, as demonstrated on the basis of real-world and hypothetical simulation examples. Full article

(This article belongs to the Special Issue Advances in Multivariate Analysis of Environmental Phenomena: Celebrating the 15th Anniversary of Copulas in Hydrology)

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15 pages, 1098 KiB

Open AccessArticle

Hazard Assessment under Multivariate Distributional Change-Points: Guidelines and a Flood Case Study

by Gianfausto Salvadori, Fabrizio Durante, Carlo De Michele and Mauro Bernardi

Water 2018, 10(6), 751; https://doi.org/10.3390/w10060751 - 8 Jun 2018

Cited by 16 | Viewed by 3869

Abstract

One of the ultimate goals of hydrological studies is to assess whether or not the dynamics of the variables of interest are changing. For this purpose, specific statistics are usually adopted: e.g., overall indices, averages, variances, correlations, root-mean-square differences, monthly/annual averages, seasonal patterns, maximum and minimum values, quantiles, trends, etc. In this work, a distributional multivariate approach to the problem is outlined, also accounting for the fact that the variables of interest are often dependent. Here, the Copula Theory, the Failure Probabilities, and suitable non-parametric statistical Change-Point tests are used in order to provide an assessment of the hazard. A hydrological case study is utilized to illustrate the issue and the methodology (viz., assessment of a dam spillway), considering the bivariate dynamics of annual maximum flood peak and volume observed at the Ceppo Morelli dam (located in the Piedmont region, Northern Italy) over a 50-year period. In particular, several problems—often present in hydrological analyses—are debated: namely, (i) the uncertainties due to the presence of heavy tailed random variables, and (ii) the hydrological meaning/interpretation of the results of statistical tests. Furthermore, the suitability of the procedures proposed to fulfill the goals of the study (viz., detecting and interpreting non-stationarity) is discussed. Overall, the main recommendation is that statistical (multivariate) investigations may represent a necessary step, though they may not be sufficient to assess hydrological (environmental) hazards. Full article

(This article belongs to the Special Issue Advances in Multivariate Analysis of Environmental Phenomena: Celebrating the 15th Anniversary of Copulas in Hydrology)

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23 pages, 3153 KiB

Open AccessArticle

Design Flood Estimation Methods for Cascade Reservoirs Based on Copulas

by Shenglian Guo, Rizwan Muhammad, Zhangjun Liu, Feng Xiong and Jiabo Yin

Water 2018, 10(5), 560; https://doi.org/10.3390/w10050560 - 26 Apr 2018

Cited by 23 | Viewed by 4649

Abstract

Reservoirs operation alters the natural flow regime at downstream site and thus has a great impact on the design flood values. The general framework of flood regional composition and Equivalent Frequency Regional Composition (EFRC) method are currently used to calculate design floods at downstream site while considering the impact of the upstream reservoirs. However, this EFRC method deems perfect correlation between peak floods that occurred at one sub-basin and downstream site, which implicitly assumes that the rainfall and the land surface process are uniformly distributed for various sub-basins. In this study, the Conditional Expectation Regional Composition (CERC) method and Most Likely Regional Composition (MLRC) method based on copula function are proposed and developed under the flood regional composition framework. The proposed methods (i.e., CERC and MLRC) are tested and compared with the EFRC method in the Shuibuya-Geheyan-Gaobazhou cascade reservoirs located at Qingjiang River basin, a tributary of Yangtze River in China. Design flood values of the Gaobazhou reservoir site are estimated under the impact of upstream cascade reservoirs, respectively. Results show that design peak discharges at the Gaobazhou dam site have been significantly reduced due to the impact of upstream reservoir regulation. The EFRC method, not taking the actual dependence of floods occurred at various sub-basins into account; as a consequence, it yields an under-or overestimation of the risk that is associated with a given event in hydrological design. The proposed methods with stronger statistical basis can better capture the actual spatial correlation of flood events occurred at various sub-basins, and the estimated design flood values are more reasonable than the currently used EFRC method. The MLRC method is recommended for design flood estimation in the cascade reservoirs since its composition is unique and easy to implement. Full article

(This article belongs to the Special Issue Advances in Multivariate Analysis of Environmental Phenomena: Celebrating the 15th Anniversary of Copulas in Hydrology)

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18 pages, 5218 KiB

Open AccessArticle

Probability Analysis of the Water Table and Driving Factors Using a Multidimensional Copula Function

by Qiying You, Yan Liu and Zhao Liu

Water 2018, 10(4), 472; https://doi.org/10.3390/w10040472 - 12 Apr 2018

Cited by 8 | Viewed by 5053

Abstract

The relationship between the water table and driving factors is a reliable theoretical reference for the reasonable planning of surface water resources and the water table. Previous research has neglected the distribution and probabilities of the water table. However, this paper analyzes the relationship between the water table and driving factors from a statistical perspective by correcting the variables and introducing the Kernel Distribution Estimation and the Copula Function. The average data of the buried depth of the phreatic water, annual irrigation volume of the surface water, and precipitation in the Jinghui Irrigation District in China from 1977 to 2013 were adopted. We precisely obtained the two-dimensional (2D) and three-dimensional (3D) Joint Distribution Function of each driving factor and the marginal distribution of the water table, calculate the conditional probability in different ranges, and exactly predict the design value of surface water irrigation giving set conditions. Eventually, we emphasize the importance of probability analysis and prediction in groundwater planning. Full article

(This article belongs to the Special Issue Advances in Multivariate Analysis of Environmental Phenomena: Celebrating the 15th Anniversary of Copulas in Hydrology)

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