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Compound flooding is a multidimensional consequence of the joint impact of multiple intercorrelated drivers, such as oceanographic, hydrologic, and meteorological. These individual drivers exhibit interdependence due to common forcing mechanisms. If they occur simultaneously or successively, the probability of their joint occurrence will be higher than expected if considered separately. The copula-based multivariate joint analysis can effectively measure hydrologic risk associated with compound events. Because of the involvement of multiple drivers, it is necessary to switch from bivariate (2D) to trivariate (3D) analyses. This study presents an original trivariate probabilistic framework by incorporating multivariate hierarchal models called asymmetric or fully nested Archimedean (or FNA) copula in the joint analysis of compound flood risk. The efficacy of the derived FNA copulas model, together with symmetric Archimedean and Elliptical class copulas, are tested by compounding the joint impact of rainfall, storm surge, and river discharge observations through a case study at the west coast of Canada. The obtained copula-based joint analysis is employed in multivariate analysis of flood risks in trivariate and bivariate primary joint and conditional joint return periods. The estimated joint return periods are further employed in estimating failure probability statistics for assessing the trivariate (and bivariate) hydrologic risk associated with compound events. The statistical tests found the fully nested Frank copula outperforms symmetric 3D copulas. Our work confirms that for practical compound flood risk analysis together with bivariate or univariate return periods, it is essential to account for the trivariate joint return periods to assess the expected compound flood risk and strength of influence of different variables if they occur simultaneously or successively. The bivariate (also univariate) events produce a lower failure probability than trivariate analysis for the OR-joint cases. Thus, ignoring the compounding impacts via trivariate joint analysis can significantly underestimate failure probability and joint return period.

Floods are among the most severe natural hazards, causing substantial damage and affecting millions of lives. These events are inherently multi-dimensional, requiring analysis across multiple factors. Traditional research often uses a bivariate framework relying on historical data, but climate change is expected to influence flood frequency analysis and flood system design in the future. This study assesses the projected changes in flood characteristics based on eight downscaled and bias-corrected General Circulation Models (GCMs) that participated in the Coupled Model Intercomparison Project Phase 6. The analysis considers two emission scenarios, including SSP2-4.5 and SSP5-8.5 for far-future (2070–2100), mid-term future (2040–2070), and historical (1982–2012) periods. Downscaled GCM outputs are utilized as predictors of the machine learning model to simulate daily streamflow. Then, a trivariate copula-based framework assesses flood events in terms of duration, volume, and flood peak in the Kan River basin, Iran. These analyses are carried out using the hierarchical Archimedean copula in three structures, and their accuracy in estimating the flood frequencies is ultimately compared. The results show that a heterogeneous asymmetric copula offers more flexibility to capture varying degrees of asymmetry across different parts of the distribution, leading to more accurate modeling results compared to homogeneous asymmetric and symmetric copulas. Also it has been found that climate change can influence the trivariate joint return periods, particularly in the far future. In other words, flood frequency may increase by approximately 50% in some cases in the far future compared to the mid-term future and historical period. This demonstrates that flood characteristics are expected to show nonstationary behavior in the future as a result of climate change. The results provide insightful information for managing and accessing flood risk in a dynamic environment.

In this study, a new method was proposed to model the occurrence of related variables based on the conditional density of copula functions. The proposed method was adopted to investigate the dynamics of meteorological and hydrological droughts in the Zarinehroud basin, southeast of Lake Urmia, during the period 1994–2015. For this purpose, the modified precipitation anomaly percentage and streamflow drought indices were extracted. Finally, the joint frequency analysis of duration-duration and severity-severity characteristics of meteorological and hydrological droughts was analyzed. Analysis of 7 different copulas used to create the joint distribution in the Zarinehroud basin indicated that the Frank copula had the best performance in describing the relationship between the meteorological and hydrological drought severities and durations. By examining the results of the bivariate analysis of duration-duration of meteorological and hydrological droughts at different stations, the expected meteorological and hydrological drought durations were estimated in the years ahead. For example, at Chalkhmaz station, 4- to 7-month duration for the hydrological drought and 9- to 12-month duration for the meteorological drought is expected in the years ahead. The joint frequency analysis of drought characteristics allows to determine the meteorological and hydrological drought characteristics at a single station at the same time using joint probabilities. Also, the results indicated that by knowing the conditional density, the hydrological drought characteristics can be easily estimated for the given meteorological drought characteristics. This could provide users and researchers useful information about the probabilistic behavior of drought characteristics for optimal operation of surface water.

The accurate determination of sediment amount is crucial for the design and operation of reservoirs. The sediment rating curve (SRC) is the most widely used method for determining sediment amount. The SRC was derived from streamflow and sediment amount measurements taken at hydrometric monitoring stations. However, when measurements cannot be made at these stations (such as flooding), sediment amount determination becomes difficult. Therefore, in recent years, researchers have used copula functions to determine the relationship between streamflow and sediment amount. In this study, the joint distribution functions of streamflow and sediment amount at five different stations (Avşar, Bulukabaşı, İnözü, Söğütlühan and Yamula) in the Kızılırmak Basin, one of the most important basins of Turkey, were determined. Initially, the relationship between streamflow and sediment amount was examined and a positive correlation was found between the parameters. Then, the marginal distributions of each dataset were determined and joint distributions were generated using the Burr, Roch-Alegre, BB1 and Tawn Copula functions. The root mean square error of the mean square error (RMSE) and Nash–Sutcliffe efficiency (NSE) were used to select the optimal joint function. According to the optimal joint function determined for each station, the common return periods for both the AND and OR scenarios were calculated. When joint return periods were analyzed, the amount of sediment exceeded the average amount of sediment at all stations.

The interaction between oceanographic, meteorological, and hydrological factors can result in an extreme flooding scenario in the low-lying coastal area, called compound flooding (CF) events. For instance, rainfall and storm surge (or high river discharge) can be driven by the same meteorological forcing mechanisms, tropical or extra-tropical cyclones, resulting in a CF phenomenon. The trivariate distributional framework can significantly explain compound events’ statistical behaviour reducing the associated high-impact flood risk. Resolving heterogenous dependency of the multidimensional CF events by incorporating traditional 3D symmetric or fully nested Archimedean copula is quite complex. The main challenge is to preserve all lower-level dependencies. An approach based on decomposing the full multivariate density into simple local building blocks via conditional independence called vine or pair-copulas is a much more comprehensive way of approximating the trivariate flood dependence structure. In this study, a parametric vine copula of a drawable (D-vine) structure is introduced in the trivariate modelling of flooding events with 46 years of observations of the west coast of Canada. This trivariate framework searches dependency by combining the joint impact of annual maximum 24-h rainfall and the highest storm surge and river discharge observed within the time ±1 day of the highest rainfall event. The D-vine structures are constructed in three alternative ways by permutation of the conditioning variables. The most appropriate D-vine structure is selected using the fitness test statistics and estimating trivariate joint and conditional joint return periods. The investigation confirms that the D-vine copula can effectively define the compound phenomenon’s dependency. The failure probability (FP) method is also adopted in assessing the trivariate hydrologic risk. It is observed that hydrologic events defined in the trivariate case produce higher FP than in the bivariate (or univariate) case. It is also concluded that hydrologic risk increases (i) with an increase in the service design life of the hydraulic facilities and (ii) with a decrease in return periods.

Drought is a complex natural disaster phenomenon. It is of great significance to analyze the occurrence and development of drought events for drought prevention. In this study, two drought characteristic variables (the drought duration and severity) were extracted by using the Theory of Runs based on four drought indexes (i.e., the percentage of precipitation anomaly, the standardized precipitation index, the standardized precipitation evapotranspiration index and the improved comprehensive meteorological drought index). The joint distribution model of drought characteristic variables was built based on four types of Archimedean copulas. The joint cumulative probability and the joint return period of drought events were analyzed and the relationship between the drought characteristics and the actual crop drought reduction area was also studied. The results showed that: (1) The area of the slight drought and the extreme drought were both the zonal increasing distribution from northeast to southwest in Yunnan Province from 1960 to 2015. The area of the high frequency middle drought was mainly distributed in Huize and Zhanyi in Northeast Yunnan, Kunming in Central Yunnan and some areas of Southwest Yunnan, whereas the severe drought was mainly occurred in Deqin, Gongshan and Zhongdian in Northwest Yunnan; (2) The drought duration and severity were fitted the Weibull and Gamma distribution, respectively and the Frank copula function was the optimal joint distribution function. The Drought events were mostly short duration and high severity, long duration and low severity and short duration and low severity. The joint cumulative probability and joint return period were increased with the increase of drought duration and severity; (3) The error range between the theoretical return period and the actual was 0.1–0.4 a. The year of the agricultural disaster can be accurately reflected by the combined return period in Yunnan Province. The research can provide guidelines for the agricultural management in the drought area.

An in-depth understanding of drought frequency analysis in a river basin is possible only with a drought characterisation study. Multiple drought characteristics associated with the drought make it essential to analyse their joint behaviour in drought frequency analysis. Conventional univariate frequency analysis may produce overestimated or underestimated drought risk magnitudes. This study utilised the capability of bivariate copulas to construct the joint dependency amongst four drought characteristics (severity, duration, peak, and interarrival time) derived from the drought indices (Standardised Precipitation Index (SPI), Standardised Precipitation Evapotranspiration Index (SPEI), and Standardized Streamflow Index (SSI)) in a tropical river basin, the Bharathapuzha, India, during the historic period and climate change scenarios (RCP 4.5 and RCP 8.5). Appropriate distributions were selected for modelling the drought characteristics to capture the probabilistic behaviour. The best marginal distribution of each characteristic is obtained from the goodness of fit measures. Various copulas from the Archimedean and Elliptical families were applied to construct the four-dimensional joint distributions. Subsequently, the best-fit copula obtained the joint return periods. The results of joint dependence show that the Clayton and Gaussian copulas best fit with meteorological and hydrological drought, respectively, and the spatial investigation at the median threshold of the joint return period provides the hotspots of drought recurrences in the river basin with return periods in the range of 2 to 8 years during the historic period, greater than four years and greater than six years for RCP 4.5 and 8.5 scenarios.

Socioeconomic drought occurs when a water shortage is caused by an imbalance between the supply and demand of water resources in natural and human socioeconomic systems. Compared with meteorological drought, hydrological drought, and agricultural drought, socioeconomic drought has received relatively little attention. Hence, this study aims to construct a universal and relatively simple socioeconomic drought assessment index, the Standardized Supply and Demand Water Index (SSDWI). Taking the Jianjiang River Basin (JJRB) in Guangdong Province, China, as an example, we analyzed the socioeconomic drought characteristics and trends from 1985 to 2019. The return periods of different levels of drought were calculated. The relationships among socioeconomic, meteorological, and hydrological droughts and their potential drivers were discussed. Results showed that: (1) SSDWI can assess the socioeconomic drought conditions well at the basin scale. Based on the SSWDI, during the 35-year study period, 29 socioeconomic droughts occurred in the basin, with an average duration of 6.16 months and average severity of 5.82. Socioeconomic droughts mainly occurred in autumn and winter, which also had more severe droughts than other seasons. (2) In the JJRB, the joint return periods of “∪” and “∩” for moderate drought, severe drought, and extreme drought were 8.81a and 10.81a, 16.49a and 26.44a, and 41.68a and 91.13a, respectively. (3) Because of the increasing outflow from Gaozhou Reservoir, the occurrence probability of socioeconomic drought and hydrological drought in the JJRB has declined significantly since 2008. Reservoir scheduling helps alleviate hydrological and socioeconomic drought in the basin.

The return period is a probabilistic criterion used to measure and communicate the random occurrence of geophysical events such as floods in risk assessment studies. Since an individual risk may be strongly affected by the degree of dependence amongst all risks, the need for the provision of multivariate design quantiles has gained ground. Consequently, several recent studies have focused on estimation of multi-hazard risk resulted from different hazard types. In this study, multi-hazard return periods are derived for riverine and pluvial floods in city of Monza, Italy, based on different copula dependence structures. It is shown that ignoring statistical dependence among different inter-correlated hazards may cause significant misestimation of risks.

This paper analyzes the multivariate and spatial distribution of heavy precipitation disasters and proposes a method for estimating disaster risk using a joint statistical model. We tested the model with hourly precipitation data from 122 meteorological stations in Shandong from 1990 to 2023. Different marginal distribution functions were used to fit precipitation duration and amount. A Copula joint distribution model established relationships between these variables to analyze heavy precipitation recurrence periods and disaster characteristics. Compared to univariate approaches, the Copula function more reasonably simulates natural disaster occurrence. The joint return period (JRP) estimated by the Copula function reveals that the JRP of 1-hour heavy rainfall is 89% higher than 6-hour rainfall, indicating significantly increased risk from short-term heavy rainfall in Shandong. This method provides a more scientific description of heavy precipitation disaster risk in different scenarios, particularly for short-term events, offering a robust foundation for disaster prevention planning and risk management.