Extreme hydrological events (e.g., floods, rainstorms, droughts) have had disastrous effects on society and the environment in recent years. Specifically, floods, as one of the most frequent and costly natural disasters, have posed a serious threat to the human life and economic development [1
]. A report issued by UNISDR (2015) highlights the fact that, between 1995 and 2015, floods affected 2.3 billion people, worldwide, accounting for 56% of the people affected by weather-related disasters [4
]. Flood risk analysis can provide extremely valuable information by estimating the occurrence of floods for flood control and disaster mitigation, hydraulic structure design, reservoir management, and so on [6
]. It is widely known that, in rain-dominant watersheds, river floods are commonly triggered by extreme precipitation events [8
]. Therefore, in practice, reducing the flood risk also requires information on extreme precipitation [10
]. Consequently, the present work focuses on exploring the bivariate risk of annual maximum flood discharge (AMF
) and associated extreme precipitation (Pr
Up until now, copula functions have been used extensively to evaluate the bivariate risk of hydro-meteorological events [13
]. For instance, Chen et al. (2013) constructed four-dimensional copulas to model the behaviors of drought events; She et al. (2016) applied copula-based severity-duration-frequency curves to evaluate the spatio-temporal variability of dry spells and wet spells. Compared with traditional bivariate hydrologic modeling, the main advantage of copulas is that they allow the joint dependence structure to be modeled, without any restrictions on marginal distributions [18
]. Given this, practitioners can flexibly choose marginal and joint probability functions [19
]. Consequently, the selection of marginal distribution is of crucial importance as it strongly impacts the performance of the copula in modeling bivariate variables [6
However, distributions that model the univariate hydro-meteorological series are diverse. In terms of hydrologic frequency analysis, the most widely used distributions are parametric ones, such as the general extreme value distribution, normal distribution, lognormal distribution, Pearson type 3 distribution, Log Pearson type 3 distribution, Gamma distribution and so on [6
]. When utilizing these distributions, one obvious drawback is that selecting the appropriate distribution from a variety of candidates is time-consuming [12
]. Worse, if the univariate probability distribution is misidentified, results derived from the copulas tend to be underestimated/overestimated [24
]. Hence, a widely applicable probability distribution with high accuracy is urgently needed. The maximum entropy principle (MEP), first expounded by [25
], offers a methodology for deriving probability distribution functions (PDFs) with a minimum of bias from limited information in a more objective way [7
]. The MEP proposes a criterion for selecting the most appropriate PDF on the basis of the rationale that the desired PDF possesses maximum uncertainty, subject to a set of constraints [27
]. As Zhang and Singh (2012) stated, an entropy-based methodology is able to reach a universal solution, and can better capture the shape of the probability density function, without first knowing the format of the a priori distribution [24
]. More moments of observations, beyond just the second moment, can be accounted for in the MEP approach. Additionally, various generalized distributions, such as Pearson type 3 distribution, Gamma distribution, etc., can be derived from the MEP-based distribution using different constraints [28
]. Attracted by the splendid performance of MEP distribution, therefore, it has been extensively used in the hydrology field [6
]. For instance, Mishra et al. (2009) employed the entropy concept to investigate the spatial and temporal variability of precipitation time series for the State of Texas, USA [31
]; Rajsekhar et al. (2013) used the entropy concept to identify the homogenous regions based on drought severity and duration [32
Given the above, the present work takes advantage of the outstanding performance of MEP distribution, and subsequently develops a framework based on a coupled MEP-copula model for bivariate hydrological risk analysis in terms of AMF and Pr.
Also of note is that uncertainty accompanies the copula-based hydrological risk analysis. As Michailidi and Bacchi (2017) stated, flood risk evaluation without accounting for uncertainty is deceptive [33
]. Serinaldi (2013) also stressed that the uncertainty of multivariate design event estimation should be considered carefully for practical application, rather than speculation [34
]. However, previous studies have paid considerably less attention to the impact of uncertainty on hydrological risk analysis [34
]. Therefore, another contribution of this paper is to present a framework aiming to reveal the impact of marginal distribution selection uncertainty and sampling uncertainty on hydrological risk analysis. The two sources of uncertainty are often overlooked in spite of their widely recognized importance; particularly sampling uncertainty, due to its difficult estimation and interpretation [35
The Loess Plateau (LP) is known as the “cradle of Chinese civilization”, and is also one of the most serious soil erosion areas worldwide. On the LP, annual average soil erosion reaches to around 2000–2500 t/km2
, and the area suffering severe soil and water loss covers more than 60% [37
]. Sparse vegetation cover, highly intense rainfall events, and the long history (over 5000 years) of human activities are generally considered to be the principal factors causing severe soil loss on the LP [38
]. Due to the arid and semi-arid continental monsoon climate, however, most previous studies have primarily focused on low flow and drought conditions [39
]. Studies investigating the bivariate risk of flood and extreme precipitation events for the LP are still few.
Consequently, the present study primarily aims to advance the coupled MEP-copula model for bivariate risk analysis, and to reveal the impact of the marginal distribution selection uncertainty and sampling uncertainty on hydrological risk analysis. The developed framework is exemplarily applied for two catchments of the LP. The remainder of the paper is constructed as follows. Section 2
describes the study area and data. Section 3
introduces the methods adopted in this study. The results and discussion are presented in Section 4
. Section 5
shows the main conclusions drawn from this study.
5. Summary and Conclusions
In this study, one general framework, aiming to analyze bivariate hydrological risk through a coupled maximum entropy-copula method and to reveal the impact of marginal distribution selection uncertainty and sampling uncertainty on hydrological risk analysis, is proposed.
The framework excels previous studies in applying the maximum entropy principle-based marginal distribution for modeling random variables and accounting for the impact of different uncertainty sources on hydrological risk analysis. The joint return periods, risk reliability, and bivariate design events are derived based on the coupled maximum entropy-copula method. For the purpose of practical engineering design applications, the so-called most-likely design event is identified to characterize the bivariate design event. To reveal the impact of marginal distribution selection uncertainty and sampling uncertainty on the bivariate design event identification, we designed a corresponding experiment project and specific Monte Carlo-based algorithm to achieve the two goals, respectively. Here, to elucidate the impact of marginal distribution uncertainty on the bivariate design event identification, 6 candidate distributions were combined with each other to produce 36 combinations for fitting univariate flood and extreme precipitation series. Then, these combinations concerning the marginal distributions of flood and extreme precipitation events were utilized to derive the bivariate design event. For the second goal, the Monte Carlo-based algorithm was designed to disclose the impact of sampling uncertainty on the bivariate design event identification.
Two sub-catchments of Loess Plateau, which were typical eco-environmentally vulnerable regions, were selected as the study regions. The primary conclusions are drawn as follows:
The maximum entropy principle (MEP)-based distributions outperform the conventional distributions (i.e., P3, Logn, Norm and Gam at least in this study) in quantifying the probability of flood and extreme precipitation events. Results of this study indicate the better performance of MEP distribution, suggesting that it could be an attractive alternative for quantifying the marginal probability of random variables.
The Gumbel and Frank copulas were suitable dependence models for quantifying the joint probabilities of flood and extreme precipitation events in the upper catchments of Linjiacun and Huaxian stations, respectively.
The bivariate return periods, risk and reliability of flood and extreme precipitation events for the two study regions were calculated based on the coupled maximum entropy-copula models, which were expected to provide practical support for the local flood control and disaster mitigation.
The bivariate design realizations were estimated for the study regions. Comprehensive uncertainty analysis revealed that the fitting performance of univariate distribution is closely related to the bivariate design event identification. If the difference of the fitting performance between two marginal distributions is small, values of the bivariate design events are similar, and vice versa. Therefore, advanced univariate distribution is critical for the bivariate design event selection.
Most importantly, the uncertainty related to the limited sample size is considerable, and should arouse critical attention. The bivariate design events of a specific return period exhibit significant variation. In other words, the return periods of the most-likely design events overlap. The 95% confidence regions of bivariate design events for flood and extreme precipitation with a return period of 30 years could reach between the values for flood and extreme precipitation with return periods of 10 and 50 years. The overlap phenomenon poses great challenges for practical engineering design applications, flood control, and so on. To enable a more reliable estimation of the design realization, increasing the information content by expanding the temporal, spatial or causal data is desirable, as proposed by Merz and Blöschl (2008) [70