The outputs of Rainfall-runoff models are inherently uncertain and quantifying the associated uncertainty is crucial for flood control, reservoir operation and relevant water resources management activities. This study presents the uncertainty quantification of rainfall-runoff simulations using the copula-based Bayesian processor (CBP) in Danjiangkou Reservoir basin, China. The variable infiltration capacity (VIC) model was established to simulate daily flows of Danjiangkou Reservoir basin. The Nash-Sutcliffe efficiency (NSE) and Relative Error (RE) were used as performance criteria for deterministic simulations, while quantile-quantile (QQ) plot, reliability (α-index), resolution (π-index) and continuous rank probability score (CRPS) for probabilistic simulations. The seasonality of uncertainty in rainfall-runoff modeling is explored, and impacts of copula selection and correlation coefficient on uncertainty quantification results are investigated. Results show that the overall performance of the CBP is satisfactory, which provides a useful tool for estimating the uncertainty of rainfall-runoff simulations. It is also demonstrated that the dry season has higher reliability and greater resolution compared with wet season, which illustrates that the CBP captures the actual uncertainty of rainfall-runoff simulations more accurately in dry season. Moreover, the performance the CBP highly depends on the selected Copula function and considered Kendall tau correlation coefficient. As a result, great attention should be paid to selecting the appropriate Copula function and effectively capturing the actual dependence between observed and simulated flows in the CBP-based uncertainty quantification of rainfall-runoff simulations practice.
Research Article
Uncertainty quantification of rainfall-runoff simulations using the copula-based Bayesian processor：impacts of seasonality，copula selection and correlation coefficient
https://doi.org/10.21203/rs.3.rs-1626906/v1
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rainfall-runoff simulations
uncertainty quantification
Bayesian processor
copula function
seasonality.
Rainfall-runoff models play an important role in flood control, reservoir operation, climate change impact assessment and other relevant water resources management activities (Beven and Binley 1992; Liu et al. 2018). However, the outputs of the rainfall-runoff model are inevitably subject to errors and inherently uncertain resulting from various sources including errors associated with input data and data for calibration, imperfection in model structure, and uncertainty in model parameters (Li et al. 2014; Ragab et al. 2020). Therefore, understanding and quantification of modeling uncertainty is crucial for risk-based decision-making and has received increasing attention from researchers and practitioners (Thyer et al. 2009; Madadgar and Moradkhani 2014; Yang et al. 2020; Zhou et al. 2020).
Numerous approaches for quantifying the uncertainty in rainfall-runoff simulations have been proposed in the literatures, and these approaches are generally fall into four categories according to uncertainty category concerned (Moges et al. 2021): (1) input uncertainty (Maskey et al. 2004; Jiang et al. 2019), (2) model parameters uncertainty (Xiong and O’Connor 2008; Ghaith and Li 2020), (3) model structure uncertainty (Knoben et al. 2020) and (4) integrated uncertainty (Kavetski et al. 2006; Ajami et al. 2007; Zhou et al. 2020). Precipitation is the most important input of hydrological models, and its uncertainty has a crucial impact on the simulation results of the hydrological model. Gabellani et al. (2007) used a stochastic rainfall generator to explore the impact of uncertainties in the spatial-temporal distribution of rainfall on the prediction of peak discharge in a typical mountain basin. Strauch et al. (2012) used precipitation data ensemble for uncertainty analysis in SWAT streamflow simulation. Zeng et al. (2018) analyzed the impacts of rain gauge density and distribution on lumped hydrological modelling uncertainty with different catchment sizes. Jiang et al. (2019) derived a new method of rainfall calculation uncertainty (RCU) assessment using an inverse sampling gauge approach, in which the influencing factors of the RCU are addressed in the calculated area precipitation and standard deviation.
Generalized likelihood uncertainty estimation (GLUE) and Bayes-based Markov Chain Monte Carlo (MCMC) are the two most commonly used hydrological model parameters uncertainty analysis methods. Beven and Binley (1992) developed an informal Bayesian approach called GLUE method which applies likelihood measures and accepts possible equifinality of models via Monte Carlo sampling to assess predictive uncertainty. Vrugt et al. (2008) presented the differential evolution adaptive Metropolis (DREAM) sampler that is especially designed to efficiently estimate the posterior probability density function of hydrologic model parameters in complex, high-dimensional sampling problems. The comparison of GLUE and Bayesian MCMC is also a hot topic in parameters uncertainty quantification of hydrological model (Jin et al. 2010; Li et al. 2014; Li et al. 2017). Bayesian model averaging (BMA) method is put forward to analyze the structural deficiencies of the specific model (Raftery et al. 2005). Madadgar and Moradkhani (2014) integrated copula functions into BMA to remove the need to assume the form of the posterior distribution and eliminate the data-transformation procedure.
Actually, there is usually interaction among the uncertainties of model inputs, model parameters and model structure, which jointly determine the uncertainty of model outputs. Many hydrologists have devoted their efforts to assess the integrated uncertainty of rainfall-runoff simulations. The hydrologic Uncertainty Processor (HUP) proposed by Krzysztofowicz and Kelly (2000) that aggregates the hydrological model uncertainties is the most commonly used approach (Yang et al. 2020). Kavetski et al. (2006) developed the Bayesian total error analysis (BATEA) framework to deal with multiple sources of uncertainty, including the sampling and measurement uncertainty in the forcing/response data and by the structural error of the model conceptualization. Ajami et al. (2007) proposed a new framework called integrated Bayesian uncertainty estimator (IBUNE) whose objective is to distinguish the various sources of uncertainty including input, model structure and parameters uncertainty. Todini (2008) developed a Model Conditional Processor (MCP) for the assessment of predictive uncertainty, which allows combination of the observations with one or several models’ forecasts by means of the Normal Quantile Transform. Li et al. (2017) introduced the nonparametric kernel density estimation, Pearson-III and empirical distribution as the prior distribution of HUP and assessed the impact of distribution type in Bayes probability flood forecasting. Liu et al. (2018) proposed a copula-based Bayesian processor (CBP) for quantifying the uncertainty in hydrological forecasting, which has been tested and proven to be reliable and skillful, thus will provide a promising tool for estimating the uncertainty of rainfall-runoff simulations.
The aims and objectives of this study are: (i) presenting uncertainty quantification of rainfall-runoff simulations using the CBP by applying to a case study in Danjiangkou Reservoir basin, China; (ii) exploring the seasonality of uncertainty in rainfall-runoff modeling; (iii) investigating impacts of copula selection and correlation coefficient on uncertainty quantification results. The remainder of the paper is organized as follows. The methodology is outlined and presented in section 2. Section 3 introduces the case study. Then, the results and analysis obtained are included in section 4. The discussions are provided in section 5. Finally, section 6 draws the main conclusions of the work.
2.1 VIC model
The variable infiltration capacity (VIC) model is adopted to simulate flows in this study. The VIC model is a macro-scale distributed hydrological model based on a soil-vegetation-atmosphere transfer scheme. It considers both energy and water balances and explicitly represents the effects of multiple vegetation covers on water and energy budgets. The VIC model also incorporates the representation of subgrid spatial variability of precipitation with the representation of spatial variability of infiltration. In addition, it includes both the saturation and infiltration excess runoff processes in a model grid cell with a consideration of the subgrid-scale soil heterogeneity and the frozen soil processes for cold climate conditions (Liang et al. 1994; Guo et al. 2009). Three types of evaporation: evaporation from wet canopy, evapotranspiration from dry canopy and evaporation are considered. The one dimensional Richard equation is used to describe the vertical soil moisture movement and the moisture transfer between soil layers obeys the Darcy law. The routing model represented by the unit hydrograph method for overland flow and the linear Saint-Venant method for channel flow, allow runoff to be predicted (Zhou and Guo, 2013). The VIC model has been tested and applied to various basins of different scales with good performance (Su and Xie 2003;Yuan et al. 2004༛Guo et al. 2009).
2.2 Marginal distributions of observed and simulated flows
The probability distributions of daily observed and simulated flows used in this study refer to the flow duration curve, and the daily streamflow is assumed to be a random variable (Castellarin et al. 2004). The Gamma and Log-normal are the two commonly used distributions in hydrology to fit the daily streamflow series in the literatures (Xiong et al. 2015), thus are selected as candidate probability models for observed and simulated flows in this study. The probability density function of the Gamma and Log-normal are as follows.
where α and β represent the parameters of the Gamma distribution.
where \({\mu _Y}\) and \({\sigma _Y}\) represent the parameters of the Log-normal distribution.
L-moment method was used to estimate the distribution parameters for given data series (Hosking 1990). The Kolmogorov- Smirnov statistic D is the measure of the goodness of fit between the hypothesized distribution and the empirical distribution obtained by Weibull plotting position formula (Liu et al. 2016). In this study, the 5% significance level was selected to reject or accept a fitted distribution. The probability distribution which provides the smaller D value is chosen as the better fitting distribution.
2.3 Joint distributions of observed and simulated flows
The dependence between observed and simulated flows is modelled with the copula function. In this study we concentrate on the bivariate case. Using a copula function, the joint distribution of observed flow H and simulated flow S is given by
$${F_{H,S}}(h,s)={C_\theta }(G(h),F(s))={C_\theta }(u,v)$$
3
where \(u=G(h)\) and \(v=F(s)\) represent the marginal distribution functions; θ is the dependence parameter of copula.
Three widely used bivariate Archimedean copulas in hydrology, including the Gumbel-Hougaard (G-H), Frank and Clayton copulas described in Table 1, are adopted as candidates (Nelsen 2006). The relationship between their dependence parameters and Kendall correlation coefficient tau (τ) are also presented in Table 1. The p-value associated to the Cramér-von Mises test statistic Sn is computed based on the parametric bootstrap technique to formally assess whether the selected model is suitable (Dung et al. 2015). The selected copula should have the lower value of the statistic Sn with an admissible p-value (i.e. larger than 0.05).
2.4 Copula-based Bayesian processor
Let H be the observed flow whose realization h is being simulated. Let estimator S be the output flow generated by a deterministic rainfall-runoff model whose realization s constitutes a point estimate of H. The posterior density function of conditional on a deterministic simulated flow \(S=s\), is derived via Bayes’ theorem:
$$\phi (h|s)=\frac{{f(s|h) \cdot g(h)}}{{\int_{{ - \infty }}^{{+\infty }} {f(s|h) \cdot g(h)} dh}}$$
4
4
The prior density used in the CBP is directly the probability density function of H which expressed as \(g(h)\). It can be estimated based on historical data of observed flows using the procedure in section 2.2 in the original space. Using the copula function, the likelihood function is defined as (Nelsen 2006)
$$f(s|h)={c_\theta }(u,v) \cdot f(s)$$
5
where \({c_\theta }(u,v)={\partial ^2}{C_\theta }(u,v)/\partial u\partial v\) is the density function of \({C_\theta }(u,v)\); \(f(s)\) is the PDF of . Given \(S=s\), the likelihood function of can be calculated by Eq. (5).
The posterior density function \(\phi (h|s)\) can be rewritten as follows
$$\phi (h|s)=\frac{{{c_\theta }(u,v)}}{{\int_{0}^{1} {{c_\theta }(u,v)du} }} \cdot g(h)$$
6
The Monte Carlo sampling technique is applied to estimate the numerical solution of the posterior density function \(\phi (h|s)\). The corresponding posterior distribution function \(\Phi (h|s)\) is defined by
$$\Phi (h|s)=\int_{{ - \infty }}^{h} {\phi (h|s)dh}$$
7
Once \(\Phi (h|s)\) has been determined, various information and products (e.g., quantiles with specified exceedance probabilities, simulation intervals with specified inclusion probabilities, and probabilities of exceedance for specified thresholds) could be provided for decision makers.
2.5 Performance criteria used
For deterministic simulations, the Nash-Sutcliffe efficiency (NSE) and Relative Error (RE) are used to evaluate the performance. NSE can range from −∞ to 1. Essentially, the closer the model efficiency is to 1, the more accurate the model is (Chen et al. 2015). A value of RE closes to zero indicates a good agreement between observed and simulated runoff volume.
For probabilistic simulations, the simulative quantile-quantile (QQ) plot, α-index (reliability), π-index (resolution) and continuous rank probability score (CRPS) are adopted to evaluate probabilistic simulations (Laio and Tamea 2007; Thyer et al. 2009; Madadgar et al. 2014; Liu et al. 2018). The metric α-index assesses the reliability of simulations and varies between 0 (worst reliability) and 1 (perfect reliability), while the π-index indicates the resolution (sharpness) of the simulative distribution, and greater value of π-index indicates greater resolution (lower uncertainty) of simulations. The smaller the CRPS value is, the better the simulation performance. For a deterministic simulation, the CRPS reduces to the mean absolute error (MAE) (Pappenberger et al. 2015). Its minimal value of zero is only achieved in the case of a perfect deterministic simulation.
3.1 Danjiangkou Reservoir
Hanjiang River (Fig. 1) is the longest tributary of the Yangtze River, which rises on the southern slope of the Qin Ling Mountains and flows through Shaanxi and Hubei provinces, falls into the Yangtze River at Wuhan, with a length of 1577km and a drainage area of about 159,000km2 (Yang et al. 2016). The basin has a sub-tropical monsoon climate and as a result has dramatic diversity in its water resources. Annual precipitation varies from 700mm to 1100mm, and 70%-80% of the total amount occurs in the wet seasons from May to October (Guo et al. 2009).
The Danjiangkou Reservoir lies in the upper Hanjiang basin and is the source of water for the middle route of the South-to-North Water Diversion Project (SNWDP) in China. It has a catchment area of 95,200km2 and the average annual runoff to the reservoir is 40.85 billion m3. It is a multipurpose reservoir with main functions, including flood control, water supply, hydropower generation, and irrigation (Zhou and Guo, 2013). Hence, accurate simulation of flows to the Danjiangkou Reservoir is of great importance.
3.2 Data
The dataset include hydrological data, forcing, Digital Elevation Model (DEM), soil and vegetation data for VIC model calibration. Daily streamflow and meteorological (precipitation and air temperature) data from 1980–1990 are available. DEM data of 0.009° (around 1×1 km2 cell size) spatial resolution for the Hanjiang Basin were derived and used to delineate the sub-basin boundary and stream network. Vegetation types data were taken from the global land cover classification collection generated by University of Maryland with a one kilometer per pixel resolution. Vegetation parameters were based on the vegetation from Land Data Assimilation System. The soil parameters were derived from the soil classification information of global 5 min data provided by the National Atmospheric and Oceanic Administration.
4.1 Simulation results of VIC model
The whole data set (1980–1990) was divided into a calibration period (1980–1986) and a verification period (1987–1990). Main parameterization procedures of the VIC model were described as follows: the model parameters were firstly calibrated for gauged sub-basins, and then the calibrated parameters were used as the initial values for the corresponding grids; finally, hydrological control stations were selected in the main streams of the Hanjiang River to test and optimize the grid parameters through a trial and error method. The values of NSE and RE are equal to 89.97% and − 3.40% respectively during calibration period, 79.32% and − 10.86% respectively during verification period. These results demonstrated that the VIC distributed hydrological model was proved to be quite efficient in simulating the flows to the Danjiangkou Reservoir. In order to explore the seasonality of simulation uncertainty, the available data from 1980 to 1990 were divided into flow data in wet (May 1 to October 31) and dry (November 1 to April 30) seasons. To keep number of days in each year constant (365 days), for simplicity, February 29 is excluded. The sample sizes of daily flows in wet and dry seasons were 2024 and 1991, respectively. These divided dataset were separately used to calibrate the CBP in wet and dry seasons.
4.2 Calibration of CBP
Marginal distributions for four random variables (observed and simulated flow in two seasons) needed to be estimated. The parameters of five candidate distributions were estimated by the L-moment method and the results were listed in Table 2. The K-S tests were used to verify the null hypothesis and the corresponding statistic D values were also reported in Table 2. Table 2 shows that the null hypothesis could not be rejected at the 95% confidence level only for Log-normal distribution. For all the four hydrological variables, Log-normal distribution provides the minimum D value and is chosen as the best fitting distribution, respectively.
For each season, the joint distribution of observed and simulated flows was constructed and determined using the three candidate bivariate copula functions. The Kendall correlation coefficient τ between observed and simulated flows results in a value of 0.760 and 0.658 in wet and dry season, respectively. Then the copula parameter θ was calculated by the method based on the inversion of τ presented in Table 1 and the results were listed in Table 3. Furthermore, the results of Cramér-von Mises goodness-of-fit test were also given in Table 3 where the statistic Sn and the associated p-value were derived based on 10000 parametric bootstrap samples. It indicates that the null hypothesis could not be rejected for Gumbel-Hougaard and Frank copulas. Moreover, Gumbel-Hougaard copula performs better due to its lower Sn value for each season and was selected.
|
Season |
Copula |
Parameter |
Statistical test |
|
|---|---|---|---|---|
|
\(\theta\) |
Sn |
p-value |
||
|
Gumbel-Hougaard |
4.17 |
0.0247 |
0.920 |
|
|
Wet |
Frank |
14.83 |
0.1822 |
0.754 |
|
Clayton |
6.34 |
0.6774 |
0.045 |
|
|
Gumbel-Hougaard |
2.92 |
0.0925 |
0.887 |
|
|
Dry |
Frank |
9.70 |
0.3329 |
0.712 |
|
Clayton |
3.84 |
1.122 |
0.008 |
|
4.3 Performance assessment of the CBP
For a given significance level \(\alpha\), the (\(1 - \alpha\)) confidence interval of observed flow which offers the lower and upper limits can be evaluated. For instance, the 5% quantile and 95% quantile compose the 90% simulation interval when \(\alpha =0.1\). As typical examples, the observed and median discharges, and 90% simulation intervals in wet and dry seasons estimated by CBP in 1983 (wet year), 1981 (normal year) and 1988 (dry year) were presented in Fig. 2. It can be seen that most observed flows are contained within the 90% simulation intervals, which indicates that these 90% simulation intervals can effectively capture the simulation uncertainty and provide more information for decision-making in flood control, reservoir operation and other water resources management activities.
The simulative QQ plot, α-index, π-index and CRPS were adopted to further evaluate the probabilistic simulations. Figure 3 presents the simulative QQ plots with respect to the CBP in wet and dry seasons. It is encouraging that the overall performances of all simulative QQ plots are pretty good. However, the behavior is found to be different in wet and dry seasons. The CBP somewhat underestimates simulative uncertainty in wet season, while overestimates simulative uncertainty in dry season. The QQ plot for dry season is a bit closer to the 1:1 line than wet season, which demonstrates the CBP performs slightly better in dry season in terms of reliability.
The results of α-index, π-index and CRPS were summarized in Table 4. It is clearly shown that both the α-index and π-index values in dry season are larger than those of the wet season. Compared with wet season, the dry season has higher reliability and greater resolution, which illustrates that the CBP captures the actual uncertainty of rainfall-runoff simulations more accurately in dry season. In terms of the CRPS, both wet and dry seasons have lower values than those of deterministic simulations and indicate the effectiveness of probabilistic simulations. The CRPS value in wet and dry seasons is improved (decreased) by 34.6% and 43.3%, respectively. All these results reveal that the performance of the CBP in dry season is superior to the wet season and the improvement in simulation capability benefited from the CBP is also greater in dry season.
|
Season |
Deterministic model |
CBP |
||
|---|---|---|---|---|
|
CRPS/MAE |
α-index |
π-index |
CRPS |
|
|
Wet |
649 |
0.9523 |
3.28 |
425 |
|
Dry |
150 |
0.9767 |
3.88 |
85 |
| Note: α-index and π-index are dimensionless; the unit of CRPS is m3/s. | ||||
Further discussions are carried out in order to investigate impacts of copula selection and correlation coefficient on uncertainty quantification results, taking the wet season as a typical representative.
5.1 Impacts of the Copula selection
Two other Archimedean copulas, i.e. Frank and Clayton copulas are also used to construct the joint distribution of observed and simulated flows, and quantify the uncertainty of rainfall-runoff simulations in wet season using the identical procedure for comparison purpose. Figure 4 presents the simulative QQ plot, α-index, π-index and CRPS associated with three copulas. It can be seen that compared with the G-H CBP, the Frank CBP has slightly higher reliability but much lower resolution, while the Clayton CBP has lower reliability and resolution simultaneously. The CRPS value in wet season associated with the Frank CBP and Clayton CBP is worsened (increased) by 17.2% and 31.0% compared with the G-H CBP, respectively. This consequence is agreement with the copula fitting results. Thus, these results highlight the importance of selecting the appropriate Copula function in the uncertainty quantification of rainfall-runoff simulations using the CBP.
5.2 Impacts of the correlation coefficient
Ten scenarios of Kendall tau correlation coefficients of observed and simulated flows are set up, i.e., 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 0.95. The simulative QQ plot, α-index, π-index and CRPS in wet season associated with different Kendall tau correlation coefficients are shown in Fig. 5. It is indicated from the simulative QQ plot that the CBP underestimates simulative uncertainty in wet season for the scenarios where the Kendall tau correlation coefficient is larger than 0.7, otherwise overestimates simulative uncertainty. With the increasing of the Kendall tau correlation coefficient, the α-index decreases firstly, then increases and finally decreases and reaches its maximum at the actual Kendall tau correlation coefficient (0.760). The π-index increases as the Kendall tau correlation coefficient increases. Moreover, The CRPS value firstly decreases, and then increases as the Kendall tau correlation coefficient increases, and reaches its minimum at the actual Kendall tau correlation coefficient. It is demonstrated that effectively capturing and considering the actual dependence between observed and simulated flows is crucial for the CBP to quantify the uncertainty of rainfall-runoff simulations.
Accurate and reliable rainfall-runoff simulation is important for flood control, reservoir operation and relevant water resources management activities. This paper presents the uncertainty quantification of rainfall-runoff simulations using the copula-based Bayesian processor (CBP) in Danjiangkou Reservoir basin, China. The seasonality of uncertainty in rainfall-runoff modeling is explored, and impacts of copula selection and correlation coefficient on uncertainty quantification results are investigated. The main conclusions were summarized as follows:
(1) The CBP quantifies the total uncertainty in terms of a posterior distribution of observed flows conditional on model output. It can be implemented in the original space directly without a data transformation procedure into Gaussian space and allows for any form of marginal distribution of observed and simulated flows, and a nonlinear and heteroscedastic dependence structure. The overall performance of the CBP is satisfactory, which provides a useful tool for estimating the uncertainty of rainfall-runoff simulations.
(2) The behavior of the CBP is found to be different in wet and dry seasons. The CBP underestimates simulative uncertainty in wet season, while overestimates in dry season. Compared with wet season, the dry season has higher reliability and greater resolution, which illustrates that the CBP captures the actual uncertainty of rainfall-runoff simulations more accurately in dry season. In general, the performance of the CBP in dry season is superior to the wet season and the simulation capability improvement benefited from the CBP is also greater in dry season.
(3) The performance the CBP highly depends on the selected Copula function and considered Kendall tau correlation coefficient. As a result, great attention should be paid to selecting the appropriate Copula function and effectively capturing the actual dependence between observed and simulated flows in the CBP-based uncertainty quantification of rainfall-runoff simulations practice.
Acknowledgements This research was funded by the National Natural Science Foundation of China (51909112), Major Discipline Academic and Technical Leaders Training Program of Jiangxi Province (20212BCJ23016) and Jiangxi Provincial Water Resources Science and Technology Project (202023ZDKT02).
Authors Contribution Conceptualization, methodology, formal analysis, investigation, and writing original draft preparation, Zhangjun Liu; Data curation, Writing, review and editing, Jingwen Zhang; Resources, Supervision and updating some parts, Tianfu Wen and Jingqing Cheng.
Data Availability Data available after request from the corresponding author.
Ethical Approval The authors declare that they have no conflict of interest.
Consent to Participate The authors declare that they are aware and consent with their participation on this paper.
Consent to Publish The authors declare that they are consent with the publication of this paper.
Competing Interests There is not any competing interest.
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