Binquan Li

Bayesian flood frequency analysis in the light of model and parameter uncertainties

The specific objective of the paper is to propose a new flood frequency analysis method considering uncertainty of both probability distribution selection (model uncertainty) and uncertainty of parameter estimation (parameter uncertainty). Based on Bayesian theory sampling distribution of quantiles or design floods coupling these two kinds of uncertainties is derived, not only point estimator but also confidence interval of the quantiles can be provided. Markov Chain Monte Carlo is adopted in order to overcome difficulties to compute the integrals in estimating the sampling distribution. As an example, the proposed method is applied for flood frequency analysis at a gauge in Huai River, China. It has been shown that the approach considering only model uncertainty or parameter uncertainty could not fully account for uncertainties in quantile estimations, instead, method coupling these two uncertainties should be employed. Furthermore, the proposed Bayesian-based method provides not only various quantile estimators, but also quantitative assessment on uncertainties of flood frequency analysis.

Evaluation of drought and wetness episodes in a cold region (Northeast China) since 1898 with different drought indices

Drought identification and drought severity characterization are crucial to understand water scarcity processes. Evolution of drought and wetness episodes in the upper Nen River (UNR) basin have been analyzed for the period of 1951–2012 using meteorological drought indices and for the period of 1898–2010 using hydrological drought indices. There were three meteorological indices: one based on precipitation [the Standardized Precipitation Index (SPI)] and the other two based on water balance with different formulations of potential evapotranspiration (PET) in the Standardized Precipitation Evapotranspiration Index (SPEI). Moreover, two hydrological indices, the Standardized Runoff Index and Standardized Streamflow Index, were also applied in the UNR basin. Based on the meteorological indices, the results showed that the main dry period of 1965–1980 and wet periods of 1951–1964 and 1981–2002 affected this cold region. It was also found that most areas of the UNR basin experienced near normal condition during the period of 1951–2012. As a whole, the UNR basin mainly had the drought episodes in the decades of 1910, 1920, 1970 and 2000 based on hydrological indices. Also, the severity of droughts decreased from the periods of 1898–1950 to 1951–2010, while the severity of floods increased oppositely during the same periods. A correlation analysis showed that hydrological system needs a time lag of one or more months to respond to meteorological conditions in this cold region. It was also found that although precipitation had a major role in explaining temporal variability of drought, the influence of PET was not negligible. However, the sole temperature driver of PET had an opposite effect in the UNR basin (i.e., misestimating the drought detection) and was inferior to the SPI, which suggests that the PET in the SPEI should be determined by using underlying physical principles. This finding is an important implication for the drought research in future.

A modified weighted function method for parameter estimation of Pearson type three distribution

In this paper, an unconventional method called Modified Weighted Function (MWF) is presented for the conventional moment estimation of a probability distribution function. The aim of MWF is to estimate the coefficient of variation (CV) and coefficient of skewness (CS) from the original higher moment computations to the first-order moment calculations. The estimators for CV and CS of Pearson type three distribution function (PE3) were derived by weighting the moments of the distribution with two weight functions, which were constructed by combining two negative exponential-type functions. The selection of these weight functions was based on two considerations: (1) to relate weight functions to sample size in order to reflect the relationship between the quantity of sample information and the role of weight function and (2) to allocate more weights to data close to medium-tail positions in a sample series ranked in an ascending order. A Monte-Carlo experiment was conducted to simulate a large number of samples upon which statistical properties of MWF were investigated. For the PE3 parent distribution, results of MWF were compared to those of the original Weighted Function (WF) and Linear Moments (L-M). The results indicate that MWF was superior to WF and slightly better than L-M, in terms of statistical unbiasness and effectiveness. In addition, the robustness of MWF, WF, and L-M were compared by designing the Monte-Carlo experiment that samples are obtained from Log-Pearson type three distribution (LPE3), three parameter Log-Normal distribution (LN3), and Generalized Extreme Value distribution (GEV), respectively, but all used as samples from the PE3 distribution. The results show that in terms of statistical unbiasness, no one method possesses the absolutely overwhelming advantage among MWF, WF, and L-M, while in terms of statistical effectiveness, the MWF is superior to WF and L-M.

Uncertainty Assessment of Hydrological Frequency Analysis Using Bootstrap Method

The hydrological frequency analysis (HFA) is the foundation for the hydraulic engineering design and water resources management. Hydrological extreme observations or samples are the basis for HFA; the representativeness of a sample series to the population distribution is extremely important for the estimation reliability of the hydrological design value or quantile. However, for most of hydrological extreme data obtained in practical application, the size of the samples is usually small, for example, in China about 4050 years. Generally, samples with small size cannot completely display the statistical properties of the population distribution, thus leading to uncertainties in the estimation of hydrological design values. In this paper, a new method based on bootstrap is put forward to analyze the impact of sampling uncertainty on the design value. By bootstrap resampling technique, a large number of bootstrap samples are constructed from the original flood extreme observations; the corresponding design value or quantile is estimated for each bootstrap sample, so that the sampling distribution of design value is constructed; based on the sampling distribution, the uncertainty of quantile estimation can be quantified. Compared with the conventional approach, this method provides not only the point estimation of a design value but also quantitative evaluation on uncertainties of the estimation.

Estimation of design flood using EWT and ENE metrics and uncertainty analysis under non-stationary conditions

Concepts of Expected Waiting Time (EWT) and Expected Number of Exceedances (ENE) have been presented in much literature for estimating the Design Flood (DF) under non-stationary conditions. The parameters of the EWT and ENE are generally no less than four, which inevitably leads to the uncertainty of the DF estimation. In this paper, the Bayesian method is proposed to analyze the impact of parameter estimation uncertainty on the EWT- and ENE-based estimation of the DF and Corresponding Design Reliability (CDR). In addition, a comparison analysis between the EWT and ENE is conducted in terms of the DF and CDR with or without a consideration being given to the impact of parameter uncertainty. In the case of giving no consideration to the impact of parameter uncertainty, the experiment results indicate that the EWT-based estimations are less than that of ENE in terms of DF and CDR in the case of a decreasing trend. While in the case of an increasing trend, the EWT-based estimations are bigger than that of ENE. In the case of considering the impact of parameter uncertainty, results in the case study show that the distribution of the EWT-based estimations of DF and CDR are left shifted compared to that of the ENE. Overall, the EWT-based estimations are significantly different from that of ENE in terms of DF and CDR. Therefore, it is necessary and open for further discussions about which metric will be optimal between the EWT and ENE for estimating the DF under non-stationarity.

A data-driven SVR model for long-term runoff prediction and uncertainty analysis based on the Bayesian framework

Accurate and reliable long-term forecasting plays an important role in water resources management and utilization. In this paper, a hybrid model called SVR-HUP is presented to predict long-term runoff and quantify the prediction uncertainty. The model is created based on three steps. First, appropriate predictors are selected according to the correlations between meteorological factors and runoff. Second, a support vector regression (SVR) model is structured and optimized based on the LibSVM toolbox and a genetic algorithm. Finally, using forecasted and observed runoff, a hydrologic uncertainty processor (HUP) based on a Bayesian framework is used to estimate the posterior probability distribution of the simulated values, and the associated uncertainty of prediction was quantitatively analyzed. Six precision evaluation indexes, including the correlation coefficient (CC), relative root mean square error (RRMSE), relative error (RE), mean absolute percentage error (MAPE), Nash-Sutcliffe efficiency (NSE), and qualification rate (QR), are used to measure the prediction accuracy. As a case study, the proposed approach is applied in the Han River basin, South Central China. Three types of SVR models are established to forecast the monthly, flood season and annual runoff volumes. The results indicate that SVR yields satisfactory accuracy and reliability at all three scales. In addition, the results suggest that the HUP cannot only quantify the uncertainty of prediction based on a confidence interval but also provide a more accurate single value prediction than the initial SVR forecasting result. Thus, the SVR-HUP model provides an alternative method for long-term runoff forecasting.

Multisource hydrologic modeling uncertainty analysis using the IBUNE framework in a humid catchment

Hydrologic cycle is a complex system associated with both certain and uncertain constituents. The propagation of confidence bounds from different uncertainty sources to model output is of great significance for hydrologic modeling. In this paper, we applied the integrated bayesian uncertainty estimator to quantify the effects of parameter, input and model structure uncertainty on hydrologic modeling progressively. Two hydrologic models (Xinanjiang model and TOPMODEL) were applied to a humid catchment under three scenarios. Case I: the shuffled complex evolution metropolis (SCEM-UA) algorithm was conducted to determine the posterior parameter distribution of hydrologic models and analyze the corresponding forecast uncertainty. Case II: input uncertainty was also considered by assuming rain depth bias follows a normal distribution, and integrated with SCEM-UA. Case III: Simulations from two models were combined by the Bayesian model averaging to fully quantify multisource uncertainty effects. Results suggested that, from Case I to II, the containing ratio (percentage of observed streamflow enveloped by 95% confidence interval) obviously increased by an average magnitude of 10% for the study period 2000–2006. Besides, it also found that the width of 95% confidence interval became wider and narrower for Xinanjiang model and TOPMODEL, respectively, from Case I to II. This may indicate that the uncertainty of TOPMODEL results was more remarkable than Xinanjiang model in Case I. By combining results from two models, model structure uncertainty was also considered in Case III. The accuracy of uncertainty bounds further improved with the containing ratio of 95% confidence interval >95%. In addition, the optimized deterministic results from the uncertainty analysis showed that the average Nash–Sutcliffe coefficient increased continually from Case I to II and III (0.82, 0.84 and 0.90, respectively) for the study period. The analysis demonstrated the improvement of modeling accuracy when extra uncertainty sources were also quantified, and this finding also proved the applicability of IBUNE framework in hydrologic modeling.