基于贝叶斯统计的水文模型不确定性研究
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摘要
我国水资源短缺、水旱灾害频发，对水文预报和预测的需求巨大。水文模型不仅是水文预报和预测的主要工具，也是研究其他水问题的重要工具，但水文模型中广泛存在参数、输入和结构等不确定性，降低了其应用的价值。本论文研究旨在构建水文模拟的贝叶斯统计推断模型，用于评估和诊断水文模型不确定性。
论文构建了水文模型系统的层次贝叶斯模型，并提出了基于马尔可夫链蒙特卡洛混合采样策略的求解方法。采用层次贝叶斯模型描述输入误差的层次分布特征，根据流域中各雨量观测站的相对误差来估计降雨的不确定性。为了推导实测流量数据的似然函数，通过径流划分降低数据的异方差性，采用一阶自回归模型描述时间序列的自相关性。贝叶斯模型求解的混合采样策略综合了MH采样、DRAM采样和Gibbs采样等方法，提高了贝叶斯模型的求解效率。
为了验证贝叶斯模型的有效性，论文选择华北平原典型农田，估计了一维分层土壤水分参数的不确定性。利用农田原位多层土壤含水率的观测数据，利用贝叶斯模型反演得到了分层土壤的持水和导水特征参数，根据土壤参数后验分布中值模拟得到的土壤水分运动过程，比基于实测参数的模拟结果具有更高精度。
采用论文构建的层次贝叶斯模型研究了分布式水文模型（GBHM）在山洪预报中的不确定性。基于合成数据试验和实际数据的应用结果表明，贝叶斯模型有效地估计了GBHM参数的不确定性，并合理给出了基于GBHM的洪水预报不确定性。基于考虑输入不确定性反演得到的参数后验分布，GBHM模型预报得到的流量系列的概率分布与实测数据具有更好的一致性。降雨误差的后验估计值存在空间负相关关系，说明合理估计分布式模型的输入不确定性还需要更多的信息。
为了分析水文模型结构不确定性，在贝叶斯框架下引入了可行域为[0,1]区间的残差独立性置信度系数（RIC），建立了经验贝叶斯模型。试验研究表明，RIC的最优值既是对模型结构不确定性的评估，也可作为参数反演中判断匹配不足与过度匹配的依据。当模型系统误差增大时，RIC系数的最优值随之降低。基于贝叶斯模型，采用新安江模型和GBHM分别在滁洲流域和赣江流域开展了洪水概率预报，预报不确定性与实测流量数据中反映的不确定性较为一致。
论文的主要创新点在于发展了层次贝叶斯模型和经验贝叶斯模型并成功应用于水文模型参数率定和洪水概率预报，为水文模型中的应用推广提供了理论基础。
China is suffering from water shortage problems and frequent flood and droughtdisasters, which places high demand for hydrological forecasting and prediction.Hydrological models are the major tool of hydrological forecasting and prediction, aswell as other water-related problems. However, uncertainties of parameters, inputs andmodel structures involved in hydrological modeling have undermined its applicationvalue. The objective of this dissertation is to develop a Bayesian framework ofhydrological model system to estimate and diagnose the uncertainties involved inhydrological models.
A hierarchical Bayesian model is first developed for the hydrological model system.A hybrid sampling strategy that employs Markov chain Monte Carlo methods is thendeveloped to explore the Bayesian model. We introduced the hierarchical Bayesianmodel for the hierarchical characters of input uncertainty, and specified inputuncertainty as relative errors in rainfall measurement at the gauges. To conduct thelikelihood function of observed discharge data, we divided the data into low and highflow dataset to reduce the heteroscedasticity, and adopted the first order autoregressivemodel to describe the autocorrelation of the time series. The hybrid sampling strategyfor exploration of the Bayesian model employs Metropolis-Hastings, delay rejectionadaptive Metropolis, and Gibbs sampling methods to improve the sampling efficiency.
To evaluate the validity of the Bayesian model, we applied it to the dataset in anagricultural land in North China to estimate the uncertainty associated with onedimension soil hydraulic parameters. Multilayer parameters of soil water retention andhydraulic conductivity functions were estimated from in situ measurements of soil watercontent at several depths. The medians of posterior distributions of soil hydraulicparameters led to higher accuracy in simulation of soil moisture variation than theparameters obtained by laboratory test did.
The developed hierarchical Bayesian model was adopted to estimate theparameters associated with a geomorphology-based hydrological model (GBHM) formountainous flood forecasting. Synthetic case study and field data applicationdemonstrated the effectiveness of the Bayesian model in estimating the uncertainty ofGBHM parameters and in giving reasonable predictive uncertainty for flood period. Jointly estimating input uncertainty with hydrological model parameters led to a lowererror criterion of probabilistic forecasting and higher consistency of the predictivedistribution with the observed data compared with estimating parameter uncertaintyseparately. The negative spatial correlation of estimated input errors suggested thatfurther information was required to improve input uncertainty estimation of distributedmodels.
To analyze the model structure uncertainty of hydrological models, we developedan informal Bayesian model under the Bayesian framework by introducing a residualindependence coefficient (RIC) with a feasible zone of [0,1]. Case study result showedthat the best RIC coefficient was related to model structure errors and was determined toavoid both overfitting and underfitting. When systematic errors of a hydrological modelincreased, the best RIC coefficient declined. The developed Bayesian model wasapplied separately to a Xinanjiang model of Chuzhou catchment and a GBHM model ofGanjiang catchment for probabilistic flood forecasting. The predictive distributions arewell consistent with the observed data.
The major innovation points of this dissertation lie in the development of thehierarchical Bayesian model and the empirical Bayesian model, and the application ofthe Bayesian models to the parameter estimation of hydrological models and to thehydrological flood forecasting. The dissertation demonstrates a theoretical base for thewidely application of Bayesian models in hydrological models.
论文构建了水文模型系统的层次贝叶斯模型，并提出了基于马尔可夫链蒙特卡洛混合采样策略的求解方法。采用层次贝叶斯模型描述输入误差的层次分布特征，根据流域中各雨量观测站的相对误差来估计降雨的不确定性。为了推导实测流量数据的似然函数，通过径流划分降低数据的异方差性，采用一阶自回归模型描述时间序列的自相关性。贝叶斯模型求解的混合采样策略综合了MH采样、DRAM采样和Gibbs采样等方法，提高了贝叶斯模型的求解效率。
为了验证贝叶斯模型的有效性，论文选择华北平原典型农田，估计了一维分层土壤水分参数的不确定性。利用农田原位多层土壤含水率的观测数据，利用贝叶斯模型反演得到了分层土壤的持水和导水特征参数，根据土壤参数后验分布中值模拟得到的土壤水分运动过程，比基于实测参数的模拟结果具有更高精度。
采用论文构建的层次贝叶斯模型研究了分布式水文模型（GBHM）在山洪预报中的不确定性。基于合成数据试验和实际数据的应用结果表明，贝叶斯模型有效地估计了GBHM参数的不确定性，并合理给出了基于GBHM的洪水预报不确定性。基于考虑输入不确定性反演得到的参数后验分布，GBHM模型预报得到的流量系列的概率分布与实测数据具有更好的一致性。降雨误差的后验估计值存在空间负相关关系，说明合理估计分布式模型的输入不确定性还需要更多的信息。
为了分析水文模型结构不确定性，在贝叶斯框架下引入了可行域为[0,1]区间的残差独立性置信度系数（RIC），建立了经验贝叶斯模型。试验研究表明，RIC的最优值既是对模型结构不确定性的评估，也可作为参数反演中判断匹配不足与过度匹配的依据。当模型系统误差增大时，RIC系数的最优值随之降低。基于贝叶斯模型，采用新安江模型和GBHM分别在滁洲流域和赣江流域开展了洪水概率预报，预报不确定性与实测流量数据中反映的不确定性较为一致。
论文的主要创新点在于发展了层次贝叶斯模型和经验贝叶斯模型并成功应用于水文模型参数率定和洪水概率预报，为水文模型中的应用推广提供了理论基础。
China is suffering from water shortage problems and frequent flood and droughtdisasters, which places high demand for hydrological forecasting and prediction.Hydrological models are the major tool of hydrological forecasting and prediction, aswell as other water-related problems. However, uncertainties of parameters, inputs andmodel structures involved in hydrological modeling have undermined its applicationvalue. The objective of this dissertation is to develop a Bayesian framework ofhydrological model system to estimate and diagnose the uncertainties involved inhydrological models.
A hierarchical Bayesian model is first developed for the hydrological model system.A hybrid sampling strategy that employs Markov chain Monte Carlo methods is thendeveloped to explore the Bayesian model. We introduced the hierarchical Bayesianmodel for the hierarchical characters of input uncertainty, and specified inputuncertainty as relative errors in rainfall measurement at the gauges. To conduct thelikelihood function of observed discharge data, we divided the data into low and highflow dataset to reduce the heteroscedasticity, and adopted the first order autoregressivemodel to describe the autocorrelation of the time series. The hybrid sampling strategyfor exploration of the Bayesian model employs Metropolis-Hastings, delay rejectionadaptive Metropolis, and Gibbs sampling methods to improve the sampling efficiency.
To evaluate the validity of the Bayesian model, we applied it to the dataset in anagricultural land in North China to estimate the uncertainty associated with onedimension soil hydraulic parameters. Multilayer parameters of soil water retention andhydraulic conductivity functions were estimated from in situ measurements of soil watercontent at several depths. The medians of posterior distributions of soil hydraulicparameters led to higher accuracy in simulation of soil moisture variation than theparameters obtained by laboratory test did.
The developed hierarchical Bayesian model was adopted to estimate theparameters associated with a geomorphology-based hydrological model (GBHM) formountainous flood forecasting. Synthetic case study and field data applicationdemonstrated the effectiveness of the Bayesian model in estimating the uncertainty ofGBHM parameters and in giving reasonable predictive uncertainty for flood period. Jointly estimating input uncertainty with hydrological model parameters led to a lowererror criterion of probabilistic forecasting and higher consistency of the predictivedistribution with the observed data compared with estimating parameter uncertaintyseparately. The negative spatial correlation of estimated input errors suggested thatfurther information was required to improve input uncertainty estimation of distributedmodels.
To analyze the model structure uncertainty of hydrological models, we developedan informal Bayesian model under the Bayesian framework by introducing a residualindependence coefficient (RIC) with a feasible zone of [0,1]. Case study result showedthat the best RIC coefficient was related to model structure errors and was determined toavoid both overfitting and underfitting. When systematic errors of a hydrological modelincreased, the best RIC coefficient declined. The developed Bayesian model wasapplied separately to a Xinanjiang model of Chuzhou catchment and a GBHM model ofGanjiang catchment for probabilistic flood forecasting. The predictive distributions arewell consistent with the observed data.
The major innovation points of this dissertation lie in the development of thehierarchical Bayesian model and the empirical Bayesian model, and the application ofthe Bayesian models to the parameter estimation of hydrological models and to thehydrological flood forecasting. The dissertation demonstrates a theoretical base for thewidely application of Bayesian models in hydrological models.
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