基于信息熵和互信息的流域水文模型不确定性分析
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摘要
随着对水文过程研究的日益深入，水文模拟日益精细；与此同时，水文模拟和预报不确定性受到愈来愈广泛的关注。目前，水文预报的不确定性研究普遍认为模型输入、结构和参数是不确定性的三个主要来源。由于并非所有不确定性均与模型有关，已有的研究不足以区分独立于和依赖于水文模型的不确定性。论文引入信息熵和互信息的概念，通过计算水文数据中的信息量，以区分独立于模型的随机不确定性，依赖于模型的认知不确定性；在此基础上，提出了基于信息熵和互信息的水文模拟不确定性分析框架，并进行了初步应用。
首先，论文借鉴了信息论的最新研究成果，度量三种信息量：径流模拟需要的信息量、数据提供的可用信息量、水文模型利用的信息量。针对研究流域和采用的水文模型而言，实测流量系列的信息熵就是该流域径流模拟需要的信息量，气象、前期径流及流域土壤含水量等数据与实测径流数据之间的互信息即为该流域水文数据为径流模拟提供的可用信息量，模拟径流与实测径流系列之间互信息即为水文模型利用的信息量。前两者之差是径流模拟需要而水文数据没有提供的信息量，称为随机不确定性；后两者的差是数据已经提供而模型没有利用的信息量，称为认知不确定性。
针对高维水文变量互信息计算中的维数灾现象，论文首次尝试了采用Leonenko方法，绕过联合分布直接计算信息熵。结果表明，Leonenko方法在低维和高维低相关情况下表现稳定，但不适用于高维高相关情况。论文进一步尝试了ISOMAP方法和SVM方法，研究表明ISOMAP方法对水文数据信息量的相对大小具有识别能力；SVM方法对确定性降雨径流关系有较强的拟合能力。使用人工合成数据测试，结果表明这两种方法能够稳定的识别水文数据提供的信息量。
论文的方法应用于多个数据质量不同的流域和结构复杂度不同的模型，结果表明该方法能有效识别随机和认知不确定性，能够判断水文模拟不确定性在多大程度上来源于模型结构的不完备性，对改进水文模型具有指导意义。
论文的主要创新性在于，提出了基于信息熵和互信息的水文模拟不确定性分析框架；以现有方法为基础，提出了适用于高维高相关数据的互信息计算方法；通过在不同流域和采用不同模型的应用分析，提出了水文预报中模型选择和改进的策略。
With the growing knowledge of hydrological processes and finer hydrologicalsimulation, the issue of hydrological model uncertainty attracts more and more attention.Currently the hydrological model uncertainty is classified into three sources: input,parameter and structure uncertainty. However, not all “model uncertainty” depends on“model”. Current uncertainty qualification methods depend on one (or multiple) specificmodel, which are lack of the ability to distinguish the two categories thatindependent/depend on model. The concepts of information entropy and mutualinformation is introduced in order to qualify the information content of hydrologicalvariables and distinguish aleatory uncertainty that independent to model and epistemicuncertainty that dependent on model. A framework of uncertainty analysis based oninformation theory is proposed and primary case studies are carried out in order to showthe applicability of the new framework.
First, the proposed framework uses the most up to date methods to explicitlycompute three kinds of information: the information required by streamflow simulation,the information offered by currently available hydrological observation data, and theinformation expressed by a specific model. For a specific catchment and model, theinformation entropy of observed streamflow is the information required by streamflowsimulation; the mutual information that meteorological data, previous streamflow andsoil moisture data contributes to streamflow observation data is the availableinformation for streamflow simulation offered by currently available data; the mutualinformation between simulated and observed streamflow is the amount of informationused by a model. The difference between the first two is aleatory uncertainty, which theinformation is required by simulation but not offered by data; the difference between thelast two is epistemic uncertainty, which the information has been offered by data but notproperly utilized by the model.
To mitigate the influence of curse of dimensionality on computing mutualinformation of high-dimensional hydrological variables, Leonenko’s method that cancompute information content without estimating joint probability density function isintroduced and applied for the first time in the context of hydrology. The result shows that Leonenko’s method performs well in low-dimension and high-dimensionlow-correlation case, but not applicable to the high-dimensional high-correlation case.Further tests are carried out and another two methods: ISOMAP and SVM are able toqualify information content of hydrological variables. ISOMAP can identify the relativeinformation content of hydrological datasets, while SVM is good at regressing to theinformation while eliminating the random error so that one can compute the informationcontent from the regression result. The result of numerical experiment based onhydrological model demonstrated that both of the two methods are capable ofidentifying the information content of hydrological observation data.
The framework is applied in multiple catchments that have different data qualityand multiple models that have different model structure complexity. The result ofapplication shows that the framework is able to distinguish aleatory and epistemicuncertainty, identify how much of the uncertainty comes from model structureinadequacy, and potentially guide model structure improvement in the future.
The novel contribution of this thesis is1) proposing a hydrological modeluncertainty qualification framework based on information entropy and mutualinformation;2) improving current methods of entropy and mutual informationestimation by proposing new method that applicable to high-dimensionalhigh-correlated cases;3) applying the framework to various catchments and models andproposed a model selection and improvements strategy in hydrological prediction.
首先，论文借鉴了信息论的最新研究成果，度量三种信息量：径流模拟需要的信息量、数据提供的可用信息量、水文模型利用的信息量。针对研究流域和采用的水文模型而言，实测流量系列的信息熵就是该流域径流模拟需要的信息量，气象、前期径流及流域土壤含水量等数据与实测径流数据之间的互信息即为该流域水文数据为径流模拟提供的可用信息量，模拟径流与实测径流系列之间互信息即为水文模型利用的信息量。前两者之差是径流模拟需要而水文数据没有提供的信息量，称为随机不确定性；后两者的差是数据已经提供而模型没有利用的信息量，称为认知不确定性。
针对高维水文变量互信息计算中的维数灾现象，论文首次尝试了采用Leonenko方法，绕过联合分布直接计算信息熵。结果表明，Leonenko方法在低维和高维低相关情况下表现稳定，但不适用于高维高相关情况。论文进一步尝试了ISOMAP方法和SVM方法，研究表明ISOMAP方法对水文数据信息量的相对大小具有识别能力；SVM方法对确定性降雨径流关系有较强的拟合能力。使用人工合成数据测试，结果表明这两种方法能够稳定的识别水文数据提供的信息量。
论文的方法应用于多个数据质量不同的流域和结构复杂度不同的模型，结果表明该方法能有效识别随机和认知不确定性，能够判断水文模拟不确定性在多大程度上来源于模型结构的不完备性，对改进水文模型具有指导意义。
论文的主要创新性在于，提出了基于信息熵和互信息的水文模拟不确定性分析框架；以现有方法为基础，提出了适用于高维高相关数据的互信息计算方法；通过在不同流域和采用不同模型的应用分析，提出了水文预报中模型选择和改进的策略。
With the growing knowledge of hydrological processes and finer hydrologicalsimulation, the issue of hydrological model uncertainty attracts more and more attention.Currently the hydrological model uncertainty is classified into three sources: input,parameter and structure uncertainty. However, not all “model uncertainty” depends on“model”. Current uncertainty qualification methods depend on one (or multiple) specificmodel, which are lack of the ability to distinguish the two categories thatindependent/depend on model. The concepts of information entropy and mutualinformation is introduced in order to qualify the information content of hydrologicalvariables and distinguish aleatory uncertainty that independent to model and epistemicuncertainty that dependent on model. A framework of uncertainty analysis based oninformation theory is proposed and primary case studies are carried out in order to showthe applicability of the new framework.
First, the proposed framework uses the most up to date methods to explicitlycompute three kinds of information: the information required by streamflow simulation,the information offered by currently available hydrological observation data, and theinformation expressed by a specific model. For a specific catchment and model, theinformation entropy of observed streamflow is the information required by streamflowsimulation; the mutual information that meteorological data, previous streamflow andsoil moisture data contributes to streamflow observation data is the availableinformation for streamflow simulation offered by currently available data; the mutualinformation between simulated and observed streamflow is the amount of informationused by a model. The difference between the first two is aleatory uncertainty, which theinformation is required by simulation but not offered by data; the difference between thelast two is epistemic uncertainty, which the information has been offered by data but notproperly utilized by the model.
To mitigate the influence of curse of dimensionality on computing mutualinformation of high-dimensional hydrological variables, Leonenko’s method that cancompute information content without estimating joint probability density function isintroduced and applied for the first time in the context of hydrology. The result shows that Leonenko’s method performs well in low-dimension and high-dimensionlow-correlation case, but not applicable to the high-dimensional high-correlation case.Further tests are carried out and another two methods: ISOMAP and SVM are able toqualify information content of hydrological variables. ISOMAP can identify the relativeinformation content of hydrological datasets, while SVM is good at regressing to theinformation while eliminating the random error so that one can compute the informationcontent from the regression result. The result of numerical experiment based onhydrological model demonstrated that both of the two methods are capable ofidentifying the information content of hydrological observation data.
The framework is applied in multiple catchments that have different data qualityand multiple models that have different model structure complexity. The result ofapplication shows that the framework is able to distinguish aleatory and epistemicuncertainty, identify how much of the uncertainty comes from model structureinadequacy, and potentially guide model structure improvement in the future.
The novel contribution of this thesis is1) proposing a hydrological modeluncertainty qualification framework based on information entropy and mutualinformation;2) improving current methods of entropy and mutual informationestimation by proposing new method that applicable to high-dimensionalhigh-correlated cases;3) applying the framework to various catchments and models andproposed a model selection and improvements strategy in hydrological prediction.
引文
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