The use of regularized inversion in groundwater model calibration and prediction uncertainty analysis
The use of regularized inversion in groundwater model calibration and prediction uncertainty analysis
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正则反演在地下水模型标定和预测不确定性分析中的应用
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发表时间：
2006
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通讯作者：
C. Moore
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作者：
C. Moore
地下水模型有一个与其他环境模型相同的特点，即精确描述所模拟的物理现象所需的参数数目往往大于可用数据估计的数目。需要表示物理细节以准确建模系统行为与通常可用的稀疏数据集之间的这种冲突导致模型校准问题的欠定性质。在欠定地下水问题中，通常使用某种简化装置进行模型参数化，以获得唯一的模型解。这样的简化装置是：使用恒定参数值的区域的参数集总;或者使用数学正则化，其允许在校准之前定义多个参数，但是其对参数关系或值施加约束，使得校准的场仍然是现实的简化版本。这两个简化设备将欠定问题转换为可以用标准参数估计方法解决的超定问题。由这些简化模型所做的模型预测总是有误差，这是由两个来源引起的：（i）观测误差，以及（ii）模型结构误差，来自模型不足，具体地说，包括欠定模型中发生的参数细节水平不足。经典的预测不确定性分析方法忽略了第二个来源，但这往往是欠定问题的最重要的误差源。上述超定、简化方法的替代方案是保留真实尺度参数化细节，以允许精确表示被建模的物理过程。这种方法的缺点是，非唯一的参数解决方案的结果，所以蒙特卡洛方法是必需的，从而该模型运行的多个参数字段已产生的基础上，先前的水力特性的变化性描述，并所有荣誉校准约束。这种方法的输出是一个完整的预测累积密度函数，它包含了真实的预测，因此定义了预测误差。蒙特卡罗方法是计算非常昂贵的详细地下水参数化。因此，如果模型预测不确定性的报告要成为标准做法，就需要一种方法来描述单一校准地下水模型的参数和预测不确定性，因为它应该避免模型结果的误导性报告。正则化使得能够对真实、详细的水力特性和简化的校准场之间的关系进行数学定义。这种关系体现在所谓的分辨率矩阵中。这种关系形成了欠定模型的预测不确定性分析的基础，如下所述。本文采用分辨率分析方法研究了水力学特性变异性对地下水污染物迁移预测不确定性的影响。特别是，这种先验知识的水力性质的变化是量化的变异函数作为地质统计学描述符。这种方法提供了：一个数学描述的可能错误的预测所作出的校准模型。本说明考虑到模型预测误差的两个来源，即观测误差和参数简化;在进行校准过程之前评估模型预测不确定性的能力;调整校准过程的能力，以便可以进行校准以减少需要进行的预测的方差;基于校准参数场的结构来推断真实水力特性变化的性质或替代地推断测量误差的能力;通过本文中提出的四种替代算法来获得非线性背景下的预测误差方差方程的解的能力。这些结果表明，在校准之前，在预测所依赖的细节水平上离散化参数具有深远的益处;尽管无法唯一地参数化该细节。这些好处中的一个重要方面是使决策者能够考虑模型预测不确定性的根本原因;判断在减少预测不确定性方面从校准工作中可以获得什么;确定这种工作是否具有成本效益;并更好地判断今后数据采集工作的最佳重点。
Groundwater models have a characteristic, common to other environmental models, whereby the desirable number of parameters required to accurately describe the physical phenomenon being modelled is often greater than the number which can be estimated with the available data. This conflict between the need to represent physical detail to accurately model system behaviour, and the sparse datasets that are typically available, leads to the underdetermined nature of model calibration problems. In underdetermined groundwater problems model parameterisation is conventionally undertaken using some kind of simplifying device, to allow a unique model solution to be obtained. Such simplifying devices are either: parameter lumping using zones of constant parameter value; or the use of mathematical regularisation, which allows a multiplicity of parameters to be defined prior to calibration, but which places constraints on parameter relationships or values, such that the calibrated field is a still a simplified version of reality. Both these simplifying devices convert the underdetermined problem to an overdetermined problem that can be solved with standard parameter estimation methods. Model predictions made by these simplified models always have error, which is caused by two sources: (i) observation errors, and (ii) model structural errors, from model inadequacy including, specifically, inadequate levels of parameter detail that occur in underdetermined models. Classical predictive uncertainty analysis methods ignore the second of these sources, however this is often the most significant error source for underdetermined problems. An alternative to the overdetermined, simplifying, approaches described above, is to retain true scale parameterisation detail, to allow accurate representation of the physical process being modelled. The drawback of this method is that nonunique parameter solutions result, and so Monte Carlo methods are required; whereby the model is run with multiple parameter fields that have been generated on the basis of a prior hydraulic property variability description, and which all honour calibration constraints. The output of such methods is a full prediction cumulative density function, which encompasses the true prediction and hence defines the prediction error. Monte Carlo methods are computationally very expensive for detailed groundwater parameterisations. Consequently a method which describes the parameter and prediction uncertainty of a single calibrated groundwater model is required if reporting on model prediction uncertainty is to become standard practice, as it should, to avoid the misleading reporting of model results. Regularisation enables a mathematical definition of the relationship between true, detailed, hydraulic properties and a simplified calibrated field to be made. This relationship is embodied in the so-called Resolution matrix. This relationship forms the basis for predictive uncertainty analysis for underdetermined models, as is discussed below. This thesis employs resolution analysis to examine the effect of hydraulic property variability on the prediction uncertainty of contaminant transport in groundwater. In particular, this prior knowledge of hydraulic property variability is quantified by means of the variogram as a geostatistical descriptor. This approach provides: a mathematical description of the likely error of a prediction made by a calibrated model. This description takes account of the two sources of model prediction error, viz. observation errors, and parameter simplification; the ability to assess the predictive uncertainty of a model prior to the calibration process being undertaken; the ability to tailor the calibration process such that it can be undertaken to reduce the variance of the prediction it is required to make; the ability to infer either the nature of true hydraulic property variability, or alternatively to infer measurement error, on the basis of the structure of the calibrated parameter field; the ability to obtain solutions to the predictive error variance equation in a nonlinear context, via four alternative algorithms presented herein. These results demonstrate that there are far reaching benefits to discretising parameters at the level of detail on which a prediction depends, prior to calibration; despite being unable to uniquely parameterise this detail. Not least of these benefits is the enabling of decision makers to consider the root causes of a models predictive uncertainty; to judge what is to be gained from a calibration exercise in terms of prediction uncertainty reduction; and to determine whether such an exercise is cost-effective; and to make better judgements on where best to concentrate future data acquisition efforts.