Coping With Conceptual Uncertainty: A Maximum Likelihood Bayesian Model Averaging Approach

目的：(1)巩固PI(Neuman，2002,2003)最近提出的最大似然贝叶斯模型平均(MLBMA)方法的理论基础，该方法通过几个相互竞争的模型及其联合预测不确定性的评估来呈现最优的水文预报。(2)在亚利桑那州中部阿帕奇跃迁研究场地(ALRS)早先在非饱和裂隙岩石中收集的水文地质数据上实施、探索和演示MLBMA，这些数据分布在三维时空中。问题：水文分析通常依赖于单一的概念性数学模型，即地质或流域组成和相应的水文过程。然而，水文环境是开放和复杂的，这使得它们容易受到多种解释和数学描述。无论可用数据的数量和质量如何，这都是正确的。基于单一水文概念的不确定性预测和分析容易出现统计偏差(由于依赖不充分的模型而导致第二类错误)和对不确定性的低估(通过对相关模型空间进行欠采样而导致第一类错误)。由于依赖不充分的概念性数学模型而产生的偏差和不确定性往往比不适当地选择模型参数值而产生的偏差和不确定性要大得多。然而，大多数水文不确定性分析忽略了前者，只关注后者。这往往导致对模型预测能力的过度自信，而现有的水文数据很少证明这一点。事实上，对水文分析的批评和对它们的科学/监管/法律挑战通常集中在基本概念模型(以及隐含的数学模型)的有效性上。在PIFS看来，处理这一问题的现有方法，最著名的是Beven和Binley的广义似然不确定性估计(GLUE)方法(1992年；也见Beven和Freer，2001)是有用的，但不一定是最优的。有必要对这一问题采取一种创新的方法，这种方法以严谨的理论为基础，并在实践中可行。方法：在目标1下，我们建议通过理论探索和综合数值研究来巩固MLBMA的理论基础：(A)MLBMA与通过马尔可夫链蒙特卡罗模拟实现的贝叶斯模型平均(BMA)的精度和计算可行性的比较(Hoting等人，1999)；(B)有无先前水文参数测量对这种比较的影响；(C)使用Kashyap fs(1982)贝叶斯信息准则KIC(Neuman，2002,2003)与渐近贝叶斯准则BIC(由Raftery，1993提出)或非贝叶斯信息理论准则(如Akaike fs(1974)AIC(Burnham and Anderson，2002))计算后验模型概率之间的差异；(D)如何将先验概率分配给各种模型的悬而未决的问题；以及(E)后验参数估计和模型概率对先验参数和模型概率的选择的敏感度随着水文数据的信息量(数量和质量)而减小的速度。其中一些同样的问题也将在目标2下对照实际数据加以处理。目标2是实施、探索和展示MLBMA对早些时候在ALRS收集的水文地质数据的预测能力。这些数据包括现场6个垂直和倾斜钻孔在1米长间隔内进行的气动注入试验的空气渗透率和充气孔隙率数据，以及这些和另外10个钻孔跨孔气动注入试验的瞬变压力数据。这些数据在很大程度上代表了相互关联的裂缝的连续体。他们的分析将分三个阶段进行。在第一阶段，我们建议考虑替代的地质-地质统计学模型，即仅基于1米尺度的测量，1米尺度的对数渗透率()10logk和对数孔隙度()10的测井数据如何在空间上变化。在第二阶段，我们将研究这些模型的一个子集，以及BMA和MLBMA，在多大程度上能够预测现场跨孔测试期间的钻孔之间的气流，仅基于关于lo、10g k 10log和强迫项的替代表示的先验信息。先验信息将包括测量、统计、地质-地质统计模型、这些量在包含所有钻孔的区域内的投影和投影协方差，在第一阶段建立。在第三阶段，我们建议根据在ALRS的一次跨孔测试中观察到的压力数据来校准(通过ML)具有替代参数结构的气流模型，并检查它们以及MLBMA预测在其他此类(验证)测试中观察到的压力的能力。将选择跨孔测试，以便向每个测试中的不同钻孔注入。智力优势和广泛影响：提供最佳水文预测的坚实理论和实用方法，以及对预测不确定性的评估，这些预测不确定性共同考虑模型结构(概念-数学框架)和参数中的不确定性。该方法适用于代表无处不在的开放和复杂的地球和环境系统中的自然过程的广泛模型。结果将通过各种方式广泛传播给研究人员和从业者。

0407123NeumanGoal: Allow hydrologists to cope quantitatively with conceptual model uncertainty via a well-founded andwell-researched methodology, incorporating maximum likelihood parameter estimation in a multimodel Bayesian updating framework, which is feasible to implement in practice.Objectives: (1) To firm up the theoretical basis of a Maximum Likelihood Bayesian Model Averaging(MLBMA) method recently proposed by the PI (Neuman, 2002, 2003) for the rendering of optimum hydrologic predictions by means of several competing models and the assessment of their joint predictive uncertainty. (2) To implement, explore and demonstrate MLBMA on hydrogeologic data, distributed in three-dimensional space and time, collected earlier in unsaturated fractured rock at the Apache Leap Research Site (ALRS) in central Arizona. The Problem: Hydrologic analyses typically rely on a single conceptual-mathematical model of geologic or watershed makeup and corresponding hydrologic processes. Yet hydrologic environments are open and complex, rendering them prone to multiple interpretations and mathematical descriptions. This is true regardless of the quantity and quality of available data. Predictions and analyses of uncertainty based on a single hydrologic concept are prone to statistical bias (by committing a Type II error through reliance on an inadequate model) and underestimation of uncertainty (by committing a Type I error through under sampling of the relevant model space). The bias and uncertainty that result from reliance on an inadequate conceptual-mathematical model are often much larger than those introduced through an inadequate choice of model parameter values. Yet most hydrologic uncertainty analyses ignore the former and focus exclusively on the latter. This often leads to overconfidence in the predictive capabilities of the model, which the available hydrologic data seldom justify. Indeed, critiques of hydrologic analyses and scientific/regulatory/legal challenges to them typically focus on the validity of the underlying conceptual (and by implication mathematical) model. Existing method of dealing with the problem, most notably the Generalized Likelihood Uncertainty Estimation (GLUE) approach of Beven and Binley (1992; see alsoBeven and Freer, 2001) are, in the PIfs view, useful but not necessarily optimal for the purpose. There is a need for an innovative approach to the problem that rests on rigorous theory and is feasible to implement in practice. Approach: Under Objective 1 we propose to firm up the theoretical basis of MLBMA by exploringtheoretically and through synthetic numerical studies (a) the accuracy and computational feasibility of MLBMA in comparison to Bayesian Model Averaging (BMA) implemented via Markov Chain Monte Carlo simulation (Hoeting et al., 1999); (b) the impact that availability or lack of prior hydrologic parameter measurements have on this comparison; (c) the difference between computing posterior model probabilities using Kashyap fs (1982) Bayesian information criterion KIC (Neuman, 2002, 2003) versus the asymptotic Bayesian criterion BIC (proposed by Raftery, 1993) or non-Bayesian information theoretic criteria such as Akaike fs (1974) AIC (proposed by Burnham and Anderson, 2002); (d) the unresolved issue of how to assign prior probabilities to various models; and (e) the rate at which the sensitivity of posterior parameter estimates and model probabilities to the choice of prior parameter andmodel probabilities diminishes with the information content (quantity and quality) of hydrologic data. Some of these same issues will also be addressed vis-a-vis real data under Objective 2. Objective 2 is to implement, explore and demonstrate the predictive capabilities of MLBMA on hydrogeologic data collected earlier at the ALRS. These include air permeability and air-filled porosity data from pneumatic injection tests in 1-m-length intervals along six vertical and inclined boreholes at the site, and transient pressure data from cross-hole pneumatic injection tests in these and ten additional boreholes. The data represent largely a continuum of interconnected fractures. Their analysis will be conducted in three stages. At Stage 1 we propose to consider alternative geological-geostatistical models of how the 1-m-scale log permeability ()10 log k and log porosity ()10 log data vary in space, based solely on 1-m-scale measurements. At Stage 2, we will examine the extent to which a subset of these models, as well as BMA and MLBMA, are capable of predicting air flow between boreholes during cross-hole tests at the site basedsolely on prior information about lo , 10 g k 10 log and alternative representations of forcing terms. The prior information will consist of measurements, statistics, geological-geostatistical models, projections and projection covariances of these quantities across a domain containing all boreholes, established at Stage 1. At Stage 3 we propose to calibrate (via ML) airflow models having alternative parameter structures against pressure data observed during one cross-hole test at the ALRS and examine their ability, as well as that of MLBMA, to predict pressures observed during other such (validation) tests. The cross-hole tests will be selected so that injection takes place into a different borehole in each of them.Intellectual Merit and Broad Impacts: A solid theory and a practical methodology of rendering optimumhydrologic predictions and an assessment of predictive uncertainty that account jointly for uncertainties in model structure (conceptual-mathematical frameworks) and parameters. The approach applies to a broad range of models representing natural processes in ubiquitously open and complex earth and environmental systems. Results will bedisseminated broadly to researchers and practitioners through various means.