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

0407123纽曼目标：允许水文学家通过一个有根据的和经过充分研究的方法来定量地科普概念模型的不确定性，将最大似然参数估计纳入多模型贝叶斯更新框架，这在实践中是可行的。目标：（1）巩固首席研究员最近提出的最大似然贝叶斯模型平均法的理论基础（Neuman，2002，2003），通过几个竞争模型和评估其联合预测的不确定性来进行最佳水文预测。(2)为了实现，探索和演示MLBMA水文地质数据，分布在三维空间和时间，收集早期在不饱和裂隙岩石在Apache飞跃研究站点（ALRS）在亚利桑那州中部。问题：水文分析通常依赖于地质或流域组成和相应水文过程的单一概念数学模型。然而，水文环境是开放的和复杂的，使他们倾向于多种解释和数学描述。无论现有数据的数量和质量如何，情况都是如此。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（通过依赖不充分的模型而犯下II型错误）和低估不确定性（通过对相关模型的采样不足而犯下I型错误）空间）。由于依赖不适当的概念数学模型而产生的偏差和不确定性往往比由于模型参数值选择不当而产生的偏差和不确定性大得多。然而，大多数水文不确定性分析忽略了前者，只关注后者。这往往导致过度自信的预测能力的模型，其中可用的水文数据很少证明。事实上，对水文分析的批评和对它们的科学/监管/法律的挑战通常集中在基本概念（以及隐含的数学）模型的有效性上。现有的方法来处理这个问题，最显着的广义似然不确定性估计（GLUE）的方法Beven和Binley（1992;也见Beven和Freer，2001），在PIFS的观点，有用的，但不一定是最佳的目的。有必要对这一问题采取一种创新的办法，这种办法应建立在严格的理论基础上，并在实践中切实可行。方法：在目标1下，我们提出通过理论上和通过综合数值研究探索来巩固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）相比的准确性和计算可行性（Hoeting等人，（1999年）;（B）先前水文参数测量的可用性或缺乏对这种比较的影响;（c）使用Kashyap fs（1982）贝叶斯信息准则KIC计算后验模型概率之间的差异（Neuman，2002年，2003年）与渐近贝叶斯准则BIC（由Raftery，1993年提出）或非贝叶斯信息理论标准，如Akaike fs（1974年）AIC（Burnham和安德森提出，2002年）;（d）如何为各种模型分配先验概率的未决问题;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性随着水文数据的信息内容（数量和质量）而减小的速率。其中一些问题也将在目标2下针对真实的数据加以处理。目标2是实施、探索和展示MLBMA对ALRS早期收集的水文地质数据的预测能力。这些包括空气渗透率和充气孔隙度数据，从气动注射测试在1米长的间隔沿着六个垂直和倾斜的钻孔在现场，和瞬态压力数据，从跨孔气动注射测试在这些和十个额外的钻孔。这些数据在很大程度上代表了一个连续的相互连接的裂缝。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-地质统计模型，说明1米尺度的测井渗透率（）10 log k和测井孔隙度（）10 log k 仅根据1米尺度的测量，数据在空间上各不相同。在第二阶段，我们将检查这些模型的子集以及BMA和MLBMA在现场跨孔测试期间能够仅基于关于lo，10 g k 10 log的先验信息来预测钻孔之间的气流的程度 和强迫项的替代表示。先验信息将包括测量、统计、地质-地质统计模型、投影和这些量在包含所有钻孔的域中的投影协方差（在第1阶段建立）。在第3阶段，我们建议校准（通过ML）气流模型具有替代参数结构对压力数据观察期间，在ALRS的一个跨孔测试，并检查他们的能力，以及MLBMA，预测在其他这样的（验证）测试期间观察到的压力。跨孔测试将被选择，以便注入到不同的钻孔在每个theme.Intellectual优点和广泛的影响：一个坚实的理论和一个实用的方法，渲染optimumhydrologic预测和评估预测的不确定性，占共同的模型结构（概念数学框架）和参数的不确定性。该方法适用于广泛的模型，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。研究结果将通过各种方式广泛传播给研究人员和从业人员。

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.