FRG: Collaborative Research: Prediction and Risk of Extreme Events Utilizing Mathematical Computer Models of Geophysical Processes

该项目的目标是开发必要的数学、统计和计算工具，以评估和预测与火山火山碎屑流等地球物理灾害有关的风险。基于初步的数据分析，研究人员开发了随机模型，从稳定的独立增量过程开始，对超过某个观测阈值的火山碎屑流的体积采用(可能是渐变的)帕累托分布，在控制多个较小喷发的总流量的阿尔法稳定过程的吸引范围内。流动起始角采用冯·米塞斯分布。采用确定性的TITAN2D二维计算环境，利用现有的数字高程图来预测特定体积和起始角的水流对不同地点的影响。TITAN2D是一种基于自适应网格Godunov求解器的深度平均薄层计算流体力学程序，适用于模拟地球物理质量流。基于简单的高斯随机场近似的TITAN2D模型的快速模拟器使研究人员能够模拟数十万次TITAN2D运行，并构建可能导致重大影响的一组可能的流量和起始角的估计；然后，分层贝叶斯统计模型反映在特定时间段内发生这种影响的可能性。最近在计算能力和算法方面的进步导致数学和计算机模拟应用于诸如风暴、洪水、地震和火山喷发等高度复杂的现象。人们越来越认识到，建立这些现象的数学模型只是对风险作出可靠估计和预测所需的更为复杂的过程的一部分。该项目开发了评估和预测与此类自然灾害相关的风险所需的数学、统计和计算工具。这项工作的一个特别重点是研究这些风险在空间和时间上如何变化，以及它们的不确定性。这一方法是在火山雪崩和火山碎屑流(所谓的地球物理物质流)这一具体问题的背景下制定的，但它的大部分内容将更广泛地适用于具有空间变异性和模型不确定性的问题中的风险分析和量化问题。它汇集了一支具有火山学专业的独特的科学家团队，以指导正在研究的地球物理过程的现实模型的开发；在随机过程中，以现实和可验证的方式反映关于初始条件、流动频率和其他特征的不确定性和变异性；在确定性计算机建模中，用于对最可能和最危险的可能事件的后果进行详细的空间预测的困难任务；在计算机模型仿真中，将预测各种可能情景下罕见事件的风险所需的计算速度提高数千倍；在统计建模和分析中，诚实地反映这一分析中所有不同的不确定性和变异性来源，从而全面量化危险事件的风险。只有拥有如此广泛的专业知识，研究人员才能构建所需的科学织锦，以开发研究毁灭性自然灾害的工具。

The goal of this project is to develop the mathematical, statistical and computational tools needed to assess and predict the risk associated with geophysical hazards such as volcanic pyroclastic flows. Based on a preliminary data analysis, the investigators develop stochastic models beginning with stationary independent increment processes employing (possibly tapered) Pareto distributions for the volumes of pyroclastic flows exceeding some observational threshold, in the domain of attraction of an alpha-stable process governing the aggregate flow volume of multiple smaller eruptions. Von Mises distributions are used for flow initiation angles. The deterministic TITAN2D two-dimensional computational environment is employed, which uses available digital elevation maps to predict the impact at various sites of interest from flows of specified volume and initiation angles. TITAN2D is a depth-averaged, thin-layer computational fluid dynamics code based on an adaptive grid Godunov solver, suitable for simulating geophysical mass flows. A rapid emulator based on a simple Gaussian random-field approximation to the TITAN2D model enables the investigators to emulate hundreds of thousands of TITAN2D runs and construct an estimate of the set of possible flow volumes and initiation angles that would lead to significant impact; a hierarchical Bayesian statistical model then reflects the probability of such an impact over a specified period of time.Recent advances in computing power and algorithms have led to the application of mathematical and computer modeling to such highly complex phenomena as storms, floods, earthquakes and volcanic eruptions. It is increasingly being understood that development of mathematical models of these phenomena is only one part of a much more complex process needed for making reliable estimates and predictions of risk. This project develops the mathematical, statistical and computational tools needed for assessing and predicting the risk associated with such natural hazards. A particular focus of the work is the study of how these risks vary in space and time, and of how uncertain they are. This methodology is developed in context of the specific problem of volcanic avalanches and pyroclastic flows (so-called geophysical mass flows), but much of it will be applicable more broadly to problems in the analysis and quantification of risk in problems featuring spatial variability and model uncertainty. It brings together a unique team of scientists with specialties including volcanology, to guide the development of realistic models for the geophysical processes under study; in stochastic processes, to reflect uncertainty and variability about initial conditions, flow frequencies, and other features in realistic and verifiable ways; in deterministic computer modeling, for the difficult task of making detailed spatial predictions of the consequences of the most probable and of the most hazardous possible events; in computer model emulation, to accelerate many thousand-fold the computations necessary for predicting the risk of rare events under a wide range of possible scenarios; and in statistical modeling and analysis, to reflect honestly all the different sources of uncertainty and variability in this analysis, leading to a full quantification of the risk of hazardous events. Only with such a broad range of expertise can investigators build the tapestry of science that is required to develop tools for studying devastating natural hazards.