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

##### Abstract

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.