Statistical theory of probabilistic hazard maps: a probability distribution for the hazard boundary location

David M Hyman,Marcus I Bursik,Andrea Bevilacqua

doi:10.5194/nhess-19-1347-2019

Abstract

Abstract. The study of volcanic flow hazards in a probabilistic framework centers around systematic experimental numerical modeling of the hazardous phenomenon and the subsequent generation and interpretation of a probabilistic hazard map (PHM). For a given volcanic flow (e.g., lava flow, lahar, pyroclastic flow, ash cloud), the PHM is typically interpreted as the point-wise probability of inundation by flow material. In the current work, we present new methods for calculating spatial representations of the mean, standard deviation, median, and modal locations of the hazard's boundary as ensembles of many deterministic runs of a physical model. By formalizing its generation and properties, we show that a PHM may be used to construct these statistical measures of the hazard boundary which have been unrecognized in previous probabilistic hazard analyses. Our formalism shows that a typical PHM for a volcanic flow not only gives the point-wise inundation probability, but also represents a set of cumulative distribution functions for the location of the inundation boundary with a corresponding set of probability density functions. These distributions run over curves of steepest probability gradient ascent on the PHM. Consequently, 2-D space curves can be constructed on the map which represents the mean, median, and modal locations of the likely inundation boundary. These curves give well-defined answers to the question of the likely boundary location of the area impacted by the hazard. Additionally, methods of calculation for higher moments including the standard deviation are presented, which take the form of map regions surrounding the mean boundary location. These measures of central tendency and variance add significant value to spatial probabilistic hazard analyses, giving a new statistical description of the probability distributions underlying PHMs. The theory presented here may be used to aid construction of improved hazard maps, which could prove useful for planning and emergency management purposes. This formalism also allows for application to simplified processes describable by analytic solutions. In that context, the connection between the PHM, its moments, and the underlying parameter variation is explicit, allowing for better source parameter estimation from natural data, yielding insights about natural controls on those parameters.

Highlights

The probabilistic study of volcanic hazards is part of an emerging paradigm in physical volcanology, one that openly admits and seeks to calculate and characterize the uncertainty inherent to physical modeling of hazardous volcanic processes (Bursik et al, 2012; Connor et al, 2015; Bevilacqua, 2016; Sandri et al, 2016)
This model used by Quick et al (2017) is a modification of the famous model due to Huppert (1982) of the axisymmetric spread of viscous gravity currents, allowing for time-dependent viscosity where the height of the current is governed by g1 ∂t h − ν(t) r ∂r rh3∂r h where g is gravitational acceleration
Using the example of the freezing viscous gravity current highlighted above (Quick et al, 2017), we show that an ensemble of natural examples of such flows can be used to invert the probability density function (PDF) of input parameters directly from the definition of the probabilistic hazard map (PHM)

Summary

Introduction

The probabilistic study of volcanic hazards is part of an emerging paradigm in physical volcanology, one that openly admits and seeks to calculate and characterize the uncertainty inherent to physical modeling of hazardous volcanic processes (Bursik et al, 2012; Connor et al, 2015; Bevilacqua, 2016; Sandri et al, 2016). This approach is founded on the notion that physical parameters related to these processes can be represented by a probability distribution. While this is the most obvious method it is just one of many options for declaring this categorical boundary

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