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

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 pointwise 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.