Statistical theory of probabilistic hazard maps: a probability density function for inundation edge location

Abstract

The study of volcanic mass flow hazards in a probabilistic framework centers around systematic experimental numerical modelling of the hazardous phenomenon and the subsequent generation and interpretation of a probabilistic hazard map (PHM). For a given volcanic mass flow phenomenon (e.g., lava flow, lahar, pyroclastic flow, etc.), the PHM is typically interpreted as giving the point-wise probability of flow material inundating a given location.

By formalizing the generation and properties of the PHM, we show that a PHM may be used to generate additional statistical measures of the hazard, which have been unrecognized in probabilistic hazard analysis, and may be of interest to analysts, planners, emergency managers, and exposed populations. Our formalism shows that a typical PHM not only gives the inundation probability at every location, but also represents a type of cumulative distribution function for the location of the inundation boundary with a corresponding probability density function. This distribution runs over contours of steepest gradient ascent on the PHM. Consequently, 2D curves can be constructed which represent the mean, median and modal locations of the inundation edge. Additionally, methods of calculation for the standard deviation and confidence intervals are presented that take the form of regions of the map surrounding the mean and median edge locations, respectively. These additional measures of central tendency and variance add significant value to probabilistic hazard analyses, giving a richer statistical description of the probability distributions underlying PHMs.

The theory may be used to construct improved hazard maps, which could prove useful for planning and emergency management purposes. Additionally, these methods can help evaluate common problems that arise in numerical models of geophysical mass flows, such as artificially thin and fast-propagating flow boundaries. The formalism also allows for application to processes describable by analytic solutions. The connection between the PHM, its derived measures, and the underlying parameter variation is explicit, allowing for better parameter estimation from natural data for use in flow modeling.