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Discussion papers
https://doi.org/10.5194/nhess-2018-344
© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/nhess-2018-344
© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.

Research article 16 Nov 2018

Research article | 16 Nov 2018

Review status
This discussion paper is a preprint. It is a manuscript under review for the journal Natural Hazards and Earth System Sciences (NHESS).

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

David M. Hyman1,2, Andrea Bevilacqua2,3,4, and Marcus I. Bursik2 David M. Hyman et al.
  • 1Collaborative Institute for Meteorological Satellite Studies (CIMSS), University of Wisconsin – Madison, WI
  • 2Department of Geological Sciences, University at Buffalo, NY
  • 3Computational Data Science and Eng., University at Buffalo, NY
  • 4Istituto Nazionale di Geofisica e Vulcanologia (INGV), Sezione di Pisa, Pisa, Italy

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 flow (e.g., lava flow, lahar, pyroclastic flow, etc.), the PHM is typically interpreted as the point-wise probability of flow material inundation.

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 mass 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 ascent on the PHM. Consequently, 2D space curves can be constructed on the map which represent 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 construct 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.

David M. Hyman et al.
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Short summary
In this work, we present new methods for calculating the mean, standard deviation, median, and modal locations of the boundaries of volcanic hazards. These calculations are based on a new, mathematically rigorous definition of a probabilistic hazard map – a way to map the probabilities of inundation by a given hazard. We apply this analysis to several models of volcanic flows: simple models of viscous flows, complex models of a tabletop granular flow, and a complex model of a volcanic mud flow.
In this work, we present new methods for calculating the mean, standard deviation, median, and...
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