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.

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

At present, a probabilistic hazard map (PHM) for a given hazard is easily constructed from numerical modeling; however, its most commonly used statistical product in hazard communication is just the set of probability contours. Consequently if a binary volcanic hazard map were to be constructed from a PHM showing the region of no likely impact versus the region of likely impact, the analyst would have to decide where to draw the line: possibly at the 5 % probability contour (a safe, conservative choice), the median contour (50 %) (a good estimate for the most likely impact–no-impact boundary), or some other choice of percentile. Often the choice of the contour level is based on agreements or official protocols made by civil protection authorities. Such potentially high-consequence decisions would benefit from additional statistical analyses which generate other tools for this task. For example, it may be useful to know the average geographical limits of a particular hazard in addition to the point-wise probability of hazard impact. For any sample data set or theoretical probability distribution, the mean and variance as well as one or more modal values are calculable in addition to the median and other percentiles. An important deficiency in the analysis of the PHM is that, previously, scientific estimates of the likely hazard boundary (a single curve on the map) have lacked the formal framework required for unique determination of statistical estimates such as the mean, variance, and higher moments of the distribution of hazard boundary locations.

For any given volcanic hazard analysis which maps probabilities of a given event, the probability values may represent a wide array of concepts in probability theory including cumulative probabilities, probability densities, conditional probabilities, and others. Here, the point-wise probability is interpreted as the relative frequency of hazard impact as predicted by repeated simulation using a deterministic physical model or ensemble of models equipped with an ensemble of inputs spanning the realistic range of underlying uncertain parameters. In the following theory and analysis, we only consider spatial probability distributions which map the probability of a given hazard impacting a given location where there is at least one point in space with a modeled probability of unity. Although this excludes the study of many possible PHMs, it covers a sufficiently wide range of models which are regular or regularized and span a realistically limited subspace of the general parameter space; that is, they depend continuously on the inputs or, in this case, maintain some level of spatial correlation across a realistically narrow, continuous range of inputs. In general, the methods detailed here are agnostic with respect to the type of uncertainty which is modeled. For any probabilistic geophysical model, there exist epistemic and aleatory uncertainties. The division between aleatory uncertainty and epistemic uncertainty corresponds to the distinction between intrinsic randomness of the system and the additional uncertainty that affects its representation, the latter originating from incomplete information about the underlying physical processes. Clearly PHMs constructed purely from aleatory uncertainties in the inputs have the most concrete meanings. However, PHMs constructed from a mix of aleatory and epistemic uncertainties can be considered as a single member of a time-dependent sequence of such assessments, each of which captures the types of uncertainty encoded in the generation of the PHM that is understood at that time. In general, as epistemic uncertainty decreases over time, the later assessments will become more accurate. As long as the uncertainties can be described as a probability space, these analyses hold up to current geophysical knowledge. Although varying the parameters in a single model gives the most concrete meaning, these methods could be applied to PHMs constructed from an ensemble of models provided that the ensemble shows regular behavior. Here, we focus on analyses of single hazards, although the boundary-finding methods could be justified to analyze PHMs constructed for multiple hazards.

Throughout this work, we realize these methods with examples using geophysical mass flow (GMF) models of inundation hazards. This is done primarily because the flow boundary is easily definable and conceptualized in these models, giving a consistent intuition to the complicated mathematical quantities discussed here. However, these concepts extend to many other types of hazard models where a particular threshold is of interest in defining the region impacted by a hazard in volcanology and other fields including concentration thresholds in volcanic clouds

The work of

The goal of the present work is to extend the probability theory of PHMs to construct statistically meaningful answers to this question. To this end, we give an explicit mathematical definition of the PHM, defining explicitly the connection between the cumulative exceedance probabilities and probability densities in a PHM and estimates of the likely hazard boundary. More specifically, we calculate spatial representations of the PHM's moments including the mean and variance of the hazard boundary location as well as its modality. We stipulate that the methods detailed here apply only to the scientific analysis component of any hazard-mapping project and must be considered as just one piece of the hazard-map-making process. This effort combines mathematical tools and concepts from probability theory, the study of dynamical systems, and differential geometry.

Solutions to flow models which involve depth-integrated partial differential equations (e.g., Saint-Venant-type shallow water equations) typically include solving for the distribution of flow thicknesses over position and parameter space:

The construction of any PHM requires the definition of a joint probability distribution of the inputs given by a probability density function (PDF)

If the range of realistic parameter values were confined to a single point

If the parameter space is measured by a uniform distribution (maximal entropy distribution), the probabilistic hazard map (PHM) is constructed by mean value integration of the indicator function through the parameter space:

In the context of a typical probabilistic hazard modeling effort, the solution space and parameter space are discrete with a uniform distribution in parameter space, and these steps are accomplished merely by cell-wise averaging of the indicator functions derived from each model run to generate a discretized 2-D hazard map

there exists a (possibly unconnected) set

We will further assume that

Formally,

Schematic representation of a hypothetical probabilistic hazard map (PHM) where contours (blue) are the percent of deterministic models inundating a given point in space. Steepest ascent integral curves on the PHM (

Along each integral curve, profiles of the PHM are denoted as

If local maxima less than unity are present, then the integral curves that ascend these maxima will not be valid CDFs (

To find the PDF d

In the notional example given above for a PHM generated by a parameter space Dirac delta, given a PHM that is identically the indicator function, the PHDM is defined only in the sense of distributions because of the apparently infinite gradient at

Using these definitions, three measures of central tendency (mode, mean, median) can be generated for each integral curve, generating a locus of points (curve) for each measure parameterized by a new parameter (

By defining a density function for the inundation edge probability distribution as in Eq. (

Furthermore, we may find the mean location of the inundation by considering the first moment of each univariate PDF

Finally, we may define a curve representing the median value of the probability distribution for inundation front location. This is simply the 50 % probability contour:

Additionally, a measure of the variance may also be constructed for each PDF

To highlight the features of the new theory, we present three examples of its application in order of increasing complexity and realism.

A simple application of this theory is to a flow which can be calculated analytically. Here, we show the application of the above theory to the problem of time-dependent solidification of a viscous gravity current. This model used by

To use this simple model to illustrate the probability distribution properties above, consider the case where all parameters are known precisely except for the viscosity freezing time coefficient

These relationships lead naturally to the estimates of the center of the distribution (Fig.

These calculations can be made analytically because of the ability to write the equation of the boundary and because that equation is invertible for the unknown parameter; that is, it can be written as

This example was greatly simplified by the assumption of radial symmetry, allowing the distribution to be cast onto one random variable, the radius of the flow boundary, giving integral curves of constant azimuthal angle which are linear in map view. However, in most realistic cases, this is a very poor assumption and the distribution will vary between integral curves.

To see the full realization of these concepts, a true two-dimensional problem is required. A simple example flow which had been well-studied is a lab-scale granular flow of sand down an inclined plane with a Mohr–Coulomb (MC) friction relation analogous to a natural-scale debris avalanche or landslide

The uncertain parameters in the problem for a given granular medium are the internal and basal friction angles of the medium,

To construct the indicator functions for this experiment, a thickness threshold of

The PHM for this problem has many important characteristics, notably that the greatest dispersion in the probability contours occurs in the runout zone, owing principally to the uncertainty in

Following from the previous application of the PHM statistics to a simple flow, here we present a more advanced application: construction and analysis of a PHM for a more complex flow over natural topography. This example consists of numerical modeling detailed in

We analyze an ensemble of TITAN2D simulations of this flow that were generated by

As in the previous example, the solution variable of interest is the maximum flow height over time; however, here the height threshold

This complex example highlights the need for consideration of additional probability measures than simply the probability contours. Where the flow is not narrowly confined or where spillover has occurred, the shape of the distribution becomes relevant and may yield significantly different results than estimating the likely flow boundary as one of the probability contours.

As shown by the above three examples, our theory can be applied to give consistent results between continuous, analytic models and discrete, numerical models of flows. The following discussion highlights several consequences of the theory as applied to analytic models in research and more complex numerical models in hazard assessments.

Although an analytic model

If the parameter space of a given inundation hazard scenario and a realistic physical model of the process are known, a simple numerical experiment can be run using classical Monte Carlo methods or other Monte Carlo variants as was done for the granular flow on an inclined plane

As illustrated in the examples, the mean boundary may differ significantly from the median boundary owing to the asymmetric statistics that are inherent to typical nonlinear flows. Furthermore, this asymmetry suggests that the modal or maximum likelihood boundary may be of interest to users of these assessments as well. For the scientific component of probabilistic hazard assessments to have maximum impact and accuracy, these analyses should incorporate the full statistics of the PHM.

With new tools to calculate the moments of any given hazard edge distribution from a PHM, hazard map makers and analysts become able to estimate the likely location of the hazard edge and the uncertainty in that estimate. As numerical modeling capabilities continue to grow, the increased ease and speed of this type of analysis will allow the scientific community to quickly construct the PHM, PHDM, the mean hazard boundary curve, standard deviation region, and higher moments of the hazard boundary distribution, giving improved estimates of the true hazard zone. Although this theory is useful for many applications where probabilities are given spatially, further work is needed to constrain which types of hazard assessments and models are most suitable for this analysis and how best to blend this scientific information within existing hazard-map-making procedures. Similarly, additional work is required to study the probabilistic inverse problem further than the simple example given and to evaluate the use of this theory in a time-dependent context. Overall, the analysis put forth here significantly enhances the study of spatial probabilistic hazards, yielding new estimates of the likely hazard boundary and the uncertainty in those estimates.

Data are available in the references, specifically in

Using the definition of the integral curves in the text as steepest ascent integral curves

There exists a (possibly unconnected) set

If

The right hand side of Eq. (

From this result, integration of Eq. () will yield

The following procedure is designed to simplify and condense the many steps involved in the above analysis. This recipe has been written as a blueprint for carrying out the analysis using a numerical model of the process in question. This procedure assumes that a numerical model of the process has already been selected.

Identify the collection of uncertain input parameters (

Generate random sample input vectors

Identify the threshold of interest for a given model observable and generate the indicator function for each model run according to the threshold. In practice, this can be realized as a sequence of grids, each of which is an element-wise indicator.
In a MATLAB environment, this could be accomplished for each grid with the following syntax in which the indicator and the variable of interest are arrays of double-precision floating-point:

The PHM

Generate the level set unit normal vector field for

Generate the gradient-ascending integral curves

Generate the valid regions of the PHM by eliminating all regions in which integral curves terminate at local maxima. In practice, this could be accomplished by generating an indicator grid in which all array elements within a certain neighborhood of these curves are set to zero and all other elements are set to

Generate the PHDM using the gradient magnitude

To compute the mean on each curve, treat the sequence

DMH and MIB conceived of the main concepts. DMH developed the mathematical results and wrote the manuscript. AB implemented and performed the numerical simulations and contributed to the mathematics. All authors interpreted and discussed the results, provided comments and feedback for the manuscript, and approved the publication.

The authors declare that they have no conflict of interest.

We acknowledge the support of NASA grant NNX12AQ10G; the JPSS PGRR program under NOAA–University of Wisconsin CIMSS Cooperative Agreement number NA15NES4320001; NSF awards 1339765, 1521855, and 1621853; and private donations to the University at Buffalo Foundation. Additionally, we thank the University at Buffalo Center for Computational Research as well as the helpful comments of two anonymous reviewers.

This research has been supported by the National Aeronautics and Space Administration (grant no. NNX12AQ10G), the University at Buffalo Foundation, the National Science Foundation (grant nos. 1339765, 1521855, and 1621853), and the National Oceanic and Atmospheric Administration (grant no. NA15NES4320001).

This paper was edited by Giovanni Macedonio and reviewed by two anonymous referees.