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# [SVC30-04] A probabilistic hazard mapping method combining numerical simulations of pyroclastic density currents with a frequency function of eruption magnitudes

Keywords:Pyroclastic density current (PDC), Run-out area, Probabilistic hazard map, Numerical simulation, Eruption magnitude, Frequency function

*M*(=log

_{10}(erupted mass [kg])-7; Hayakawa, 1993, Kazan; Pyle, 1995, Geophys. Res. Lett.) varies greatly from ~0 to ~8 depending on the events and significantly changes the run-out area of PDCs, the treatment of

*M*differs among hazard map authors. To eliminate arbitrariness in the treatment of

*M*, we propose a probabilistic hazard mapping method that combines a frequency function of

*M*(

*N*(

*M*)) with the results of numerical PDC simulations.

Our probabilistic hazard mapping method consists of the following four procedures.

[Procedure 1] The lower and upper limits of

*M*are determined; then, in this range, the values of

*M*are sampled as

*M*at intervals d

_{i}*M*

_{ i}

_{±1/2}=|

*M*-

_{i}*M*

_{i}

_{±1}|.

[Procedure 2] For each of the sampled values

*M*, a numerical simulation of PDCs is conducted to predict the run-out area of PDCs; the scores of 1 and 0 are given to each grid of simulation area in the run-out area and the other grids, respectively.

_{i}[Procedure 3] The scores for each

*M*are weighted by multiplying the probability distribution of

_{i}*M*:

*N*/Σ

_{i}*(*

_{i}*N*) (or the frequency distribution

_{i}*N*), where

_{i}*N*=0.25{

_{i}*N*(

*M*)+

_{i}*N*(

*M*-0.5d

_{i}*M*

_{i-}_{1/2})}d

*M*

_{i}_{-1/2}+0.25{

*N*(

*M*)+

_{i}*N*(

*M*+0.5d

_{i}*M*

_{i}_{+1/2})}d

*M*

_{i}_{+1/2}.

[Procedure 4] The weighted scores are summed up for all of

*M*and its result is plotted on the map.

_{i}These procedures produce a probabilistic hazard map that combines the run-out area of numerical PDCs and the frequency function

*N*(

*M*) for the set range of

*M*. Compared with existing methods (e.g., the Monte Carlo-type sampling-based approaches), our method has the advantage that we can restart from [Procedure 3] when performing hazard mapping with a different

*N*(

*M*). This means that we can save a lot of time because we do not need to redo the numerical simulations (i.e., [Procedure 2]) every time we perform hazard mapping with different

*N*(

*M*).

As an example, we made a probabilistic hazard map of the run-out area of PDCs generated from the Showa Crater of Sakurajima volcano. The lower and upper limits of

*M*were set to 1 and 3.5, respectively, and the values of

*M*were sampled at equal intervals of 0.5. In the numerical simulations, a dense granular flow model was used to describe the dynamics of the dense basal region of PDCs (Shimizu, 2021, Volcanological Society of Japan Fall Meeting), and a high-resolution digital elevation model provided by NHK was used for generating the surface mesh. The frequency function

*N*(

*M*) is assumed to be two distributions: one is a uniform distribution (

*N*=const), and the other is an empirical power-law distribution (

*N*∝-0.75

*M*; Nakada, 2015, Kazan), which is valid for each volcano on a geological time scale. Compared with the case of

*N*=const, the hazard map with

*N*∝-0.75

*M*strongly reflects the general tendency that small-scale PDCs with narrow run-out areas occur at high frequency and large-scale PDCs with wide run-out areas occur at low frequency. In the future, we will sophisticate our method by introducing a method to set the lower and upper limits of

*M*and by evaluating also the frequency functions of eruption conditions other than

*M*(e.g., location of the volcanic vent).