# Bayesian Inference in Snow Avalanche Simulation with r.avaflow

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*in.ge.na.*, 39100 Bozen, Italy

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## Abstract

**:**

## 1. Introduction

## 2. Simulation and Postprocessing

## 3. Avalanche Data

## 4. Back Calculation

#### 4.1. Mathematical Framework

- The prior probability density ${\pi}_{\mathrm{prior}}\left(\theta \right)$ encoding the prior knowledge about the model parameters;
- The likelihood function $\pi \left({y}_{\mathrm{obs}}\right|\theta )$ expressing the probability of the observed data when the parameter has a given value $\theta $.

#### 4.2. Application—Kerngraben Avalanche

## 5. Forward Calculation and Prediction

#### 5.1. Mathematical Framework

#### 5.2. Forward Calculation—Application to the Kerngraben Avalanche

#### 5.3. Prediction—Application to the Wolfsgruben Avalanche

## 6. Conditional Runout Probabilities

#### 6.1. Mathematical Framework

#### 6.2. Application to the Kerngraben and Wolfsgruben Avalanche

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Trace plots and histograms showing parameter values for the 2000 iterations and total counts in the marginal posterior distribution ${\theta}_{\mathrm{post}}$ of the Markov chain. The trace plot indicates a good mixing for both parameters, which is reflected by the acceptance rate of 0.48 (967 elements of 2000). The candidates for ${\delta}_{0}$ mainly concentrate around the mean value ${\overline{\delta}}_{0}={11.3}^{\circ}$, while observed $\xi $ values span the entire parameter space around $\overline{\xi}=1714\phantom{\rule{3.33333pt}{0ex}}\frac{\mathrm{m}}{{\mathrm{s}}^{2}}$.

**Figure 2.**Scatterplots and histograms (counts are re-scaled to the maximum count for a better visibility) for the Monte Carlo sample of the posterior distribution ${\theta}_{\mathrm{post}}^{\mathrm{pred}}$ with $N=500$ elements. Forward calculations for the Kerngraben and predictive simulations for the Wolfsgruben avalanche are performed with these Monte Carlo samples.

**Figure 3.**Runout statistics of the forward and predictive simulations with the Monte Carlo sample (${\theta}_{\mathrm{post}}^{\mathrm{pred}}$, $N=500$) for the Kerngraben (red) and Wolfsgruben (blue) avalanche with the observed runouts (dashed black line). Solid lines (red, blue) depict the quantiles of the resulting runout distribution and transparent bars the corresponding histograms. The observed runout for the Kerngraben avalanche is $r=$1741 m, with the 95–5% quantiles ranging from 1701 to 1831 m and $r=$ 2103 m for the Wolfsgruben avalanche with a range of 1985m to 2233m for the 95–5% quantiles.

**Figure 4.**Visualization of the conditional probability $P({p}_{\mathrm{peak}}\left(\mathfrak{x}\right)>{p}_{\mathrm{lim}}|{y}_{\mathrm{obs}})$ for the forward calculations (left, Kerngraben avalanche) and predictive simulations (right, Wolfsgruben avalanche) with the Monte Carlo sample ${\theta}_{\mathrm{post}}^{\mathrm{pred}}$. The observed affected area ${A}_{\mathrm{obs}}$ is shown as reference for the true positive ($tp$) and false positive $fp$ values in the runout area (black shading, solid line). The color map indicates the probability $P({p}_{\mathrm{peak}}\left(\mathfrak{x}\right)>{p}_{\mathrm{lim}}|{y}_{\mathrm{obs}})$, that a respective area of the simulation raster is affected by an avalanche, given the considered data. The $P=95\%$ and $P=5\%$ probability isolines are highlighted (dotted and dashed-dotted lines), allowing to identify areas that exceed the threshold peak pressure ${p}_{\mathrm{peak}}\left(\mathfrak{x}\right)>{p}_{\mathrm{lim}}$ in most simulation runs (dark red) or outliers that appear with probabilities $P<5\%$ (light blue).

**Table 1.**Avalanche observation data ${y}_{\mathrm{obs}}$ (runout r, observed affected area ${A}_{\mathrm{obs}}$, maximum velocity ${u}_{\mathrm{max}}$) and median (50% quantiles) values of the corresponding simulation results y (${r}_{50},t{p}_{50},f{p}_{50},{u}_{\mathrm{max},50}$). True positive $tp$ and false positive $fp$ values for each simulation are noted in % with respect to the observed affected area ${A}_{\mathrm{obs}}$.

r [m] | ${\mathit{r}}_{50}$ [m] | ${\mathit{A}}_{\mathbf{obs}}$ [ ${\mathbf{m}}^{2}$] | ${\mathit{tp}}_{50}$ [%] | ${\mathit{fp}}_{50}$ [%] | ${\mathit{u}}_{\mathbf{max}}$ [m/s] | ${\mathit{u}}_{\mathbf{max},50}$ [m/s] | |
---|---|---|---|---|---|---|---|

Kerngraben avalanche | 1741 | 1751 | 241,272 | 0.99 | 0.53 | 55 | 25 |

Wolfsgruben avalanche | 2103 | 2071 | 550,992 | 0.97 | 0.32 | 58 | 35 |

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**MDPI and ACS Style**

Fischer, J.-T.; Kofler, A.; Huber, A.; Fellin, W.; Mergili, M.; Oberguggenberger, M.
Bayesian Inference in Snow Avalanche Simulation with r.avaflow. *Geosciences* **2020**, *10*, 191.
https://doi.org/10.3390/geosciences10050191

**AMA Style**

Fischer J-T, Kofler A, Huber A, Fellin W, Mergili M, Oberguggenberger M.
Bayesian Inference in Snow Avalanche Simulation with r.avaflow. *Geosciences*. 2020; 10(5):191.
https://doi.org/10.3390/geosciences10050191

**Chicago/Turabian Style**

Fischer, Jan-Thomas, Andreas Kofler, Andreas Huber, Wolfgang Fellin, Martin Mergili, and Michael Oberguggenberger.
2020. "Bayesian Inference in Snow Avalanche Simulation with r.avaflow" *Geosciences* 10, no. 5: 191.
https://doi.org/10.3390/geosciences10050191