## Beta Distribution in Wildfire Spreading

##### Abstract

This paper is the result of research work carried out in collaboration with the Statistical Physics team of the Basque Center for Applied Mathematics in Bilbao, supervised by Dr. Gianni Pagnini.
Main subject is PROPAGATOR, an algorithm for fires simulation based on a cellular automata model (CA).
The choice of such a modelling approach is not random, the cellular automata, in fact, thanks to their modular nature, are able to simplify the physical processes that influence the propagation of fires, while retaining the ability to achieve whatever you want level of complexity and accuracy. They are also one of the most widely known examples of stochastic lattice models.
The PROPAGATOR model, in fact, is based on raster implementation, which discretizes the space in a grid composed of rectangular cells of arbitrary length, and the propagation is modeled as a contamination process between adjacent cells of the considered domain.
The aim is to identify how the burned area is distributed in a limited observation interval.
For this purpose, a simplified case of fire propagation is considered. In fact, fuels and possible intervention of fire-fighting helicopters, which can affect fire size, intensity and duration, are not being studied.
In addition, the fire-spotting phenomenon is excluded. It consists in the propagation of fires outside the perimeter of the main fire caused by burning particles that, raised in the air by convective currents and driven by the wind, generate secondary fires with distances of the order of tens meters.
The focus is therefore on the following parameters: observation interval; propagation perimeter; vegetation type; wind intensity and direction; inclination of the territory.
In this thesis, in particular, we report the results obtained by studying the phenomenon of propagation as the slope of the territory changes and fixing the remaining parameters. To obtain these results, a modification to the PROPAGATOR algorithm was made, in that the latter is programmed to return in output, for each instant of time, the arithmetic mean over the number of realizations of the burned area values, whereas for the analysis these values were needed for each realization, since they were interested, not only in the mean, but also in the variance, skewness, kurtosis and more generally in their distribution. This accuracy has cost in terms of computational effort. For a large number of realizations, in fact, it was necessary to use the Hypatia server.
The work is organized as follow:
* In the first chapter, after a first introduction to the special functions and their history, with references to those most known and used, are defined, starting from the Beta function and the Gamma function, the Beta Distribution and the General Logistics Distribution.
* The second chapter introduces the PROPAGATOR model, the story of the algorithm development, and then arrive at the role of the territory inclination in the fires propagation.
* The third chapter describes in detail the data analysis carried out and gives the graphic results for each case studied: slope 10º, slope 15º, slope 20º, slope 30º, slope 40º e slope 50º.
* In Appendix A some preliminary work on the simulation of known stochastic processes, to develop a kind of critical sense for the results, in order to get prepared for the use of PROPAGATOR.
* Appendix B shows the codes developed for the analysis.
* Appendix C lists the software, apps and routines used.
Finally, we inform that the PROPAGATOR algorithm is currently in use by the Department of National Civil Protection and that the version used in the following is of 2020, even if an update to 2022 is already available.