Wildfire Spreading: a new application of the Beta distribution

Abstract

This dissertation is in the mathematical physics area, more specifically, applications in the statistics field. The thesis, under the supervision of Dr. Gianni Pagnini, was carried out at the BCAM - Basque Centre for Applied Mathematics in Bilbao, Spain. It is the result of the continuous interaction with the team of Statistical Physics, characterised by an international, stimulating and constantly growing environment. The subject of this thesis is PROPAGATOR: a stochastic cellular automaton model for forest fire spread simulation, conceived as a rapid method for fire risk assessment. The reason behind the popularity of cellular automata can be traced to their simplicity, and to the enormous potential they hold in modeling complex systems, in spite of their simplicity. Cellular automata can be viewed as a simple model of a spatially extended decentralized system made up of a number of individual components: cells. The communication between constituent cells is limited to local interaction. PROPAGATOR is a cellular automata model which simulates wildfire spread through empirical laws that guarantee probabilistic outputs. This algorithm, whose first version was released in 2009, is currently in use, along with other software, although it is constantly being updated. In fact, the first version was requested by the Italian Civil Protection, but later it became part of the ANYWHERE project. This project, active from June 2016 to December 2019, was funded under the EU’s research and innovation funding program Horizon 2020 (H2020), which aimed to improve emergency management and response to high-impact weather and climate events such as floods, landslides, swells, snowfalls, forest fires, heat waves and droughts. As part of the ANYWHERE project, Propagator was rewritten in Python. The version we worked with is the 2020 version, but an updated 2022 version is already available. The main aim of this work was to understand the distribution of the wildfire propagation. As can be seen from Propagator input parameters, the propagation depends on different factors: ignition point, wind speed and direction, as well as fuel moisture content and firebreaks-fire fighting strategies. Wind is recognized to be by far the most important factor in the entire problem of forest fire propagation. In this paper, we analyzed four different situations varying initial conditions, in particular we changed wind speed: 0 km/h, 10 km/h, 20 km/h, 30 km/h. However, the phenomenon of fire spotting and firebreaks-fire fighting strategies were not taken into consideration. By modifying the code, it was possible to obtain the output required to achieve the desired result. The conclusion we came to is that the distribution of a wildfire spreading is described by the beta distribution. This allows us, for the first time, to attribute a new application of the beta function: describing the propagation of a process studied using a cellular automaton algorithm.

The thesis is organised as follows:

In the first chapter, there is an introduction to special functions. In particular, their role in applied mathematics is analyzed, followed by a discussion of the two most commonly used special functions: the Gamma function and the Beta function.

• In the second chapter, the PROPAGATOR model was introduced following the article "PROPAGATOR: An Operational Cellular-Automata Based Wildfire Simulator" by A. Trucchia.

• The third chapter contains the analysis carried out on the output data. A discussion of the obtained results and suitable observations can be found in the conclusions.

• There are three appendixes containing:

– Appendix A: the lines of code we wrote to carry out the analysis.

– Appendix B: explanation of the software, apps and routines used, with particular reference to the Hypathia server.

– Appendix C: discussion on stochastic processes carried out as an approach and preparation for the subsequent work with Propagator.