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Probability Density Function of a Random Area and Its Application to Wildfires †

Alvaro Crespo-Santiago
Andrea Trucchia
Paolo Fiorucci
3 and
Gianni Pagnini
BCAM—Basque Center for Applied Mathematics, 48009 Bilbao, Spain
Department of Physics, Stockholm University, AlbaNova University Center, SE-106 91 Stockholm, Sweden
CIMA Research Foundation, 17100 Savona, Italy
Ikerbasque, Basque Foundation for Science, 48011 Bilbao, Spain
Author to whom correspondence should be addressed.
Presented at the Third International Conference on Fire Behavior and Risk, Sardinia, Italy, 3–6 May 2022.
Environ. Sci. Proc. 2022, 17(1), 75;
Published: 15 August 2022
(This article belongs to the Proceedings of The Third International Conference on Fire Behavior and Risk)


We show that the probability density function (PDF) of a burned area enclosed by a random fire perimeter is driven by the PDF of the bounding-box sides. In particular, the random value of the area emerges to be proportional to the random position of the bounding-box sides times an averaged coefficient dependent on the geometry of the burned area. Therefore, the two PDFs are functionally equal. This means that the PDF of the burned area is driven and functionally equal to the PDF of the position of the head of the fire. The displacement of the head of the fire is given by the rate of spread (ROS); thus, the PDF of the burned area is driven and equal to the PDF of the ROS. This result holds in general whenever a fire exhibits an advancement in a main direction. The main theoretical result has been tested by different families of stochastic processes and also by using the operational fire simulator PROPAGATOR, which is based on a cellular automata approach. By using PROPAGATOR, the criteria for the validity of the derived result in realistic cases has been established by analyzing different configurations of orography and wind. This study can be understood as a start for the development of a theory of stochastic dynamics of wildfire propagation with the aim, for example, to provide physically grounded initial perturbations of wildfire perimeters for ensemble forecasting.

Author Contributions

All authors contributed equally. All authors have read and agreed to the published version of the manuscript.


This research is supported by the Basque Government through the BERC 2018–2021 program; by the Spanish Ministry of Economy and Competitiveness MINECO through the BCAM Severo Ochoa excellence accreditation SEV-2017-0718 and also through the project PID2019-107685RB-I00, and by the European Regional Development Fund (ERDF) and the Department of Education of the regional government, the Junta of Castilla y Léon, (Grant contract SA089P20).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.
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MDPI and ACS Style

Crespo-Santiago, A.; Trucchia, A.; Fiorucci, P.; Pagnini, G. Probability Density Function of a Random Area and Its Application to Wildfires. Environ. Sci. Proc. 2022, 17, 75.

AMA Style

Crespo-Santiago A, Trucchia A, Fiorucci P, Pagnini G. Probability Density Function of a Random Area and Its Application to Wildfires. Environmental Sciences Proceedings. 2022; 17(1):75.

Chicago/Turabian Style

Crespo-Santiago, Alvaro, Andrea Trucchia, Paolo Fiorucci, and Gianni Pagnini. 2022. "Probability Density Function of a Random Area and Its Application to Wildfires" Environmental Sciences Proceedings 17, no. 1: 75.

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