Multifractal point processes and the spatial distribution of wildfires in French Mediterranean regions

TL;DR

This paper introduces a multifractal Cox process model for spatial point patterns, applies it to wildfire data in French Mediterranean regions, and demonstrates the multifractal nature of wildfire risk distribution.

Contribution

It develops a maximum likelihood estimation method for multifractal spatial point processes and applies it to real wildfire data to reveal their multifractal properties.

Findings

Wildfire spatial distribution exhibits multifractal scaling behavior.

The multifractal spectrum of wildfire risk is consistent over time and regions.

The proposed model accurately captures the scaling laws of wildfire occurrence.

Abstract

We introduce a simple and wide class of multifractal spatial point patterns as Cox processes which intensity is multifractal, i.e., the class of Poisson processes with a stochastic intensity corresponding to a random multifractal measure. We then propose a maximum likelihood approach by means of a standard Expectation-Maximization procedure in order to estimate the distribution of these intensities at all scales. This provides, as validated on various numerical examples, a simple framework to estimate the scaling laws and therefore the multifractal properties for this class of spatial point processes. The wildfire distribution gathered in the Prom\'eth\'ee French Mediterranean wildfire database is investigated within this approach that notably allows us to compute the statistical moments associated with the spatial distribution of annual likelihood of fire event occurence. We show that for each order $q$, these moments display a well defined scaling behavior with a non-linear spectrum of scaling exponents $\zeta_q$. From our study, it thus appears that the spatial distribution of the widlfire ignition annual risk can be described by a non-trivial, multifractal singularity spectrum and that this risk cannot be reduced to providing a number of events per $km^2$. Our analysis is confirmed by a direct spatial correlation estimation of the intensity logarithms whose the peculiar slowly decreasing shape corresponds to the hallmark of multifractal cascades. The multifractal features appear to be constant over time and similar over the three regions that are studied.