Statistical Characterization of Wildfire Dynamics: Studying the Relation Between Burned Area and Head of the Fire

We show that the probability density function (PDF) of an area enclosed by a random perimeter is driven by the PDF of the integration bounds and the mean value of the perimeter function. With reference to wildfires, if the integration interval is aligned with the main direction of propagation, the PDF of the burned area is driven by the PDF of the position of the head of the fire times the mean value of the fire front. We show that if the random fire front perimeter is modelled as an ellipse-like curve with stochastic noise, the relation between the probability distribution of the burned area and the one of the head of the fire is linear, with the constant mean value of the perimeter as a factor. Different random noise models have been developed and implemented for this purpose. Different behaviours of the stochastic position of the head of the fire have been studied, including Viegas’ rate of spread model. We show that for the realistic propagation model given by the operational wildfire cellular automata simulator Propagator [46], the PDF of the burned area is still driven by the position of the head of the fire and the mean value of the fire front perimeter, and the shape of the perimeter is not relevant. This has been shown for 5 case studies, ranging from easy ones to realistic complicated cases

3/2 Firefighters are not enough

The firefighter problem is a monotone dynamic process in graphs that can be viewed as modeling the use of a limited supply of vaccinations to stop the spread of an epidemic. In more detail, a fire spreads through a graph, from burning vertices to their unprotected neighbors. In every round, a small amount of unburnt vertices can be protected by firefighters. How many firefighters per turn, on average, are needed to stop the fire from advancing? We prove tight lower and upper bounds on the amount of firefighters needed to control a fire in the Cartesian planar grid and in the strong planar grid, resolving two conjectures of Ng and Raff.Comment: 8 page

On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape

We study the lengths of curves passing through a fixed number of points on the boundary of a convex shape in the plane. We show that for any convex shape $K$, there exist four points on the boundary of $K$ such that the length of any curve passing through these points is at least half of the perimeter of $K$. It is also shown that the same statement does not remain valid with the additional constraint that the points are extreme points of $K$. Moreover, the factor $\frac12$ cannot be achieved with any fixed number of extreme points. We conclude the paper with few other inequalities related to the perimeter of a convex shape.Comment: 7 pages, 8 figure