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An Approximated Model of Wildfire Propagation on Slope

Hengameh R. Dehkordi

TL;DR

The paper addresses accurate wildfire propagation on inclined terrain by integrating the classical Rothermel model with Huygens' principle within a Finsler geometric framework, enabling slope aware propagation analysis. It develops a Finsler metric $F$ and a distance function $\rho$ so that fire fronts are given by $\rho^{-1}(t)$ and fire rays are geodesics orthogonal to fronts, satisfying Huygens' principle. Four propagation models are defined for homogeneous and nonhomogeneous vegetation distributions, with and without wind, yielding explicit expressions for fire rays and fronts such as $F(p,v)=\frac{||v||}{R_0(p)(1+\phi_s(p)\cos\theta)}$ in no wind cases and wind-adjusted forms involving $a,b,c$ and $\hat\theta$; these models are validated through flowcharts and examples. MATLAB simulations illustrate practical applicability, and the framework supports improved firefront/path predictions that can inform firebreak planning and wildfire management within simulators like FARSITE.

Abstract

The increasing frequency and intensity of wildfires underscore the need for accurate predictive models to enhance wildfire management. Traditional models, such as Rothermel and FARSITE, provide foundational insights but often oversimplify the complex dynamics of wildfire spread. Advanced methods, employing sophisticated mathematical techniques, offer more precise modeling by accounting for real-world complexities and dynamic environmental factors. This paper focuses on wildfire propagation over inclined terrains and combines the Rothermel model, Huygens' principle, and advanced mathematical techniques to provide a more precise model of propagation. Environmental parameters and vegetation factors are directly incorporated into formulas and equations to improve the reliability and effectiveness of wildfire management strategies. The practical application of these results is demonstrated through MATLAB simulations, specifically examining wildfire spread under wind conditions that do not impede upwind fire advancement. The findings of this work contribute to both wildfire research and the development of more effective management strategies.

An Approximated Model of Wildfire Propagation on Slope

TL;DR

The paper addresses accurate wildfire propagation on inclined terrain by integrating the classical Rothermel model with Huygens' principle within a Finsler geometric framework, enabling slope aware propagation analysis. It develops a Finsler metric and a distance function so that fire fronts are given by and fire rays are geodesics orthogonal to fronts, satisfying Huygens' principle. Four propagation models are defined for homogeneous and nonhomogeneous vegetation distributions, with and without wind, yielding explicit expressions for fire rays and fronts such as in no wind cases and wind-adjusted forms involving and ; these models are validated through flowcharts and examples. MATLAB simulations illustrate practical applicability, and the framework supports improved firefront/path predictions that can inform firebreak planning and wildfire management within simulators like FARSITE.

Abstract

The increasing frequency and intensity of wildfires underscore the need for accurate predictive models to enhance wildfire management. Traditional models, such as Rothermel and FARSITE, provide foundational insights but often oversimplify the complex dynamics of wildfire spread. Advanced methods, employing sophisticated mathematical techniques, offer more precise modeling by accounting for real-world complexities and dynamic environmental factors. This paper focuses on wildfire propagation over inclined terrains and combines the Rothermel model, Huygens' principle, and advanced mathematical techniques to provide a more precise model of propagation. Environmental parameters and vegetation factors are directly incorporated into formulas and equations to improve the reliability and effectiveness of wildfire management strategies. The practical application of these results is demonstrated through MATLAB simulations, specifically examining wildfire spread under wind conditions that do not impede upwind fire advancement. The findings of this work contribute to both wildfire research and the development of more effective management strategies.

Paper Structure

This paper contains 11 sections, 2 theorems, 25 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Propagation on an Inclined Plane
  4. Homogeneous Distribution of Vegetation Characteristics
  5. Nonhomogeneous distribution of vegetation characteristics
  6. Flowcharts
  7. Flowchart for finding fire fronts for homogeneous vegetation characteristics
  8. Flowchart for Determining Propagation in Nonhomogeneous Vegetation Characteristics
  9. Examples
  10. Conclusion and Final Remarks
  11. Appendix: The formulas to calculate the frame

Key Result

Theorem 2.1

dehkordi2019huygens Assume $(M, F)$ is a Finsler space across which a wildfire spreads and $A$ is a compact subset of $M$. We define $\rho: M \to \mathbb{R}$ by $\rho(p) = d_F(A, p)$ and assume $\rho^{-1}([s, r]) = U$ and there is no cut locus in $\rho^{-1}(s, r)$. If $\rho^{-1}(s)$ is the fire fron

Figures (2)

  • Figure 1: Modeling the propagation with smooth distribution of vegetation and uniform environmental conditions.
  • Figure 2: Comparing the methods of modeling using Huygens' principle and rays with $\Delta t=2$ hours.

Theorems & Definitions (6)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 3.1
  • Remark 3.2
  • Example 4.1
  • Example 4.2