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A review of wildland fire spread modelling, 1990-present 3: Mathematical analogues and simulation models
A. L. Sullivan
TL;DR
Wildland fire modelling needs landscape-scale predictions, but existing spread models were often one-dimensional and recent reviews had not comprehensively assessed the field. This paper reviews simulation and mathematical-analogue models, emphasizing how spread models are propagated across GIS landscapes and how analogue approaches represent fire spread. It finds that simulation quality depends on the underlying spread model and input data, while different representations trade perimeter realism against heterogeneous-fuel handling.
Problem
The paper addresses the need to understand models that simulate wildland fire spread across landscapes beyond predominantly one-dimensional spread predictions and superficial prior reviews.
Method
The paper critically reviews simulation and mathematical-analogue models, including GIS-based landscape propagation, raster and vector representations, and mathematical concepts applied by analogy to fire spread.
Results
Vector simulations produce a more realistic fire perimeter, particularly for off-axis spread, while raster simulations are more proficient with heterogeneous fuels.
Takeaways & Limitations
Simulation quality depends on the veracity, verification, and validation of the underlying spread model and on the quality of required geographic and environmental input data.
Takeaways & Limitations
Coupled fire-atmosphere propagation using Rothermel’s model away from prevailing wind can produce odd perimeter deformations in nonuniform terrain or fuels.
Abstract
from arXiv · showhide
In recent years, advances in computational power and spatial data analysis (GIS, remote sensing, etc) have led to an increase in attempts to model the spread and behvaiour of wildland fires across the landscape. This series of review papers endeavours to critically and comprehensively review all types of surface fire spread models developed since 1990. This paper reviews models of a simulation or mathematical analogue nature. Most simulation models are implementations of existing empirical or quasi-empirical models and their primary function is to convert these generally one dimensional models to two dimensions and then propagate a fire perimeter across a modelled landscape. Mathematical analogue models are those that are based on some mathematical conceit (rather than a physical representation of fire spread) that coincidentally simulates the spread of fire. Other papers in the series review models of an physical or quasi-physical nature and empirical or quasi-empirical nature. Many models are extensions or refinements of models developed before 1990. Where this is the case, these models are also discussed but much less comprehensively.
1 Introduction
The paper reviews fire-spread modelling in the context of improved computing and spatial-data capabilities, distinguishing physically grounded, empirical, simulation, and mathematical-analogue approaches. This final review focuses on landscape simulation and mathematical analogues, while providing less comprehensive context for pre-1990 work.
- 1 Introduction: Fire-behaviour prediction developed from one-dimensional forward-spread estimates toward landscape-scale fire-spread simulation as technology advanced.Early models were easy to plot and extrapolate, whereas later modelling benefited from GIS, remote sensing, and increased computing power.
- 1 Introduction: The review series classifies models as physical and quasi-physical, empirical and quasi-empirical, or simulation and mathematical analogue models.The categories distinguish whether models represent fire physics and chemistry, use statistical descriptions, or simulate spread through analogous mathematical concepts.
- 1 Introduction: The review is limited primarily to work published since 1990, with earlier studies included less comprehensively for context.The cutoff reflects rapid development in spatial data analysis, including GIS and remote sensing, after 1990.
- 1 Introduction: This paper examines landscape fire simulations and mathematical analogues, separating GIS-based perimeter propagation from mathematical concepts not derived from fire-behaviour understanding.Simulation models generally implement an existing spread model across a landscape, whereas mathematical analogues exploit similarities between fire spread and other mathematical behaviours.
- 1 Introduction: The review is organized by modelling approach rather than by author or organization because many authors adopted similar approaches.This structure accommodates the broad range of contributors and overlapping methods discussed in the paper.
2 Fire Spread Simulations
Fire-spread simulations convert one-dimensional spread models into two-dimensional landscape propagation using raster or vector representations and perimeter-expansion algorithms. Huygens-based methods are widely used, while raster and vector approaches trade perimeter realism against heterogeneous-fuel handling and data demands.
- 2 Fire Spread Simulations: Most fire-spread simulations convert one-dimensional forward-spread models into two-dimensional perimeter propagation across landscapes.The fire is represented either as contiguous raster cells or as a closed vector curve of linked perimeter points.
- 2 Fire Spread Simulations: Huygens’ wavelet principle is the most widely used propagation method, with each perimeter point spawning a new fire according to local spread conditions.The spawned wavelets commonly use ellipse templates, although alternative shapes have also been proposed.
- 2.1 Huygens wavelet principle: In homogeneous fuels and weather, an elliptical Huygens method suitably models fire spread with only small fire-shape distortion.The approach derives flank and rear spread from ellipse geometry, allowing a single forward rate of spread to drive the full perimeter.
- 2.2 Raster-based simulation: Raster simulations are computationally less intensive and better suited to heterogeneous fuels and weather, but higher spatial resolution increases storage and access requirements.Each raster cell stores fuel information and a state such as unburnt, burning, or burnt.
- 2.3 Other propagation methods: A coupled fire-atmosphere model predicts wind direction around the perimeter to drive spread, but use of Rothermel’s model away from prevailing wind produces questionable deformations in nonuniform terrain or fuels.The model’s wind-field interaction can generate complex horizontal and vertical vortices, while coupling weakens above 5 m s−1 at 15 m above ground.
3 Mathematical Analogues
Mathematical analogues simulate wildland fire spread through abstract computational rules, while simulation models commonly adapt existing fire-spread models to landscapes. Cellular automata, percolation, fractal, diffusion-limited aggregation, and related approaches provide varied representations but often simplify or incompletely reproduce observed fire behaviour.
- Scope and model types: Mathematical analogue models use functions such as cellular automata and self-organised criticality that resemble fire spread without being physically based.Simulation models generally implement pre-existing spread models across landscapes, whereas mathematical analogues use abstract mathematical conceits.
- Cellular automata: Cellular automata discretise space and time into cells with finite states, using neighbourhood rules to represent ignition and spread.Wildland-fire CA range from isotropic neighbour heating to models incorporating wind, slope, fuel, moisture, fuzzy logic, or convection.
- Cellular automata: Albinet et al.’s isotropic CA required a critical density of burnable cells for successful spread, with the threshold decreasing as more neighbours contributed heat.The resulting fire front was fractal, with dimension approximately 1.8; later extensions introduced anisotropic wind and slope effects.
- Self-organised criticality: The original forest-fire CA was later found not to exhibit true critical behaviour because it was not scale invariant, requiring scaling corrections on larger lattices.This challenges its use as a faithful representation of self-organised criticality rather than merely a useful analogy.
- Cellular automata: Fire-shape reproduction improved when CA incorporated convection: Sullivan and Knight generated grassland-like parabolic headfire shapes, although rate of spread was not investigated.Duarte’s moisture-driven CA instead placed windless and windy variants in undirected and directed percolation universality classes, respectively, while noting that parabolic headfire shape remained unexplained.
- Percolation and fractals: Percolation models captured qualitative threshold behaviour but failed quantitatively when radiant heating, convective plume narrowing, or beyond-neighbour interactions mattered.Laboratory and field studies found simple nearest-neighbour or directed-percolation rules inadequate under relevant conditions, despite observed fractal burned areas with D_f ≃1.9.
4 Discussion
The review contrasts simulation methods that extend one-dimensional fire-spread models across landscapes with mathematical analogues such as cellular automata. It emphasizes that model validity, input-data quality, computational feasibility, and prediction purpose jointly constrain useful fire-spread simulation.
- Simulation models: Simulation techniques primarily convert one-dimensional forward-spread models into two-dimensional landscape simulations, most commonly using Huygens’ wavelet principle.The spawned wavelet is usually an ellipse, although alternative template shapes are also applicable.
- Simulation models: Vector simulations produce more realistic fire perimeters, especially for off-axis spread, whereas raster simulations handle heterogeneous fuels more effectively.Historically, raster resolution and storage costs influenced the balance between approaches, while GIS platforms increasingly drive the choice.
- Simulation models: Alternative propagation methods can be highly model-specific, replacing generalized template ellipses with variable spread directions around the perimeter.The Clark et al. approach couples a three-dimensional atmospheric model with a one-dimensional fire-spread model and requires subgrid perimeter propagation.
- Mathematical analogue models: Mathematical analogue models, especially cellular automata and percolation approaches, provide spatial propagation frameworks used to investigate critical behaviour and self-organisation.Reaction-diffusion models offer a more physical basis but generally treat fire as spatially continuous and therefore omit discontinuities such as spotting unless specialized components are added.
- Mathematical analogue models: Mathematical analogues can be more computationally feasible than physical or quasi-physical models and can represent fuel heterogeneity, but realistic fire simulation requires both local and larger-scale propagation rules.These rules must account for non-local processes such as convection, spotting, and radiative heating.
- Practical constraints: Prediction precision should be matched to the end purpose because increasing model precision also increases the required geographical, topographical, meteorological, and fuel-data precision.For many operational purposes, understanding input and prediction error ranges may be more effective than pursuing highly precise forecasts, whose computational and data costs can eventually exceed cost-effective suppression costs.