# HexFire: A Flexible and Accessible Wildfire Simulator

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model Overview

_{contagion}= P

_{proximity}× P

_{flammability},

_{proximity}and P

_{flammability}are hexagon-specific values derived in part from HexFire model state (proximity) and input maps (flammability). For every unburned cell that has at least one neighbor presently on fire, we compute the value of P

_{contagion}, draw a uniformly distributed random number R between 0 and 1, and initiate a new fire if P

_{contagion}> R. To accomplish this, we define

_{proximity}= 1 − (1 − N / 6)

^{α},

_{flammability}= 1 − (1 − F)

^{β},

_{proximity}and P

_{flammability}will generate families of nontrivial probability curves (Figure 1).

_{contagion}is the product of P

_{proximity}and P

_{flammability}, HexFire’s local fire spread dynamics can vary widely (Figure 2). Through judicious choice of the proximity and flammability exponents, users can make contagious fire spread more or less sensitive to a cell’s relative flammability, or its number of burning neighbors. Alternatively, users may simply replace our functions governing contagious fire spread with their own (see Supplemental Materials).

**ε**.

**ε**is a parameter we refer to as the Ember Creation Rate, and F (discussed above) represents a cell’s relative flammability. The Ember Creation Rate sets the maximum number of embers that can be created per iterate, for each burning cell. The number of embers actually created in a cell (within a single iterate) will range between zero and this maximum value. E is always rounded to an integer. Each ember will then move a distance D (in hexagons) defined by

**δ**,

**δ**, our Ember Max Distance parameter, sets the maximum distance in hexagons that embers are allowed to travel before settling into a distant cell. The distance that any given ember actually travels will range between zero and this maximum value. D is always rounded to an integer. As embers move, they take successive steps along the wind gradient, and then in a random direction, until they reach their assigned movement distance. The lengths of these gradient-following and random steps are controlled by the Ember Step Length—Wind and Ember Step Length—Random parameters. The addition of random steps keeps the ember’s movements from becoming overly deterministic. When embers land, they initiate new fires at a rate specified by P

_{flammability}, as defined above. The mechanisms that HexFire uses to simulate fire spotting may be easily modified (see Supplemental Materials).

#### 2.2. Example 1—Model Parameters

#### 2.3. Example 2—Fire Suppression

#### 2.4. Example 3—Coupled Models

## 3. Results

#### 3.1. Example 1—Model Parameters

#### 3.2. Example 2—Fire Suppression

#### 3.3. Example 3—Coupled Models

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Morandini, F.; Silvani, X. Experimental investigation of the physical mechanisms governing the spread of wildfires. Int. J. Wildland Fire
**2010**, 19, 570–582. [Google Scholar] [CrossRef] [Green Version] - Finney, M.A.; Cohen, J.D.; Forthofer, J.M.; McAllister, S.S.; Gollner, M.J.; Gorham, D.J.; Saito, K.; Akafuah, N.K.; Adam, B.A.; English, J.D. Role of buoyant flame dynamics in wildfire spread. Proc. Natl. Acad. Sci. USA
**2015**, 112, 9833–9838. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cruz, M.G.; Alexander, M.E.; Sullivan, A.; Gould, J.S.; Kilinc, M. Assessing improvements in models used to operationally predict wildland fire rate of spread. Environ. Model. Softw.
**2018**, 105, 54–63. [Google Scholar] [CrossRef] - Gould, J.S.; Sullivan, A.L. Two methods for calculating wildland fire rate of forward spread. Int. J. Wildland Fire
**2020**, 29, 272. [Google Scholar] [CrossRef] - Sahila, A.; Zekri, N.; Clerc, J.-P.; Kaiss, A.; Sahraoui, S. Fractal analysis of wildfire pattern dynamics using a Small World Network model. Phys. A Stat. Mech. Its Appl.
**2021**, 583, 126300. [Google Scholar] [CrossRef] - Li, S.; Banerjee, T. Spatial and temporal pattern of wildfires in California from 2000 to 2019. Sci. Rep.
**2021**, 11, 8779. [Google Scholar] [CrossRef] - Rothmell, R.C. A Mathematical Model for Predicting Fire Spread; Forest Service Research Paper; US Department of Agriculture: Fort Collins, CO, USA, 1972. [Google Scholar]
- Grishin, A. Mathematical Modeling of Forest Fires and New Methods of Fighting Them; Publishing House of the Tomsk State University: Tomsk, Russia, 1988. [Google Scholar]
- Linn, R.; Reisner, J.; Colman, J.J.; Winterkamp, J. Studying wildfire behavior using FIRETEC. Int. J. Wildland Fire
**2002**, 11, 233–246. [Google Scholar] [CrossRef] - Morvan, D.; Dupuy, J. Modeling the propagation of a wildfire through a Mediterranean shrub using a multiphase formulation. Combust. Flame
**2004**, 138, 199–210. [Google Scholar] [CrossRef] - Sullivan, A.L. Wildland surface fire spread modelling, 1990–2007. 1: Physical and quasi-physical models. Int. J. Wildland Fire
**2009**, 18, 349–368. [Google Scholar] [CrossRef] [Green Version] - Sullivan, A.L. Wildland surface fire spread modelling, 1990–2007. 2: Empirical and quasi-empirical models. Int. J. Wildland Fire
**2009**, 18, 369–386. [Google Scholar] [CrossRef] [Green Version] - Hong, H.; Jaafari, A.; Zenner, E.K. Predicting spatial patterns of wildfire susceptibility in the Huichang County, China: An integrated model to analysis of landscape indicators. Ecol. Indic.
**2019**, 101, 878–891. [Google Scholar] [CrossRef] - Jain, P.; Coogan, S.C.P.; Subramanian, S.G.; Crowley, M.; Taylor, S.W.; Flannigan, M.D. A review of machine learning applications in wildfire science and management. Environ. Rev.
**2020**, 28, 478–505. [Google Scholar] [CrossRef] - Zigner, K.; Carvalho, L.M.V.; Peterson, S.; Fujioka, F.; Duine, G.-J.; Jones, C.; Roberts, D.; Moritz, M. Evaluating the Ability of FARSITE to Simulate Wildfires Influenced by Extreme, Downslope Winds in Santa Barbara, California. Fire
**2020**, 3, 29. [Google Scholar] [CrossRef] - Finney, M.A. FARSITE: Fire Area Simulator-Model Development and Evaluation; US Department of Agriculture, Forest Service, Rocky Mountain Research Station: Fort Collins, CO, USA, 1998. [Google Scholar]
- Finney, M.A. An Overview of FlamMap Fire Modeling Capabilities. In Proceedings of the Fuels Management—How to Measure Success, Portland, Ore, USA, 28–30 March 2006. USDA Forest Service Proceedings RMRS-P-41. [Google Scholar]
- Tymstra, C.; Bryce, R.W.; Wotton, B.M.; Taylor, S.W.; Armitage, O.B. Development and Structure of Prometheus: The Canadian Wildland Fire Growth Simulation Model; Information report NOR-X-417; Natural Resources Canada: Edmonton, AB, Canada, 2010. [Google Scholar]
- Finney, M.A.; McHugh, C.W.; Grenfell, I.C.; Riley, K.L.; Short, K.C. A simulation of probabilistic wildfire risk components for the continental United States. Stoch. Environ. Res. Risk Assess.
**2011**, 25, 973–1000. [Google Scholar] [CrossRef] [Green Version] - de Groot, W.J.; Cantin, A.S.; Jurko, N.; Newbery, A. Modeling fire behaviour and carbon emissions. In Advances in Forest Fire Research; University of Coimbra: Coimbra, Portugal, 2014. [Google Scholar]
- Gaudreau, J.; Perez, L.; Drapeau, P. BorealFireSim: A GIS-based cellular automata model of wildfires for the boreal forest of Quebec in a climate change paradigm. Ecol. Inform.
**2016**, 32, 12–27. [Google Scholar] [CrossRef] - Linn, R.; Goodrick, S.; Brambilla, S.; Brown, M.; Middleton, R.; O’Brien, J.; Hiers, J. QUIC-fire: A fast-running simulation tool for prescribed fire planning. Environ. Model. Softw.
**2020**, 125, 104616. [Google Scholar] [CrossRef] - Katan, J.; Perez, L. ABWiSE v1.0: Toward an agent-based approach to simulating wildfire spread. Nat. Hazards Earth Syst. Sci.
**2021**, 21, 3141–3160. [Google Scholar] [CrossRef] - Schumaker, N.H.; Brookes, A. HexSim: A modeling environment for ecology and conservation. Landsc. Ecol.
**2018**, 33, 197–211. [Google Scholar] [CrossRef] - Lyons, A.L.; Gaines, W.L.; Singleton, P.H.; Kasworm, W.F.; Proctor, M.F.; Begley, J. Spatially explicit carrying capacity estimates to inform species specific recovery objectives: Grizzly bear (Ursus arctos) recovery in the North Cascades. Biol. Conserv.
**2018**, 222, 21–32. [Google Scholar] [CrossRef] - Messager, M.L.; Olden, J.D. Individual-based models forecast the spread and inform the management of an emerging riverine invader. Divers. Distrib.
**2018**, 24, 1816–1829. [Google Scholar] [CrossRef] [Green Version] - Snyder, M.N.; Schumaker, N.H.; Ebersole, J.L.; Dunham, J.B.; Comeleo, R.L.; Keefer, M.L.; Leinenbach, P.; Brookes, A.; Cope, B.; Wu, J.; et al. Individual based modeling of fish migration in a 2-D river system: Model description and case study. Landsc. Ecol.
**2019**, 34, 737–754. [Google Scholar] [CrossRef] [PubMed] - Heinrichs, J.A.; O’Donnell, M.S.; Aldridge, C.L.; Garman, S.L.; Homer, C.G. Influences of potential oil and gas development and future climate on Sage-grouse declines and redistribution. Ecol. Appl.
**2019**, 29, e01912. [Google Scholar] [CrossRef] [PubMed] - Ward, E.M.; Wysong, K.; Gorelick, S.M. Drying landscape and interannual herbivory-driven habitat degradation control semiaquatic mammal population dynamics. Ecohydrology
**2020**, 13, e2169. [Google Scholar] [CrossRef] - Ward, E.M.; Solari, K.A.; Varudkar, A.; Gorelick, S.M.; Hadly, E.A. Muskrats as a bellwether of a drying delta. Commun. Biol.
**2021**, 4, 750. [Google Scholar] [CrossRef] - Pacioni, C.; Kennedy, M.S.; Ramsey, D.S.L. When do predator exclusion fences work best? A spatially explicit modelling approach. Wildl. Res.
**2020**, 48, 209–217. [Google Scholar] [CrossRef] - Penteado, H.M. Urban open spaces from a dispersal perspective: Lessons from an individual-based model approach to assess the effects of landscape patterns on the viability of wildlife populations. Urban Ecosyst.
**2021**, 24, 753–766. [Google Scholar] [CrossRef] - Andersen, D.; Yi, Y.; Borzée, A.; Kim, K.; Moon, K.-S.; Kim, J.-J.; Kim, T.-W.; Jang, Y. Use of a spatially explicit individual-based model to predict population trajectories and habitat connectivity for a reintroduced ursid. Oryx
**2022**, 56, 298–307. [Google Scholar] [CrossRef] - D’Elia, J.; Schumaker, N.H.; Marcot, B.G.; Miewald, T.; Watkins, S.; Yanahan, A.D. Condors in space: An individual-based population model for California condor reintroduction planning. Landsc. Ecol.
**2022**, 37, 1431–1452. [Google Scholar] [CrossRef] - Schumaker, N.; Watkins, S. Adding Space to Disease Models: A Case Study with COVID-19 in Oregon, USA. Land
**2021**, 10, 438. [Google Scholar] [CrossRef] - Wolfram, S. Cellular automata as models of complexity. Nature
**1984**, 311, 419–424. [Google Scholar] [CrossRef] - Gardner, M. Mathematical Games—The Fantastic Combinations of John Conway’s New Solitaire Game “Life”. Sci. Am.
**1970**, 223, 120–123. [Google Scholar] [CrossRef] - Daniel, C.J.; Frid, L.; Sleeter, B.M.; Fortin, M. State-and-transition simulation models: A framework for forecasting landscape change. Methods Ecol. Evol.
**2016**, 7, 1413–1423. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Shapes of the probability curves derived from specific values of N (neighbors on fire) or F (relative flammability), and α or β (the proximity and flammability exponents). Exponent values are superimposed on the curves they produce.

**Figure 2.**Six examples of how the probability of contagious fire spread changes as a function of a cell’s relative flammability, and its number of burning neighbors (N).

**Figure 3.**Example 2 spatial inputs, including relative flammability (

**left**), wind gradient (

**center**) and relative wind speed (

**right**). The emergent simulated burn perimeter (see Results) is shown as a black outline. The arrow at the top-center indicates the location at which the fire was initiated.

**Figure 4.**Example 3 spatial inputs, including (

**A**) relative flammability, (

**B**) wind gradient, and (

**C**) marten habitat quality.

**Figure 5.**Mean burn frequencies produced by 100 replicates of six model variants: (

**A**) the baseline model, (

**B**) increased influence of proximity to fire, (

**C**) increased influence of fuel flammability, (

**D**) longer ember distances, (

**E**) enhanced wind influence, (

**F**) diminished wind influence. The checkerboard patterns derive from the model’s flammability map, which was made up from alternating blocks of low and high flammability fuels.

**Figure 6.**Mean burn frequencies (

**A**) without and (

**B**) including fuel breaks. Fuel breaks, shown in white, indicate areas where the fuel flammability was set to zero. Black arrows show the location at which the fires were initiated. These results were assembled from 100 replicate simulations.

**Figure 7.**The emergent distribution of burn areas observed from 1000 fire simulations. Replicates that generated a fire size of less than 100 hexagons (of 119 K hexagons total) were replaced with the results from an additional simulation.

**Figure 8.**Mean burn frequency generated from 1000 fire simulations (

**left**), and nine illustrative non-overlapping fires selected from this collection (

**right**). Mean burn frequency is displayed using the color ramp of Figure 4, and the black outline indicates the extent of marten habitat. Burn sizes indicate the proportion of the landscape consumed by a single fire. The color assigned to individual fires (black vs. blue) has no significance.

**Figure 9.**Estimates of population size from marten life history simulator, without ((

**left**), 100 replicates) and including ((

**right**), 1000 replicates) impacts from wildfire. The dark blue and red lines at the plot centers indicate mean population size, while the pairs of light blue and orange lines bracket the range of observed values.

Parameter Name | Parameter Interpretation |
---|---|

Burn Iterations per Time Step | The number of times that the contagious and ember-driven fire spread algorithms are run per time step. |

Iterates to Burn Completely | The number of burn iterations during which an ignited cell will continue to burn. |

Flammability Exponent | The exponent that influences how fuel flammability affects contagious wildfire spread. |

Proximity Exponent | The exponent that influences how the number of burning neighbors affects contagious wildfire spread. |

Ember Creation Rate | The maximum number of embers that can be created, per iterate, within each burning cell. |

Ember Max Distance | The maximum distance, in hexagons, that an individual ember may travel. |

Ember Step Length—Wind | The step length, in hexagons, assigned to embers moving along a wind gradient. |

Ember Step Length—Random | The step length, in hexagons, assigned to embers moving in a random direction. |

**Table 2.**Input maps used by the HexFire model. Variable suggests that map construction effort will range from minimal to significant, depending on study design. Automatic implies that a method or utility is available to construct the map. Optional indicates that a map may be left blank.

Map Name | Level of Effort | Map Function |
---|---|---|

Relative Flammability | Variable | Provides the relative flammability of each cell in the landscape. Values must range between 0 and 1. |

Ignition Sites | Variable | Controls the time and location at which fires are initiated, including back burns. |

Hexagon ID | Automatic | Contains the individual ID of each cell. This map is trivial to create in HexSim. |

Patch Maps (A-D) | Automatic | A collection of four patch maps for which the union of all patches is space-filling, and each patch slightly overlaps its neighbors. We provide a utility for constructing these patch maps, which are used to improve model performance. |

Relative Wind Speed | Variable | Provides the wind speed for each cell in the landscape. Values must range between 0 and 1. |

Wind Gradient | Variable | Indicates the directions that embers will travel. We provide a utility that builds wind gradient maps from maps of wind direction. |

Fuel Breaks | Optional | Indicates where and when fuels should be removed from the flammability map. |

Fuel Barriers | Optional | Specifies the location of fuel barriers, which can block the movement of embers. |

**Table 3.**Example 1 input parameters. Labels used to distinguish the six experiments are shown in parentheses. Experiments B–F were conducted using the baseline model (experiment A) parameters, except where indicated.

Baseline | Model Variants | |||||
---|---|---|---|---|---|---|

(A) | (B) | (C) | (D) | (E) | (F) | |

Burn Iterations per Time Step | 2 | |||||

Iterates to Burn Completely | 3 | |||||

Flammability Exponent | 2 | 5 | 0.2 | |||

Proximity Exponent | 0.5 | 0.2 | 5 | |||

Ember Creation Rate | 5 | |||||

Ember Max Distance | 10 | 50 | ||||

Ember Step Length—Wind | 1 | 10 | ||||

Ember Step Length—Random | 1 | 10 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Schumaker, N.H.; Watkins, S.M.; Heinrichs, J.A.
HexFire: A Flexible and Accessible Wildfire Simulator. *Land* **2022**, *11*, 1288.
https://doi.org/10.3390/land11081288

**AMA Style**

Schumaker NH, Watkins SM, Heinrichs JA.
HexFire: A Flexible and Accessible Wildfire Simulator. *Land*. 2022; 11(8):1288.
https://doi.org/10.3390/land11081288

**Chicago/Turabian Style**

Schumaker, Nathan H., Sydney M. Watkins, and Julie A. Heinrichs.
2022. "HexFire: A Flexible and Accessible Wildfire Simulator" *Land* 11, no. 8: 1288.
https://doi.org/10.3390/land11081288