Predicting wildfire propagation across landscapes
2026-02-27
On the afternoon of 16 August 2003, the Okanagan Mountain Park fire in British Columbia jumped Highway 33 and ran 25 km before midnight. By the time it stopped, 30,000 residents of Kelowna were under evacuation order and 239 homes had been destroyed. Prometheus, Canada’s operational wildfire growth model, was running simulations of the fire’s likely spread as incident commanders made decisions about where to deploy suppression resources and which communities to evacuate first. The simulations did not tell the commanders exactly where the fire would be — fire behaviour is too sensitive to local wind gusts and terrain for that — but they produced probabilistic spread envelopes that captured where the fire was likely to reach under a range of weather scenarios.
The physics of fire spread is encoded in the Rothermel equations, first published in 1972 and still the backbone of every operational fire behaviour model in North America. The equations predict the rate at which a fire front moves through fuel of given type, loading, and moisture, under given wind and slope conditions. The spread rate varies with direction — wind and slope both push the fire, so headfire spreads far faster than backfire — producing an elliptical fire shape that grows over time. Combining spread rate, direction, and an initial perimeter with a cellular automaton or vector propagation scheme produces a fire perimeter evolution that can be computed in near-real time. This model derives the Rothermel rate-of-spread equation, implements elliptical fire growth, and shows how the cellular automaton approach simulates fire spread across heterogeneous landscapes.
Where will this fire be in 6 hours, and what communities are at risk?
Fire spread modelling:
Predicts fire perimeter expansion over time.
Inputs: - Fuel type and loading - Fuel moisture - Wind speed and direction - Topography (slope, aspect) - Weather forecast
Outputs: - Rate of spread (m/min) - Fire perimeter at time t - Fireline intensity - Flame length - Arrival time at locations
Models:
Empirical: Rothermel (1972)
Physical: FIRETEC, WFDS (computational fluid dynamics)
Statistical: Machine learning on historical fires
Hybrid: Couple multiple approaches
Operational systems: - FARSITE (USA Forest Service) - Phoenix RapidFire (Australia) - Prometheus (Canada) - FlamMap (landscape fire potential)
Basic principle:
Fire spreads when radiant/convective heat preheats adjacent fuel to ignition.
Rate of spread:
ROS = \frac{I_R \xi (1 + \Phi_w + \Phi_s)}{\rho_b \epsilon Q_{ig}}
Where: - ROS = rate of spread (m/min) - I_R = reaction intensity (kW/m²) - \xi = propagating flux ratio - \Phi_w = wind coefficient - \Phi_s = slope coefficient - \rho_b = bulk density (kg/m³) - \epsilon = effective heating number - Q_{ig} = heat of preignition (kJ/kg)
Reaction intensity:
Energy release rate per unit area.
I_R = \Gamma' w_n h
Where: - \Gamma' = optimum reaction velocity (min⁻¹) - w_n = net fuel loading (kg/m²) - h = heat content (kJ/kg, ~18,600 for wood)
Wind-driven fires: Elongated downwind.
Head fire: Fastest spread (downwind)
Flank fire: Perpendicular to wind
Back fire: Upwind (slow)
Ellipse parameters:
a = \frac{ROS_h + ROS_b}{2}
b = \frac{ROS_h + ROS_b}{2 \times LB}
Where: - a = semi-major axis - b = semi-minor axis - ROS_h = head fire rate - ROS_b = backing fire rate - LB = length-to-breadth ratio (~3-4 typical)
Area at time t:
A(t) = \pi a b t^2
Energy release per unit length:
I = h w ROS
Where: - I = fireline intensity (kW/m) - h = heat content (kJ/kg) - w = fuel consumed (kg/m²) - ROS = rate of spread (m/s)
Byram’s flame length:
L = 0.0775 I^{0.46}
Where L = flame length (m)
Suppression difficulty: - I < 500 kW/m: Direct attack possible - I = 500-2000 kW/m: Indirect attack - I = 2000-4000 kW/m: Very difficult - I > 4000 kW/m: Suppression ineffective
Wind coefficient:
\Phi_w = C \left(\frac{\beta}{\beta_{op}}\right)^{-E} \left(\frac{U_m}{60}\right)^B
Where: - C, E, B = fuel-specific constants - \beta = packing ratio (fuel density / particle density) - \beta_{op} = optimum packing ratio - U_m = midflame wind speed (ft/min)
Typical: B \approx 1.5 (spread ∝ wind^1.5)
Slope coefficient:
\Phi_s = 5.275 \beta^{-0.3} \tan^2\theta
Where \theta = slope angle
Example: 20° slope
\Phi_s = 5.275 \times (0.01)^{-0.3} \times \tan^2(20°)
= 5.275 \times 2.15 \times 0.133 = 1.51
Slope multiplies spread by factor of 2.5 (1 + 1.51)!
Discretize landscape:
Grid cells (30-100 m resolution).
State: Unburned / Burning / Burned
Transition rules:
P_{\text{ignite}} = f(\text{fuel, weather, distance, time})
Simplified:
P = P_0 \times \exp\left(-\frac{d}{ROS \times \Delta t}\right)
Where: - P = ignition probability - P_0 = base probability - d = distance to burning cell - ROS = rate of spread - \Delta t = time step
Algorithm:
For each time step:
For each burning cell:
For each neighbor:
Calculate P_ignite
If random() < P_ignite:
Ignite neighbor
Burning cells → BurnedAdvantages: - Computationally fast - Handles complex landscapes - Stochastic variation
Disadvantages: - Requires calibration - Less physical basis than Rothermel
Minimum time path:
T(x,y) = \min_{\text{path}} \int_{\text{path}} \frac{ds}{ROS(s)}
Where: - T = arrival time - ds = path element - ROS(s) = rate of spread along path
Solved by:
Level-set methods or Dijkstra’s algorithm on grid.
Isochrones:
Contours of equal arrival time.
Problem: Calculate fire spread and perimeter.
Conditions: - Fuel: Grass (short, dry) - Fuel load: 0.5 kg/m² - Fuel moisture: 6% - Wind speed: 30 km/h (8.3 m/s) - Slope: 15° downwind - Initial ignition: Point source
Constants (grass fuel model): - \xi = 0.4 (propagating flux ratio) - \epsilon = 0.7 (effective heating) - Q_{ig} = 300 kJ/kg (heat of preignition) - h = 18,600 kJ/kg (heat content)
Calculate ROS, fire perimeter after 1 hour.
Step 1: Wind coefficient
Simplified: \Phi_w \approx 0.15 U^{1.5} where U in m/s
\Phi_w = 0.15 \times 8.3^{1.5} = 0.15 \times 23.9 = 3.6
Step 2: Slope coefficient
\Phi_s = 5.275 \times (0.01)^{-0.3} \times \tan^2(15°)
= 5.275 \times 2.15 \times 0.072 = 0.82
Step 3: Reaction intensity
Assume \Gamma' = 15 min⁻¹ (typical grass):
I_R = 15 \times 0.5 \times 18600 = 139,500 \text{ kW/m}^2
Step 4: Rate of spread
ROS = \frac{139500 \times 0.4 \times (1 + 3.6 + 0.82)}{500 \times 0.7 \times 300}
= \frac{139500 \times 0.4 \times 5.42}{105000}
= \frac{302,472}{105000} = 2.88 \text{ m/min}
Head fire ROS = 2.88 m/min = 173 m/h
Step 5: Backing fire
Assume ROS_b = 0.1 \times ROS_h = 0.29 m/min
Step 6: Ellipse parameters
a = \frac{2.88 + 0.29}{2} = 1.59 \text{ m/min}
Length-to-breadth ratio LB = 3:
b = \frac{1.59}{3} = 0.53 \text{ m/min}
Step 7: Perimeter after 1 hour
Semi-axes at t = 60 min:
a_{60} = 1.59 \times 60 = 95.4 \text{ m}
b_{60} = 0.53 \times 60 = 31.8 \text{ m}
Area:
A = \pi \times 95.4 \times 31.8 = 9,540 \text{ m}^2 \approx 1 \text{ hectare}
Fire spread 95 m downwind, 32 m on flanks in 1 hour.
Step 8: Fireline intensity (head)
I = 18600 \times 0.5 \times \frac{2.88}{60} = 446 \text{ kW/m}
Flame length:
L = 0.0775 \times 446^{0.46} = 0.0775 \times 12.3 = 0.95 \text{ m}
Moderate intensity (direct attack still possible if resources available quickly)
Below is an interactive fire spread simulator.
<label>
Wind speed (km/h):
<input type="range" id="wind" min="0" max="60" step="5" value="20">
<span id="wind-val">20</span>
</label>
<label>
Slope (°):
<input type="range" id="slope" min="0" max="30" step="5" value="10">
<span id="slope-val">10</span>
</label>
<label>
Fuel moisture (%):
<input type="range" id="moisture" min="4" max="20" step="2" value="8">
<span id="moist-val">8</span>
</label>
<label>
Time elapsed (hours):
<input type="range" id="time" min="1" max="12" step="1" value="4">
<span id="time-val">4</span>
</label>
<div class="spread-info">
<p><strong>Head ROS:</strong> <span id="ros-head">--</span> m/min</p>
<p><strong>Fire area:</strong> <span id="fire-area">--</span> ha</p>
<p><strong>Fireline intensity:</strong> <span id="intensity">--</span> kW/m</p>
<p><strong>Suppression:</strong> <span id="suppression">--</span></p>
</div><canvas id="spread-canvas" width="700" height="400" style="border: 1px solid #ddd;"></canvas>Observations: - Fire spreads fastest downwind (head fire) - Elliptical shape elongated in wind direction - Higher wind dramatically increases spread rate - Slope amplifies spread when aligned with wind - Low moisture enables rapid spread - Fireline intensity determines suppression feasibility
Key insights: - Exponential area growth over time - Wind and slope effects multiplicative - Even moderate winds create elongated fires - Suppression window closes quickly with high intensity
FARSITE workflow:
Outputs: - Fire perimeter evolution (hourly) - Arrival time maps - Intensity maps - Flame length maps
Used for: - Evacuation planning - Resource pre-positioning - Backfire/burnout strategy - Structure triage
Example - 2018 Camp Fire (California):
FARSITE predicted Paradise threatened within 6 hours.
Enabled evacuation (though still 85 deaths, rapid spread).
Objective: Reduce fuel loads safely.
Model used to:
Determine burn window (weather conditions for control).
Required: - Low intensity (< 1000 kW/m) - Slow spread (< 1 m/min) - Favorable wind direction (away from structures)
Example conditions: - Fuel MC: 12-15% - Wind: 10-15 km/h - Atmospheric stability: Stable - RH: 40-60%
Model confirms burn will remain controllable.
Reconstruct fire behavior:
Given final perimeter, weather records.
Calibrate models:
Adjust parameters (fuel models, moisture, wind reduction factors).
Improve predictions:
Better local fuel characterization.
Identify extreme fire behavior:
Rates of spread >> model predictions indicate: - Spotting (long-range ember transport) - Crown fire (not surface fire) - Fire whirls (tornadic circulation)
Surface fire models (Rothermel) don’t predict crown fire.
Crown fire: Spreads through tree canopy, 10-100× faster.
Transition: Occurs when:
I > I_{\text{critical}}
Critical intensity:
I_{\text{crit}} = \frac{(460 + 25.9 \times MC)^{3/2}}{60} \times 0.01 \times \rho_{\text{canopy}}
Once in crown:
ROS can exceed 100 m/min (vs 2-10 m/min surface).
Solution: Van Wagner crown fire models, FIRETEC 3D physics.
Embers travel ahead of fire front.
Distance: Function of ember size, height lofted, wind.
Model (Albini):
d_{\text{spot}} = f(\text{flame length, wind, terrain})
Typically: 0.5-2 km, extreme cases 10-30 km.
Creates secondary ignitions ahead of modelled perimeter.
Catastrophic when embers land in receptive fuel.
Solution: Spotting submodels, but highly stochastic.
Models assume uniform fuel in each type.
Reality:
Variable loading, moisture, species within same classification.
Result: Under/overprediction locally.
Solution: - Fine-scale fuel mapping (LiDAR-derived) - Stochastic variation in parameters
Fire spread depends on forecast wind, temperature, humidity.
Forecast errors propagate:
10% wind error → 15% ROS error → 25% perimeter error.
Solution: - Ensemble forecasts (multiple weather scenarios) - Probabilistic fire prediction
Data-driven approaches:
Train on historical fires (perimeter progression, weather, fuels).
Neural networks:
Input: Fuel, weather, terrain
Output: ROS or probability of burning
Advantages: - Capture complex nonlinear relationships - No need to specify physics - Fast inference
Disadvantages: - Requires large training datasets - Difficult to extrapolate beyond training conditions - Less interpretable than physics-based
Hybrid: Use physics model as baseline, ML for corrections.
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
Where: - a = semi-major axis (larger) - b = semi-minor axis (smaller)
Area:
A = \pi a b
Perimeter (approximate):
P \approx \pi (3(a+b) - \sqrt{(3a+b)(a+3b)})
e = \sqrt{1 - \frac{b^2}{a^2}}
Circle: e = 0 (a = b)
Elongated: e \to 1 (b \ll a)
Fire ellipses:
Typical e = 0.8-0.95 (highly elongated in strong wind).