Lean-FIRe: Hybrid Wildfire Modeling Framework

• Lean-FIRe is a hybrid wildfire simulation framework combining a deterministic drift term with a stochastic fluctuation component to capture turbulence and fire-spotting.
• It decomposes fire-front propagation into a classical modeling part from simulation engines and a random term drawn from physics-based probability density functions.
• The framework is applicable to both Eulerian (LSM) and Lagrangian (DEVS) approaches, enabling simulation of complex fire behaviors such as pre-heating, secondary ignition, and flank fire development.

Lean-FIRe is a mathematical and computational framework designed to enrich the traditional deterministic modeling of wildland fire spread by incorporating stochastic effects arising from turbulence and fire-spotting. The central principle of Lean-FIRe is to decompose fire-front propagation into a superposition of a classical drift term—provided by existing simulation engines—and a fluctuating term statistically characterized by physically motivated probability density functions. This hybridization allows operational wildfire simulators to emulate key physical phenomena such as pre-heating by hot air, ignition by firebrands, overcoming fuel breaks, flank fire development, and secondary ignition, all of which are challenging for deterministic approaches to reproduce. Both Eulerian (Level Set Method, LSM) and Lagrangian (Discrete Event System Specification, DEVS) front propagation schemes are supported, enabling broad applicability across common fire modeling platforms (Kaur et al., 2016).

1. Combined Deterministic–Stochastic Mathematical Formulation

The Lean-FIRe paradigm prescribes a formal splitting of the fire-front kinematics:

• Drifting (Deterministic) Component: The interface propagation dictated by the underlying deterministic fire simulator—either grid-based (LSM) or marker-based (DEVS)—models fire spread according to fuels, wind, and slope-dependent rate-of-spread.
• Random Fluctuating Component: Superimposed is a stochastic displacement sampled from a composite probability density function (PDF) that captures the diffusive spread due to atmospheric turbulence and the nonlocal, wind-oriented transport of firebrands causing spot fires.

For an evolving burning point $x(t)$, the position update is expressed as: $$x(t) = x_\text{drift}(t) + x_\text{fluctuation}(t),$$ where $x_\text{fluctuation}(t)$ is drawn from the PDF $f(x;t\,|\,x_0)$. The ensemble-averaged indicator function for burned area, $\varphi_e(x, t)$, is produced via: $$\varphi_e(x, t) = \int_S \varphi(x, t) f(x; t\,|\,x_0) dx,$$ where $\varphi(x, t)$ is the deterministic burned set indicator and $f(x; t\,|\,x_0)$ is the spatial PDF.

The PDF $f$ combines:

• Turbulence: Represented by a bi-variate Gaussian,

$$G(x; t) = \frac{1}{4\pi Dt} \exp\left\{ -\frac{(x - \bar{x})^2 + (y - \bar{y})^2}{4Dt} \right\}$$

with turbulent diffusion coefficient $D$.

• Fire-spotting (Downwind Direction): Landing distances $\ell$ of firebrands follow a lognormal distribution:

$$q(\ell;t) = \frac{1}{\sqrt{2\pi} \sigma(t) \ell} \exp\left\{ -\frac{[\ln(\ell/\ell_0) - \mu(t)]^2}{2\sigma(t)^2} \right\}.$$

For $x$ in the wind direction ($e_U \cdot (x-x_0) \geq 0$), the full PDF is a convolution:

$$f(x; t\,|\,x_0) = \int_0^\infty G(x - \ell \cdot e_U; t) q(\ell; t) d\ell.$$

A time-accumulation (heating–before–burning) mechanism is modeled through: $$\psi(x, t) = \int_0^t \varphi_e(x, \eta) \frac{d\eta}{\tau}$$ with ignition occurring when $\psi(x, t) \geq 1$, where $\tau$ encodes the effective delay due to pre-heating and firebrand ignition.

2. Implementation in Eulerian LSM and Lagrangian DEVS Models

Lean-FIRe can be applied to both classical front-tracking schemes:

• LSM (Eulerian):
  • Employs an implicit level-set function $\gamma(x, t)$, evolving by $$\partial \gamma/\partial t = \mathcal{V}(x, t) |\nabla \gamma|$$.
  • The (binary) indicator $\varphi(x, t)$ is defined over the spatial domain; postprocessing with $f$ yields $\varphi_e(x, t)$.
  • Supports direct application to grid-based fire simulators such as WRF-SFIRE.
• DEVS (Lagrangian):
  • Represents the fire front by explicit markers or particles advancing asynchronously.
  • Marker movement is event-driven, with propagation normals estimated via local geometric bisectors.
  • For Lean-FIRe enrichment, marker outputs are rasterized to a grid to form $\varphi(x, t)$; the stochastic envelope is applied, and new ignition markers spawn when the cumulative effect $\psi(x, t)$ exceeds the ignition threshold.
  • Integration with simulators such as ForeFire is facilitated by post-processing routines.

Table: Key Methodological Distinctions

Aspect	LSM (Eulerian)	DEVS (Lagrangian)
Fire-front	Implicit (level-set, fixed grid)	Explicit (markers, continuous space)
Normal Comput.	Finite-differences	Marker neighbour bisector
Stochastic step	Direct integration	Rasterization + Monte Carlo
Time-stepping	Global CFL restricted	Local, asynchronous

The principal differences in simulated outcomes arise from geometry-specific propagation (flank fire, direction)^*, computational efficiency, and the asynchrony of DEVS versus the globally synchronized LSM.

3. Numerical Experimentation and Phenomena Captured

A suite of idealized simulations validates the capability of Lean-FIRe to express key phenomenology not accounted for in classical deterministic models:

• In absence of stochasticity, neither LSM nor DEVS permits fire to cross fuel breaks.
• Introduction of turbulence ($D = 0.15, 0.30$ m²/s) enhances lateral spread and enables the fire to bridge no-fuel zones through pre-heating and ignition accumulation.
• Fire-spotting characterized by lognormal distributions (e.g., $\mu = 2.69, \sigma = 1.25$) further increases rate and heterogeneity of fire advance, generating secondary fires downwind.
• Simulation comparisons indicate both LSM and DEVS, when equipped with the Lean-FIRe overlay, yield similar large-scale behavior (e.g., head and rear fire agreement, development of secondary ignitions), although DEVS may exhibit stronger lateral development due to its normal estimation method.

These results collectively demonstrate that Lean-FIRe not only reproduces critical real-world effects—such as spot fires, overcoming suppression lines, and dynamic flank fire behavior—but also remains modular and robust across numerical platforms.

4. Practical Integration and Operational Utility

The Lean-FIRe approach is explicitly constructed as a postprocessing module, requiring only minimal modifications to existing wildfire simulation workflows:

• For LSM-based tools (e.g., WRF-SFIRE), stochastic enrichment is applied directly to the spatial indicator field.
• For DEVS-based engines (e.g., ForeFire), marker-to-grid rasterization feeds the Lean-FIRe pipeline, with the possibility of marker generation for new spot-fires and pre-heating effects.

This architecture enables straightforward operational deployment, providing more physically comprehensive forecasts of fire behavior for incident management and resource allocation, especially in scenarios involving rapid fire propagation, wind-driven spot fires, and heterogeneous landscapes.

5. Parameterization, Limitations, and Future Research

The robustness of Lean-FIRe is dependent on the specification of key parameters:

• Turbulent diffusion coefficients are initially estimated from canonical non-dimensional analogies (Nusselt, Rayleigh), but require location- and scale-specific refinement.
• Fire-spotting parameters ($\mu, \sigma$ in the lognormal) are critical and may need adjustment for different fuel or meteorological regimes; the use of alternative landing distance distributions (e.g., Weibull, Rayleigh) is suggested as a focus for future work.
• Validation against empirical event data has not been completed; current results serve as proof-of-concept, and systematic benchmarking against observed fires is explicitly identified as a research priority.

Potential extensions include coupling with full atmosphere–fire models, topography-aware PDFs, and explicit modeling of ignition delays as functions of environmental conditions.

6. Summary and Significance

Lean-FIRe constitutes a versatile, physically grounded augmentation for wildfire simulators, fusing deterministic fire-front propagation with stochastic modeling of turbulence and spot-fire generation. It enables both Eulerian and Lagrangian wildfire simulation platforms to overcome classical limitations, notably in scenarios involving rapid, heterogeneous spread, fuel breaks, and multiscale ignition sources. The approach is compatible with operational fire modeling systems, supports modular integration, and paves the way for more realistic, physics-informed fire management simulations. Critical future directions focus on experimental validation and improved stochastic parameter modeling to further refine predictive fidelity (Kaur et al., 2016).