The spread of wildland fire involves the complex interactions of many physical phenomena. The reaction between combustible fuel and oxygen is exothermic, resulting in the formation of hot smoke and combustion products. The decrease in local density due to the elevated temperatures results in a buoyancy-driven plume rising through the air, which can affect the atmospheric boundary layer, local burning conditions, and local wind speeds. Some of the energy released from the reaction may be transferred to additional sources of fuel in the environment, which can result in the dehydration, pyrolysis, and ignition of additional vegetative fuels. The rate of spread of a wildland fire is the speed at which additional fuel ignites along an existing fire perimeter, and is often used to characterize wildland fires .
However, realistic fires in the wildland occur over diverse topography, fuels, and weather conditions. Two main approaches have been developed to adapt Rothermel’s model to account for these spatial variations. The first approach decomposes the flame front into a series of points that propagate independently , ; whereas, the second approach implements the fire perimeter as a solution of a level set function , . Figure 2 provides an example of wildland fire spread over heterogeneous terrain using the series of points approach. While these approaches provide a framework to predict the spatially varying fire behavior, the limitations in the fundamental fire spread model make it difficult to generate accurate forecasts over diverse landscapes. In practice, accurately reconstructing wildfire behavior requires careful modification of vegetation and weather inputs on a case-by-case basis.
Typical operational models are limited to simplified representations of the wind over the landscape, which neglects the feedback between the fire plume and the atmosphere. Higher fidelity operational models use two-way coupling between a surface fire spread model and a numerical weather prediction model to account for this feedback , , –. Figure 3 shows that these models can resolve fire-induced vertical velocities associated with experimental burns. However, incorporating these physics into the models comes at a cost, where the computational requirements of the model are significantly increased.
An alternative approach to reduce the uncertainties in wildfire modeling and reduce the computational time of these models is to leverage data-driven simulations. There are four main approaches which have been presented for data-driven wildfire simulation. Data assimilation models fuse a forward prediction scheme (such as Rothermel’s fire spread model) with observed data (such as satellite observed fire perimeters) to improve the model predictions over time. One approach is to directly overwrite model predictions of fire perimeter at specific times when data is available . A more complex approach is to fuse the observations and predictions with a filter, such as particle filters – and ensemble Kalman filters –. In either approach, the error in the predicted fire perimeters is low after a correction since the new observations update the incorrect aspects of the model. However, the inputs to the forward model are not updated between model predictions which can result in the model error growing between assimilated observations. Figure 4 shows an example comparing model predictions with and without data assimilation .
Another data-driven approach is to apply an inverse modeling technique to update unknown model parameters (such as wind speed, moisture content, or fuel loading) using observation data rather than directly updating the fire perimeter. A key advantage of this approach over data assimilation is the potential to improve forecasts of future behavior by reducing uncertainty in model inputs. However, this potential improvement in forecasts depends on the fundamental accuracy of the forward model as well as the assumption that the statistics of the invariant parameters in the model are not changing. An example calculated using the inverse modeling approach is illustrated in Figure 5 .
A third data-driven approach to improve wildfire modeling is to use statistical tools to reduce the computational time of the physical models. Reduced-order modeling uses mathematical tools to project complex non-linear equations to a reduced order space. Figure 6 shows an example presented by Lattimer et al. to predict the temperature front of a one-dimensional wildfire based on an advection-reaction-diffusion equation . The existing formulations of the projected equations demonstrated the capability of the approach in homogeneous terrain; however, a new formulation would be needed to incorporate different fuel loadings and reaction rates in the model for fire spread over heterogeneous terrain.
The final data-driven approach to improve wildfire modeling is to use machine learning to generate the forecasts directly. Machine learning models use historic data from wildfires to make predictions from new inputs, often using artificial neural networks (ANNs) to make rapid predictions. ANNs are a complex system of equations that store information about a problem space in the free parameters which are learned during training such that the model error is minimized. McCormick presented an ANN-based approach to predict the final spatially-resolved perimeter of a fire given heterogeneous landscape and fixed wind conditions . While the model provided promising results, it did not predict the time-resolved profile, nor did it consider changing weather conditions.
Hodges et al. adapted the approach presented by McCormick to predict the time evolution of the wildfire front using convolutional neural networks, which are a modern machine learning technique that leverages key spatial features to improve the quality of model predictions , see Figure 7 for an example. The authors found that the spatial features present in the neural network design were able to represent the fire spread over heterogeneous landscapes at a computational cost which was 100-10,000 times less than the empirical model. Burge et al. extended this modeling approach using a convolutional long short-term memory model to predict the spread of a wildfire , see Figure 8, for example, fire perimeters from this model. This approach improves upon the previous models by integrating recent history information in the modeling approach. While each of these model derivations support changing weather conditions in the predictions, the models are not directly coupled with a numerical weather prediction model. Applying similar technologies to coupled atmosphere-wildfire behavior models would make it more feasible to use these models to support real-time decision-making.
One of the key difficulties with the first three discussed approaches is the fundamental fire spread forecast is based on the empirical formulation of fire spread. As a result, the forecasts from each model inherit the same difficulties as the empirical model where the predictions are based on experiments of fire spread in homogeneous terrain and weather conditions and neglect the impact of heterogeneous conditions on the local fire spread. Although the results presented by Hodges et al.  and Burge et al.  were trained on simulated fire perimeters, the fundamental approaches used could instead be trained on historic fire perimeters directly which has the potential to achieve more accurate forecasts than can be achieved from empirical models based on experimental data in homogeneous conditions.
Looking to the future, integration of data-driven models in operational tools has the potential to improve the accuracy and computational efficiency of existing wildfire models. Development of standardized databases of historic vegetation, fire perimeters, and weather will be critical to the long-term success of these initiatives.
Jonathan L. Hodges, PhD is with Jensen Hughes
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