The FARSITE model (Fire Area Simulator) simulates fire growth as a spreading elliptical wave. The fire is propagated over a finite time step using points, that define the fire front, as independent sources of small elliptical wavelets. These small ellipses can be thought of as forming an envelope around the original perimeter, the outer edge representing the new fire front. This process has been referred to as Huygens' principle (Anderson et al. 1982). Huygens' principle is named for the 17th century Dutch mathematician Christian Huygens who proposed it for describing the travel of light waves.
The reliance on an assumed fire shape, in this case an ellipse, is necessary because the spread rate of only the heading portion of a fire is predicted by the present fire spread model (Rothermel 1972). Fire spread in all other directions is inferred from this forward spread rate using the mathematical properties of the ellipse. An elliptical shape would not have to be assumed if the spread rate in all directions could be computed independently from the fuels, weather, and topography.
Two examples of Huygens' principle with elliptical fire shapes. Winds are uniform from the southwest.
Homogenous fuels.
Mosaic of four cover types (fuel, wind speed) that change the size and shape of the ellipses.
The metaphor of a fire front spreading as a wave is intuitively attractive for several reasons:
The fire front at most scales of perception is continuous along its active portions in time and space and can naturally be represented as a series of iso-chrons (time contours),
The wave propagation technique is distinct from the data resolution or data type used for its solution (e.g. not limited to raster-type or vector-type landscape information).
Radiation from flames is an important means of fire propagation and is itself a wave,
Existing models of surface fire behavior (Rothermel 1972), crown fire behavior (Van Wagner 1977), and spotting (Albini 1979) are formulated as point-vectors (giving rates and distances of spread from a given location) that translate naturally to the vector fire perimeters produced by Huygens' principle.
This approach differs considerably in concept from the cellular models, or cellular automata, that simulate fire spread as a contagion process between cells (Kourtz and O'Regan 1971, Kourtz et al. 1977, Green 1983). Cellular models solve for ignition times of cells at known regular spacing. The position of the fire front at a given time must be interpreted from cells having similar arrival times. This type of model has some inherent difficulties. Among the most critical, appear to be the distortion of fire shapes caused by the gridded landscape geometry (Ball and Guertin 1992), and absence of information on spread between cells; the latter becomes vital to synchronizing effects of temporal changes in weather or fuel moisture around the fire perimeter. None of the cellular models adequately simulated fire spread under test conditions with spatial and temporal heterogenities (French 1992). Techniques recommended by (French 1992) for reducing geometric distortion have included enlarging the "search radius" for interacting cells (increasing the number of "adjacent cells"), and decreasing the time step (depending on the specific cellular algorithm in use). Cellular models may be deterministic, or have probabilistic or fractal modifications (Clark et al. 1994) to spread rates and/or directions. Non-deterministic models would require multiple runs to generate a risk map (event-probability map) for a given scenario because outputs change between individual simulations having identical input parameters.
The concept of applying Huygens' principle to model fire growth simply involves using the fire environment at each perimeter point to dimension and orient an elliptical wave around each point on a fire front at each time step. The shape and direction of the ellipse is determined by wind-slope vector while the size (e.g. spread rate) is determined by the fuel conditions. The implementation of this in a practical fire growth model is, however, considerably more complex.
Huygens' principle has been applied to fire growth modeling in various forms. The earliest published application was the "radial fire propagation model" by Sanderlin and Sunderson (1975). This was a computerized method for projecting fire perimeter growth over complex landscapes. It used a 3 dimensional wind field, a rasterized landscape of fuels and topography, and provided reasonable projections of fire growth (Sanderlin and Sunderson 1975, Sanderlin and Van Gelder 1977). The essential mathematics and many of the complications of this approach were first identified here. Anderson et al. (1982) brought the terminology and concept of Huygens' principle to the fire literature. They described the mathematics and applied Huygens' principle to perimeter data from a test fire, finding it suitable as a fire growth model. French et al. (1990) and French (1992) employed a graphical technique which used computer graphics block-copy techniques to produce fire fronts. The "four-point" technique (Beer 1990, French 1992) uses 4 points on an elliptical perimeter that correspond to its major and minor axes as the propagation points that form the new fire perimeters. Richards (1990) analytically derived a differential equation that propagates any point using an elliptical fire shape. Richards (1990) technique is employed in the FARSITE model and uses the vertices of the fire perimeter polygon as the propagation points. The same result is achieved by the method of Roberts (1989; discussed by French 1992) in which the line segments between the vertices are the objects of propagation. Knight and Coleman (1993), Dorrer (1993), and Wallace (1993) also developed procedures for computing fire perimeter positions based on Huygens' principle of wave propagation. Recently, Richards (1995) has extended his equations to expand fire shapes different from the simple ellipse (lemniscate, double ellipse etc).
Essential as it is, the method chosen to implement Huygens' principle really becomes a minor part of the whole simulation process. The outline of the process control used in FARSITE illustrates this; Richards' (1990) equation is used only in step 5 of the surface fire calculations and step 7 of the crown fire calculations.