## Abstract

Despite the many complexities concerning their initiation and propagation, forest fires exhibit power-law frequency-area statistics over many orders of magnitude. A simple forest fire model, which is an example of self-organized criticality, exhibits similar behavior. One practical implication of this result is that the frequency-area distribution of small and medium fires can be used to quantify the risk of large fires, as is routinely done for earthquakes.

Frequency-size distributions of natural hazards provide important information on calculating risk and are used in hazard mitigation (1). Robust power-law frequency-size distributions are associated with self-organized critical behavior. Examples of this behavior are found in a number of computer models: the sandpile model (2), the slider-block model (3), and the forest fire model (4). The slider-block model is considered to be an analog for earthquakes. Earthquakes exhibit a power-law dependence of occurrence frequency on rupture area and are considered to be the type example of self-organized critical behavior in nature (5). We found that, under a wide variety of circumstances, forest fires exhibit a power-law dependence of occurrence frequency on burn area over many orders of magnitude and that actual forest fires can be directly associated with the forest fire model. The only previous major application of the forest fire model was to epidemics of measles in isolated populations (6).

The forest fire model consists of randomly planting trees on a square grid at successive time steps and, at a specified number of time steps, randomly dropping a match on the grid. A maximum of one tree can occupy each grid site. The sparking frequency (* f*
_{s}) is the inverse of the number of attempts to plant trees on the grid before a model match is dropped on a randomly chosen site. If *f*
_{s} = 1/100, there have been 99 attempts to plant trees (some successful, some unsuccessful) before a match is dropped at the 100th time step. If the match is dropped on an empty site, nothing happens. If it is dropped on a tree, the tree ignites, and a model fire consumes that tree and all adjacent (nondiagonal) trees. Many variations on this basic forest fire model have been proposed (7).

Having specified the number of squares in the grid (*N*
_{g}) and the sparking frequency, a computer simulation was run for a number of time steps (*N*
_{S}), and the number of fires (*N*
_{F}) with area (*A*
_{F}) was determined; *A*
_{F} is the number of trees that were burned in each fire. We examined the resulting noncumulative frequency-area distributions for three forest fire model simulations. The number of fires per time step (*N*
_{F}/*N*
_{S}) with area (*A*
_{F}) is given as a function of*A*
_{F} for a grid size of 128 by 128 squares at three sparking frequencies, *f*
_{S} = 1/125, 1/500, and 1/2000 (Fig. 1). The different sparking frequencies represent short and long time intervals between match drops. For all three sparking frequencies, there is a range of small to large fires, with many more small fires than larger ones. The small and medium fires correlate well with the power-law (fractal) relation(1)with α = 1.0 to 1.2. The results for large fires are influenced by the finite-size effect of the grid. A value of α ≈ 1 in Eq. 1indicates that, over the range where the relation holds, small and large fires contribute equally to the total number of trees burned by all fires.

Large forest fires are dominant when the sparking frequency is small (Fig. 1). This dominance is easily explained on physical grounds. For small sparking frequencies or small grid sizes, the grid becomes full before a match sparks a fire. The areas of the fires will generally involve a large number of trees, and in most cases, the fires will span the grid. This transition can be termed the “Yellowstone effect.” Until 1972, Yellowstone National Park had a policy of suppressing many of its fires, resulting in a large accumulation of dead trees, undergrowth, and very old trees (8). This accumulation is analogous to a small sparking frequency in the forest fire model. The grid becomes full, and the likelihood of very large fires is much higher than that in forest fire models with larger sparking frequencies. In 1988, a series of fires in Yellowstone burned 800,000 acres. These very large fires might have been prevented or reduced if, before 1972, the sparking frequency in Yellowstone had been larger (that is, if there had not been a policy of fire suppression). Many individuals in the forest fire community now recognize that the best way to prevent the largest forest fires is to allow the small and medium fires to burn.

We next assessed the frequency-area distributions of actual forest fires and wildfires using four data sets from the United States and Australia. The data sets are from a variety of geographic regions with different vegetation types and climates. The first data set (9) includes 4284 fires on U.S. Fish and Wildlife Service lands during 1986–1995. The second set (10) includes 120 of the largest forest fires in the western United States during 1150–1960; the areas of the fires were interpreted by means of tree rings. The third set (11) includes the areas of 164 fires in the Alaskan boreal forests during 1990 and 1991. The fourth set (12) includes 298 fires in the Australian Capital Territory (ACT) during 1926–1991.

Frequency-area distributions for the four data sets are given inFig. 2. The noncumulative number of fires per year (−*dN˙*
_{CF}/*dA*
_{F}) with area (*A*
_{F}) is given as a function of*A*
_{F} (13). The results shown in Fig. 2are in very good agreement with the power-law relation (Eq. 1) with α = 1.3 to 1.5. A value of α > 1 means that the smallest fires contribute the most to the total area burned by all fires. The relation (Eq. 1) is a straight line in log-log space and is characterized by two parameters: the slope (−α) and the *y* intercept. A knowledge of both parameters for a specific geographic area allows us to characterize the area's frequency-area distribution; given enough data for several years, we can forecast how large the average 10- or 50-year fire will be. The occurrence frequency of small and medium fires can then be used to quantify the risk of large fires.

The actual forest fires (Fig. 2) have good power-law (fractal) distributions over many orders of magnitude, which is consistent with the model data (Fig. 1). However, the model data have slopes (−α) that are somewhat less than the slopes that were derived from the actual data. The environmental and human-related variables that affect the size of wildfires are many and include the proximity and type of combustible material, meteorological conditions, and fire-fighting efforts to extinguish certain fires. Despite these complexities, the application of the statistics associated with the forest fire model appears to be robust.

Our results have a number of practical implications. First, the occurrence frequency of small and medium fires can be used to quantify the risk of large fires. Second, the behavior of the forest fire model can be used to assess the role of controlled burns to reduce the hazard of very large fires.

↵* To whom correspondence should be addressed. E-mail: Bruce{at}Malamud.Com