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Characterizing wildfire regimes in the United States

Communicated by Donald L. Turcotte, University of California, Davis, CA, February 2, 2005 (received for review August 12, 2004)
Abstract
Wildfires statistics for the conterminous United States (U.S.) are examined in a spatially and temporally explicit manner. We use a highresolution data set consisting of 88,916 U.S. Department of Agriculture Forest Service wildfires over the time period 19702000 and consider wildfire occurrence as a function of ecoregion (land units classified by climate, vegetation, and topography), ignition source (anthropogenic vs. lightning), and decade. For the conterminous U.S., we (i) find that wildfires exhibit robust frequencyarea powerlaw behavior in 18 different ecoregions; (ii) use normalized powerlaw exponents to compare the scaling of wildfireburned areas between ecoregions, finding a systematic change from east to west; (iii) find that wildfires in the eastern third of the U.S. have higher powerlaw exponents for anthropogenic vs. lightning ignition sources; and (iv) calculate recurrence intervals for wildfires of a given burned area or larger for each ecoregion, allowing for the classification of wildfire regimes for probabilistic hazard estimation in the same vein as is now used for earthquakes.
Over the last decade, highprofile wildfires (1, 2) have resulted in numerous fatalities and loss of infrastructure. Wildfires also have a significant impact on climate and ecosystems; recently, several researchers (37) have emphasized the need for regionallevel examinations of wildfireregime dynamics and change, and the factors driving them. With implications for hazard management, climate studies, and ecosystem research, there is, therefore, significant interest in appropriate analysis of historical wildfire databases. Insightful studies using wildfire database statistics exist (524) but are often hampered by the low spatial and/or temporal resolution of their data sets. Here, we use a highresolution database of wildfires for the conterminous United States (U.S.), combined with techniques drawn from recent advances in statistical physics and complexity theory, to examine U.S. wildfire statistics in a spatially and temporally explicit manner.
Statistical physics and complexity theory have begun to be applied to a wide range of natural hazards (e.g., refs. 25 and 26). One characteristic of many of these studies is powerlaw (scaleinvariant) statistical distributions (27), in which the probability of a certain value occurring is raised to some power of that value. For instance, earthquakes follow a powerlaw relationship of the frequency (number) vs. energy released, the GutenbergRichter relationship (28). The frequencysize statistics of many other natural hazards also appear to satisfy powerlaw distributions to a good approximation under a wide variety of conditions (29). These natural hazards include asteroid impacts (30, 31), landslides (32, 33), volcanic eruptions (34), and the subject of this paper, wildfires. Powerlaws and other “heavytailed” distributions are increasingly being used by reinsurance companies and governments for probabilistic hazard analysis (35, 36) and are playing a growing role in environmental and social policy decisions.
The wildfire regime encompasses the timing, frequency, and magnitude of all wildfires that occur in a region. The term “wildfire,” as used in this paper, is taken to mean any burned area, irrespective of size or ignition source. Recent studies of wildfire regimes suggest frequencyarea probability distributions that are powerlaw (813) or otherwise “heavytailed” (1419) over many orders of wildfire area. The powerlaw takes the form
with frequency density ḟ(A _{F}), the number of wildfires in “unit” bins with A _{F} burned area, and β and α constants. Local and broad regional studies of wildfires in the U.S. (9), Australia (9), Italy (10), and China (12), for example, have shown powerlaw behavior with exponents ranging from β = 1.11.8, over two to five orders of magnitude of wildfire area, for 1209,000 individual wildfire events. However, the low spatial and temporal resolution of the data sets used in these studies has made analyses as a function of wildfire regime drivers difficult. Therefore, in this paper, we use a highresolution data set (discussed in the next section) consisting of U.S. Department of Agriculture Forest Service (USFS) wildfireoccurrence records [ref. 37; the wildfire data discussed in ref. 37 were obtained through personal communication with T. J. Brown (University of Nevada, Reno)] for N _{FT} = 88,916 wildfires (A _{F} ≥ 0.004 km^{2} = 1 acre) between 1970 and 2000. We then examine the resultant wildfire burnedarea statistics both spatially and temporally at regional scales as a function of ecological and anthropogenic driving factors. First, we will discuss the data and methods; next, the results of analyses; and, finally, the general implications of having robust powerlaw behavior for wildfire statistics in each ecoregion.
Data and Methods
Data Quality and Completeness. Using 657,949 wildfires recorded between 1970 and 2000 by the USFS and the U.S. Department of the Interior (DOI), Brown et al. (37) compiled an inventory and performed a coarse assessment of the quality of historical federal wildland wildfireoccurrence records. They found 90.7% (324,122 wildfires) of USFS records and 71.0% (214,687 wildfires) of the DOI records “useable” in terms of their completeness and spatial coordinates. Furthermore, they note that DOI wildfire reporting was not continuous for most of the 1970s and that for both the USFS and DOI data, counts of very small wildfire sizes are often incomplete because wildfires may go undetected or unreported. Therefore, in our study we use only those Brown et al. (37) records (i) from the USFS and (ii) with areas A _{F} ≥ 0.004 km^{2} = 1 acre, giving in our data set (for 19702000) a total number of wildfires N _{FT} = 88,916 and total area burned A _{FT} = 68,994 km^{2} (see Table 1 for numbers of wildfires and area burned per ecoregion). Wildfires in our study were classified as “anthropogenic” (64% by number) or “lightning” (36%) according to the description given by Brown et al. (37). All nonanthropogenic fires are here termed “lightning,” because of the very small occurrence (<0.01%) of “other” natural causes (e.g., volcanism) (37).
Bailey's Ecoregions. To allow spatial analyses with regard to the biophysical factors that drive wildfire regimes, we classify the USFS wildfire data into the ecoregion divisions developed by Bailey (38) (Fig. 1A ). Ecoregions distinguish geographic regions that share common biophysical characteristics. In Bailey's classification (38), a threelevel hierarchy is used: domains (based primarily on climate), divisions (climate, vegetation, and soils), and provinces (climate, vegetation, soils, landsurface form, and fauna). Mountainous areas within specific divisions and provinces are also classified. In the conterminous U.S., there are three domains, subdivided into 19 divisions and further divided into 34 provinces. Ecoregion divisions are shown in Fig. 1 A , with division names, codes, and areas in Table 1. Using Bailey's ecoregion division level in our analyses allows spatial disaggregation of the data, at a resolution suitable for the study of potential wildfireregime drivers, while ensuring that an adequate number of wildfires are available for statistical analyses. Wildfires occurring on USFS lands are used as a representative sample of the ecoregion division in which they are located. USFS land area per ecoregion is given in Table 1, with the spatial distribution in the conterminous U.S. shown in Fig. 1B . Of the 18 ecoregion divisions with wildfire data, USFS lands range from 0.2% (Prairie) to 51.3% (Temperate Steppe Mtns.) of the respective ecoregion's area.
FrequencyArea Statistics. In each ecoregion division, we examine the frequencyarea statistics of USFS wildfires. Because the wildfire inventories used are not “complete” (there are many more “smaller” wildfires than measured), probability densities are not appropriate; instead, we use frequency densities f(A _{F}) defined as
where A _{F} is the wildfire burned area and δN _{F} is the number of wildfires in a “bin” of width δA _{F}. The frequency densities f(A _{F}) are then the number of wildfires per “unit” bin. We use frequency densities because cumulative frequencies can obscure underlying trends in finite data sets (39). Because there are many more small wildfires than larger ones, we increase our bin width δA _{F} with increasing area A _{F}, to give approximately equal bin widths in logarithmic coordinates. The USFS areas over which the wildfires occur change from one ecoregion to another (Table 1 and Fig. 1B ). Therefore, to allow specific comparison between ecoregions, we take the frequency densities ḟ(A _{F}) (fires·km^{2}) and normalize them by (i) the USFS area (km^{2}) within each ecoregion and (ii) the period of observation (yr), to give normalized frequency densities ḟ(A _{F}) (fires·yr^{1}·km^{4}).
For each ecoregion division, the normalized frequency densities ḟ(A _{F}) are plotted as a function of wildfire area A _{F}. As we will show later (see Results), an excellent fit in each case is given by the inverse powerlaw distribution (Eq. 1), a twoparameter distribution that forms a straight line in loglog space. In each case, we estimate the best fit for log[ḟ(A _{F})] = β log[A _{F}] + logα, where β is the gradient and logα is the yintercept. The powerlaw exponent β quantifies the ratio of the number of large to small wildfires in a given area, with β = 0 indicating the same number of large as small wildfires (per “unit” size bins). As β increases, large events become rarer with respect to small ones. The yintercept, or logα, is the normalized number of wildfires per unit bin; in our case, A _{F} = 1 km^{2}.
Calculation of Recurrence Intervals. An extension of having the two parameters α and β in the inverse powerlaw distribution given in Eq. 1 is the calculation of the recurrence interval T(≥A _{F}), based on the probability that in a defined spatial area (i.e., some specific region A _{R}), a given size event with area A _{F} will be equaled or exceeded in any given year. For example, if in a defined region there is an average 1 in 100 chance per year that a wildfire of area ≥10 km^{2} will occur, this size event (10 km^{2}) is said to have a 100yr recurrence interval. In other words_{,} assuming that the events are uncorrelated in time (Poissonian), then for this region there will be, on average, 100 yr between wildfires of size 10 km^{2} or greater. In calculating the recurrence intervals associated with different wildfire areas A _{F}, we first take the integral of Eq. 1 to arrive at N _{CF}(≥A _{F}), the cumulative number of wildfires with areas greater than or equal to A _{F}, giving (for β > 1)
where α and β are constants from Eq. 1, τ is the data set duration in years, and A _{R} is the spatial area over which the probabilistic “hazard” is to be considered. As A _{R} increases, the number of wildfires of a given size or larger N _{CF}(≥A _{F}) in that region also increases. We could terminate the integral in Eq. 3 at some A _{max} instead of infinity, because the powerlaw distribution given in Eq. 1 must have some upper bound. However, because the numbers of the most extreme events are few, the shape of the tail at these extreme values is unclear. Therefore, we choose to consider “small” and “medium” values of A _{F}, well below the maximum wildfire area that might occur in any given ecoregion.
Using a Weibull equation, the recurrence interval T(≥A _{F}) associated with a wildfire of a given area or larger is τ + 1, the total number of years in the data set plus one, divided by N _{CF}(≥A _{F}), giving (for β > 1)
The length of our data set is τ = 31 yr, and we consider the recurrence interval of wildfires in spatial areas of A _{R} = 1,000 km^{2}, giving (for β > 1)
with A _{F} in km^{2} and T(≥A _{F}) in years. In Results, we present recurrence intervals by ecoregion for A _{F} = 0.01 km^{2} and A _{F} = 10 km^{2} for spatial areas within each ecoregion of A _{R} = 1,000 km^{2}.
Results
FrequencyArea Statistics of Wildfires. In Table 1, for each ecoregion division, we give the results of our normalized frequencyarea analyses. Each ecoregion division exhibits excellent frequencyarea powerlaw behavior (r ^{2} ≥ 0.96) over more than five orders of magnitude for burned area. In each case, the wildfire statistics have been examined as a function of ecoregion division, regardless of ignition source (lightning vs. anthropogenic). Also included in Table 1 are ±2σ error bars (see legend) equivalent to lower/upper 95% confidence intervals for both β and logα. Two example analyses are given in Fig. 2, with the Mediterranean Ecoregion (β = 1.30 ± 0.05) and the Subtropical Ecoregion (β = 1.81 ± 0.07) exhibiting the smallest and largest β values for the 18 ecoregions, respectively. Note that care should be taken in interpreting parameter values for ecoregion divisions with very small wildfire numbers (e.g., N _{FT} < 100; Marine and Warm Continental Mtns.), because these have increased uncertainty in estimates for β and logα.
Spatial Distribution of β. We next map the spatial distribution of β values by ecoregion (Fig. 3A ), where β is the result of each frequencyarea analysis shown in Table 1. Fig. 3A suggests an easttowest gradient of highertolower β values across the conterminous U.S. We note that Fig. 3A does not take into account the ±2σ error bars on β (95% confidence intervals) as given in Table 1. However, even taking the error bars into account, there is still strong evidence of the easttowest gradient of β values across the conterminous U.S.
Lightning vs. Anthropogenic Wildfires. To explore alternative hypotheses given for the easttowest gradient of β, we examine wildfire records with reference to ignition source, whether natural (lightning) or anthropogenic. The numbers of anthropogenic vs. lightning wildfires in our data set varies as a function of ecoregion division, with the ratio (N _{F anthropogenic}/N _{F lightning}) = 0.20.8 in the seven Temperate and Tropical/Subtropical ecoregions (all divisions with codes >300); 1.24.7 in Marine, Marine Mtns., Mediterranean Mtns., and Subtropical Mtns.; and 1143 in the remaining seven ecoregion divisions.
Within each ecoregion division, we examine β_{anthropogenic} and β_{lightning} and find (β_{anthropogenic}/β_{lightning}) > 1 in the eastern third of the U.S. (35% by area), where β_{anthropogenic}/β_{lightning} = 1.30 (Hot Continental), 1.21 (Warm Continental), 1.14 (Hot Continental Mtns.), and 1.12 (Subtropical). Most other areas have (β_{anthropogenic}/β_{lightning}) ≈ 1 (within ±2σ error bars, as described in Table 1 legend), except for the Temperate Steppe division, where (β_{anthropogenic}/β_{lightning}) = 0.88. At Bailey's ecoregion “domain” level (38), where divisions with related climates are grouped, we find that ecoregion divisions with (β_{anthropogenic}/β_{lightning}) > 1 generally fall into the Humid Temperate domain.
Wildfire Statistics as a Function of Decade. In each ecoregion we also examine the wildfire data by decade (19701979, 19801989, and 19901999), both by different ignition source and for all fires irrespective of ignition source. We find similar results for β and logα as for the entire 31yr period (Table 1). However, there is a small (statistically nonsignificant) decrease of 312% in β values from the decades 1970s to 1990s for all ecoregions except Warm Continental Mtns., Marine, and Prairie (each of which have scant data).
Wildfire Recurrence Intervals. For each ecoregion we use Eq. 5 to calculate recurrence intervals T(≥0.01 km^{2}) and T(≥10 km^{2}). This probabilistic hazard analysis gives us the average time between events with burned areas greater than or equal to 0.01 and 10 km^{2}, respectively, occurring in a defined spatial “area” within each ecoregion. For comparison between ecoregions, we will consider relatively small spatial areas of size 1,000 km^{2}.
To examine the strength of temporal correlation in wildfire areas (also see ref. 24), we examined the time lags between successive wildfire areas for specific ecoregion divisions, taking different lower cutoff bounds for the wildfire areas used. We find that the wildfire events exhibit shortterm but not longterm memory; the smallest wildfire areas are correlated in time, but the medium and large ones are approximately uncorrelated (i.e., Poissonian). For medium and large events, this allows us to calculate recurrence intervals based on the results of the frequencyarea wildfire statistics found earlier in Table 1.
Using Eq. 5, the recurrence intervals T(≥A _{F}) for each ecoregion division are given in Table 1, including ±2σ error bars as calculated from the error bars on β and logα. Because of the small amount of data used to fit the medium/upper tail of the distribution in Eq. 1, the ±2σ error bars on T(≥A _{F}) are large, averaging 3060% of the actual recurrence interval value. Despite this, there are clear differences between ecoregions. For example, the Mediterranean Ecoregion has T(≥10 km^{2}) = 2 ± 1 yr. In other words, for any 1,000km^{2} “area” in this ecoregion, we “expect” on average one wildfire with burned area A _{F} ≥ 10 km^{2} every 13 yr (33100% probability of occurring in any year). By contrast (Table 1), the Warm Continental Ecoregion has T(≥10 km^{2}) = 203 ± 99 yr; the occurrence probability for a wildfire with A _{F} ≥ 10 km^{2} has dropped significantly to 0.31.0% in any given year, a factor of ≈100 between the two ecoregion divisions. A spatial mapping for T(≥10 km^{2}) is given in Fig. 3B . For both the eastern and western thirds of the U.S., there is a gradient from large to small recurrence intervals (i.e., lower to higher hazard) going from north to south, with the largest recurrence intervals of wildfires (lowest hazard) in the northeast U.S. Our method for calculating wildfirerecurrence intervals gives a simple and quick way of determining approximate quantitative hazard assessments of given size wildfires (or larger) occurring across the conterminous U.S.
Discussion
Spatial Distribution of β. The easttowest gradient in β values (Fig. 3A ) observed at Bailey's ecoregion division level suggests that the ratio of the number of large to small wildfires decreases from east to west across the conterminous U.S. Controls on the wildfire regime (e.g., climate and fuels) vary temporally, spatially, and at different scales (3), so it is difficult to attribute specific causes to this easttowest gradient. For example, the observed reduced contribution of large wildfires to total burned area (i.e., β large) in eastern ecoregion divisions may be due to greater human population densities that increase forest fragmentation compared with western ecoregions (40). Alternatively, the observed gradient may have natural drivers, with climate and vegetation producing conditions more conducive to large wildfires in some ecoregions compared with others.
In other studies, gradients similar to that observed here have been described and related to climate and vegetation. Turner et al. (41) describe wildfireoccurrence gradients, as a function of altitude and latitude, in crown fire ecosystems in continental northern North America. They attribute these gradients to broad climatic variations and note that western and central regions tend to have relatively frequent fires with forest stand structures dominated by younger individuals, whereas the eastern region experiences longer interfire intervals and older stand structures. Other studies (5, 22, 23, 42) have found climatic variability to be a dominant factor affecting wildfire regimes at large temporal and spatial resolutions and extents. A broadscale gradient in wildfireregime characteristics across potential natural vegetation types has also been observed in the U.S. (42) and Spain (6). Potential natural vegetation is the successional endpoint (”climax”) vegetation of an area, in the absence of climate change or human disturbance.
To examine potential drivers of the easttowest gradient in β values observed, we also examined wildfire records with reference to ignition source. The ratio β_{anthropogenic}/β_{lightning} > 1 in the eastern third of the conterminous U.S. suggests a potential influence of human activity on the relative scaling of wildfireburned areas, because these areas are more populated. It has been suggested (20) that increased landscape heterogeneity decreases disturbance (e.g., wildfire) spread. Historic anthropogenic forest clearance, resulting in forests fragmented by agricultural and urban land cover, has increased the heterogeneity of eastern landscapes (40). This may have reduced the relative number of large to small fires in the east compared to the west. In western landscapes, forests have also become fragmented, but via replacement by shrublands and grasslands (40) that are more conducive to wildfire spread than the agricultural and urban land covers that have fragmented eastern forests.
We suggest that the use of the powerlaw exponent β to characterize wildfire regimes, in a similar fashion used in this paper, will be useful in future wildfireregime studies. In our analyses, the influence of climatic factors on the observed easttowest gradient in β across the conterminous U.S. is not clear. However, examining the relationships between β and past/current climates could aid estimates of future wildfire regime behavior (specifically the scaling of fire sizes) by linking these relationships to models of future climate change. Furthermore, research considering the spatial relationship between β and net primary production (the total stored energy/biomass produced by photosynthesis, minus the energy used in autotrophic respiration, in a unit area) may be useful in examining human impacts on the relative scaling of wildfireburned areas due to management or manipulation of vegetated lands.
SelfOrganization in Wildfire Regimes. Wildfires influence vegetation, which, in turn, influences future wildfire activity (43). Within ecosystems, the feedbacks between process (the wildfires) and the ecological patterns (vegetation type, age, physiognomy, etc.) produce both spatial complexity and temporal memory effects (44, 45). Both spatially and temporally, a “structured” pattern results, with some degree of spatial and temporal autocorrelation. The patterns in vegetation both constrain, and are in turn constrained by, the processes that generate them. For an area of forest that has never been burned, after a wildfire occurs there will be a pattern, the “mosaic,” of burned and unburned patches of vegetation (e.g., ref. 46). These patterns influence the next wildfire that occurs. Recently burned areas may be less flammable (for example) than older ones. The next wildfire is constrained by the old pattern but will create a new one, and the feedback continues (47).
In this article, we have discussed powerlaw or scaleinvariant statistical distributions for wildfireburned areas. Bolliger et al. (48) queried whether, in nature, selforganizing dynamics would overcome environmental gradients to ensure powerlaw behavior. We certainly acknowledge that there are always upper and lower cutoffs in nature for any powerlaw behavior, and the same is true for wildfires. For instance, in the case of wildfires, an upper boundary might be the barrier presented by a mountain range or high topography dividing drainage networks. A lower boundary might be the partial burning of a single tree or bush, or finescale discontinuities in the fuel bed. However, despite these feedbacks and upper/lower boundaries, we have shown that wildfires within each of 18 different ecoregion divisions have frequencyarea relationships that are robustly scaleinvariant over more than five orders of burned area.
This robust powerlaw (scaleinvariant) behavior of wildfire areas might be taken as evidence that ecosystems selforganize through the feedbacks described above to ensure that energy is dissipated at the maximum rate across all scales. Selforganization has previously been discussed for both real and model ecosystems (912, 25, 47, 4953). Thus, variation in β between ecoregions may indicate differences in energy balance and rates of energy release in ecosystems according to differences in climate. An examination of β and its spatial and temporal relationships with the net primary production and the potential natural vegetation, in conjunction with more accurate estimates of energy released during combustion currently being developed (54), would contribute to understanding the role of wildfire in ecosystem energetics and the organization within open, dissipative systems (such as ecosystems) in general.
Recurrence Intervals of Wildfires. Using both α and β, we have calculated the recurrence intervals T(≥0.01 km^{2}) and T(≥10 km^{2}) for each ecoregion division (Table 1 and Fig. 3B ). This type of mapping, and extensions of it, will prove very useful for government agencies and reinsurance groups when examining wildfire hazard.
Conclusions
Wildfires are costly in terms of damage and fatalities. The ways in which they begin and propagate are complex, with an individual wildfire depending strongly on meteorological conditions, topography, vegetation, and, of course, wildfirefighting efforts. Despite this complexity, the frequencyarea of wildfires in conterminous U.S. ecoregions appear to robustly follow powerlaw (heavytailed) distributions, over more than five orders of burned area in each ecoregion. The simplicity of accepting powerlaw (or similar heavytailed) distributions, which exhibit scaleinvariant behavior with excellent fits over many orders of magnitude, allows the use of the parameters β and α to describe the relative contribution and hazard of large wildfires within a wildfire regime. In turn, normalized β and α values allow for the explicit comparison and examination of wildfireregime dynamics and change between ecoregions, as illustrated here.
Acknowledgments
We thank Timothy J. Brown, for access to the USFS wildfire data set. We also thank the two reviewers, Ian Main (University of Edinburgh) and Malcolm Gill (Centre for Plant Biodiversity Research, Canberra, Australia), for their constructive and comprehensive comments. The contributions of B.D.M. were partially supported by United Kingdom Natural Environment Research Council/Engineering and Physical Sciences Research Council Grant NER/T/S/2003/00128.
Footnotes

↵ † To whom correspondence should be addressed. Email: bruce{at}malamud.com.

Author contributions: B.D.M., J.D.A.M., and G.L.W.P. designed research; B.D.M. and J.D.A.M. performed research; B.D.M. contributed new reagents/analytic tools; B.D.M. and J.D.A.M. analyzed data; and B.D.M., J.D.A.M., and G.L.W.P. wrote the paper.

Abbreviations: U.S., United States; USFS, U.S. Department of Agriculture Forest Service.
 Copyright © 2005, The National Academy of Sciences
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