Analysis of Area Burned by Wildfires Through the Partitioning of a Probability Model

An analysis of forest fires by using a partitioned probability distribution is presented. Area burned during afire is fitted to a probability model. This model is partitioned into small, medium, and large fires. Conditional expected values are computed for each partition. Two cases are presented: the two-parameter Weibull and the Truncated Shifted Pareto probability models. The methodology allows a comparison of area burned and cost for small, medium, and large fires among different attack strategies. The partitioned functions may also be used as a basis to refocus fire management optimization with a multiple objective formulation. Introduction Fire management decisions are frequently made on the basis of many uncertain and highly variable factors, such as fire behavior, weather forecasts, fire effects, and performance of initial attack force. Although most of the modeling of these components have been deterministic, fire managers probably balance their judgments about the uncertain factors against their preferences for possible consequences or outcomes, deviating from the optimal solution provided by a decision-making model. All of these deviations from an optimum solution implicitly recognize the effect of large fires in deciding what level of effort or resources to allocate in fire management. Despite that forest fires are highly stochastic and complex events, most of the decision models that are uses are based on techniques that use the expected value of the damage function, and a few others are deterministic models. Fire management decisions obviously are influenced by the concern about the occurrence of large fires. Large fires with low probability of occurrence have a large impact in natural, social, and economic systems. However, most fires are extinguished in the initial stages and remain small. These smaller fires have a large probability of occurrence, but the resulting damage is practically negligible on an individual basis. Because of these probabilities, the expected value of a damage function misrepresents both extremes. Also, the use of expected value does not adequately represent the consequences associated with different fire management policies. Traditional decision-making and budgeting in forest fire management have been based on the postulate that resources must be allocated in proportion to the resource value to be protected. However, it has been mentioned in several forums that the high costs of fire control are not justified when a strict economic analysis is made because the optimization criterion consists largely of costs (Gale 1977). Moreover, actions that are seen to reduce the risk of the occurrence of large wildfires justify high expenditures made on those large fires, as well as those made on smaller fires. This paper presents a methodology to incorporate different damage levels in fire management decision-making by analytically incorporating large fires. The methodology partitions a probability model to describe area burned, to address different damage levels and their probability of occurrence. The assumption is An abbreviated version of this paper was presented at the Sym­ posium on Fire Economics, Planning, and Policy: Bottom Lines. April 5-7, 1999. San Diego, California. Research Scientist, Field Station for Protected Area Research, College of Forest Resources, Box 352100, University of Washington, Seattle, WA 98195. E-mail: [email protected]. Supervisory Biologist. USDA Forest Service, Pacific Northwest Research Station, 3200S.W. Jefferson Way, Corvallis, OR 97331. e-mail: Sandberg_Sam/ [email protected]. Professor. College of Forest Resources, Box 352100, Universi ty of Washington, Seattle, WA 98195. e-mail: [email protected] USDA Forest Service Gen. Tech. Rep. PSW-GTR-173. 1999. 59 Session II Analysis of Area Burned by Wildfire----Alvarado, Sandberg, Bare that partitioning the probability distribution of fire damage will enable comparisons of the cost of different initial attack strategies for different damage levels. The probability distribution of fire damage will be partitioned to segregate the small fires that have a large probability of occurrence but cause little damage, the large fires that cause extensive damage but have low probability of occurrence, and the intermediate events. By using this approach, more information about fire damage variability is accounted for in the decision process and large fires are analytically included. Modeling the partition containing the large fires is emphasized because they are responsible for most of the damage caused to the forest. A great amount of resources are devoted to fighting or preventing large fires and on restoration. The few large fires, as opposed to the many small fires, are of interest to the public and have more influence on the policies of a fire organization. Partition of a Probability Distribution Show (1921) stated that large fires have been misrepresented in the current fire size classification systems because they have been pooled into a single fire class. He pointed out the need of subdividing the fire classes similarly to a geometric progression according to the fire size. In this paper, we propose that partitioning the distribution will reflect more accurately the fire occurrence distribution. The distribution function fitted to fire sizes is partitioned in three ranges. They represent three damage levels: the small fires with high probability of occurrence; the medium damage fires; and the large fires with a low probability of occurrence but high consequences. For illustration purposes of the partition methodology, we used the twoparameter Weibull and the Truncated Shifted Pareto (TSP) distributions (Alvarado 1992). Those two distributions were fitted to the fire occurrence records from 1961 to 1988 for the Provincial Forest Service of Alberta, Canada; table 1 includes the parameter estimates for those two distributions. The data was also separated by resources used in the initial attack: fires in which only man power was reported, those in which air attack was used, and those fires in which man and other equipment different than aircrafts were used. The number of partitions into which the probability axis is divided depends on the nature of the problem and concerns of the decision-makers. In cases where risk is involved, three partitions are usually made: one is the range for high frequency events with low damage; one for intermediate events; and a third one consisting of the events that represent large losses with low probability of occurrence. Table 1Parameter estimates for the two-parameter Weibull and the Truncated Shifted Pareto distributions. Initial attack strategy Parameter estimates Two-parameter Weibull distribution c ase a ase All fires 0.3263 0.0014 1.718 0.0382 Manpower only 0.3442 0.0193 1.3028 0.0364 Air attack 0.3044 0.0024 2.4165 0.11 Ground attack 0.3344 0.0036 2.4496 0.1349 Truncated Shifted Pareto distribution All fires Manpower only Air attack Ground attack a 0.0605 0.0553 0.0727 0.0769 ase b ase 0.0012 2.1585 0.0197 0.0013 2.0033 0.0243 0.0029 2.2821 0.0404 0.0041 2.3241 0.0551 1 Asymptotic standard error of the estimate. 60 USDA Forest Service Gen. Tech. Rep. PSW-GTR-173. 1999. Analysis of Area Burned by Wildfire----Alvarado, Sandberg, Bare There is no consensus in the literature on the level at which events should be considered as extremes. In some instances the cut-off is made for practical reasons on the probability axis. These are referred to as the "events that exceed a quantile." Examples are sea wave extremes that exceed the level 1:10,000 (Dekkers and Haan 1987) or those with large return periods, e.g., rain depths with a return period of more than 100 years (Foufoula-Georgiou 1989). Large forest fires are usually referred to in terms of area burned. The variance may also be used if the normal distribution is used as a risk function, e.g., one or two times the variance. Another approach is to use past experience regarding damage levels, e.g. fire size classes, emergency state declarations, or others. The partitions mapped on the probability axis can represent three damage levels (fig. 1). The 1 to 1-αi range include the events with large probability of occurrence but cause the lowest damage (0 to βij interval on the damage axis). The events bounded by αi and αij are those events causing an intermediate damage. Extreme value theory and risk management focus on events known as low probability/high consequence events. These are the events in the lower partition of the probability axis that cause the most damage (fig. 1). The partitions used in this study are based on the fire size classes defined by Canada's Province of Alberta (Alberta Forest Service 1985). The first partition includes the fire size classes A and B, which includes the fires that burned from 0.1 to 4.0 hectares. For the second and third partitions we presented two cases. For the second partition, the upper limit depends on the lower limit of the large fire level. It consists either of those in size classes C and D, i.e., from 4 to 200 hectares; or it includes C and D and part of E classes, with the lower limit larger than 4 hectares and the upper limit set to exclude the upper 1 percent of the fires. The third partition encompasses the large fires. Two cases of large fires are studied: fires in the size class E, i.e., larger than 200 hectares, and fires in the upper one percentile of the observed distribution. Once the partition values are defined either in terms of probability or damage, they are mapped onto both the probability and damage axis. Asbeck and Haimes (1984) and Karlsson and Haimes (1988b) consider that the problem is to find a βij for each partition point αi for i=1......,n, and values of sj for j=1,...,q, such that P(βij; sj)= αi. Then, the existence of a unique inverse Px (x;sj) is guaranteed from the following standard probability assumptions: the px(x;s) is nonnegative and: −∞ px (x; s)dx = 1 [1] ∫∞ with the probability of (a<X<b) given by: b P(a < X < b) = ∫a px (x; s)dx [2] The probability and damage axes are partitioned into a set of n ranges, three USDA Forest Service Gen. Tech. Rep. PSW-GTR-173. 1999. Session II