INTRODUCTION 



In the process of planning and evaluating forest fire control activities, it is 

 often desirable to have available quantitative expressions relating the effort expended 

 in fire suppression and the results to be expected under various conditions. It is the 

 purpose of this paper to introduce a mathematical method for the analysis of fire sup- 

 pression (or fire containment! activity which can be used to derive such relationships. 

 We show, by example, how the method can be applied to determine the burned area and the 

 time required for suppression, using a simple but flexible expression for the shape of 

 the fire. 



In addition, we explore the sensitivity of the burned area and the time required 

 for suppression to the following factors: 



(]] The size of the fire at the start of suppression. 



(2) The rate of spread of the fire in the front, flank, and back directions, and 

 a parameter describing the shape of the fire. 



(3) The rate of suppression (or rate of control line construction). 



(4) The tactics employed. 



In the following section we introduce the basic concepts of and the limitations on 

 the method of analysis and present the general analytical forms. In subsequent sections 

 we introduce simplifying assumptions, present examples, and extend the analysis to the 

 situations requiring a change of tactics to stop a rapidly spreading head fire. 



BASIC CONCEPTS 

 AND LIMITATIONS OF METHODS 



We seek to describe analyt ical 1>' tlie rate and direction of progress of a fire- 

 suppression force as a function of the shape and rate of growth of a fire. For this 

 purpose, we assume that the work proceeds, if ])ossible, at tlie edge of tlie fire. It is 

 not important wliethcr the effort is directed toward extinguishing tlie flames (direct 

 suppression in U.S. fire control tcrminolog)') , or toward construction of a barrier 

 across which the fire will not spread. Tlic matliemat ical description of the rate of 

 containment is the same. But it is necessary to restrict our attention to fires which 

 do not spread by spotting and whicli have a perimeter tliat is a smooth curve. 



If the fire meets the conditions described and the work of sup]iress i on (or contain- 

 ment) proceeds at tlie fire edge, it is possible to determine the boundary of burned area 

 and the time that the work will retjuirc, if the jiosition of the fire edge can be 

 expressed analytically as a function of position and time. 



1 



