# The Spotting Distribution of Wildfires
## 1 Introduction
### Ignition of Fuel Beds by Firebrands  
One of the most challenging problems in modelling the spotting process, is to determine the ignition probability \(E\), which is the probability for ignition to occur once a firebrand has made contact with a given fuel bed. Since the fire landscape is very heterogeneous, the ignition probability \(E\) may vary spatially, and can depend on a variety of factors, like:  
* The species of plants emitting firebrands. * Landed firebrand characteristics like diameter, length, and mass. * The travel time \(t^{*}\) from launch to landing. * The moisture content of the fuel bed and local weather. * The surface area, and thermal conductivity between firebrand and fuel. * Whether the firebrand is in a \"glowing\" or \"flaming\" state upon landing. * Variability of firebrand type within the launching stand (e.g., a coniferous tree might emit both small brands or cones). * Whether there is a \"re-settling\" after landing due to slope or wind. * Whether there is a shading effect from the sun due to the presence of the convection column. An ideal model for ignition would account explicitly for all factors just mentioned. Several physics-based models have attempted to do just that (e.g., [31]), but necessarily include many equations whose analysis is mostly limited to computer simulation. Since we are interested in fire fronts, which occur at the macroscopic scale, we can ignore some of the smaller-scale details in our development of ignition models. Prior to 2006 the experimental investigation of this topic was limited to qualitative descriptions of ignition. The experiments show that fuel bed moisture, firebrand mass and geometry are the most important characteristics determining ignition probability [5; 9; 42]. The Fine Fuel Moisture Code (FFMC) used by the FBP system [34], which provides a numerical measure for the moisture contained in forest litter, has been a useful standard for determining ignition probabilities. It is determined in turn by the Fire Weather Index, another component of the FBP system [34]. The lower the FFMC, the higher the fine fuel moisture content. Experiments by the Aerospace Corporation [16; 43] found that for high fine moisture contents only large flaming brands cause ignition, while for low fine fuel moisture content glowing embers may easily ignite a fuel bed. [PERSON] [13] reports that spotting can be significant when fine fuel moisture content is below ten percent, and confirmed the earlier results by [43] that glowing embers can be sources of ignition. Following these experiments, in about 2006, [PERSON] [44] began qualitative experiments on firebrand ignition, for brands emitted during controlled laboratory burning of pine and fir trees. This work is a collaboration between the National Institute of Standards and Technology in the USA, and the Building Research Institute in Japan [45], where ongoing experiments may help further quantify the ignition process. One important result following from the combustion experiments of [PERSON], is that firebrands are not produced if the dead fuel moisture content exceeds thirty percent. One can further postulate that there will always be a maximum moisture above which spotting does not occur. Together these results suggest, as is common knowledge, that lower fine fuel moisture content (high FFMC) can correspond to greater spotfire risk. A still more recent paper [46], is the first to describe ignition probabilities using regression analysis on systematic experimental data. The authors performed a number of controlled lab experiments to determine the time of ignition, rate of spread, rate of combustion, maximum and mean flame heights, and ignition frequency of fuel beds for a variety of fuel beds, representative of the Mediterranean. Examples of fuel beds include a variety of pine, eucalyptus, and grass beds, which could also be representative of fine fuels from forests in North America and Australia.