# The Spotting Distribution of Wildfires
## Appendix. Ember Release, Burning, Flying and Fuel Ignition
### Distribution of Launched Masses  
A recent series of studies by [PERSON] and colleagues [5; 44; 45] investigated firebrands emitted from the controlled burning of either pine or fir trees. In the [PERSON] experiments, trees both 2.6 m and 5.2 m tall were investigated. For each tree, more than 70 firebrands were collected. These were all cylindrical in shape. The average firebrand length and diameter for the 2.6 m class was 40 mm in length and 3 mm in diameter, while for the 5.2 m class the average was 53 mm in length and 4 mm in diameter. The most recent of the experiments, on Korean pine [44], confirmed that the distribution was approximately the same. The total number of firebrands collected numbered almost 1000.  
With respect to mass, all three of [PERSON]'s studies indicate that between 60 and 80 percent of the firebrands have masses less than 0.1 g. Further, for both pine and fir taller trees produce larger firebrands, with the largest found at about 5 g. In addition, between 58 and 65 percent of the firebrand mass is needles, which are insignificant in long-range spotting, but may be effective at igniting local fuel only in short-range spotting [46]. We remove the needles from [PERSON]'s mass distribution, to obtain an 'effective mass' distribution.  
**Model G1: Regression analysis on [PERSON]'s data.** In order to determine a functional form for the effective mass distribution, we reproduced the histograms from [44; 45], an example of which is shown in Figure A2. Between 60 to 70 percent of the mass is needles which are negligible for long-range spotting. Hence we first removed the needles lying in the 0.1 g mass class, to obtain an effective firebrand distribution. We use non-linear regression of the functional form:  
\[\mu(m)=am^{-b},\hskip 28.452756 pt0\leq m\leq\bar{m}, \tag{10}\]  
where \(\bar{m}=4\), corresponding to the maximum firebrand mass. Regression analysis gives \(b=1/2\), and \(a=1/4\), which gave a better fit than an exponential form.  
**Model G2: Models obtained from burning slash.** Another firebrand distribution was suggested in [32], which relates the possible radius \(r\) of a firebrand to the mass consumption rate \(f\), in the form:  
\[p(r)=\frac{\alpha f}{r\sigma\sqrt{2\pi}}\exp\left(\frac{-1}{2\sigma^{2}}\log \left(r/r_{0}\right)^{2}\right), \tag{11}\]  
where \(r_{0}\) and \(\sigma\) were parameters determined by regression analysis, and \(\alpha\) represents the number of firebrands generated per unit mass. If we assume a relationship of the form \(m=m(r)=\rho V(r)\), where \(\rho\) is the density and \(V(r)\) is the firebrand volume, then from Equation (11) we find the mass density:  
\[\mu(m)=\int_{0}^{\infty}\delta(m-m(r))p(r)dr, \tag{12}\]  
where \(\delta\) denotes the Dirac delta distribution.  
From the experiments of [PERSON] described in this chapter, it was found that approximately one percent of the total mass lost during combustion appeared as firebrands. This could inform our parameter \(\alpha\), and in turn determine the total mass and total number of firebrands released when a given number of trees begin to spot.