# The Spotting Distribution of Wildfires
## 2 A Transport Model for Firebrand Transport and Combustion
### From Landed Firebrands to the Spotting Distribution  
The \"landed brands\" are those that reach the canopy at height \(z=H\) (where we assume that they fall straight to the ground--meaning we neglect within-stand winds and resettling, although both effects can be considered [49]). We can track these explicitly according to the solution Equation (19) of the impulse-release IBVP for the transport Model (5) described earlier in this Section. We recall from the introduction to Section 2 the distribution keeping track of all brands that have landed before time \(t\), namely the _landing distribution_  
\[L(t,x,m):=\int_{0}^{t}p(s,x,H,m)ds \tag{20}\]  
and recall that the limit as \(t\to\infty\) the _asymptotic landing distribution_  
\[\mathbb{L}(x,m)=\lim_{t\to\infty}L(t,x,m). \tag{21}\]  
Figure 2: **(a)** Schematic of the method of characteristics. A typical characteristic \(Y(t,Y_{0})\) is shown as a curved red line; **(b)** Spatial characteristics (solid lines) for the power-law wind profile, with constant vertical velocity, described in Equation (34). The vertical-axis represents height in metres, while the \(x\)-axis represents downwind distance in metres. Here \(w_{H}=5\), \(\beta=0.5\), \(H=0.5\) and \(v=-1\).  
To determine the spotting distribution, we must first integrate the asymptotic landing distribution appropriately, to obtain the landed masses in kilograms. Employing the inverse combustion operator \(C^{-1}\) and the landing time \(t^{*}(x)\), the mass in kilograms at location \(x\) is given by  
\[\int_{0}^{m}m\ \mathbb{L}\left(x,C^{-1}(m,t^{*}(x))\ \ dm.\right. \tag{22}\]  
Finally we can determine the spotting distribution in terms of the latter, by employing the ignition operator \(E\), which we recall maps mass in kilograms to values in \([0,1]\), as follows:  
\[\mathbb{S}(x)=E\left(\int_{0}^{m}m\ \mathbb{L}\left(x,C^{-1}(m,t^{*}(x)) \right)\ \ dm\right). \tag{23}\]  
Our use of the spotting distribution is similar to the Green's-function approach that is used in the study of linear partial differential equations [50]. The Green's function describes the evolution of an impulse release, and the full solution to a PDE can be found by integrating the Green's function against initial and boundary conditions.  
In the examples in the next Section, we show some illustrative but simple spotting distributions. We have seen that the spotting distribution depends on various physical submodels for firebrand launching, flight, burning, landing and ignition. In the Appendix, we provide a detailed review of the most commonly used submodels, which we summarize in Table 11. Each combination of these submodels leads to a reasonable spotting distribution. This gives more than 500 cases, which we cannot cover in this paper. Hence, in the next Section, we focus on some easy to understand examples, which we identify by their corresponding model number as used in Table 11. We hope that the present work will generate interest, especially in the experimental community, in order to determine the most physically plausible spotting distributions for a variety of important spread scenarios.