# The Spotting Distribution of Wildfires
## 3 Examples of the Spotting Distribution
### Case (W1,V1, F0, L3, I3): A Family of Simple Spotting Kernels  
Let us consider the threshold ignition law (I3) presented in the Appendix, in the case where \(\underline{m}\ \to\ 0\) (see Equation (A35)). Then any firebrand landing on a location which is not burning, will instantly generate a fire. We again have constant \(w\) and \(v\), and further we will suppose that no mass is lost during transport corresponding to the constant burning law \(f=const\) with zero rate of combustion (the transport process being assumed very rapid). Let's assume that the firebrand vertical launching distribution \(\phi(z,m)\) satisfies the assumption (L3), so that  
\[\phi(z,m)=\mathbb{Z}(z)\ \mu(m). \tag{48}\]  
This says the lofting heights \(z\) are independent of the masses \(m\). This may occur, for example, in the case of well-mixed line thermals. We can choose the mass distribution \(\mu(m)\) in accordance with experiments such as in the experiments by [PERSON] [5], or out of mathematical curiosity we could consider any other probability distribution. What is most important is to assume that \(\mathbb{Z}(z)\) is not exponentially bounded, in order to obtain a fat-tailed kernel. For example, one could assume:  
\[\mathbb{Z}(z)=e^{-z\beta}, \tag{49}\]  
for \(\beta\in(0,1)\). This kernel is of particular interest here, since it decays sub-exponentially, and has been shown in integro-difference models [26] to give rise to accelerating propagation of the corresponding solution in space [26]. Physically this could correspond to extreme spotting conditions like in the presence of fire whirls. From Equation (46), with the form Equation (49) appearing in the landed mass distribution \(M(x)\), we find that for a release of \(N\) firebrands the landed distribution \(M(x)\) of masses at location \(x\) is:  
\[M(x) = N\int_{0}^{\underline{m}}m\ \mathbb{Z}\left(|v|\frac{x}{w} \right)\ \mu(m)\ dm \tag{50}\] \[= N\int_{0}^{\underline{m}}m\ e^{-(|v|\frac{x}{w})^{\beta}}\ \mu(m)\ dm\] (51) \[= e^{-(|v|\frac{x}{w})^{\beta}}\ N\int_{0}^{\underline{m}}m\ \mu(m)\ dm\] (52) \[= he^{-(|v|\frac{x}{w})^{\beta}}, \tag{53}\]  
where \(h>0\) is the total landed mass at \(x\), namely  
\[h:=N\int_{0}^{\underline{m}}m\ \mu(m)\ dm<\infty. \tag{54}\]  
Since we are assuming instant ignition, the spotting distribution is the same as the landed mass distribution. Referring to Equation (50), we can then write the spotting kernel in this case:  
\[\mathbb{S}(x)=he^{-(\frac{|v|x}{w})^{\beta}}. \tag{55}\]  
This whole procedure can be generalized. If instead of the kernel Equation (49) we use an arbitrary kernel \(\mathbb{Z}(z)\), fat-failed or not, repeating the steps leading to Equation (55), we obtain the spotting distribution:  
\[\mathbb{S}(x)=h\mathbb{Z}(|v|\ \frac{x}{w}). \tag{56}\]  
We can also extend the latter formula to include our other combustion models, by altering \(h\) defined in Equation (54) to read  
\[h:=N\int_{0}^{\underline{m}}m\ \mu\left(C^{-1}(m,\frac{x}{w})\right)\ dm, \tag{57}\]  
To obtain the spotting distribution, we need an ignition law. Here we use the piecewise linear ignition function with lower ignition threshold \(\underline{m}\) and we obtain  
\[\mathbb{S}(x)=E(M(x))=\left\{ \begin{array}{ll}\frac{M(x)}{\underline{m}}&M(x) \leq\underline{m}\\ 1&M(x)>\underline{m}\end{array} \right. \tag{71}\]  
with \(M(x)\) given by Equation (70). Hence we find an explicit formula for the spotting distribution. As examples we chose the parameters as in Table 1 and we plot the landed mass \(M(x)\) and the spotting distributions \(\mathbb{S}(x)\) for these cases in Figure 3.  
The parameters in Table 1 have been chosen to model realistic physical scenarios, at the lower limit where spotting might begin to have an impact. The windspeed \(w\) of two metres per second is on the low end of observed values [3]. Terminal speed of one metre per second is also low, but on the same order of magnitude as firebrands with diameter, mass and length as described in [PERSON]'s study.