# The Spotting Distribution of Wildfires
## 3 Examples of the Spotting Distribution
### Case (W1,V1, F0, L3, I3): A Family of Simple Spotting Kernels
The upper and lower bounds on the mass, measured in kilograms, correspond to the values found in [PERSON]'s combustion experiments [5]. The rate of combustion, chosen between 0.03 and 0.05 g per second, results in flaming burnout in under two minutes. Choosing 1000 firebrands, corresponds to combustion of about ten trees--a moderate estimate for the width of our front. The decay rates \(\lambda\) were chosen so that firebrand launching would drop off appreciably beyond several hundred metres. Finally the parameter \(a\) is a normalization constant for our mass distribution. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Parameter** & \(w\) & \(v\) & \(\hat{m}\) & \(\underline{m}\) & \(a\) & \(\kappa\) & \(N\) & \(\lambda\) & \(x_{max}\) \\ \hline base case (thick solid) & 2 & \(-1\) & 0.004 & 0.001 & 7.91 & 0.00005 & 1000 & 0.01 & 160 \\ slow burner (dotted) & 2 & \(-1\) & 0.004 & 0.001 & 7.91 & 0.00003 & 1000 & 0.01 & 266.66 \\ lower release height (dashed) & 2 & \(-1\) & 0.004 & 0.001 & 7.91 & 0.00005 & 1000 & 0.05 & 160 \\ \hline
**Parameter** & \(w\) & \(v\) & \(\bar{m}\) & \(\underline{m}\) & \(a\) & \(\eta\) & \(N\) & \(\lambda\) & \(x_{max}\) \\ \hline [PERSON]’s case (thin blue) & 2 & \(-1\) & 0.004 & 0.001 & 7.91 & 0.000286 & 1000 & 0.01 & \(\infty\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the model parameters for the examples (in metric units). Figure 3: **(a)** Examples of landed mass distributions, with mass in kilograms along the \(y\)-axis and distance in metres along the \(x\)-axis; **(b)** Corresponding spotting distributions, dimensionless values on the \(y\)-axis and distance in metres along the \(x\)-axis. The parameters are from Table 1, with the base case in thick solid black, the slow burning case in dotted, the lower release height case in dashed and [PERSON]’s model in thin solid blue. Next we consider the effect of changing combustion model, choosing instead our modified version of [PERSON]'s model (F2), while keeping all other parameters fixed (see Table 1). It is thus important to reconsider the inverse combustion operator. According to the model (F2), we have  
\[C^{-1}(m,t^{*}(x))=m\ \left(1+\eta\frac{x^{2}}{w^{2}}\right), \tag{72}\]  
since this is the initial mass, which travelled through the air after launching during a time of \(t^{*}=\frac{x}{w}\). We can start with Equation (65) and replace the term \(m+\kappa\frac{x}{w}\) by \(m\left(1+\eta\frac{x^{2}}{w^{2}}\right)\). Then the landed mass is computed to be  
\[M(x)=\frac{2}{3}N\lambda e^{-\lambda|v|\frac{x}{w}}a\left(1+\eta\frac{x^{2}}{ w^{2}}\right)^{-2}\hat{m}^{\frac{3}{2}} \tag{73}\]  
and the spotting distribution \(S(x)\) is again explicitly given by Equation (71). As an example we use parameters as in Table 1 and plot the landed mass and the spotting distribution as thin blue lines in Figure 3. Notice that the inverse combustion operator in Equation (72) is always positive for all \(x>0\). Hence in this case we have no maximum spotting range and \(x_{max}=\infty\), even though it is effectively zero beyond 200 m.  
It is important to determine the minimum spatial extent of the spotting distribution required for spotting to be important. Below this extent, the main fire will outrun spot fires before they can develop into a separate front. As a specific example, [PERSON] has estimated such an effective distance for crown fires in North America to be approximately 300 m (in the most extreme ignition conditions [3]). In Figure 3 we consider [PERSON]'s crown fire situation. In comparing the Tarifa-case with the dotted curve (base case with slower burning), both distributions extend at least approximately to the 300 m cutoff--though in the Tarifa-case the distribution is essentially zero there. We further note that, while not shown, just halving the combustion rate in the base case pushes the distribution past the 300 m mark. By changing the combustion model or parameters, we obtain very different outcomes for the importance of spotting.