# The Spotting Distribution of Wildfires
## Appendix. Ember Release, Burning, Flying and Fuel Ignition
### Firebrand Combustion  
Several combustion models have been proposed in the literature, and the interested reader is invited to peruse the review in [7]. They are usually modeled as continuous-time processes, so we will present functional forms for the density, effective radius, or simply the mass as functions of time. Suppose the mass \(m(t;m_{0})\) is the unique solution of the initial value problem  
\[\frac{dm}{dt}=f,\hskip 28.452756 ptm(0)=m_{0}. \tag{101}\]  
where \(f\) is the rate of combustion. We introduce the _combustion operator_\(C\) as the unique solution  
\[C(m_{0},t):=m(t;m_{0}). \tag{102}\]  
In our spotting problem, we need to compute backwards. If a fire brand of mass \(m\) lands, then we like to know its initial mass \(m(-t)\). Hence we define the inverse combustion operator \(C^{-1}(m,t)\), which is the unique solution to the initial value problem \(\frac{dm}{dt}=-f,m(0)=m\) at time \(t\geq 0\). Before continuing, it is important to note that the combustion process does not continue past the point where the firebrands have masses less than or equal to zero. Hence the combustion operator is only defined up to the burnout time \(t_{b}\), and similarly for its inverse. **Model F0: Constant burning rate. ** Suppose the burning rate is a constant \(f=-\kappa<0\), then we have \(m(t)=m(0)-\kappa t<m(0)\), and \(C(m_{0},t)=m_{0}-\kappa t\). The inverse satisfies \(C^{-1}(m,s)=m+\kappa s\) for \(s>0\). **Model F1: [PERSON]'s original experiments and models. ** The first experiments on the density and shape changes in combusting firebrands were carried out by [PERSON] and collaborators at the Aerospace Corporation [1]. This data was fit by regression analysis obtained from windtunnel experiments, where both spherical and cylindrical firebrands were examined under a variety of ambient windspeeds. Suppose \(\rho(t)\) represents the density of a firebrand at time \(t\), and \(\rho(0)\) is the initial density. Then [PERSON] found the density varied as:  
\[\rho(t)=\frac{\rho(0)}{1+\eta t^{2}}, \tag{103}\]  
where the constant \(\eta=2.86\times 10^{-4}\) was determined from regression analysis. **Models F2: Caricature of [PERSON]'s model. ** We employ an analogue of Tarifa's density evolution for the mass \(m(t)\), namely  
\[m(t)=\frac{m(0)}{1+\eta t^{2}}. \tag{104}\]  
**Model F3: Constant mass. ** If we take case F2 to its extreme, sending \(\eta\to 0\), we get  
\[m(t)=m(0). \tag{105}\]  
**Model F4: [PERSON] and [PERSON]'s improvements. ** [PERSON] and [PERSON] revisited Tarifa's data set [75], and determined the model which best fit the data for the effective radius evolution is:  
\[r(t)^{4}=r(0)^{4}-\frac{\chi\beta^{2}t^{2}}{16}. \tag{106}\]  
Clearly the latter is only defined while \(r(t)>0\), which implies a finite burnout time. Here the parameter \(\chi\) depends on the wood species and moisture content of the firebrand, and \(\beta\) is described below in the derivation of Equation (101). Based on Tarifa's data, the parameter \(\chi=3.5\) gave the best fit [69]. To determine the mass \(m(t)\), we first employ the simple relation:\[m(t)=\rho(t)\text{Vol}(r(t)), \tag{100}\]  
where \(\text{Vol}(r(t))\) is the volume of the firebrand, and hence dependent on the radius \(r(t)\). As a simple example, suppose the firebrand is approximately spherical, so that \(\text{Vol}(r(t))=\frac{4}{3}\pi r(t)^{3}\). Then we combine Equations (101) and (102), so that the mass evolution Equation (100) is given by  
\[m(t)=\rho(t)\frac{4}{3}\pi r(t)^{3}=\frac{4}{3}\frac{\rho(0)}{1+\eta t^{2}} \left(r(0)^{4}-\frac{\chi\beta^{2}}{16}t^{2}\right)^{3/4}. \tag{101}\]  
Taking for example cedar wood as representative of forest fuel, as did the authors in [22], the initial density \(\rho(0)\) of the firebrands would be \(513\) kg/m\({}^{3}\), while in the experiments of [PERSON] [44] he found radii ranging from one half to five centimetres.