## 2 Material and Methods
### Probability of Smoldering Fire Ignition in Organic Soils  
The probabilistic method was designed by [PERSON] [8]. This method relates the probability of ignition with the content of inorganic matter, the bulk density and the moisture content of the sample Equation (1):  
\[P=1/(1+exp^{-(B_{0}+B_{1}xH+B_{2}xIM+B_{3}xBD)}) \tag{1}\]  
where \(B_{0}\), \(B_{1}\), \(B_{2}\) and \(B_{3}\) are coefficients that vary according to the probabilistic distribution of each sample group. \(H\) is the moisture content, in %, \(IM\) is the mineral content, in %, and \(BD\) is the bulk density kg/m\({}^{-3}\). The calculation of the moisture content was done at a probability of ignition of 50% (P\({}_{50}\)) for each sample. Then, the result is expressed as the maximum moisture content given for ignition to occur. Isolating the percentage of maximum moisture content at a probability of 50% results in Equation (2):  
\[H\left(\%\right)=(B_{0}+B_{2}xIM+B_{3}xBD)/B_{1} \tag{2}\]  
Given the lack of tools to experimentally analyze ignition on the samples collected, the probability coefficients used (\(B_{0}\), \(B_{1}\), \(B_{2}\) and \(B_{3}\)) are Frandsen [8] for the sample group _'Pine duff (Seney)'_ (Table 3). This sample group have similar mean values of inorganic matter (IM, \(36.5\pm 16\%\)), organic matter (OM, \(63.5\pm 16\%\)) and bulk density (BD, \(190\pm 19\) kg/m\({}^{-3}\)) at a 5 cm depth as the samples collected in Rocallaura (Table 2).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Study Site** & _Pinus resinosa_[8] & _Pinus ponderosa_[12] & **Commercial Peat [9]** \\ \hline Depth (cm) & 4.98 & 5 & 4–8 & 6 \\ IM (\%) & 35.5 & 36.5 & 25 & 3 \\ BD (kg/m\({}^{3}\)) & 160 & 190 & 50–275 & 50–150 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of mean values of soil properties with past studies of modelling organic soil combustion.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(B_{0}\) & \(B_{1}\) & \(B_{2}\) & \(B_{3}\) \\ \hline Pine duff (Seney) & 45.1778 & \(-\)0.3227 & \(-\)0.3644 & \(-\)0.362 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Coefficients used for the probabilistic model [8].  
Experimental Design to Examine the Effect of Soil Moisture and Bulk Density on the Smoldering Spread Rate  
The spread rate of samples that could smolder is estimated using the linear regression model proposed by [PERSON] et al. [9] (Equation (3)). This model estimates the spread rate in relation to the moisture content and bulk density by experimental analysis of commercial peat smoldering ignition at different bulk densities. The samples used in [PERSON] et al. [9] had a depth of 6 cm, an average mineral content of 3% and bulk density ranges that were representative of peat and humus in peat bogs (50 kg/m\({}^{3}\) to 150 kg/m\({}^{3}\)) and, therefore, comparable to the bulk densities of the organic stratum (Oe + Oa) of this study (48 kg/m\({}^{3}\) and 190 kg/m\({}^{3}\)). The greatest difference with the samples from this study lay in the mineral content (mean 35%).  
\[Log\left(Vp\right)=A_{0}+A_{1}\times MC+A_{2}\times BD+A_{3}\times MC\times BD +\varepsilon \tag{3}\]  
where \(Vp\) is the velocity of propagation or spread rate (cm/h), \(MC\) the soil moisture, estimated using dry mass basis (%), \(BD\) the bulk density (kg/m\({}^{3}\)), \(A_{0}\), \(A_{1}\), \(A_{2}\) and \(A_{3}\) are the coefficients of the dependent variables and \(\varepsilon\) is the residual term. Given that we do not have the necessary tools to perform the same experiments and so obtain the model coefficients, we opted to use those that provide the best fit with the linear model proposed by [PERSON] et al. [9] (\(R^{2}=0.77\)) and as is detailed in Table 4.