# The Spotting Distribution of Wildfires
## 1 Introduction
### The Primary Questions of This Article  
Motivated by all the problems which spotting causes, the central questions which we address here or in future work are:  
1. What is the _spotting distribution_, or the probability of spot fire ignition, at each location downwind of an existing fire front?In this paper we will focus primarily on a mechanistic approach to the derivation of the spotting distribution. Of equal importance is the development of experimental approaches to _measure_ the spotting distribution; we discuss this problem in Section 4.2. As soon as the spotting distribution is found, we can ask important follow-up questions, such as:  
What is the probability that a fire will _breach_ an obstacle?  
What role does spotting play on the _rate of spread_ of a fire front? Can spotting cause a wildfire to quickly traverse a region across which it would spread slower with purely local spread?  
Can spotting _accelerate_ a fire's advance?  
In this paper we will address Question 1 in Sections 1.4 and 2. Here we will borrow heavily from methodologies proposed in plant, insect and animal dispersal [27; 28]. Questions 2, 3 and 4 have been discussed in detail in the PhD thesis of [PERSON] [29], while Problem 3 has been further discussed in the more general mathematical framework of _birth-jump processes_ in [30]. The problem of connecting theory to experimental measurement is taken up in the Discussion.  
Our approach is based on existing physical principles and experimental results, summarized in Table 13 from the Appendix, to derive the spotting distribution. In Section 2 we present a new transport model for firebrand transport and combustion. We model the time-mean behaviour of trajectories and combustion, since turbulence creates high variability in the atmospheric paths of individual firebrands. The authors have not been able to find published quantitative spotting distributions from more complicated physics-based models (e.g., fire-atmosphere models, or the extensive physics-based model by [PERSON] [31]). In fact, the only known article presenting a spotting distribution is that of [PERSON] [32], though our methodology is much more general. Our ability to provide spotting distributions, incorporating realistic sub-processes as discussed in the Appendix, is the greatest strength of the present article.  
We also introduce, in Section 2, a transport and combustion model in the form of a hyperbolic partial differential equation (PDE), based on the _launching distribution_, the _horizontal wind profile_, the _terminal falling velocity_, the _burning rate_ of the flying brands, ending with a discussion of the _ignition probability_ for a landed firebrand to generate a spot fire.  
In Section 3 we discuss how the models of Section 2 can be used to determine the spotting distribution. We provide analytical solutions of the transport PDE from Section 2, and employ these to generate examples of the spotting distribution. The analytical solutions are used to illustrate how the characteristic components of the model, such as horizontal wind, terminal falling velocity or combustion model may influence the spotting distribution.  
In our discussion of measuring the spotting distribution in Section 4 and in the Appendix, we describe a number of successes which have been achieved in describing some of the above subprocesses. Most of these are fairly recent, as the wildfire research community has become increasingly interested in understanding spotting. We also draw attention to components of the spotting process which are relatively poorly understood, taking the lead from experimentalists in Dispersal Ecology [27]. We suggest potential methodologies which may be employed in experimentally measuring spotting distributions.