The Cell

The cell is the object that fixes the instances of fuel component models to a geographic location. The grid cell knows its surface area, geographic coordinates, landcover type, slope, aspect, and elevation. It also contains any number of fuel component objects that represent its ability to burn. Since, in this simulation, a grid cell is 30m resolution, or 900m$^2$ area, the 1 hour fuel component that it contains is a model of all of the grassy material in that 900m$^2$ area. The 10 hour object represents all of the twigs, and so on.

While the actual burning of material in each component is conducted within the object, the spatial dynamics of the burn is dependent on messages that are sent between objects. The cell is responsible for coordinating messages between its various fuel components, forwarding messages from its burning fuel components to other cells, and delivering messages from other cells to its fuel components. The important communication is between fuel components; since the cell bundles all messages from components together to send them to other cells, it is convenient to refer to this message passing as communication between cells. This abstraction will be used for the remainder of this discussion; the reader should be aware that references to ``messages between cells'' are in fact ``messages between fuels components in cells''.

In this simulation, all of the messages are requests to burn. In a flat, homogeneous landscape, the urgency $U$ of the request from one cell $i$ to another cell $j$ to burn would simply be a function of the intensity of the burn in cell $i$ ($I_i$), and inversely proportional to the distance $d_{ij}$ between the cell centroids.

$$U_{ij} \sim \frac{I_i}{d_{ij}}$$ (1)

If a cell is burning intensely, a neighboring cell should receive a correspondingly urgent request to burn. The neiboring cell's decision to act on that request will be a function of its internal state; if it's burnable, it will likely honor the request to burn. If it's just not flammable, or has no fuel left to burn, it will stay dormant regardless of the urgency of the requests it recieves. In an alternate formulation, the distance parameter may be replaced with a simple connectivity function $c_{ij}$, reflecting whether cells are adjoining or not.

Fire spread is highly influenced by wind and topography. In this simulation, these factors are implemented by adjusting the urgency of requests from a burning cell to its neigbor according to windspeed, wind direction, slope, and aspect. It was considered important, in keeping with the principles of cellular modeling, that the cell sending the request be responsbile for adjusting the urgency of the request; the recieving cell should have no notion of the request other than it being ``urgent'' or ``not urgent''.

  

Figure 3: The inverse cosine function

Scaling of urgency according to wind and topography was accomplished by taking the inverse cosine of the difference in direction to the to cell $j$ (azimuth $A_{ij}$) and to wind direction $W$ and slope aspect $a$, respectively.

$$-\cos (A_{ij} - W) \hspace{1cm} -\cos (A_{ij} - a)$$ (2)

The inverse cosine gives a value of $-1$ ``head on'' (0$^0$ or 360$^0$), 1 in the opposite direction (180$^0$), and 0 at right angles to the wind or slope (figure 3). This inverse cosine of wind and slope direction gives the effect of deflating the level of urgency upwind (down slope) and correspondingly inflating the urgency downwind (up slope). The degree (amplitude) to which the urgency of the request gets deflated or inflated is a function of windspeed $V_w$ and steepness of slope $S$, respectively. The function that the cell uses to adjust the urgency of the request from cell $i$ to $j$ ($U_{ij}$) then becomes

$$U_{ij} = \frac{\alpha I_i - \beta V_w \cos (A_{ij} - W) - \gamma S_i \cos(A_{ij} - a_i)} {d_{ij}^\delta}$$ (3)

where $I$ is the intensity of the burn in cell $i$, $V_w$ is the wind velocity, $A_{ij}$ is the azimuth to cell $j$, $W$ and $a$ is the wind direction and slope aspect, $S$ is slope percent, $d_{ij}$ is the distance between cells $i$ and $j$, and $\alpha$, $\beta$, $\gamma$, and $\delta$ are scaling parameters. Determination of the appropriate values of $\alpha$, $\beta$, $\gamma$, and $\delta$ is an ongoing problem beyond the scope of this discussion.

In the abscence of wind or slope (when $V_w, S \rightarrow 0$), this equation reduces to

$$U_{ij} = \frac{\alpha I_i}{d_{ij}^\delta}$$ (4)

which is a simply parameterized version of equation 1.

This urgency is calculated for every burning cell at every time step. If any parameter were to change from one time step to the next, the cell would simply respond by changing the urgency of its request to its neighbor at the next time step. This way, if there were to be a sudden change in windspeed or direction, every cell would simply react accordingly.

In summary, the cell acts as a broker for the fuel components, taking the burn messages, adjusting the magnitude of the message for localized conditions, and transmitting the messages to the appropriate neighbors.