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Equilibrium moisture content (EMC)
Equation 
The equilibrium moisture content (EMC), represented as $d_h$, refers to the moisture level of a fuel particle that has had sufficient time to reach equilibrium with its surroundings, as described by Bradshaw et al. in 1983. This moisture content value is dependent on the relative humidity ($RH$) and temperature ($T$) of the surrounding environment. Simard, in 1968, defined $d_h$ as follows:
 $ d_h = \begin{cases}
d_{1,1} + d_{1,2} RH - d_{1,3} RH \cdot T & RH < 0.1 \\
d_{2,1} + d_{2,2} RH - d_{2,3} T & 0.1 < RH < 0.5 \\
d_{3,1} - d_{3,2} RH + d_{3,3} RH ^2 - d_{3,4} RH \cdot T & RH > 0.5 \\
\end{cases}$ |
The relationship between the equilibrium moisture content (EMC) of vegetation and the probability of ignition, as well as the intensity and rate of fire spread, is direct and significant.
Bradshaw, L.S., J.E. Deeming, R.E. Burgan, and J.D. Cohen. 1983. The 1978 National Fire-Danger Rating System: Technical Documentation. USDA Forest Service
Simard, A.J. 1968. The moisture content of forest fuels - 1. A review of the basic concepts. Forest Fire Research Institute, Department of Forestry and Rural Development.
ID:(649, 0)
Moisture damping coefficient
Equation
The presence of moisture in vegetation has a significant effect on the ease of ignition and the ability to sustain a fire, making it a retarding factor in fire spread. Therefore, it is modeled based on the equilibrium moisture content of the vegetation and is represented as a damping coefficient.
The moisture damping coefficient is calculated as follows and represents the fraction by which the flammability of vegetation is reduced:
| $ \eta = 1 - 2\displaystyle\frac{ d_h }{ d_{h0} }+1.5\left(\displaystyle\frac{ d_h }{ d_{h0} }\right)^2-0.5\left(\displaystyle\frac{ d_h }{ d_{h0} }\right)^3$ |
To calculate the damping coefficient using the following equation:
| $ \eta = 1 - 2\displaystyle\frac{ d_h }{ d_{h0} }+1.5\left(\displaystyle\frac{ d_h }{ d_{h0} }\right)^2-0.5\left(\displaystyle\frac{ d_h }{ d_{h0} }\right)^3$ |
First, it's necessary to compute the equilibrium moisture content using the following expression:
| $ d_h = \begin{cases} d_{1,1} + d_{1,2} RH - d_{1,3} RH \cdot T & RH < 0.1 \\ d_{2,1} + d_{2,2} RH - d_{2,3} T & 0.1 < RH < 0.5 \\ d_{3,1} - d_{3,2} RH + d_{3,3} RH ^2 - d_{3,4} RH \cdot T & RH > 0.5 \\ \end{cases}$ |
ID:(648, 0)
Fosberg fire weather index
Equation
The Fosberg Fire Weather Index ($FFWI$) is a fire danger index developed by Fosberg (1978). It is based upon equilibrium moisture content and wind speed, and requires hourly observations of temperature, relative air humidity and wind speed as input data (Fosberg, 1978, Goodrick 2002, Sharples 2009a). It was designed for assessing the impacts of small-scale/short term weather variations on fire potential and is highly sensitive to changes in fine fuel moisture (Goodrick 2002, Crimmins 2005). The $FFWI$ has been found to be correlated with fire occurrence in the northeastern and southwestern United States (Haines et al. 1983, Roads et al. 1997, Sharples 2009a).
The $FFWI$ formulation is divided into a fuel moisture component, corresponding to the equilibrium moisture content defined by Simard (1968), and a rate of spread component based on the Rothermel (1972) model (Goodrick 2002):
| $ FFWI = \displaystyle\frac{10}{3} \eta \sqrt{1 + \left(\displaystyle\frac{ U }{ U_0 }\right)^2}$ |
Fosberg, M.A. 1978. Weather in wildland fire management: the fire weather index. In Proceedings of the Conference on Sierra Nevada Meteorology, Lake Tahoe, California. Boston.
Goodrick, S.L. 2002. Modification of the Fosberg fire weather index to include drought. International Journal of Wildland Fire 11, Nr. 3: 205-211.
Sharples, J.J., R.H.D. McRae, R.O. Weber, and A.M. Gill. 2009a. A simple index for assessing fire danger rating. Environmental Modelling and Software 24, Nr. 6 : 764-774.
Crimmins, M.A. 2006. Synoptic climatology of extreme fire-weather conditions across the southwest United States. International Journal of Climatology 26, Nr. 8 (6): 1001-1016. doi:10.1002/joc.1300.
Haines, D.A., W.A. Main, J.S. Frost, and A.J. Simard. 1983. Fire-danger rating and wildfire occurrence in the Northeastern United States. Forest Science 29: 679-696.
Roads, J.O., F. Fujioka, H. Juang, and M. Kanamitsu. 1997. Global to Regional Fire Weather Forecasts. International Forest Fire News 17.
Simard, A.J. 1968. The moisture content of forest fuels - 1. A review of the basic concepts. Forest Fire Research Institute, Department of Forestry and Rural Development.
Rothermel, R.C. A mathematical model for predicting fire spread in wildland fuels. Intermountain forest and range experiment station.
ID:(647, 0)
Modified Fosberg Fire Weather Index
Equation
The fact that the $FFWI$ does not take rainfall into account was considered as problematic, in particular for capturing spatial variations in fire potential in regions where spatial variability of rainfall is important (Goodrick 2002). Therefore, a rainfall component, in the form of a fuel availability factor ($FAF$) was added by Goodrick (2002) to the $FFWI$ in order to take the impact of drought on fuels into account.
The modified $mFFWI$ is then obtained by multiplying the fuel availability factors $FAF$ with the $FFWI$:
| $ mFFWI = FAF \cdot FFWI $ |
Goodrick, S.L. 2002. Modification of the Fosberg fire weather index to include drought. International Journal of Wildland Fire 11, Nr. 3: 205-211.
ID:(650, 0)
Fuel availability factor
Equation
The fuel availability factor ($FAF$), introduced by Goodrick in 2002 to correct the fact that the Fosberg Index does not take into account the effect of precipitation, is a function of the Keetch-Byram Drought Index ($KBDI$) and is calculated as follows:
| $ FAF = FAF_0 + \left(\displaystyle\frac{ KBDI }{ KBDI_0 }\right)^2$ |
By utilizing the Keetch-Byram-Drought Index (KBDI), this factor reflects fuel availability, as it directly depends on the moisture retained in the soil:
• The type of vegetation and organic material, as well as their flammability, are conditioned by the moisture levels in the soil.
• The quantity of available fuel in the area is significantly impacted by soil moisture levels, as dryness can reduce the fuel's availability for a fire.
• The moisture content within the fuel can diminish its flammability and slow down fire propagation.
• The distribution of fuel, such as its concentration and continuity, plays a role in accelerating fire spread.
• Terrain topography also plays a crucial role, as slopes can accelerate fire propagation, while valleys tend to reduce fuel availability.
Goodrick, S.L. 2002. Modification of the Fosberg fire weather index to include drought. International Journal of Wildland Fire 11, Nr. 3: 205-211.
ID:(651, 0)
Variation of humidity deficit
Equation
The soil moisture deficit $Q$ is calculated by subtracting the available water $w$ from the capacity $w_c$. Since the variation in available water is a function of evapotranspiration, which depends on temperature $T$ and past precipitation $P$, we can establish an equation for the variation of moisture deficit in terms of temperature and previous precipitation.
Assuming, based on experimental data [1], that the evapotranspiration function for temperature takes the form:
$f_1(T) = q_{1,1}e^{T/q_{1,2}}-q_{1,3}$
And for precipitation, it follows the form:
$f_2(P) = \displaystyle\frac{1}{1+q_{2,1}e^{-P/q_{2,2}}}$
We can derive an equation for the variation of the deficit as follows:
| $ \Delta Q = ( w_c - Q )\displaystyle\frac{ q_{1,1} e^{ T / q_{1,2} }- q_{1,3} }{1+ q_{2,1} e^{- P / q_{2,2} }} \Delta t $ |
[1] Keetch, J.J., and G.M. Byram. 1968. A Drought Index for Forest Fire Control. Southeastern Forest Experiment Station, Asheville, North Carolina.
ID:(654, 0)
Net rainfall
Equation
Precipitation reduces moisture deficit. Therefore, Keetch-Byram (1968) defines a net precipitation $P_{net,t}$ that should be subtracted from the moisture deficit. In this context, a base loss of 5 mm is assumed, so the procedure is as follows:
• If the rain lasts only one day and does not exceed 5 mm, it is not considered as rain for that day.
• If it rains for more than one day, only the days after the first one that exceed 5 mm are considered, and 5 mm are subtracted once.
Keetch, J.J., and G.M. Byram. 1968. A Drought Index for Forest Fire Control. Southeastern Forest Experiment Station, Asheville, North Carolina.
ID:(653, 0)
Keetch-Byram drought index
Equation
The Keetch-Byram Drought Index ($KBDI$) was developed in the United States by Keetch and Byram in 1968 to measure drought and facilitate wildfire control operations.
This index is cumulative and requires daily temperature and daily and annual precipitation data as input. Its primary purpose is to reflect the dryness, and hence the flammability, of organic material in the soil, taking into account the effects of rain and evapotranspiration on moisture deficiency in both shallow and deep soil layers (Keetch and Byram, 1968).
$KBDI$ has been used in practical applications in the southeastern United States and, to some extent, in the northeastern United States (Burgan, 1988). It has also been employed in research studies in other regions, such as the Hawaiian Islands and northern Eurasia (Dolling et al., 2005; Groisman et al., 2007). $KBDI$ has been incorporated into the United States' National Fire Danger Rating System (NFDRS; Burgan, 1988; Melton, 1989).
Since the $KBDI$ index represents the accumulated moisture deficit at a given moment, it is calculated by summing the day-to-day variation from the last day $k_0$ when there was no deficit:
| $ KBDI_k = KBDI_{k-1} - P_{net,k} + ( w_c - KBDI_{k-1} + P_{net,k} )\displaystyle\frac{ q_{1,1} e^{ T_k / q_{1,2} }- q_{1,3} }{1+ q_{2,1} e^{- P / q_{2,2} }} \Delta t $ |
To calculate the $KBDI_t$ index, we should first consider the previous deficit $KBDI_{t-1}$ minus the net precipitation $P_{net,t}$:
$Q = KBDI_{t-1} - P_{net,t}$
Then, we add the variation in moisture deficit $\Delta Q$:
$KBDI_t = KBDI_{t-1} - P_{net,t} + \Delta Q$
The variation in moisture deficit is calculated using the following expression:
| $ \Delta Q = ( w_c - Q )\displaystyle\frac{ q_{1,1} e^{ T / q_{1,2} }- q_{1,3} }{1+ q_{2,1} e^{- P / q_{2,2} }} \Delta t $ |
This value is obtained using the expression for $Q$:
| $ KBDI_k = KBDI_{k-1} - P_{net,k} + ( w_c - KBDI_{k-1} + P_{net,k} )\displaystyle\frac{ q_{1,1} e^{ T_k / q_{1,2} }- q_{1,3} }{1+ q_{2,1} e^{- P / q_{2,2} }} \Delta t $ |
The day $k_0$ can be considered to be when there has been a period of abundant rainfall, typically corresponding to a one-week period with significant precipitation.
Keetch, J.J., and G.M. Byram. 1968. A Drought Index for Forest Fire Control. Southeastern Forest Experiment Station, Asheville, North Carolina.
Burgan, R.E. 1988. 1988 revisions to the 1978 national fire-danger rating system. Southeastern Forest Experiment Station, Asheville, North Carolina.
Dolling, K., P.S. Chu, and F. Fujioka. 2005. A climatological study of the Keetch/Byram drought index and fire activity in the Hawaiian Islands. Agricultural and Forest Meteorology 133, Nr. 1: 17-27.
Melton, M. 1989. The Keetch/Byram Drought Index: A Guide to Fire Conditions and Suppression Problems. Fire Management Notes 50, Nr. 4: 30-34.
ID:(652, 0)
Drought factor
Equation
The drought factor, expressed in terms of the $KBDISI$ index, is defined as follows (Noble et al., 1980):
| $ DF_k = min[ f_0 ,\displaystyle\frac{ f_{1,1} (KBDI_k + f_{1,2} )( n +1)^{3/2}}{ f_{2,1} ( n +1)^{3/2} + P_n - f_{2,2} }]$ |
In this formula, $n$ represents the number of days since the last rainfall and the last amount of precipitation $R$. This factor reflects the dryness of vegetation based on the time elapsed since the last rainfall, as well as the amount of precipitation and any moisture deficit present in the soil.
It is important to note that this coefficient does not account for the effects of minimum humidity, maximum temperature, and average wind speed in the environment, which are included in the wildfire danger index.
ID:(655, 0)
Mark 5 forest fire danger index
Equation
The Mark 5 Forest Fire Danger Index (FFDI) was developed by McArthur in 1967 to assess the risk and behavior of fires in eucalyptus forest fuel types and has been widely used in eastern Australia (Noble et al., 1980; Sharples et al., 2009a). The FFDI requires maximum temperature ($T_{max}$), minimum relative humidity ($RH_{min}$), mean wind speed ($U_{mean}$), and a fuel availability index ($DF$), which represents a measure of dryness.
The drought factor is determined based on soil moisture deficit (calculated using the Keetch-Byram Drought Index, KBDI), the time since the last rainfall, and the amount of rainfall.
The FFDI is defined as follows:
| $ FFDI_k = \left(\displaystyle\frac{ DF_k }{ i_1 }\right)^{i_0} e^{- RH_{min} / i_2 + T_{max} / i_3 + U_{mean} / i_4 }$ |
The calculation of the Forest Fire Danger Index $FFDI$ is based on the following equation:
| $ FFDI_k = \left(\displaystyle\frac{ DF_k }{ i_1 }\right)^{i_0} e^{- RH_{min} / i_2 + T_{max} / i_3 + U_{mean} / i_4 }$ |
To calculate this index, we first need to compute the Drought Factor $DF$ using the following formula:
| $ DF_k = min[ f_0 ,\displaystyle\frac{ f_{1,1} (KBDI_k + f_{1,2} )( n +1)^{3/2}}{ f_{2,1} ( n +1)^{3/2} + P_n - f_{2,2} }]$ |
And to obtain the Keetch-Byram Drought Index $KBDI$, the following calculation is required:
| $ KBDI_k = KBDI_{k-1} - P_{net,k} + ( w_c - KBDI_{k-1} + P_{net,k} )\displaystyle\frac{ q_{1,1} e^{ T_k / q_{1,2} }- q_{1,3} }{1+ q_{2,1} e^{- P / q_{2,2} }} \Delta t $ |
Through this formula, fire risk is assessed, taking into account various meteorological and fuel availability factors:
| FFDI range | Fire danger class |
| 0 - 5 | low |
| 5-12 | moderate |
| 12-25 | high |
| 25-50 | very high |
| > 50 | extreme |
ID:(656, 0)