# Wind Reduction Factor
The rate of spread models often require a wind input at a certain height.
Generally, they need the wind of mid-flame height, representing the average influence of the wind on the flame.
However, it is only sometimes possible to get this information.
The wind information can come from stations at a specific height (20 ft, 10 m) or an atmospheric model.
The latter vertical resolution is often larger than the midflame height itself.
Therefore, a method is needed to estimate the midflame wind speed from wind speed information at a different height.
The wind reduction factor, noted $\alpha$ [-], is introduced to get the wind at the midflame height $U_f$ from the wind at a different height, noted $U_r$:
$$ U_f = \alpha \, U_r. $$
To compute the wind reduction factor, the method is generally based on a specific vertical profile that describe the wind speed as a function of height, considering the vegetation cover.
The following sections describe methods to compute the wind reduction factors from a vertical profile and vegetaiton properties.
To apply a wind reduction factor in a workflow, you can use the function `firebench.wind_interpolation.apply_wind_reduction_factor`. The block for this function can be found in `firebench/docs/_static/diagram_blocks/process/apply_wind_reduction_factor.svg`.
## Baughman and Albini (1980)
This section is based on [1, 2].
The wind profile above the vegetation cover is given by:
$$U (z) = \frac{U_*}{\kappa} \ln \left ( \frac{z - D_0}{z_0} \right ),$$
where $U(z)$ is the wind speed at the height above ground level $z$, $U_*$ is the friction velocity, $\kappa$ is the Von Karman constant, $D_0$ is the zero-plane displacement, and $z_0$ is the reoughness length.
[1] sets $D_0 = 0.64\,h$, and $z_0 = 0.13\,h$, where $h$ is the vegetation height.
### Unsheltered wind reduction factor for 20-ft reference wind
When considering that the reference wind speed is 20 ft above the fuel top, the wind reduction factor is given by:
$$\alpha (h, h_f) = \frac{1 + 0.36 h / h_f}{\ln \left ( \frac{20 + 0.36h}{0.13h} \right )} \left [ \ln \left ( \frac{h_f/h + 0.36}{0.13} \right ) -1 \right ],$$
where $h$ is the fuel height [ft], and $h_f$ is the flame height [ft].
This wind reduction factor does not interpolate the value to midflame height but allows to calculate the average wind speed over the flame length.
**Note**
This formula can be applied to [Anderson](../21_fuel_models/02_Anderson.md) fuel model, considering $h_f/h=1$, to retrieve the wind reduction factor contained in the dataset. However, the values for fuel category 7 and 8 are different from the values in [1], where the value 0.36 is given in [1] whereas the formula gives 0.28.
### Generalized unsheltered wind reduction factor
The methodology described in [1, 2] can be applied to any reference height for the input wind, noted $h_r$.
The advantage of this formulation is that is does not force the input wind to a certain height and it works with any units (m or ft) as long as they are the same for all input variables.
The definition of $h_r$ can be different depending on the context, explicited in the following paragraphes.
In any case, the following integral needs to be computed:
$$\int_h^{h+h_f} U(z) dz = \mathcal F (h+h_f) - \mathcal F (h),$$
where
$$\mathcal F (z) = (z - D_0) \ln \left( \frac{z - D_0}{z_0} \right ) - z.$$
#### Reference height is defined above the ground level
When the reference wind height $$h_r$$ is given from the ground level, the wind reduction factor is given by:
$$\alpha (h, h_f, h_r) = \frac{1}{h_f U (h_r)} \int_h^{h+h_f} U(z) dz.$$
Fig. 1
:
Interpolation of midflame wind from a wind at height defined above ground level.
#### Reference height is defined above the vegetation level
When the reference wind height $h_r$ is given from the top of the vegetation layer, the wind reduction factor is given by:
$$\alpha (h, h_f, h_r) = \frac{1}{h_f U (h+h_r)} \int_h^{h+h_f} U(z) dz.$$
Fig. 2
:
Interpolation of midflame wind from a wind at height defined above vegetation top.
## References
[1] [Baughman, R. G., & Albini, F. A. (1980, April). Estimating midflame windspeeds. In Proceedings, Sixth Conference on Fire and Forest Meteorology, Seattle, WA (pp. 88-92).](https://www.frames.gov/catalog/8151)
[2] [Albini, F. A. (1979). Estimating windspeeds for predicting wildland fire behavior (Vol. 221). Intermountain Forest and Range Experiment Station, Forest Service, US Department of Agriculture.](https://www.frames.gov/catalog/8152)