Weather Influence Score

Basic Definition

The Weather Influence Score ($WIS$) is the measure of how much a given flight path is influenced by the weather.

Given the 4-D path of an airfract $\vec{x}=\left(t,x,y,z\right)$, the $WIS$ is computed as:

$$WIS = C_P I_P + C_C I_C + C_T I_T + C_W I_W + \dots$$

Where $C_X$ are coefficients associated to different weather quantities of interest, and $I_X$ stands for the integration of the flight path over the influence of the different weather quantities. For example:

$I_P$ is the integration of the flight path over the Precipitation Influence Function through the path

$$I_P = \int_S \Gamma_P(\vec{x})d^4x$$

Where $S$ is the path in question, and $\Gamma_P(\vec{x})$ is the Precipitation Influence Function at the time and location $\vec{x}$, which translates solely to the precipitation measure from weather stations $\Gamma_P(\vec{x}) = P(\vec{x})$.

In some cases, the Influence Function of a quantity can be more complex, for example; temperature. There's an optimal temperature at which turbines operate, too high or too little would decrease the performance, so a simple model for the Temperature Influence Function would be: $$\Gamma_T(\vec{x}) = 1 - A e^{-k\left(T\left(\vec{x}\right) - T_{opt}\right)^2}$$ Or some other efficiency function if available.

Current Influence Functions Used

Below are the current $\Gamma_x$ for the quantities of interest.

Precipitation ($\Gamma_P$)

The Precipitation Influence Function $\Gamma_P$, as stated above, is simply a direct map from the precipitation $P$ measure at $\vec{x}$ $$\Gamma_P(\vec{x}) = P(\vec{x})$$

Clouds ($\Gamma_C$)

The Clouds Influence Function $\Gamma_C$ is a mapping from the cloudiness condition $C$ at location $\vec{x}$ asserted at different heights by the cloudiness column in weather data. That column is an interpolation of the "Sky Coverage" and "Sky Location" variables skycX and skylX, translated between 0 and 1 and interpolated.

$$\Gamma_C(\vec{x}) = C(\vec{x})$$

Temperature ($\Gamma_T$)

As mentioned before, the Temperature Influence Function will be dependent on some engine efficiency model. For our case we'll simply use the aforementioned efficiency model: $$Eff(T(\vec{x})) = a e^{-k\left(T(\vec{x}) - T_{opt}\right)^2}$$, where:

$a$ is the engine's peak efficiency at $T(\vec{x})=T_{opt}$

$k$ is a measure of how fast the efficiency drops around the peak optimal temperature.

$T_{opt}$ is the optimal temperature of operation of the engine.

Thus we can model $\Gamma_T$ as:

$$\Gamma_T(\vec{x}) = 1 - Eff(\vec{x})$$

Wind ($\Gamma_W$)

The wind influence is straightforward, however it's important to note that the estimation of wind speed and direction is not a trivial matter and thus, it is likely that the final coefficient associated with it's integral $C_W$ will be small.

The wind influence will be a piecewise function, as headwind is beneficial during takeoff and landing, but not during cruising.

$$ \Gamma_W(\vec{x}) = \begin{cases} \hat{v}_a(\vec{x})\cdot\hat{v}_w(\vec{x}) \frac{\left|\vec{v}_w(\vec{x}) - \vec{v}_a(\vec{x})\right|}{\left|\vec{v}_a(\vec{x})\right|}, & \vec{x} \in cruise \ -\hat{v}_a(\vec{x})\cdot\hat{v}_w(\vec{x}) \frac{\left|\vec{v}_w(\vec{x}) - \vec{v}_a(\vec{x})\right|}{\left|\vec{v}_a(\vec{x})\right|}, & \vec{x} \in takeoff\ or\ \vec{x} \in landing \ \end{cases} $$

That is: For cruising, if the plane and wind are going in the same direction, the influence is positive, and is scaled by the relative speed of the wind and plane. If the wind is against the plane the influence is negative. The opposite applies for takeoff and landing.

Estimating the coefficients $C_x$

For the estimate of the coefficients associated to each of the quantities, it boils down to an optimization problem. That is, what combination of coefficients $C_x$ best predicts the Fuel Consumption Index ($FCI$)? That is:

$$FCI = C_o + C_P I_P + C_C I_C + C_T I_T + C_W I_W + \dots$$

And then the $WIS$ would simply be defined as $WIS = FCI - C_o$, where $C_o$ would be a measure related to fuel consumption in optimal conditions.