Generating Membership Functions

The fuzzy rules were formulated based on the properties of a region in an image that would be part of a 3D BWER structure. Each of these properties is measured quantitatively within a volume scan of the radar and based on quantities carried over from preceding volume scans.

The degree to which a rule such as ``The minimum reflectivity within the region is low'' holds is obtained by first finding the minimum reflectivity within the region and then using a fuzzy membership function for low in the context of minimum reflectivity to read out the degree to which the rule holds. The membership function, $f(x)$ is generated using two values - the value of the variable in a ``textbook case'', $x_2$, and the value, $x_1$, of the variable beyond which the rule does not hold true - as:

$$f(x) = \begin{array}{cl} \frac{x-x_1}{x_2-x_1} &if~~ 0 \leq... ...x_1} < 0 \\ 1 &if~~~ \frac{x-x_1}{x_2-x_1} > 1 \\ \end{array}$$ (7)

The membership functions for the rules ``somewhat low'' ($f^+(x)$) and ``very low'' ($f^-(x)$) are generated from $f(x)$ using:

$$\frac{1-e^{-\gamma.f(x)}}{1-e^{-\gamma}}$$ (8)

where $\gamma$ is $1.5$ for $f^+(x)$ and $-1.5$ for $f^-(x)$. The membership function for ``low minimum reflectivity'' given that the minimum reflectivity within an ideal BWER would be less than $25dBZ$ and that $35dBZ$ is quite high for the minimum value is shown in Figure 10.

Figure 10: The membership function generated for ``low minimum reflectivity'' given that minimum reflectivity in an ideal BWER would be less than $25dBZ$ and that $35dBZ$ is quite high for the minimum value.