A Fuzzy Classifier

We then use a classifier that acts on the fuzzy outputs of the following four questions

• To what degree is this 3D structure a BWER?
• To what degree is this 3D structure not a BWER?
• To what degree is this 3D structure very much a BWER?
• To what degree is this 3D structure very much not a BWER?

to decide which of the three categories the region belongs to. The four fuzzy measures are utilized by the classifier to tag a given structure as one of the three categories - BWERs, marginal BWERs and non-BWERs.

The fuzzy measures to the questions in Section 2.3 were determined using the fuzzy rule base. We will denote as $f_W$, the fuzzy measure that denotes to what degree the BWER is at the right location relative to the storm and has the right physical properties. The fuzzy measure, $f_W$, is derived by combining about twenty fuzzy sets such as the extent of capping, the intensity of updraft, and the proximity to a mesocyclone. The basic feature values are mapped using membership functions derived from Equation 7 with the values $x_1$ and $x_2$ drawn from our expectation of an ideal BWER. The fuzzy rule base consists of the positive rules that go into $f_W$ and the negative ones that determine $f_{no}$, the degree to which the structure is not a BWER.

For example, the degree to which a region is capped is obtained from four fuzzy properties of the region:

• the degree to which there are many pixels above this region with reflectivities greater than 45dBZ,
• the degree to which there are fewer pixels with reflectivites greater than 45dBZ below this region than there are above it,
• the degree to which the average reflectivity above the region is higher than the average reflectivity within the region,
• and the degree to which the average reflectivity below the region is lower than the average reflectivity above this region.

The way these four fuzzy attributes are combined to conclude the degree to which the region is capped is shown in Figure 11. Note that the properties that determine the 45dBZ capping extent correspond either to the region itself or to one of the regions above/below it and from which it can inherit attributes (see Figure 7).

Figure 11: An example of a rule in the rule base: the degree to which a region is capped is derived from the four properties at the extreme left of the figure. Partial Vertically Integrated Liquid (VIL) is correlated with the severity of storms.

The structure is a BWER to the extent $f_{yes}$ given by:

$$f_{yes} = ((f_D \wedge f_W)\vee({f_D}^- \wedge {f_W}^+) \vee {f_W}^-)$$ (9)

where $f_D$ is the degree to which this region is associated with a BWER that occurred in a previous volume scan. The structure is very much a BWER to the extent of $f_{vyes}$ given by $(f_D \wedge f_W)^-$.

The structure is not a BWER to the extent $f_{no}$ given by:

$$f_{no} = f_{gap} \vee f_{low\_gradient} \vee f_{not\_capped} \vee \bar{f}_W \vee (\bar{f}_{edge} \wedge \bar{f}_{bounded})$$ (10)

where the terms on the right are negative criteria i.e. criteria that a human observer would cite when dropping the region from consideration. It is very much not a BWER to the extent of the above criteria holding together (i.e. the intersection of all the sets on the right-hand side).

The rule base of the classifier yields fuzzy scores for a given structure being a BWER, a non-BWER and a marginal BWER. These scores are given by:

$$f_{marginal} = (f_{yes} \wedge f_{no}) \vee (f_{vyes} \wedge f_{vno})$$ (11)

$$f_{bwer} = (f_{yes} \wedge \bar{f}_{no}) \vee (f_{vyes} \wedge \bar{f}_{vno})$$ (12)

$$f_{nonbwer} = (\bar{f}_{yes} \wedge f_{no}) \vee (\bar{f}_{vyes} \wedge f_{vno})$$ (13)

The structure is tagged as the maximum of the three fuzzy scores, unless the maximum fuzzy score pertains to the ``marginal'' category. A structure that receives the maximum score for being a marginal BWER is tagged as such only if $f_{yes}$ exceeds $f_{no}$. If $f_{yes}$ does not exceed $f_{no}$, the structure is tagged as a non-BWER.