1) Classification using differential reflectivity (Z_DR)

Since the introduction of Z_DR measurement for rainfall estimation applications, Z_DR has been used to distinguish between rain and ice, as well as identify other targets. For example, Bringi et al. (1984) utilized the differential reflectivity measurements to detect hail and showed that reliable differentiation between regions of hail and rainfall is possible in convective storms. Hall et al. (1984) used differential reflectivity and reflectivity measurements to identify various hydrometeor types and ground targets.

2) Classification using linear depolarization ratio

Bringi et al. (1986) studied the profiles of Z_DR, linear depolarization ratio (LDR) and reflectivity at horizontal polarization (Z_H) through the core of convective storms and found that these three measurements are useful in identifying the graupel region. At the same time, they found that the vertical structure of Z_DR (below the melting level), LDR (above the melting level), and dual-frequency ratio can provide information on hail shaft structure and vertical extent. A fairly detailed study about the vertical profiles of Z_H, Z_DR, and the dual-wavelength ratio in terms of the size, shape, and fall behavior of the hailstones were presented by Aydin et al. (1990).

3) Classification using specific differential propagation phase (K_DP) and copolar correlation coefficient (ρ_HV)

Straka and Zrnić (1993) examined the specific differential phase, the backscatter differential phase (δ), ρ_HV, and Z_DR observed over a severe hailstorm and demonstrated that these three measurements can be used to detect hail, as well as identify and quantify mixed-phase hydrometeors.

4) Classification using difference reflectivity (Z_DP)

Tong et al. (1998) used the difference reflectivity to estimate the fraction of rain and ice in mixed-phase precipitation. The difference reflectivity Z_DP can be estimated from Z_H and Z_DR.

5) Empirical algorithm for classifying hydrometeor types

Building on the results of hydrometeor classification in the existing literature, Straka and Zrnić (1993), and Höller (1995) described classification schemes to discriminate between the different hydrometeors. They both used the decision tree method in which predefined boundaries were used to define the decision region.

Thus the polarimetric radar signatures of precipitation have shown the potential for hydrometeor classification and has set the foundation for the development of automatic classification procedure using advanced techniques such as a neural network and fuzzy logic.

1) Fuzzification

The function of the “fuzzification” block is to convert the crisp inputs (or precise measurements) to the fuzzy sets with a corresponding membership degree. A specific crisp input can belong to different fuzzy sets but with different membership degrees (or truth value). The most important component in fuzzification is the membership function, which is used to describe the relationship of the crisp input and the fuzzy sets in the input domain (Zadeh 1983). The definition of membership function is as follows: μ_A(x) is called membership function of fuzzy set A (for a fuzzy variable x), whose value is the degree to which x is a member of fuzzy set A.

2) Inference

In a fuzzy logic system, rules are used to describe linguistically the complex relationship between the input and output fuzzy variables in the form of IF–THEN statements. Typically, the rule is composed of several antecedents in the IF statement and one or several consequents in the THEN statement. The process of deducing the “strength” of these consequents from the strength of the antecedents is called rule inference. The most commonly used inference methods are correlation minimum, correlation product, and MIN–MAX (Heske and Heske 1996).

3) Aggregation

Several rules (instead of a single rule) can be used to describe a fuzzy logic system. This set of rules is called a rulebase. The complete knowledge about a fuzzy model is contained in its rulebase and the membership functions (MBFs). We can use the inference methods to derive the strength of each rule, then the aggregation method can be used to determine an overall fuzzy region. Two commonly used aggregation methods are additive aggregation and MAX Aggregation.

4) Defuzzification

The output of aggregation process is a fuzzy set, but in many applications (such as hydrometeor classification) it is necessary to find a crisp value that best represents the fuzzy output set, and this process is called defuzzification. Two commonly used defuzzification methods are the Center of Area (COA) and the Mean of Maximum (MOM).

1) Fuzzification and membership functions

The purpose of fuzzification is to convert the precise input measurements to fuzzy sets with corresponding membership degree. The specification of membership functions is critical to the classification performance. Two different sets of membership functions are used, one for summer storms and the other for winter storms. Ten fuzzy sets corresponding to the 10 hydrometeor types are specified for each of the six input variables for summer storms. Similarly, five fuzzy sets corresponding to the hydrometeor types (viz., drizzle, rain, dry snow, oriented ice crystal, and wet snow) are specified for each of the six input variables for winter snowstorms. Each fuzzy set is represented by a membership function, denoted as MBFi_j, where index i corresponds to the six inputs and index j corresponds to the fuzzy sets. Index j takes values 1–10 for summer storms and 1–5 for winter snowstorms.

Several functional forms can provide adequate representation of membership functions, such as triangular, trapezoidal, Gaussian shapes, S and Z curves, and beta functions. In this study, a beta function is chosen to describe membership function for following reasons. In a hydrometeor classification problem, we expect that most membership functions have a wide flat region in which maximum value is 1. One hydrometeor type such as rain can have a wide range of reflectivity. In other words, there is no preferred or unique value of reflectivity for rain, but there is a preferred region, such as 25 to 60 dBZ. The best MBF to represent this is by means of a flat function over the preferred region that tapers off outside the preferred range. The beta function has the desired characteristics; therefore, it is chosen as the form of the membership functions. In addition, a beta function has a long tail, which improves the robustness of FHC. The derivative of the beta function is continuous, and this feature is useful for automatic adjustment of the parmameters that will be needed for development of neuro-fuzzy system.

The beta membership function is defined as

View Expanded

As seen in (2) there are three parameters that define the shape of a beta function, namely, the center of the function m, the width a, and the slope b (shown in Fig. 4). Typical one-dimensional membership functions for Z_H are shown in Fig. 5 for summer storms and Fig. B4 in appendix B for winter storms.

Ten membership functions for Z_H are shown in Fig. 5, which are representations of the 10 fuzzy sets of Z_H. For each curve, the horizontal axis is the value of Z_H, and the vertical axis indicates the membership degree of Z_H corresponding to that fuzzy set. In a fuzzy logic system, fuzzy sets instead of precise values are used to represent input variables. For example, if we have Z_H = 40 dBZ, it belongs to the drizzle set with membership degree 0, to the rain set with membership degree 1, . . . , to the dry graupel set with membership degree 0.8, . . . , to the rain and hail mixture with membership degree 0.

If the measurements of Z_DR are physically independent of other radar measurements such as Z_H, then the one-dimensional membership functions are adequate to represent the fuzzy sets. Multidimensional membership functions can be used to represent combinational fuzzy sets. Let f_{rain_zh} be the one-dimensional membership function for rain set with respect to input variable Z_H. Then f_{rain_zh} can be written as

View Expanded

Similarly, f_{rain_zdr}, the one-dimensional membership function for rain set with respect to Z_DR, can be writ- ten as

View Expanded

If Z_H and Z_DR are treated as independent variables for rain, then in a two-dimensional Z_H–Z_DR space, the membership function for rain set with respect to Z_H and Z_DR can be expressed as the product of f_{rain_zh} and f_{rain_zdr}:

f_{rain_zhzdr}Z_HZ_DRf_{rain_zh}Z_Hf_{rain_zdr}Z_DR(5)

Here, f_{rain_zhzdr}(Z_H, Z_DR) is shown in Fig. 6a and its contours are shown in Fig. 6b.

However, the radar measurements Z_H and Z_DR are not independent in rainfall. For example, the maximum excursion of Z_DR is related to the value of Z_H for rain. This feature can be seen from the scatterplot of Z_DR versus Z_H from rain data shown in Fig. 6c as “+” (Bringi et al. 1991). All the (Z_H, Z_DR) pairs lie in a small portion of the two-dimensional (Z_H, Z_DR) domain of Fig. 6a. Therefore, we have to modify the 2D membership function given by (5). We can define a new 2D membership functions of rain set with respect to Z_H and Z_DR as shown in Fig. 7a; its contours in Z_H–Z_DR space are shown in Fig. 7b. In Fig. 7c, we can see that the new two-dimensional membership functions restrict the Z_H–Z_DR domain for rain set, incorporating the internal correlation of these two radar measurements. The membership functions of Z_DR for other hydrometeor type sets are represented by using simple one-dimensional membership functions; their waveforms are shown in Fig. 8.

One-dimensional membership functions are used to represent K_DP, LDR and ρ_HV for all the hydrometeor type fuzzy sets. They are shown in appendix B. It is easy to notice that the slope of membership functions for LDR is small compared to other radar measurement variables;this is based on the fact that LDR has relatively higher measurement error and it is vulnerable to noise contamination. Decreasing the slope is equivalent to increasing the robustness of the parameter. The membership function of altitude is dependent on location and season. The most important parameter here is the melting level. In addition, the altitude of the melting level depends on the season. There are 90 membership functions (60 for summer, 30 for winter). The membership functions will get fine-tuned over time when more in situ data are observed.

2) Inference

The strength for the six antecedent propositions could be obtained from the fuzzification block as PS_{i_j} where the index i represents the five measurements and the altitude and index j represents the classes. For example, PS_{1_2} is the strength of reflectivity for rain. In this paper,“product intersection operation” is used to get the strength of the IF-SIDE, and “correlation product” inference method is used to get the rule strength. The truth value of the IF-SIDE is used to scale the consequent fuzzy set. In this case, the strength of the rule is equal to the strength of the IF-SIDE because the output is singleton. Therefore, the strength of rule j (RS_j) can be obtained as the product of the strength of individual propositions as

View Expanded

3) Aggregation

The MAX aggregation method is used to get net fuzzy result from the individual rule inference results. Max aggregation procedures takes only the consequent with the highest truth value. Therefore, the aggregation result is the maximum rule strength of the various RS_j’s defined in (6).

4) Defuzzification

For the FHC, the output is singleton. A simple method is used to defuzzify the output in order to get a singleton result, namely, the index of the rule with maximum rule strength.

1) CSU–CHILL radar

The radar data used in this study were collected by the CSU–CHILL radar. CSU–CHILL is an S-band, dual-polarization radar, which can measure a full set of polarimetric measurements. The important characteristics of the CSU–CHILL radar relevant to this paper are given in Table 2. Most of the data analyzed in this paper were at ranges less than 70 km from the radar.

2) In situ observations

1) The severe hail storm on 7 June 1995

On 7 June 1995, the CSU–CHILL radar observed a supercell structure, and a chase van with a roof-mounted hail collector net was sent to intercept the storm core and collect in situ measurements. The 7 June 1995 storm turned out to be a severe hailstorm. Figure 11 shows the radar measurements Z_h, Z_dr, K_dp, LDR, and ρ_hυ, and Fig. 12 shows the the classification result from the neuro-fuzzy classification system. We can see from the hydrometeor classification result that the hail and rain mixture, wet graupel, and rain are found on the ground as well as at low altitude, whereas ice crystals were inferred at high altitude. The vertical structure of the storm can be seen fairly well from the classification result. The in situ ground observations were consistent with the result of the hydrometeor classifier. The hail chase van collected some hailstones and also observed rain mixed within the storm. The in situ ground observations for this case are given in Hubbert et al. (1998); they have presented detailed dual-Doppler analysis of this case. The fuzzy classification results agree very well with inference of Hubbert et al. (1998). Figure 13 shows the time series of the measurements at the hail chase van location (y = −4.0 km) for 50 min. Figure 13 also shows the hydrometeor classification inferred by the hydrometeor classification system, and the “ground truth” observed by the hail chase van. Note in Fig. 13 that the automatic classification result agrees fairly well within the limits of comparison between radar and ground observations.

2) The convective storm on 22 June 1995

A severe hailstorm occurred on 22 June 1995 near Fort Collins, Colorado. This storm grew to a height of 12.5 km and upon collapsing produced heavy rain and hail of maximum sizes 3–4 cm. The intense part of the storm was located at a distance of 45–50 km to the northeast of the CSU–CHILL radar around 1730:29 local time (LT). The radar continuously scanned the storm approximately with 2-min resolution for about an hour. At the same time the T-28 aircraft made several penetrations through the storm, collecting samples of hydrometeors. The flights were at altitudes between 2.5 and 3.5 km above ground level to collect data in small hail region. The storm was characterized by heavy rain mixed with hail. A constant altitude plan position indicator (CAPPI) of the radar measurements Z_h, Z_dr, K_dp, LDR, and ρ_hυ at the height of the aircraft is shown in Fig. 14. Figure 15 shows the automatic hydrometeor classification result. The solid line on Fig. 15 is the flight track of T-28 aircraft. As seen in Fig. 15 the T-28 track was mostly in the region of small hail and graupel. Figure 16 shows time series of automatic hydrometeor classification encountered along the T-28 flight penetration shown in Fig. 15. Note here that the time series correspond to the distance along the aircraft track [from (x: 14.1 km, y: 39.7 km) to (x:16.96 km, y: 42.01 km)]. Figures 17 and 18 show sample HVPS images for the data shown in Fig. 16. Figure 19 shows the comparison of automatic hydrometeor classification and in situ observations from T-28 HVPS data. From Fig. 19, we can see fairly good agreement between automatic classification and in situ observations.

3) The convective storm on 20 June 1995

On 20 June 1995, the CSU–CHILL radar observed a storm cell that formed 20 km east of the radar. A T-28 aircraft carrying an HVPS probe penetrated through the core of the storm to collect in situ observations. Most of the aircraft flights through the storm were at a constant altitude of 4 km above ground. During the flight, the HVPS collected samples of hydrometeors along the path. Figure 20 shows a CAPPI of the radar measurements at an altitude of 4 km (radar scan time is from 1638:17 to 1639:14 UTC). The five radar polarimetric fields are shown in Fig. 20, and the hydrometeor classification result is shown in Fig. 21. The solid line is the T-28 aircraft track. In Fig. 21, note that the aircraft tracks are in predominantly graupel region and a region of small hail. This inference agrees very well with the T-28 in situ aircraft observations obtained from the HVPS probe. Sample images of hydrometeors from HVPS for the flight at 1638:34 UTC are shown in Fig. 22, the corresponding T-28 aircraft location is at 30.33 km in the x direction and 11.80 km in the y direction with respect to the location of the CSU–CHILL radar. The presence of conical graupel particles is seen in the images.

4) The snow storm on 18 February 1997

On 18 February, there was a light snow event in the vicinity of the CSU–CHILL radar. Figures 23 and 24 show a vertical section of the CSU–CHILL radar measurements through this storm. The radar measurements of Z_H, LDR and ρ_HV and the hydrometeor classification result are shown in Figs. 23 and 24. The hydrometeor classification indicates wet snow below 1 km, and dry snow and oriented ice crystal above 1 km. We can see a transition from rain to snow on the ground. This feature of rain-to-snow transition was observed on the ground in excellent agreement with radar-based inferences.