# Research on Hail Mechanism Features Based on Dual-Polarization Radar Data

^{*}

## Abstract

**:**

## 1. Introduction

_{H}), radial velocity, and spectrum width in the horizontal polarization direction. These radar data are employed to offer information regarding the density, intensity, and three-dimensional distribution of hydrometeor particles within the surveillance area [10]. In recent years, dual-polarization (dual-pol) weather radar has emerged as a novel meteorological sensing instrument. In comparison to single-polarization weather radar, dual-pol radar has the capability to simultaneously transmit and receive radar probing and echoes in both horizontal and vertical polarization directions. This technology provides detailed microphysical features and morphological information about hydrometeor particles [11,12,13]. It introduces additional parameters, including differential reflectivity factor (Z

_{DR}), co-polar cross-correlation coefficient (ρ

_{hv}), and specific differential phase (K

_{DP}). These enhancements make it more conducive for the identification and prediction of convective weather-related disasters [9,14].

^{−2}and 50 kg m

^{−2}, there is a high probability of severe hail occurrence. Amburn et al. [17] defined the ratio of VIL to the echo top as the VIL density. This parameter is used to determine the presence of hail particles within convective systems.

_{DR}, ρ

_{hv}and K

_{DP}, has opened up new avenues for studying hail mechanisms. The K

_{DP}column [26], characterized by a relatively high K

_{DP}above the 0 °C level, is typically dominated by a concentration of raindrops or partially melted ice particles and is often associated with downdrafts. Snyder et al. [27] confirmed that the Z

_{DR}column (an extension of positive Z

_{DR}above the 0 °C level) is closely associated with the updrafts in convective systems, with high-altitude Z

_{DR}columns often leading to the formation of hail. Dawson et al. [28,29] defined the large Z

_{DR}region (typically exceeding 3 dB) frequently observed on the leading edge of the inflow sector of a single-cell storm as the Z

_{DR}arc, using it as a mechanistic feature for meteorological phenomena such as hail.

## 2. Data

- (1)
- The original base data (Z
_{H}, Z_{DR}, ρ_{hv}) were transformed into three-dimensional uniform grid format data using a bilinear interpolation algorithm; - (2)
- The composite reflectivity (CR) map was generated from three-dimensional uniform grid reflectivity data using an extremum extraction algorithm [31];
- (3)
- The convective cells were segmented on the CR images using a flood-fill algorithm with 40 dBZ as the threshold [32], resulting in the creation of individual cell masks. Subsequently, other dual-pol data for these cells were obtained.

_{H}data after the aforementioned three-step data preprocessing.

## 3. Method

#### 3.1. Mechanism Feature Construction Based on Fine-Grained Cell Data Distribution Details

_{H}echoes. These features are comparatively easier to identify on CR images. In contrast to non-hail convective cells, hail-producing cells tend to display stronger Z

_{H}with higher tops in the strong Z

_{H}region. They often exhibit an overhanging morphology with weak echo regions, or even well-defined weak echo regions [33], owing to stronger updrafts and higher near-surface moisture saturation. Leveraging these features of hail-producing cells, this study proposes the construction of five novel mechanism features for hail identification, capitalizing on the distinctive attributes of dual-pol radar data. These features focus on the fine-grained pixel value distributions within cells. A flowchart that sets out the overall process of feature construction is presented in Figure 3. The details of the symbols mentioned in this section are shown in Appendix A, Table A1.

#### 3.1.1. Gradient-Based Features

_{H}images, convective cells typically exhibit a data distribution pattern characterized by a high-value core surrounded by a gradual decrease in values. Strong updrafts often lead to rapid changes in Z

_{H}intensity on one side of the core to the boundary within developing hail-producing cells, while the change is slower on the other side [24,34]. In contrast, this disparity in Z

_{H}change is less pronounced in short-duration heavy precipitation cells without hail. This indicates the presence of high-gradient values in the Z

_{H}data within hail cell areas.

_{DR}values around 0 dB [35]. In contrast, raindrops falling through the air experience a flattening effect due to air resistance, leading to larger observed Z

_{DR}values. Consequently, within the region of hail cells containing both solid- and liquid-phase water particles, local high gradients in the Z

_{DR}data are expected to be higher than those in non-hail cells consisting solely of liquid-phase water particles. Therefore, the steps for extracting high-dimensional micro-features of high gradients within individual cells are as follows:

_{40}and Ω

_{50}within the CR images using respective lower threshold values of 40 dBZ and 50 dBZ.

_{H}images at five different height levels and the Z

_{DR}images at five different height levels. The pixel indexing used in Equation (1) is illustrated in Figure 4, and the masking types are specified in Table 1. The five height levels are sequentially defined as the 0 °C layer height (H

_{0°C}), H

_{0°C}− 1 km, H

_{0°C}+ 1 km, the −20 °C layer height (H

_{−20°C}), and H

_{−20°C}− 1 km.

_{i}are illustrated in Table 3. I(x) denotes the indicator function, which takes a value of 1 when x is true and 0 otherwise. The gradient threshold parameter, th, and the total dimensions of acquired gradient-based features are detailed in Table 1.

#### 3.1.2. Proportion-Based Features in a Specified Value Range

_{H}values detected by weather radar increase accordingly. Several statistical analyses have shown that convective cells with Z

_{H}components of 55 dBZ and higher are more likely to trigger hail [36]. Secondly, the sensitivity of dual-pol radar’s Z

_{DR}to hydrometeor morphology, along with the irregular spherical shape of hailstones and the flattened shape of raindrops, results in varying Z

_{DR}values depending on the shape of hydrometeor particles. Thirdly, in general, high-altitude pure hail tends to produce high values of ρ

_{hv}provided by dual-pol radar. However, the presence of wet hail can lower the values of ρ

_{hv}. Therefore, ρ

_{hv}aids in distinguishing between pure rain, hail, and mixed hail–rain components within the individual cells.

_{H}data, Z

_{DR}data, and ρ

_{hv}data obtained from dual-pol radar. Within the individual cell masking area (Ω

_{40}or Ω

_{50}), it calculates the proportion of values falling within the specified ranges, thus generating ‘Proportion-Based Features in a Specified Value Range’. To capture more discriminative features, this study defines multiple specified value range intervals (ω) and masking types for each of the three dual-pol radar data, as shown in Table 4, where the meanings of masking areas Ω

_{40}and Ω

_{50}are as described earlier. The steps for obtaining proportion-based features in a specified value range within individual cells are outlined as follows:

_{40}and Ω

_{50}on the CR image using lower threshold values of 40 dBZ and 50 dBZ, denoted as N

_{40}and N

_{50}, respectively.

_{H}, Z

_{DR}, and ρ

_{hv}at five height levels, all within the masking area, as specified in Table 5.

_{i}have the same meanings as in the gradient-based features. f(x, y) represents the datum value at point (x, y), while ω denotes the value range intervals corresponding to various data types, with specific values as shown in Table 4. The total dimensions of the proportion-based features in a specified value range are detailed in Table 4.

#### 3.1.3. Quantile-Based Intensity Features

_{H}is positively correlated with the strength of meteorological targets and serves as the most direct piece of data for assessing the strength of convective cells. Conversely, the Z

_{DR}and ρ

_{hv}coefficients are negatively correlated with hail events, meaning that lower values indicate a higher likelihood of hail [37,38]. Therefore, this paper directly utilizes different quantile data from three categories of data at various height levels within individual cells as intensity-based features. Additionally, for the data related to Z

_{DR}and ρ

_{hv}, due to the potential presence of significant value disparities within two-dimensional data at fixed height levels, the range (the difference between maximum and minimum intensity values) is incorporated as a feature. The specific extraction algorithm steps are outlined as follows:

_{40}and Ω

_{50}within the CR images and sort pixels within the masking areas in various data images by pixel value size.

_{H}, Z

_{DR}, and ρ

_{hv}at five height levels. In addition, calculate the range of values for Z

_{DR}images and ρ

_{hv}images, which are directly employed as percentile-based intensity-class features.

#### 3.1.4. Statistical Moment Features Based on the Gray-Level Histogram

_{H}, Z

_{DR}, and ρ

_{hv}images into grayscale using the quantification scheme outlined in Table 8.

_{40}, and the first-order, second-order, and third-order moments of the data are computed. The dimensions of the statistical moment features based on the grayscale histogram are presented in Table 9.

#### 3.1.5. Features Based on the Gray-Level Co-occurrence Matrix

_{ij}(i, j = 0, 1, 2, …, L − 1), represent the probability of a specified pixel pair occurring at a designated position in the image. The value of gij is equal to the ratio of pixel pairs meeting the conditions f(p

_{1}) = i and f(p

_{2}) = j along the θ direction with a separation of r in the image, where r = 1, and θ is set to 0°, 45°, 90°, and 135° in this study.

_{H}images, Z

_{DR}images, and ρ

_{hv}images under the cell mask region Ω

_{40}. The grayscale levels and their corresponding data intervals are detailed in Table 8. Following this, 64 GLCMs are constructed from these 16 grayscale images (1 + 5 × 3 = 16) for the four θ. Subsequently, texture features, including contrast ratio, energy, entropy, and inverse variance [41], typically used to represent image texture features, are extracted. The overall dimensionality of the features based on the GLCM is 256 (64 × 4).

#### 3.2. Construction of Traditional Mechanism-Based Features

_{H}images and the Z

_{DR}column feature [27] based on Z

_{DR}images as representatives of traditional mechanism-based features. Each feature is described as follows:

- (1)
- Kurtosis (K) [24]: Kurtosis is a fourth-order statistical measure based on histogram data from images, used to quantify the steepness of the peak in the histogram. Compared to non-hail cells, hail cells (regions with Z
_{H}of 40 dBZ and higher) have a higher proportion of high Z_{H}values. Therefore, typically, the intensity distribution histogram of non-hail cells has a steeper peak; - (2)
- Average reflectivity of nucleus (ARN) [42]: ARN is defined as the reflectivity-weighted average within contiguous regions on a radar CR image where the Z
_{H}exceeds or equals 45 dBZ. A higher ARN indicates a greater likelihood of hail precipitation; - (3)
- Strong echo ratio (SER) [24]: SER is used to describe the proportion of strong echoes above the −20 °C level and serves as a quantitative measure of the intense echo signals at higher altitudes;
- (4)
- Liquid ratio of nucleus (LRN) [42]: LRN is the feature specifically designed for hail identification on the basis of vertical integrated liquid water content (VIL). As cells pass through the freezing level, precipitation particles gradually transition from a liquid to a solid state (such as ice crystals). The reflectivity values of substances like ice crystals within hail clouds do not conform to their empirical relationship with liquid water. By establishing an attenuation coefficient, LRN attenuates liquid water content converted from the reflectivity, calculating the density of weighted vertical integrated liquid water content. This plays a significant role in distinguishing between hail and short-duration heavy rainfall events;
- (5)
- Effective thickness (ET) [42]: Severe hail-producing weather is more likely to occur with higher and more intense updraft. Effective thickness is a quantitative measure of this phenomenon;
- (6)
- Overhang (OH) [24]: Cells with the potential for hail exhibit an overhanging pattern, where the Z
_{H}structure at lower levels displays strong echoes suspended above weaker echo bodies. The presence of low-level moisture-carrying strong updrafts is a primary contributor to the weaker echo regions. Hence, the size of the weaker echo body volume can be used to quantitatively measure the intensity of the updraft, defining overhang in terms of the volume of the weaker echo region; - (7)
- Volume and Height of Z
_{DR}Column

_{DR}data provided by dual-pol radar determines the capacity of Z

_{DR}values to carry phase information about detected hydrometeors. For instance, Z

_{DR}values greater than or equal to 1 dB typically indicate the presence of liquid-phase water particles. In contrast, water particles situated above the 0 °C level for a period of time are often in solid form, resulting in Z

_{DR}values around 0 dB.

_{DR}data. Research by Snyder et al. [27] has shown that the vertical extent of the Z

_{DR}column is related to the intensity of the strongest updraft. Thus, it can be stated that the Z

_{DR}column of liquid water particles aggregated at and above the 0 °C level provides information about the position and intensity of updrafts within storm cells.

_{DR}column can indirectly serve as an indicator of the physical and dynamic structure of hail clouds, and its variations can be used to predict the development of hail clouds [43]. In this study, based on a regional growth approach, we have redefined the algorithm for extracting Z

_{DR}column features, focusing on two key aspects: maximum volume and maximum height.

_{H}values equal to or exceeding 40 dBZ were extracted from the CR image and utilized as a mask to delineate the corresponding areas in the ρ

_{hv}image and the Z

_{DR}image;

_{DR}column. The collection of all points within the column is denoted as Ω

_{ZDR}, where the growth template size is 3 × 3 × 3, and the growth criteria include Z

_{DR}values greater than or equal to 1 dB and ρ

_{hv}values between 0.8 and 1.

_{DR}column, where the value at point (x, y) is set to ${H}_{\mathrm{Z}\mathrm{D}\mathrm{R}}\left(x,y\right)=\underset{z}{\mathrm{max}}\left\{H\left(x,y,z\right)-{H}_{0\xb0\mathrm{C}}\right\}$.

_{DR}column are obtained:

## 4. Experiment and Analysis

#### 4.1. Feature Validity Assessment

#### 4.1.1. Hypothesis Testing Methods

#### 4.1.2. Hypothesis Testing Results

- (1)
- Among the 564-dimensional features constructed in this study, all 48 features in the gradient category exhibit significant differences;
- (2)
- Among the remaining 516 (564–48) features, 74 features were removed due to excessively high p-values, indicating that these features did not demonstrate a significant difference in mean distribution between hail and non-hail cells.

_{hv}at three lower altitude levels, 11 of them do not pass the hypothesis testing. (3) Percentile features are generally more effective on Z

_{H}images at lower altitude levels. (4) The selection rates for features in the categories of grayscale histogram statistical moments and GLCM texture features are 83% and 93%, respectively.

#### 4.2. Hail Identification Model Construction and Test Metrics

#### 4.2.1. Identification Model

#### 4.2.2. Testing Metrics for Identification Models

#### 4.3. Comparison of Hail Identification Models in Two Mechanism Feature Spaces

#### 4.3.1. Hail Identification Model Based on Traditional Mechanism Features

_{H}and another based on six-dimensional features from Z

_{H}combined with two-dimensional mechanistic features derived from the Z

_{DR}column. Subsequently, the models were tested using the remaining 1000 hail and 1500 non-hail samples. The test results are presented in Table 12.

_{H}features achieved a commendable CSI score of 60.5%. (2) Upon the inclusion of two-dimensional Z

_{DR}column features, the model’s POD increased by 1.6%, the FAR decreased by 0.4%, and the CSI improved by 1.2%. This indicates the positive impact of dual-pol parameters on enhancing model quality [9,48,49]. It is worth noting that dual-pol radar necessitates higher data accuracy and stability. As the calibration quality of dual-pol radar improves, the inclusion of dual-pol parameters like Z

_{DR}may potentially yield even greater enhancement in algorithm performance.

#### 4.3.2. Hail Identification Model Based on Mechanism Features from Cell Data Distribution Details

- (1)
- Principal Component Analysis (PCA) [50] was used to integrate the 490-dimensional features from five different perspectives. Specifically, from the five sub-feature spaces of gradient, specified value range ratio, quantiles, grayscale histogram statistical moments, and texture based on the GLCM, the directions of maximum variance and second maximum variance were obtained, resulting in a total of 10 comprehensive features (principal components);

- (2)
- Fisher Linear Discriminant Analysis [51] was employed to integrate and reduce the dimensionality of the 490-dimensional features from five different perspectives. In each of the five sub-feature spaces, the projection direction that maximizes the criterion function of difference between inter-class means divided by sum of intra-class variances was determined, resulting in five comprehensive features.

- (1)
- As the dimensionality of PCA comprehensive features increases, classifier performance improves. Furthermore, the use of the 10-dimensional PCA comprehensive feature scheme is more effective than directly building models in the 490-dimensional feature space. This aligns with our expectations and previous research findings [19,20,42];
- (2)
- The Fisher feature comprehensive scheme, overall, outperforms the PCA feature comprehensive scheme. This is because PCA’s “maximum variance criterion” may not necessarily align with the classification objective, while Fisher’s “intra-class cohesion, inter-class separation [52]” criterion aligns with the goals of a classification function. This further substantiates the feasibility of Fisher Linear Discriminant Analysis in hail identification [21,53];
- (3)
- After adopting the Fisher feature comprehensive scheme, the model based on the five-dimensional comprehensive features outperforms the traditional mechanism feature-based identification model. This strongly validates the significant effectiveness of the newly constructed five-dimensional mechanism features in hail identification, further emphasizing that dual-pol radar data can indeed enhance the quality of hail identification models. In the eight-dimensional traditional mechanism feature space, the utilization of dual-pol data is inadequate.

#### 4.3.3. Joint Utilization of Two Types of Mechanism Features for Hail Identification Model Construction

- (1)
- Combining the five-dimensional Fisher-comprehensive features of the second-class mechanism features with the first-class mechanism features (traditional, eight-dimensional) indeed improves the hail identification model’s scores;
- (2)
- Comparing the three models in Table 15, it is evident that among the 13-dimensional features used to describe samples, the 5-dimensional Fisher-comprehensive features play an overwhelmingly dominant role. In terms of the difficulty level of feature extraction, constructing the eight-dimensional traditional feature extraction algorithm is more challenging than the latter. This is because the former requires a deeper understanding of the overall morphology and structure of hail clouds, as well as the ability to summarize and generalize, along with precise algorithmic representation capabilities. In contrast, the latter only needs to internalize the understanding of hail formation mechanisms into determining data height layers, partitioning data value ranges, and setting thresholds. The extraction of micro-gradient operators, specified value range ratios, quantiles, grayscale histogram statistical moments, and texture feature calculation methods based on the GLCM are all generic and convenient. Therefore, its technical threshold is much lower than that of the former. When using the Fisher method to integrate the corresponding sub-feature classes, it is possible to obtain high-quality hail identification models in a lower-dimensional comprehensive feature space.

## 5. Conclusions

- (1)
- The addition of dual-pol data, specifically Z
_{DR}and ρ_{hv}, to the hail identification model has led to an improvement of nearly five percentage points in the CSI test results. This demonstrates a clear and significant enhancement in the quality of the hail identification model when utilizing dual-pol data; - (2)
- In the face of five types of high-dimensional feature spaces, Fisher linear discriminant analysis is used to obtain comprehensive features of each category, and this method of feature construction is proven to be feasible and beneficial. The CSI score of the hail identification model based on 5-dimensional features is 0.65, which is five percentage points higher than that of the model based on 490-dimensional features;
- (3)
- The construction of traditional mechanism features requires algorithm designers to have a profound understanding of the phenomena and essence of the research object and to be able to scientifically incorporate the phenomena and essence of the research object into the algorithms. This approach has a high technical threshold. In contrast, the computational methods for the features proposed in this study are both versatile and easily implementable. When combined with appropriate feature comprehensive techniques, they can achieve higher quality results. Leveraging machine learning methods, it becomes possible to achieve hail identification in a low-dimensional comprehensive feature space.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Symbol | Meaning |
---|---|

Ω_{40}, Ω_{50} | The core masking areas within the CR images using respective lower threshold values of 40 dBZ and 50 dBZ |

p_{i} | The neighborhood point of pixel (x, y) |

M | The mask used to calculate the feature |

th, th_{CR}, th_{ZH}, th_{ZDR} | The threshold used to calculate the feature |

Grad_{xy} | The gradient at pixel (x, y) |

(x, y)∈A | $A=\left\{\left(x,y\right)|{M}^{\left(x,y\right)}=1\right\}$, (x, y) is the point within the mask M |

h_{i} | The height level where (x, y) is located |

ω, ω_{CR}, ω_{ZH}, ω_{ZDR}, ω_{ρ}_{hv} | The value range intervals of various data types |

N_{40}, N_{50} | The pixel count within the masking areas Ω_{40} and Ω_{50} |

L | Gray level |

Ω_{ZDR} | The collection of all points within the Z_{DR} column |

H(x, y, z) | The height of point (x, y, z) |

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**Figure 1.**General view of study area and location map of 11 radar stations. The radar data were sourced from the ACHIVE Ⅱ base data of the Next Generation Weather Radar (NEXRAD), obtained from the National Centers for Environmental Information (NCEI) in the United States. The hail cases were recorded by the NOAA National Severe Storms Laboratory’s mPING system.

**Figure 2.**Examples of radar base data and their preprocessing results. (

**a**) Reflectivity at 14 elevations. The angle value of the elevation is indicated in the top left corner, denoted in degrees (°); (

**b**) Interpolated three-dimensional grid reflectivity; (

**c**) CR; (

**d**) Examples of segmented cells. The unit of reflectivity is dBZ.

**Figure 6.**Scatter diagram of p-value change rates. (

**a**) Scatter diagram of overall p-value change rates; (

**b**) Enlarged view of the red rectangle area in the image (

**a**).

Data Type | Mask (M) | Gradient Threshold (th) | Number of Microfeatures | Dimensions of Gradient-Based Feature |
---|---|---|---|---|

CR | Ω_{40} | 2, 3, 4 | 3 | 48 |

Z_{H} at 5 different heights | Ω_{40} | 2, 3, 4 | 3 × 5 | |

Z_{DR} at 5 different heights | Ω_{40}, Ω_{50} | 1, 1.5, 2 | 3 × 5 × 2 |

Feature Name | Calculation Formula |
---|---|

${\mathrm{T}\_\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}\_\mathrm{C}\mathrm{R}}_{M,th}$ | ${\sum}_{\left(x,y\right)\in A}I\left({Grad}_{xy}\ge {th}_{\mathrm{C}\mathrm{R}}\right)$ |

${\mathrm{T}\_\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}\_\mathrm{Z}\mathrm{H}}_{{h}_{i},M,th}$ | ${\sum}_{\left(x,y\right)\in A}I\left({Grad}_{xy}\ge {th}_{\mathrm{Z}\mathrm{H}}\right)$ |

${\mathrm{T}\_\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}\_\mathrm{Z}\mathrm{D}\mathrm{R}}_{{h}_{i},M,th}$ | ${\sum}_{\left(x,y\right)\in A}I\left({Grad}_{xy}\ge {th}_{\mathrm{Z}\mathrm{D}\mathrm{R}}\right)$ |

h_{i} | ||||
---|---|---|---|---|

h_{1} | h_{2} | h_{3} | h_{4} | h_{5} |

H_{0°C} − 1 km | H_{0°C} | H_{0°C} + 1 km | H_{−20°C} − 1 km | H_{−20°C} |

**Table 4.**Data types, range intervals, masks, and such feature dimensions that form the proportion-based features in a specified value range.

Data Type | Value Range (ω) | Mask | Feature Dimension 72 | |||
---|---|---|---|---|---|---|

CR | [45, ∞) | [55, ∞) | Ω_{40} | 2 | ||

Z_{H} at 5 heights | [45, ∞) | [55, ∞) | Ω_{40} | 2 × 5 = 10 | ||

Z_{DR} at 5 heights | [−0.5, 0.5] | [−1, 1] | [−1.5, 1.5] | Ω_{40}, Ω_{50} | 5 × 3 × 2 = 30 | |

ρ_{hv} at 5 heights | [0.85, 0.92] | [0.83, 0.94] | [0.8, 0.97] | Ω_{40}, Ω_{50} | 5 × 3 × 2 = 30 |

Feature Name | Calculation Formula |
---|---|

${\mathrm{T}\_\mathrm{R}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\_\mathrm{C}\mathrm{R}}_{M,\omega}$ | $\frac{{\sum}_{\left(x,y\right)\in A}I\left(f\left(x,y\right)\in {\omega}_{\mathrm{C}\mathrm{R}}\right)}{{N}_{\Omega}}$ |

${\mathrm{T}\_\mathrm{R}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\_\mathrm{Z}\mathrm{H}}_{{h}_{i},M,\omega}$ | $\frac{{\sum}_{\left(x,y\right)\in A}I\left(f\left(x,y\right)\in {\omega}_{\mathrm{Z}\mathrm{H}}\right)}{{N}_{\Omega}}$ |

${\mathrm{T}\_\mathrm{R}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\_\mathrm{Z}\mathrm{D}\mathrm{R}}_{{h}_{i},M,\omega}$ | $\frac{{\sum}_{\left(x,y\right)\in A}I\left(f\left(x,y\right)\in {\omega}_{\mathrm{Z}\mathrm{D}\mathrm{R}}\right)}{{N}_{\Omega}}$ |

${\mathrm{T}\_\mathrm{R}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\_\mathsf{\rho}\mathrm{h}\mathrm{v}}_{{h}_{i},M,\omega}$ | $\frac{{\sum}_{\left(x,y\right)\in A}I\left(f\left(x,y\right)\in {\omega}_{\rho \mathrm{h}\mathrm{v}}\right)}{{N}_{\Omega}}$ |

Data Type | Percentile (x%) | Mask | Feature Dimensions 140 |
---|---|---|---|

CR | 1%, 25%, 50%, 75%, 100% | Ω_{40} | 5 |

Z_{H} at 5 heights | 1%, 25%, 50%, 75%, 100% | Ω_{40} | 5 × 5 = 25 |

Z_{DR} at 5 heights | 1%, 25%, 50%, 75%, 100% | Ω_{40}, Ω_{50} | 5 × 5 × 2 + 5 = 55 |

ρ_{hv} at 5 heights | 1%, 25%, 50%, 75%, 100% | Ω_{40}, Ω_{50} | 5 × 5 × 2 + 5 = 55 |

**Table 7.**The meanings of percentiles in Table 6.

x% | 1% | 25% | 50% | 75% | 100% |
---|---|---|---|---|---|

Meaning | Minimum value | Low quartile point | Median | High quartile point | Maximum value |

Data Type | Gray Level L and Data Range | |||||||
---|---|---|---|---|---|---|---|---|

l = 0 | l = 1 | l = 2 | l = 3 | l = 4 | l = 5 | l = 6 | l = 7 | |

CR | [34, 40) | [40, 45) | [45, 50) | [50, 55) | [55, 60) | [60, 65) | ≥65 | |

Z_{H} at 5 heights | ||||||||

Z_{DR} at 5 heights | [−3, −2) | [−2, −1) | [−1, 0) | [0, 1) | [1, 2) | [2, 3) | [3, 4) | ≥4 |

ρ_{hv} at 5 heights | [0.4, 0.5) | [0.5, 0.6) | [0.6, 0.7) | [0.7, 0.8) | [0.8, 0.9) | ≥0.9 |

**Table 9.**Data type, mask area, and total dimension of statistical moment features based on the gray-level histogram.

Data Type | Mask | Feature Dimensions 48 |
---|---|---|

CR | Ω_{40} | 3 |

Z_{H} at 5 heights | Ω_{40} | 5 × 3 = 15 |

Z_{DR} at 5 heights | Ω_{40} | 5 × 3 = 15 |

ρ_{hv} at 5 heights | Ω_{40} | 5 × 3 = 15 |

Type | Feature Dimensions | |
---|---|---|

Gradient-based features | 48 | 564 |

Proportion-based features in a specified value range | 72 | |

Quantile-based intensity features | 140 | |

Statistical moment features based on the gray-level histogram | 48 | |

Features based on the GLCM | 256 |

Feature Type | Data Type | Proportion | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Z_{H} | Z_{DR} | ρ_{hv} | CR | H_{0–1} | H_{0} | H_{0+1} | H_{–20} | H_{–20–1} | ||

Proportion-based | 0 | 0 | 11 | 0 | 4 | 5 | 2 | 0 | 0 | 11/72 |

Quantile-based | 1 | 18 | 18 | 0 | 0 | 13 | 10 | 8 | 6 | 37/140 |

Statistical moments | 4 | 1 | 3 | 0 | 1 | 2 | 1 | 2 | 2 | 8/48 |

GLCM | 6 | 8 | 4 | 0 | 2 | 1 | 2 | 7 | 6 | 18/256 |

Model | Z_{H} Feature | Z_{DR} Column | POD | FAR | CSI |
---|---|---|---|---|---|

1 | √ | 0.804 | 0.290 | 0.605 | |

2 | √ | √ | 0.820 | 0.286 | 0.617 |

Model | Feature | POD | FAR | CSI |
---|---|---|---|---|

3 | 490-dimensional mechanism features | 0.701 | 0.193 | 0.600 |

Model | Comprehensive Feature | POD | FAR | CSI |
---|---|---|---|---|

4 | 5 first principal components | 0.796 | 0.316 | 0.582 |

5 | 5 first principal components + 5 s principal components | 0.799 | 0.279 | 0.610 |

6 | 5 Fisher | 0.867 | 0.278 | 0.650 |

Model | Sample Description Method | POD | FAR | CSI |
---|---|---|---|---|

1 | 8-dimensional traditional mechanism features | 0.820 | 0.286 | 0.617 |

2 | 5-dimensional Fisher-comprehensive features | 0.867 | 0.278 | 0.650 |

3 | 8-dimensional traditional mechanism features +5-dimensional Fisher-comprehensive features | 0.865 | 0.275 | 0.651 |

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## Share and Cite

**MDPI and ACS Style**

Li, N.; Zhang, J.; Wang, D.; Wang, P.
Research on Hail Mechanism Features Based on Dual-Polarization Radar Data. *Atmosphere* **2023**, *14*, 1827.
https://doi.org/10.3390/atmos14121827

**AMA Style**

Li N, Zhang J, Wang D, Wang P.
Research on Hail Mechanism Features Based on Dual-Polarization Radar Data. *Atmosphere*. 2023; 14(12):1827.
https://doi.org/10.3390/atmos14121827

**Chicago/Turabian Style**

Li, Na, Jun Zhang, Di Wang, and Ping Wang.
2023. "Research on Hail Mechanism Features Based on Dual-Polarization Radar Data" *Atmosphere* 14, no. 12: 1827.
https://doi.org/10.3390/atmos14121827