An Enhanced Hail Detection Algorithm for the WSR-88D

Arthur Witt NOAA/ERL/National Severe Storms Laboratory, Norman, Oklahoma

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Michael D. Eilts NOAA/ERL/National Severe Storms Laboratory, Norman, Oklahoma

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Gregory J. Stumpf NOAA/ERL/National Severe Storms Laboratory, Norman, Oklahoma

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J. T. Johnson NOAA/ERL/National Severe Storms Laboratory, Norman, Oklahoma

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E. De Wayne Mitchell NOAA/ERL/National Severe Storms Laboratory, Norman, Oklahoma

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Kevin W. Thomas NOAA/ERL/National Severe Storms Laboratory, Norman, Oklahoma

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Abstract

An enhanced hail detection algorithm (HDA) has been developed for the WSR-88D to replace the original hail algorithm. While the original hail algorithm simply indicated whether or not a detected storm cell was producing hail, the new HDA estimates the probability of hail (any size), probability of severe-size hail (diameter ≥19 mm), and maximum expected hail size for each detected storm cell. A new parameter, called the severe hail index (SHI), was developed as the primary predictor variable for severe-size hail. The SHI is a thermally weighted vertical integration of a storm cell’s reflectivity profile. Initial testing on 10 storm days showed that the new HDA performed considerably better at predicting severe hail than the original hail algorithm. Additional testing of the new HDA on 31 storm days showed substantial regional variations in performance, with best results across the southern plains and weaker performance for regions farther east.

* Additional affiliation: Cooperative Institute for Mesoscale Meteorological Studies, Norman, Oklahoma

Corresponding author address: Arthur Witt, National Severe Storms Laboratory, 1313 Halley Circle, Norman, OK 73069.

Email: witt@nssl.noaa.gov

Abstract

An enhanced hail detection algorithm (HDA) has been developed for the WSR-88D to replace the original hail algorithm. While the original hail algorithm simply indicated whether or not a detected storm cell was producing hail, the new HDA estimates the probability of hail (any size), probability of severe-size hail (diameter ≥19 mm), and maximum expected hail size for each detected storm cell. A new parameter, called the severe hail index (SHI), was developed as the primary predictor variable for severe-size hail. The SHI is a thermally weighted vertical integration of a storm cell’s reflectivity profile. Initial testing on 10 storm days showed that the new HDA performed considerably better at predicting severe hail than the original hail algorithm. Additional testing of the new HDA on 31 storm days showed substantial regional variations in performance, with best results across the southern plains and weaker performance for regions farther east.

* Additional affiliation: Cooperative Institute for Mesoscale Meteorological Studies, Norman, Oklahoma

Corresponding author address: Arthur Witt, National Severe Storms Laboratory, 1313 Halley Circle, Norman, OK 73069.

Email: witt@nssl.noaa.gov

1. Introduction

The Weather Surveillance Radar-1988 Doppler (WSR-88D) system contains numerous algorithms that use Doppler radar base data as input to produce meteorological and hydrological analysis products (Crum and Alberty 1993). The radar base data (reflectivity, Doppler velocity, and spectrum width) are collected at an azimuthal increment of 1° and at a range increment of 1 km for reflectivity and 250 m for velocity and spectrum width. Currently, two prespecified precipitation-mode scanning strategies are available for use whenever significant precipitation or severe weather is observed. With volume coverage pattern 11 (VCP-11), the radar completes a volume scan of 14 different elevation angles in 5 min, whereas with VCP-21, a volume scan of 9 elevation angles is completed in 6 min. In either case, the antenna elevation steps from 0.5° to 19.5° (for further details, see Brandes et al. 1991).

In the initial WSR-88D system, one set of algorithms, called the storm series algorithms, was used to identify and track individual thunderstorm cells (Crum and Alberty 1993). The storm series process begins with the storm segments algorithm, which searches along radials of radar data for runs of contiguous range gates having reflectivities greater than or equal to a specified threshold. Those segments whose radial lengths are longer than a specified threshold are saved and passed on to the storm centroids algorithm. This algorithm builds azimuthally adjacent segments into 2D storm components and then builds vertically adjacent 2D components into 3D “storms.” The storm tracking algorithm relates all storms found in the current volume scan to storms detected in the previous volume scan. The storm position forecast algorithm calculates a storm’s motion vector and predicts the future centroid location of a storm based on a history of the storm’s movement. Finally, the storm structure and hail algorithms produce output on the storm’s structural characteristics and hail potential.

The initial WSR-88D hail algorithm was developed by Petrocchi (1982). The design is based on identification of the structural characteristics of typical severe hailstorms found in the southern plains (Lemon 1978). The algorithm uses information from the storm centroid and tracking algorithms to test for the presence of seven hail indicators (Smart and Alberty 1985). After testing is completed, a storm is given one of the following four hail labels: positive, probable, negative, or unknown (insufficient data available to make a decision).

Early testing of the hail algorithm showed good performance (Petrocchi 1982; Smart and Alberty 1985). However, subsequent testing by Winston (1988) showed relatively poor performance. Irrespective of its performance, the utility of the hail algorithm is limited by the nature of its output. Since the National Weather Service (NWS) is tasked with providing warnings of severe-size hail (diameter ≥19 mm), it needs an algorithm optimized for this hail size. The aviation community, however, is interested in hail of any size. Most users would also like an estimate of the maximum expected hail size. Finally, given the general uncertainty involved in discriminating hailstorms from nonhailstorms, or severe hail storms from nonsevere hailstorms, the use of probabilities is advisable.

This has led to the design and development of a new hail detection algorithm (HDA) for the WSR-88D. In place of the previous labels, the new algorithm produces, for each detected storm cell, the following information: probability of hail (any size), probability of severe hail, and maximum expected hail size.

2. Algorithm design and development

The new HDA is a reflectivity-based algorithm and has been designed based upon the demonstrated success of the RADAP II vertically-integrated liquid water (VIL) algorithm (Winston and Ruthi 1986) and techniques used during several hail suppression experiments. The HDA runs in conjunction with the new storm cell identification and tracking (SCIT) algorithm (Johnson et al. 1998). Each cell detected by the SCIT algorithm consists of several 2D storm components, which are the quasi-horizontal cross sections for each elevation angle scanning through the cell (Fig. 1). The height and maximum reflectivity of each storm component are used to create a vertical reflectivity profile for the cell. This information is then used by the HDA to determine a cell’s hail potential. To satisfy the different needs of the NWS and the aviation community, the HDA has separate components for detecting hail of any size and severe hail.

a. Detection of hail of any size

To determine the presence of hail of any size, the height of the 45-dBZ echo above the environmental melting level is used. This technique has proven to be successful at indicating hail during several different hail suppression experiments (Mather et al. 1976; Foote and Knight 1979; Waldvogel et al. 1979). Using the data presented in Waldvogel et al. (1979), a simple relation between the height of the 45-dBZ echo above the melting level and the probability of hail at the ground was derived (Fig. 2).

b. Detection of severe hail

1) Severe hail index

To determine the presence of severe hail, an approach similar to the VIL algorithm (i.e., vertical integration of reflectivity) was adopted and changes have been made that should improve on its already successful performance. The first change involves moving from a grid-based algorithm to a cell-based algorithm, using output from the SCIT algorithm. The advantage of a cell-based system is that the problem associated with having a hail core cross a grid boundary, and therefore not being accurately measured, is eliminated. The disadvantage is that if an error occurs in the cell identification process, this may cause an error in the HDA.

The second change involves using a reflectivity-to-hail relation, instead of a reflectivity-to-liquid-water relation as VIL does. The reflectivity data are transformed into flux values of hail kinetic energy (˙E) (Waldvogel et al. 1978a; Waldvogel et al. 1978b; Federer et al. 1986) by
˙E−60.084ZWZ
where
i1520-0434-13-2-286-eq1
Here Z is in dBZ, ˙E in Joules per square meter per second, and the weighting function W(Z) can be used to define a transition zone between rain and hail reflectivities. The default values for this algorithm have initially been set to ZL = 40 dBZ and ZU = 50 dBZ (but are adaptable).1 From Fig. 3, it can be seen that, whereas the VIL algorithm filters out the high reflectivities associated with hail by having an upper-reflectivity limit of 55 dBZ, the Z˙E relation functions in the opposite way, using only the higher reflectivities typically associated with hail and filtering out most of the lower reflectivities typically associated with liquid water. Also, ˙E is closely related to the damage potential of hail at the ground.
A third change involves using a temperature-weighted vertical integration. Since hail growth only occurs at temperatures <0°C, and most growth for severe hail occurs at temperatures near −20°C or colder (English 1973; Browning 1977; Nelson 1983; Miller et al. 1988), the following temperature-based weighting function is used:
i1520-0434-13-2-286-e2
where H is the height above radar level (ARL), H0 is the height ARL of the environmental melting level, and Hm20 is the height ARL of the −20°C environmental temperature. Both H0 and Hm20 can be determined from a nearby sounding or from other sources of upper-air data (e.g., numerical model output).
All of the above leads to the following radar-derived parameter, which is called the severe hail index (SHI). It is defined as
i1520-0434-13-2-286-e3
where HT is the height of the top of the storm cell. In the HDA, SHI is calculated using information from the 2D storm components for the cell being analyzed, with at least two components required for calculation (i.e., SHI values are not calculated for storm cells with just one 2D component). Here ˙E is calculated using the maximum reflectivity value for each storm component, and this value is applied across the vertical depth (or thickness) of the storm component. For interior storm components (i.e., those having an adjacent component both above and below them), the vertical depth ΔHi of the component is given by ΔHi = (Hi+1Hi−1)/2. For the top and bottom storm components, ΔHN = HNHN−1 (N being the number of 2D components) and ΔH1 = H2H1, respectively. If the height of the base of the storm cell is above H0, then ΔH1 = (H1 + H2)/2 − H0. The units of SHI are Joules per meter per second. An example of (3) applied to a storm cell detected by the SCIT algorithm is shown in Fig. 4.

2) Initial developmental testing of SHI

To determine the utility of SHI as a severe hail predictor, WSR-88D level II data (Crum et al. 1993) were analyzed for 10 storm days from radar sites located in Oklahoma and Florida (Table 1). The process consisted of running the SCIT algorithm and HDA on the radar data and correlating the algorithm output to severe hail reports, with ground-truth verification coming from Storm Data (NCDC 1989, 1992). The following analysis procedure was used.

1) A “hail-truth” file was created relating hail reports to storm cells observed in the radar data. This involved recording the time and location of the cell that produced the hail report for a series of volume scans before and after the time of the report (e.g., Table 2). Cell locations were recorded up to 45 min prior to the report time and 15 min after the report time, for those volume scans when the cell had a maximum reflectivity >30 dBZ. Storm cells located within the radar’s cone of silence (≲30 km) or at ranges >230 km were not analyzed.

2) The algorithm was run using the level II data, and an output file was generated. For each volume scan analyzed, the locations and SHI values of all cells detected by the SCIT algorithm were saved, in decreasing order based on SHI. To avoid the detection of large numbers of relatively small-scale cells within a larger multicellular storm, an adaptable parameter (the minimum separation distance between cell detections) within the SCIT algorithm was set to 30 km. Thus, only the dominant cell within a multicellular storm would be identified by the algorithm.

3) A scoring program was then run using the hail-truth and algorithm output files. The scoring program functions as follows. A “warning” threshold is selected. Then, starting with the first volume scan, and continuing until all volume scans are examined, for each storm cell identified, if the SHI value is greater than or equal to the warning threshold, a “yes” forecast of severe hail is made for that cell; otherwise, a “no” forecast is made. The truth file is scanned to see if the given cell correlates with any of the hail reports. A match occurs if a location entry exists in the truth file for the same volume scan and the distance between the truth-file location and the algorithm location is <30 km. If the cell and a report are related, the entry in the truth file is flagged so that it cannot be associated with any other cells, and the time difference (ΔT) between the report and the current volume scan is calculated (ΔT = volume scan time minus report time). If T1 ⩽ ΔTT2, where T1 and T2 define the temporal limits of an analysis “time window” (relative to the time of the hail report), then a hit is declared if a yes forecast was made, and a miss is declared if a no forecast was made. If a yes forecast was made, and ΔT < T1 or ΔT > T2, or if the prediction is not associated with a hail report, then a false alarm is declared. If an algorithm prediction is associated with more than one hail report, resulting in multiple hits and/or misses (due to overlapping time windows from two or more hail reports), only one hit or miss is counted. Finally, any location entries in the hail-truth file that have not been matched to a storm cell, and fall within the time window of their corresponding report, are counted as misses.

Performance results were determined for two time windows of different lengths. The first time window (TW20) was 20 min in length, with T1 = −15 min and T2 = 5 min (Fig. 5a). The choice of these specific temporal limits was based on the time it takes for large hail (already grown and located at midaltitudes) to fall out of a storm, which is typically up to 10 min (Changnon 1970), and a 5-min buffer zone was added onto both ends of this initial 10-min interval (−10 min ⩽ ΔT ⩽ 0 min) to account for synchronization errors between radar and hail observation times. The second time window (TW60) was 60 min in length, with T1 = −45 min and T2 = 15 min. These temporal limits were chosen based on the time it takes for large hail to both grow and fall out of a storm, which can be up to ∼30 min (English 1973), and a 15-min buffer zone was added onto both ends of this initial 30-min interval (−30 min ⩽ ΔT ⩽ 0 min) to produce a length similar to that of typical NWS severe weather warnings. An alternate way of visualizing the time windows, relative to the time of an algorithm prediction, is shown in Fig. 5b.

This scoring methodology was used in order to deal with the verification problems caused by the highly sporadic nature of the severe hail reports in Storm Data, while still allowing for evaluation of all the algorithm’s predictions (Witt et al. 1998). Since many of the reports in Storm Data are generated through the verification of NWS severe weather warnings (Hales and Kelly 1985), the reports contained therein will often be on the same time- and space scales as the warnings, which are typically issued for one or more counties for up to 60 min in length. This led to the choice of 60 min as the length of the second time window and was an additional factor in the choice of a large minimum separation distance between cell detections. Thus, for those situations where a storm produces a long, continuous swath of large hail, but hail reports are relatively infrequent (but still frequent enough to verify severe weather warnings), a long time window effectively “fills in” the time gap between individual reports. However, since storms can gradually increase in strength before initially producing large hail, and multicellular storms can produce large hail in short, periodic bursts, it would be inappropriate to use just a single, long-time window for algorithm evaluation (because too many misses would be counted in these situations). An additional reason for using a skewed time window (i.e., a larger period before versus after the time of the report) is that this allows for the evaluation (indirectly) of algorithm lead-time capability, which is particularly important to the NWS (Polger et al. 1994).

In the process of building the hail-truth files for this dataset, there were many instances when a hail report either could not be easily correlated to a storm cell based on the radar data (e.g., hail was reported at a specific location and time, with the nearest storm cell 50 km away), or occurred at the edge of a cell, away from the higher reflectivity core (Z ≥ 45 dBZ). For the 10 storm days analyzed here, there were 115 hail reports in Storm Data, of which 33 (29%) did not correlate well with the radar data, if the location and time of the report were assumed to be accurate. One possible solution was to simply discard these reports, but given the general scarcity of ground-truth verification data, this was deemed unacceptable. Instead, an attempt was made to correct such reports. The procedure involved assuming that the location of the report was generally correct (to within a few km), but that the time of the report was in error (up to ±1 h). The radar data were perused to see if a storm cell had, in fact, passed over the location of the report, within an hour’s time of the report and, if so, the original time was changed to correlate best with the radar data and the report was added to the truth file. Of the 33 questionable hail reports, 24 were corrected in this manner. An additional questionable report was added by correcting a typographical error in the location of the report. Eight of the original 115 reports were ultimately discarded.

Using the time-window scoring method mentioned above, performance results were generated for all the storm days analyzed using a multitude of different warning thresholds. As an example, the results for the 17 February 1992 case (for TW20) are shown in Table 3. Similar tables of scoring results (not shown) were generated for each of the other storm days, and the warning threshold producing the highest critical success index (CSI) was noted. For the 8 storm days when severe hail was observed, it was found that the optimum warning threshold (leading to the highest CSI) was highly correlated with the melting level on that day [linear correlation coefficients of 0.78 and 0.81 for TW20 and TW60, respectively (Fig. 6)]. From these results, a simple warning threshold selection model (WTSM) was created and is defined as
H0
where WT (J m−1 s−1) is the warning threshold and H0 (km) is measured ARL.2 If WT < 20 J m−1 s−1, then WT is set to 20 J m−1 s−1.

Using (4), a new set of performance results were generated (Table 4). For each severe hail day, the number of hits is lower, and false alarms higher, for TW20 versus TW60. Misses are higher by a factor of at least 2 for TW60 versus TW20, except for the 28 May 1992 case. This all leads to higher probability of detection (POD) and false-alarm rate (FAR) values for TW20 versus TW60, except for the 28 May 1992 case, where POD values are identical. The corresponding CSI values are higher on 3 days and lower on 5 days for TW20 versus TW60. This raises the question of which set of results is more likely representative of actual algorithm performance. For POD, the values from TW20 are probably more accurate, as some of the no forecasts that are being counted as misses with TW60 are likely occurring at times when storms are not producing large hail. Conversely, for FAR, the values from TW60 are probably more accurate, since the yes forecasts that are counted as false alarms with TW20, but not with TW60, do correspond to a known severe hail event, the full extent of which is unknown due to deficiencies in the verification data (Witt et al. 1998). Consequently, the CSI values (for either time window) are likely understating actual algorithm performance.

For comparison, performance results were also generated for the original WSR-88D hail algorithm (using the same procedure given above) and are shown in Table 5. Although the overall number of hits and misses (and POD values) are nearly the same for the two algorithms, the original WSR-88D hail algorithm produces many more false alarms, with a FAR much higher than that of the new algorithm. Comparing CSI values for the days with severe hail, and the number of false alarms for the days with no severe hail reports, the new algorithm outperforms the original algorithm on 9 of the 10 days.

3) Development of a probability function

Given the general success of SHI and the WTSM at predicting severe hail (overall CSI values >40%), the final stage of development was to implement an appropriate probability function. Since the dataset used for development thus far was quite small, it was decided that the initial probability function should be fairly simple in nature to avoid overfitting the data. Candidate functions were first developed (by trial and error) using test results (for TW60) from only 2 storm days, those with the lowest and highest melting levels, and their calibration (for all 10 storm days) was determined using reliability diagrams (Wilks 1995). This rather limited initial analysis led to a surprisingly good (for this developmental dataset) probability function, which is given by
i1520-0434-13-2-286-e5
where POSH is the probability of severe hail (%), POSH values <0 are set to 0, and POSH values >100 are set to 100. Despite the continuous nature of (5), actual algorithm output probabilities are rounded off to the nearest 10%, in order to avoid conveying an unrealistic degree of precision. Note that when SHI = WT, POSH = 50%. The reliability diagram of (5) applied to all 10 storm days is shown in Fig. 7.

c. Prediction of maximum expected hail size

The SHI is also used to provide estimates of the maximum expected hail size (MEHS). Using data from the 8 severe hail days shown in Table 1, along with data from Twin Lakes on 18 June 1992, (yielding a total of 147 severe hail reports), an initial model relating SHI to maximum hail size was developed. The process involved comparing SHI values with observed hail sizes. For each hail report in the dataset, the maximum value of SHI within TW20 was determined. A scatterplot of these SHI values versus observed hail size is shown in Fig. 8. One thing that is clearly seen in Fig. 8 is the common practice of reporting hail size using familiar circular or spherical objects (e.g., various coins or balls) as reports tend to be clustered along discrete sizes. Concerning the relationship between SHI and hail size, it is apparent that the minimum and average SHI (for the different common size values) increase as hail size increases. However, there does not appear to be an upper-limit cutoff value for SHI as hail size increases. This is likely due to the fact that a storm producing very large hail will almost always be producing smaller diameter hail at the same time (often falling over a larger spatial area than the very large hail), and this smaller (but still severe-sized) hail will also usually be observed and reported.

Since the hail-size model being developed is meant to forecast maximum expected hail size, it was developed such that around 75% of the hail observations would be less than the corresponding predictions. As was the case with development of the probability function for POSH, it was decided that the initial hail-size prediction model should also be fairly simple in nature. This led to the following relation:
0.5
with MEHS in millimeters. Equation (6) is also shown in Fig. 8. Comparing (6) with the hail-size observations shows that it meets the 75% goal mentioned above and is close to 75% for each of the three distinct size clusters (Table 6). Again, to avoid conveying an unrealistic degree of precision, actual algorithm output size values are rounded off to the nearest 6.35 mm (0.25 in.).

3. Performance evaluation

a. Hail of any size

Evaluating the performance of the probability of hail (POH) parameter was difficult due to the lack of available ground-truth verification data (for hail of any size). However, during the summer months of 1992 and 1993, the National Center for Atmospheric Research (NCAR) conducted a hail project in the high plains of northeastern Colorado to collect an adequate dataset for algorithm verification (Kessinger and Brandes 1995). As part of the hail project, both the new HDA and the original hail algorithm were run using reflectivity data from the Mile High Radar (Pratte et al. 1991), a prototype Next Generation Weather Radar (NEXRAD) located 15 km northeast of Denver. Given the highly detailed nature of the special verification dataset that was collected, it was possible to score algorithm performance on an individual volume scan basis, instead of the time-window method that was developed for use with Storm Data. Performance results are summarized in Kessinger et al. (1995), with detailed results given in Kessinger and Brandes (1995). Two pertinent overall results are repeated here. Using 50% as a warning threshold for the POH parameter, and verifying against hail observations of any size, the following accuracy measures were obtained: POD = 92%, FAR = 4%, and CSI = 88%. A similar evaluation of the original hail algorithm using “probable” as a warning threshold gave these results: POD = 74%, FAR = 5%, and CSI = 72%.

b. Severe hail

To provide an independent test of SHI, the initial WTSM, and the initial probability function used to calculate the POSH parameter, additional testing was done using the same analysis procedures presented in section 2b. Since this algorithm testing occurred in phases spanning a period of several years, case selection was largely determined by the availability of WSR-88D level II data at the time of testing. Despite these constraints, it was still possible to obtain radar data from numerous different sites across the United States (Table 7).

To test the accuracy of SHI and the WTSM, performance statistics were again generated using the WTSM to produce categorical forecasts of severe hail for each day listed in Table 7, with results shown in Tables 8 and 9. Table 8 gives performance statistics for each individual day, and Table 9 shows overall performance statistics for cases grouped together into different geographical regions.

Algorithm performance varied widely from one storm day to another. For null cases (i.e., days with no severe hail reports), the best result possible was zero false alarms (e.g., 8 June 1992). However, on some days (e.g., 10 August 1992) the HDA produced many false alarms. For those days with reported severe hail, the HDA had CSI values that varied from a low of 3% (on 8 June 1993) to a high of 78% (on 20 June 1995). Except for two days, the HDA had POD values ≥50% (for TW20). Conversely, FAR values (for TW60) varied greatly, from a low of 0% (on 20 June 1995) to a high of 96% (on 8 June 1993). Of particular interest are the two days from Memphis (26 and 27 April 1994). The large-scale synoptic pattern was nearly identical for these two days, but algorithm performance was quite different. On 26 April 1994, the HDA performed quite well, producing a CSI of 62% (for TW60). However, on 27 April 1994 (the most active storm day in the dataset, in terms of the number of algorithm predictions), the algorithm produced a very large number of false alarms, resulting in markedly poorer performance. The reasons for this large difference in performance are not known. Comparison of overall test results between the independent and developmental datasets (Table 8 vs Table 4) shows an increase in both the POD and FAR and a decrease in the CSI.

On a regional basis, substantial performance variations exist (Table 9). The POD values exhibit the smallest amount of regional variation, with large differences in regional FAR values. These FAR differences are the primary factor leading to the corresponding regional variations in CSI (i.e., lower relative FAR corresponds with higher relative CSI). Also shown in Table 9 are overall POD values for two larger hail-size thresholds. Except for Florida (FL), the POD increases as hail size increases.

Another test of the accuracy of the WTSM is to determine if it remains highly correlated with the melting level and, if so, whether the initial model equation is still the best one to use. Therefore, for each severe hail day in Table 7, the optimum warning threshold was calculated and plotted versus the day’s melting level (Fig. 9). For both time windows, most of the days (74% for TW20 and 78% for TW60) have optimum warning thresholds (including the five-point range bars) on or close to the WTSM.3 For those days with optimum warning thresholds not close to the WTSM, these were all higher than the WTSM for TW20 and, except for one day, were all lower than the WTSM for TW60. Regional variations are shown in Figs. 10 and 11. For all regions except the Mississippi River (MR), there is a generally good match between the WTSM and the observed optimum warning thresholds.

To evaluate the POSH parameter, reliability diagrams were again used. Figure 12 shows the reliability diagram for all the days listed in Table 7. Although Fig. 7 showed a slight overforecasting bias (for medium-range probabilities) for the developmental dataset, Fig. 12 shows a pronounced overforecasting bias for the independent dataset. However, this overforecasting bias varies dramatically for the different regions (Fig. 13). For the southern plains, there is little bias and very good calibration. For the northern United States, there is a considerable overforecasting bias for probabilities of 20%–60%, and also 80%, with the remaining probability values showing good calibration. However, for FL, and especially the MR region, large overforecasting biases exist. For these two regional datasets, the initial probability function developed for the POSH parameter shows very poor calibration, and suggests the need for regionally dependent definitions of the POSH parameter.

The effect of population density on algorithm performance was investigated in a very limited study involving the two Wisconsin cases. In addition to the analysis results presented in Table 8, a second evaluation, limited to storms occurring over the Milwaukee (MKE) metropolitan area, was done (Table 10). As would be expected, limiting the analysis domain to only the MKE area greatly reduced the number of algorithm predictions available for evaluation. However, given that the remaining storm events occurred over an urban area (high population density), it is much less likely that a severe weather event would go unreported, compared to the full domain. Thus, any false alarms produced by the algorithm are more likely to be valid, and not simply because the storm occurred over an area with few, if any, storm spotters. Comparing the full and MKE domain performance results does, in fact, show large differences in the FAR values, with superior CSI values for the MKE domain. And the CSI (and POD) can be increased to even higher values for the MKE domain by lowering the algorithm’s warning threshold by 33% (MKE2). What these results seem to indicate is that some, and possibly many, of the false alarms shown in Tables 3–5 and 8 (and also affecting Figs. 6, 7, and 9–13) may be fictitious. Thus, some of the large overforecasting bias seen in Figs. 12 and 13 could be due to underreporting of actual severe hail events. However, because of the small amount of data analyzed here, further investigation is needed in order to validate this hypothesis. Initial results from a larger study of this issue (Wyatt and Witt 1997) also show improved algorithm performance for higher population density areas.

c. Maximum hail size

An independent evaluation of SHI as a hail-size predictor was done using hail reports from the days given in Table 7 (minus the reports from 18 June 1992, which were used in the initial development of the hail-size model), along with some supplemental reports (diameters >4 cm) from the days shown in Table 11 (yielding a total of 314 reports).4 Once again, the maximum value of SHI within TW20 was determined and plotted versus the size of the hail report (Fig. 14). Comparing Fig. 14 to Fig. 8, it is apparent, for sizes >33 mm, that the average value of SHI has decreased substantially, resulting in a smaller percentage of observed sizes greater than the MEHS model curve (Table 12). Also evident is an increased vertical stacking of the observations, thus reducing the discrimination capability of SHI as a hail-size predictor.

4. Discussion

The new WSR-88D HDA attempts to do considerably more than its original counterpart. Instead of simply providing a single, categorical statement on whether or not a storm is producing hail, it tries to determine the potential hail threat from multiple perspectives and provide quantitative guidance to end users. However, the ability to properly design and develop an algorithm like the new HDA (i.e., one that is empirical in nature and produces detailed quantitative information) depends greatly on the quality and quantity of ground-truth data available for development and testing. Inadequacies and errors in the ground-truth database will have a corresponding negative impact on algorithm design and performance. For development and testing of the new HDA, verification data has come both from special field projects (such as NCAR’s hail project) and from Storm Data. Field project datasets are limited in scope but are generally of high quality. On the other hand, Storm Data provides severe weather information for the entire United States, but the information is less detailed and often less accurate.

The probability of hail parameter was both developed and tested using special field project data. Hence, ground-truth deficiencies and errors should be minimal. The test results from Kessinger et al. (1995) show that the POH parameter performs very well in Colorado. However, it should be noted that the development and testing of the POH parameter exclusively involved data collected in a “high-plains”-type of geographical environment. Therefore, it is possible, and perhaps even likely, that the performance of the POH parameter will be poorer in other regions of the United States.

Unlike the POH parameter, the probability of severe hail parameter was developed using Storm Data for ground-truth verification. One thing that is obvious from the results presented in section 3 is that the POSH parameter performs considerably better in the southern plains than in other parts of the United States. There are several reasons why this may be so. One potential reason has to do with ground-truth verification efficiency, that is, the percentage of actual severe weather events that are observed and reported to the NWS. From the performance statistics shown in Table 9, it is clear that regional variations in CSI are largely a function of variations in FAR. The cause of this variation in the FAR is unknown. However, the information that appears in Storm Data is largely the result of NWS severe weather warning verification efforts (Hales and Kelly 1985) and thus is a function of both severe weather climatology and verification efficiency. Now, if verification efficiency was constant across the United States, then the regional differences in algorithm performance could be attributed solely to differences in severe weather climatology. But verification efficiency is not constant across the United States (Hales and Kelly 1985; Crowther and Halmstad 1994). Some NWS offices put a greater emphasis on severe weather verification than do others, and population density varies dramatically across the United States. Therefore, regional differences in algorithm performance are a function of both differences in severe weather climatology and differences in verification efficiency, with the largest impact of verification efficiency being on the FAR statistic.

As it is, both of these factors are likely affecting the regional performance statistics. Considering the severe weather climatology aspect, large hail is simply more common in the Great Plains compared to other parts of the United States (Kelly et al. 1985). There, hailstorms are often fairly long-lived and produce longer hailswaths (Changnon 1970), making observation of a single hailfall event more likely. Considering the verification efficiency aspect, since NWS offices in the southern plains tend to have extensive severe weather spotter networks, warning verification efforts often lead to many hail reports during severe storm events (Table 7). Changnon (1968) shows that observation density greatly affects the frequency of damaging hail reports, as well as whether or not damaging hail is observed at all on a given storm day. He states that a network comprised of one or more observation sites per square mile is necessary to adequately measure the areal extent of damaging hail. Since NWS spotter networks are much less dense than this, it is possible that many small-scale severe hail events, such as those produced by “single-pulse” type storms, are simply unreported. With single-pulse type storms relatively more common east of the Great Plains, this may be a significant factor leading to the higher algorithm FAR values in these regions.

Other potential causes of regional variation are differences in the typical storm environment. Numerical modeling studies have shown that the melting of hailstones is affected by a number of factors (Rasmussen and Heymsfield 1987). The most dominant factor is the thermal profile through which the hailstone falls. However, the RH profile also has a substantial effect. The HDA already incorporates some information on the vertical thermal profile (both the POH and POSH parameters are functions of the melting level). The impact that RH might have on the HDA was the focus of an additional study. Specifically investigated was whether the WTSM would benefit from the addition of an RH-dependent term (to its defining equation). Unfortunately, initial test results showed a minimal improvement in overall performance (the CSI increased by only 2%). However, for this study, the environmental RH was determined in a rather crude manner (using 700-mb upper-air plots), and so further investigation is needed to fully evaluate its effects.

Since the FAR is the statistic most variable on a regional basis, one might think that simply changing the WTSM to produce higher warning thresholds in those regions with higher FAR values would improve performance. Whereas this may be true in the MR region, that does not appear to be the case for the other regions (Figs. 10 and 11). Although the number of false alarms will decrease as the warning threshold increases, so too will the number of hits. And if, as the warning threshold is increased, the number of hits decreases more rapidly than the number of false alarms, the FAR will actually increase as the warning threshold rises. Therefore, despite the regional variations in overall CSI, it is not obvious that a separate WTSM is needed for each region. However, it is clear from the results shown in Fig. 13 that regional, or perhaps storm-environment-dependent, probability functions need to be developed. This will likely result in different optimum POSH thresholds (i.e., the threshold producing the highest overall CSI) for each region or storm environment, since the current model, optimized at 50%, does not appear to be appropriate for all regions or environments. And even within any one region, it will often not be best to always use just one overall, optimized threshold. For example, in situations where a storm is approaching a heavily populated area, a lower threshold may be better (given the results in Table 10).

It should be noted that all but two of the storm days used for the performance evaluation presented here came from WSR-88D sites at relatively low elevations (<∼400 m) above mean sea level (MSL). Thus, the accuracy of the WTSM for WSR-88D sites at relatively high elevations (≳1 km) is questionable, since WT [as given by Eq. (4)] is a function of H0 measured relative to the height ARL. Recent evaluation of HDA performance over Arizona (Maddox et al. 1998) indicates that, for the different WSR-88D sites located there, WT and POSH values can vary widely for constant melting level (relative to MSL) and SHI values, due solely to variations in the radar site elevation. There are also indications of a large overforecasting bias to POSH for high elevation WSR-88D sites (Kessinger and Brandes 1995;Maddox et al. 1998), due primarily to low values of WT for all seasons. Hence, until a terrain model can be added to the HDA and more extensive testing is done using data from high elevation WSR-88D sites, it may be necessary to change Eq. (4) so that H0 is measured relative to MSL instead of ARL.

The prediction of maximum expected hail size is probably the most difficult and challenging aspect of the HDA. Also difficult is proper evaluation of the performance of the MEHS predictions, given the highly sporadic nature of the hail reports in Storm Data. The deficiencies in ground-truth verification of maximum hail size make scoring this component of the HDA very problematic. Without a high-density hail-observing network, one has no way of truly knowing the size of the largest hail being produced by a storm at any given time. The extent of this problem is amply illustrated by Morgan and Towery (1975). They present observations of a hailstorm on 21 May 1973, which moved across a very high-density (hailpads every 100–200 m) network located in Nebraska. Hail was observed at every site, with maximum sizes ranging from ∼1 to 3 cm. However, the area covered by the largest hail (3 cm) was only 1% of the total area of the network, with about 80% covered by hail <2 cm in diameter. Thus, at least in this case, the probability of a single hail report providing a true measure of the maximum hail size produced by the storm is very small. Therefore, due to the large uncertainties in the verification data, no attempt was made to determine any size-error statistics. Instead, only percentages of observed sizes greater than predicted sizes were calculated. Given the tendency for many different hail sizes to be observed for the larger SHI values, providing probabilities for various hail-size categories, in addition to a maximum size estimate, seems appropriate. And although correlations between the MEHS predictions and actual observed hail sizes will likely be poor at times, using the MEHS predictions as a relative indicator of overall hail damage potential may prove to be useful.

In addition to using SHI as a predictor of maximum hail size, Doppler-radar-determined storm-top divergence has been shown to be a reliable indicator of maximum hail size (Witt and Nelson 1991). A separate algorithm, called the upper-level divergence algorithm (ULDA), has been developed by the National Severe Storms Laboratory to detect and measure the strength of these divergence signatures. During the early stages of HDA development, the ULDA was run concurrently with the HDA, and output from both algorithms was used to produce a final estimate of the maximum expected hail size. However, more extensive testing to date has shown that frequent problems associated with velocity dealiasing errors, range folding, and coarse vertical sampling when the WSR-88D is operating in VCP-21, degrade the ULDA’s performance to the point that, overall, better size estimates are produced when solely using SHI as a predictor. It is hoped that future enhancements to the ULDA, along with better dealiasing techniques, will improve its performance to the point that it can make a positive contribution to the overall performance of the HDA. The detection and quantification of other radar signatures and/or environmental parameters may also help produce better MEHS predictions (e.g., bounded weak echo regions, midaltitude rotation, midaltitude winds).

At a minimum, the new WSR-88D HDA provides more information on the hail potential of a storm than the original hail algorithm. Test results also indicate that the new HDA outperforms the original hail algorithm. Steadham and Lee (1995) indicate that the original hail algorithm was not utilized much by operational warning forecasters. This may be due to poor performance and/or the limited nature of the output. With damaging hail being a significant hazardous weather threat, it is important that a hail detection algorithm produce guidance that a warning forecaster finds useful. Although additional improvements can certainly be made to the new HDA, its operational implementation will hopefully lead to more accurate and timely hazardous weather warnings.

Acknowledgments

We thank Robert Maddox, Conrad Ziegler and two anonymous reviewers for providing many useful comments that improved the manuscript. We also thank Joan O’Bannon for drafting two of the figures used in this paper. This work was partially supported by the WSR-88D Operational Support Facility and the Federal Aviation Administration.

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Fig. 1.
Fig. 1.

Diagram illustrating the identification of 2D storm components (thick lines and circles) within a cell by the SCIT algorithm.

Citation: Weather and Forecasting 13, 2; 10.1175/1520-0434(1998)013<0286:AEHDAF>2.0.CO;2

Fig. 2.
Fig. 2.

Probability of hail at the ground as a function of (H45H0). Here H45 is the height of the 45-dBZ echo above radar level (ARL), and H0 is the height of the melting level ARL (derived from Waldvogel et al. 1979).

Citation: Weather and Forecasting 13, 2; 10.1175/1520-0434(1998)013<0286:AEHDAF>2.0.CO;2

Fig. 3.
Fig. 3.

Plot of hail kinetic energy flux (solid curve), and liquid water content (used to calculate VIL; dashed curve), as a function of reflectivity.

Citation: Weather and Forecasting 13, 2; 10.1175/1520-0434(1998)013<0286:AEHDAF>2.0.CO;2

Fig. 4.
Fig. 4.

Sample SHI values (J m−1 s−1) for a typical storm cell, along with the corresponding maximum reflectivities (dBZ) for each 2D storm component, as identified by the SCIT algorithm for five volume scans. Reflectivity values are plotted at the center height of each component. Here H0 = 3 km and Hm20 = 6 km.

Citation: Weather and Forecasting 13, 2; 10.1175/1520-0434(1998)013<0286:AEHDAF>2.0.CO;2

Fig. 5.
Fig. 5.

Diagram of the time window scoring methodology, (a) relative to the time of a hail report and (b) relative to the time of an algorithm prediction.

Citation: Weather and Forecasting 13, 2; 10.1175/1520-0434(1998)013<0286:AEHDAF>2.0.CO;2

Fig. 6.
Fig. 6.

Optimum warning threshold as a function of the melting level for the 8 days in Table 1 with hail reports for (a) TW20 and (b) TW60. Solid circles correspond to the highest CSI. Vertical bars represent the range of warning thresholds with a CSI within 5 percentage points of the maximum value (i.e., nearly optimal). The sloping line is the warning threshold selection model.

Citation: Weather and Forecasting 13, 2; 10.1175/1520-0434(1998)013<0286:AEHDAF>2.0.CO;2

Fig. 7.
Fig. 7.

Reliability diagram for the probability of severe hail for the 10 storm days in Table 1. Numerals adjacent to the plotted points indicate the number of forecasts for that POSH value. The diagonal line represents perfect reliability.

Citation: Weather and Forecasting 13, 2; 10.1175/1520-0434(1998)013<0286:AEHDAF>2.0.CO;2

Fig. 8.
Fig. 8.

Scatterplot of SHI vs observed hail size for 147 hail reports from 9 storm days. The plotted curve is the MEHS prediction model.

Citation: Weather and Forecasting 13, 2; 10.1175/1520-0434(1998)013<0286:AEHDAF>2.0.CO;2

Fig. 9.
Fig. 9.

Same as Fig. 6, except for the 31 storm days in Table 8.

Citation: Weather and Forecasting 13, 2; 10.1175/1520-0434(1998)013<0286:AEHDAF>2.0.CO;2

Fig. 10.
Fig. 10.

Same as Fig. 9, except for TW20 subdivided into four different regions: (a) southern plains, (b) Florida, (c) Mississippi River, and (d) northern United States.

Citation: Weather and Forecasting 13, 2; 10.1175/1520-0434(1998)013<0286:AEHDAF>2.0.CO;2

Fig. 11.
Fig. 11.

Same as Fig. 10, except for TW60.

Citation: Weather and Forecasting 13, 2; 10.1175/1520-0434(1998)013<0286:AEHDAF>2.0.CO;2

Fig. 12.
Fig. 12.

Same as Fig. 7, except for the 31 storm days in Table 8.

Citation: Weather and Forecasting 13, 2; 10.1175/1520-0434(1998)013<0286:AEHDAF>2.0.CO;2

Fig. 13.
Fig. 13.

Same as Fig. 12, except subdivided into four different regions: (a) southern plains, (b) Florida, (c) Mississippi River, and (d) northern United States.

Citation: Weather and Forecasting 13, 2; 10.1175/1520-0434(1998)013<0286:AEHDAF>2.0.CO;2

Fig. 14.
Fig. 14.

Same as Fig. 8, except for 314 hail reports from 30 storm days.

Citation: Weather and Forecasting 13, 2; 10.1175/1520-0434(1998)013<0286:AEHDAF>2.0.CO;2

Table 1.

List of the storm cases analyzed. Here RS is the radar site, BT and ET are the beginning and ending times of data analysis, H0 is the melting level ARL, NR is the number of hail reports used in the analysis, MS is the maximum reported hail size, NVS is the number of volume scans analyzed, NAP is the total number of algorithm predictions, and MZ is the maximum reflectivity for all the storm cells analyzed. The date corresponds to the beginning time. RS–locations: FDR is Frederick, OK; MLB is Melbourne, FL; OUN is Norman, OK;and TLX is Twin Lakes, OK.

Table 1.
Table 2.

The hail truth file for 2 June 1992. The value on the first line is the number of hail reports for the day. The second (9th) line of values contains the size and time of the first (second) report. The values on lines 3–8 (10–16) are the storm locations (azimuth and range) and volume scan times needed for algorithm scoring.

Table 2.
Table 3.

Performance results for the new HDA for 17 February 1992 using the 20-min. time window (TW20). Here WT is warning threshold, H is hits, M is misses, FA is false alarms, POD is probability of detection, FAR is false-alarm rate, and CSI is critical success index. POD, FAR, and CSI values are in percent.

Table 3.
Table 4.

Performance results for the new HDA. Cases are listed by increasing warning threshold (melting level). For those days with severe hail reports, the first row of values are for TW20, with the second row for TW60. (See Table 3 for definition of terms.)

Table 4.
Table 5.

Same as Table 4, but for the original WSR-88D hail algorithm using a warning threshold of “probable” (i.e., both “probable” and “positive” indications are used to make positive hail forecasts).

Table 5.
Table 6.

Hail-size observations compared to model predicted sizes for 9 storm days.

Table 6.
Table 7.

List of the additional storm cases analyzed. See Table 1 for definition of terms. New locations (RS): DDC is Dodge City, KS; LSX is St. Louis, MO; LWX is Sterling, VA; MKX is Milwaukee, WI; MPX is Minneapolis, MN; NQA is Memphis, TN; and OKX is New York City, NY.

Table 7.
Table 8.

Same as Table 4, but for 31 additional storm days.

Table 8.
Table 9.

Regional performance results. For each region, the first row of overall POD, FAR, and CSI values are for TW20, with the second row for TW60. Here NR is the number of hail reports, NAP is the number of algorithm predictions, SP is southern plains (DDC, OUN, TLX), FL is Florida (MLB), MR is Mississippi River (LSX, NQA), and NUS is northern United States. (MKX, MPX, OKX). The last two columns are for larger hail diameters (D). POD, FAR, and CSI values are in percent.

Table 9.
Table 10.

Same as Table 4, but for the two MKX cases for different analysis domains (AD). NAP is the number of algorithm predictions and MKE2 refers to the case where the warning threshold has been reduced by 33%.

Table 10.
Table 11.

List of the additional storm cases analyzed to increase the number of very large hail reports. See Table 1 for definition of terms. Additional RS–location: IWA is Phoenix, AZ.

Table 11.
Table 12.

Same as Table 6, but for 30 additional storm days.

Table 12.

1

These values are lower than those used by Federer et al. (1986), since severe hail is occasionally observed with storms having maximum reflectivities <55 dBZ.

2

It should be noted that, at the present state of algorithm development, a flat earth is assumed. This will undoubtedly lead to errors in the WT calculations for those radar sites where terrain height varies substantially within 230 km. Hopefully, future enhancements to the algorithm will include site-specific terrain models for those locations where this correction is needed.

3

Close to the WTSM is defined as being within 20 J m−1 s−1.

4

Some of the hail reports listed in Table 7 were not usable for the size evaluation, because they occurred during time periods without complete radar data, or at ranges >230 km or ≲30 km. They were usable in the other evaluations because of the time-window scoring methodology.

Save
  • Brandes, E. A., D. S. Zrnic, G. E. Klazura, C. F. Suprenant, and D. Sirmans, 1991: The Next Generation Weather Radar (WSR-88D) as an applied research tool. Preprints, 25th Int. Conf. on Radar Meteorology, Paris, France, Amer. Meteor. Soc., 47–50.

  • Browning, K. A., 1977: The structure and mechanisms of hailstorms. Hail: A Review of Hail Science and Hail Suppression, Meteor. Monogr., No. 38, Amer. Meteor. Soc., 1–43.

    • Crossref
    • Export Citation
  • Changnon, S. A., 1968: Effect of sampling density on areal extent of damaging hail. J. Appl. Meteor.,7, 518–521.

    • Crossref
    • Export Citation
  • ——, 1970: Hailstreaks. J. Atmos. Sci.,27, 109–125.

    • Crossref
    • Export Citation
  • Crowther, H. G., and J. T. Halmstad, 1994: Severe local storm warning verification for 1993. NOAA Tech. Memo. NWS NSSFC-40, 30 pp. [NTIS PB 94215811.].

  • Crum, T. D., and R. L. Alberty, 1993: The WSR-88D and the WSR-88D Operational Support Facility. Bull. Amer. Meteor. Soc.,74, 1669–1687.

    • Crossref
    • Export Citation
  • ——, ——, and D. W. Burgess, 1993: Recording, archiving, and using WSR-88D data. Bull. Amer. Meteor. Soc.,74, 645–653.

    • Crossref
    • Export Citation
  • English, M., 1973: Alberta hailstorms. Part II: Growth of large hail in the storm. Alberta Hailstorms, Meteor. Monogr., No. 36, Amer. Meteor. Soc., 37–98.

    • Crossref
    • Export Citation
  • Federer, B., and Coauthors, 1986: Main results of Grossversuch IV. J. Climate Appl. Meteor.,25, 917–957.

    • Crossref
    • Export Citation
  • Foote, G. B., and C. A. Knight, 1979: Results of a randomized hail suppression experiment in northeast Colorado. Part I: Design and conduct of the experiment. J. Appl. Meteor.,18, 1526–1537.

  • Hales, J. E., Jr., and D. L. Kelly, 1985: The relationship between the collection of severe thunderstorm reports and warning verification. Preprints, 14th Conf. on Severe Local Storms, Indianapolis, IN, Amer. Meteor. Soc., 13–16.

  • Johnson, J. T., P. L. MacKeen, A. Witt, E. D. Mitchell, G. J. Stumpf, M. D. Eilts, and K. W. Thomas, 1998: The Storm Cell Identification and Tracking (SCIT) algorithm: An enhanced WSR-88D algorithm. Wea. Forecasting,13, 263–276.

    • Crossref
    • Export Citation
  • Kelly, D. L., J. T. Schaefer, and C. A. Doswell III, 1985: Climatology of nontornadic severe thunderstorm events in the United States. Mon. Wea. Rev.,113, 1997–2014.

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  • Fig. 1.

    Diagram illustrating the identification of 2D storm components (thick lines and circles) within a cell by the SCIT algorithm.

  • Fig. 2.

    Probability of hail at the ground as a function of (H45H0). Here H45 is the height of the 45-dBZ echo above radar level (ARL), and H0 is the height of the melting level ARL (derived from Waldvogel et al. 1979).

  • Fig. 3.

    Plot of hail kinetic energy flux (solid curve), and liquid water content (used to calculate VIL; dashed curve), as a function of reflectivity.

  • Fig. 4.

    Sample SHI values (J m−1 s−1) for a typical storm cell, along with the corresponding maximum reflectivities (dBZ) for each 2D storm component, as identified by the SCIT algorithm for five volume scans. Reflectivity values are plotted at the center height of each component. Here H0 = 3 km and Hm20 = 6 km.

  • Fig. 5.

    Diagram of the time window scoring methodology, (a) relative to the time of a hail report and (b) relative to the time of an algorithm prediction.

  • Fig. 6.

    Optimum warning threshold as a function of the melting level for the 8 days in Table 1 with hail reports for (a) TW20 and (b) TW60. Solid circles correspond to the highest CSI. Vertical bars represent the range of warning thresholds with a CSI within 5 percentage points of the maximum value (i.e., nearly optimal). The sloping line is the warning threshold selection model.

  • Fig. 7.

    Reliability diagram for the probability of severe hail for the 10 storm days in Table 1. Numerals adjacent to the plotted points indicate the number of forecasts for that POSH value. The diagonal line represents perfect reliability.

  • Fig. 8.

    Scatterplot of SHI vs observed hail size for 147 hail reports from 9 storm days. The plotted curve is the MEHS prediction model.

  • Fig. 9.

    Same as Fig. 6, except for the 31 storm days in Table 8.

  • Fig. 10.

    Same as Fig. 9, except for TW20 subdivided into four different regions: (a) southern plains, (b) Florida, (c) Mississippi River, and (d) northern United States.

  • Fig. 11.

    Same as Fig. 10, except for TW60.

  • Fig. 12.

    Same as Fig. 7, except for the 31 storm days in Table 8.

  • Fig. 13.

    Same as Fig. 12, except subdivided into four different regions: (a) southern plains, (b) Florida, (c) Mississippi River, and (d) northern United States.

  • Fig. 14.

    Same as Fig. 8, except for 314 hail reports from 30 storm days.

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