# An Evaluation of Relationships between Radar-Inferred Kinematic and Microphysical Parameters and Lightning Flash Rates in Alabama Storms

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## Abstract

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## 1. Introduction

_{2}= NO

_{X}) to the upper troposphere, where a net increase can lead to net production of ozone, which is a powerful greenhouse gas [1,4,41]. A better understanding of the relationships between lightning and convective cloud microphysics and kinematics has the potential to improve the diagnosis and forecasting of these direct lightning impacts. For example, the lightning production of nitrogen oxides (LNO

_{X}) has been estimated indirectly in cloud-resolving chemical transport models from calculated kinematic and microphysical fields [42,43,44,45] without the use of existing explicit cloud electrification and lightning physics, which can be computationally expensive. In contrast to studies using explicit cloud electrification and lightning flash rate [46,47], the flash rate used for LNO

_{X}production in these chemical transport models is estimated from flash rate parameterization schemes, which utilize model-predicted storm kinematic and microphysical quantities and are often based on radar-observed relationships [8,9,10,13,14,25,48,49]. Similarly, the initiation, cessation, and frequency of lightning have been forecasted using flash rate parameterizations with model-predicted [50,51] or radar-observed [52,53,54,55] kinematic and microphysical parameters as inputs.

#### 1.1. Background

^{−1}) updraft volume and maximum updraft velocity. These detailed linear relationships were consistent with earlier studies that also found a high Pearson correlation coefficient between precipitation ice mass and flash rate using different observational approaches on the mesoscale [9] to global scale [10].

^{−1}, precipitation ice mass, and ice mass flux product. Based on earlier work [8,25], the more recent study [48] also explored the relationship between flash rate and two other storm parameters, namely graupel echo volume and 35 dBZ echo volume in the mixed phase zone. They [48] found that 35 dBZ echo volume had the lowest error for estimating flash rate, although graupel echo volume and precipitation ice mass performed nearly as well. A recent modeling study [47] has also concluded that graupel mass is a good estimator of the flash rate. In contrast to an earlier study [14], the more recent observational study [48] concluded that linear flash rate parameterizations based on maximum updraft and updraft volumes (>5 m s

^{−1}and >10 m s

^{−1}) in the mixed-phase zone had lower coefficients of determination and higher error than similar linear relations based on graupel echo volume, precipitation ice mass and 35 dBZ echo volume. Absence of a strong correlation between flash rate and maximum updraft velocity has also been noted in numerical modeling studies [11,47]. In more recent observational work [48], it was also determined that some of the linear relations from the earlier observational research [13,14] to estimate flash rate from precipitation ice mass and updraft volume resulted in large root mean square error, large negative bias error and frequent unphysical negative flash rates when applied to the new sample of Colorado storms, in part because of negative y-intercepts in the formulation of the earlier linear relationships. A recent modeling study [47] also raised the issue of the interpretation, suitability and performance of having a non-zero y-intercept, including positive and negative, in estimating flash rate as a dependent variable from a linear relationship based on a microphysical or kinematic storm parameter as the independent variable. For example, it was determined that the linear relation to estimate flash rate from convective (>10 m s

^{−1}) updraft volume was not satisfactory in their modeling study [47] because of high false alarms and an over-estimate of the lightning activity (i.e., positive bias error), likely associated with a large positive y-intercept (i.e., significant flash rates are estimated even when updraft volume is very small or zero). The optimal form of the linear relationship between flash rate and storm parameter is clearly an open research question.

#### 1.2. Motivation and Objectives

_{X}studies.

## 2. Datasets and Methods

^{−1}to 101.7 min

^{−1}. To be included in the sample, the storm needed to pass within the dual-Doppler lobes (Figure 1) so that dual-Doppler radar analysis could be conducted to estimate updraft vertical velocities during at least the mature phase of the storm, if not also other phases (e.g., growth, decay), for at least 5 radar volumes (or for ≥20–25 min). Additional constraints for the inclusion of storms into the sample are discussed in the sections on lightning (Section 2.1) and radar (Section 2.2) data and methods below.

#### 2.1. Lightning

^{−1}) are computed by adding the number of associated NALMA flashes during the radar volume scan time and dividing by the radar volume scan time. As such, lightning flash rates are an average storm flash rate over about 4–6 min (Section 2.2).

#### 2.2. Radar

_{h}) and Doppler radial velocity (V

_{r}) are measured by both ARMOR and KHTX. ARMOR also measures several polarimetric radar variables, including the differential reflectivity (Z

_{dr}), the co-polar correlation coefficient (ρ

_{hv}) and the differential phase (ϕ

_{dp}). During data processing, additional radar variables and products are also estimated and utilized as described in Section 2.2.1, Section 2.2.2, Section 2.2.3 and Section 2.2.4.

#### 2.2.1. Processing, Quality Control and Gridding

_{dp}-based, self-consistent method with constraints [64]. The ARMOR-specific differential phase (K

_{dp}) is estimated from the range derivative of the filtered or smoothed ϕ

_{dp}[65]. The relative calibration of Z

_{dr}is maintained utilizing vertical pointing scans of light rain and the absolute calibration of Z

_{h}is maintained using the self-consistency between Z

_{h}, Z

_{dr}and K

_{dp}in rain [66]. Aliased V

_{r}data are manually unfolded using National Center for Atmospheric Research’s (NCAR) SOLO, version 2 or 3 [67]. Other radar artifacts such as ground clutter, sidelobe and second-trip echoes are also manually removed with SOLO. Using NCAR REORDER software [68], ARMOR and KHTX radar data are interpolated onto a Cartesian grid centered on ARMOR (34.646° N, 86.771° W). Given the 70 km baseline between ARMOR and KHTX, dual-Doppler radar analysis is limited to ranges <86 km from each radar in order to optimize dual-Doppler accuracy, resolution and coverage area [31,32,69]. For consistency and data quality purposes, polarimetric PID and other radar analyses are also limited to the same maximum range from each radar. Given the <1.5 km radar resolution implied by a 1° beamwidth at radar ranges <86 km, a grid resolution of 1 km is implemented in the horizontal and vertical directions. The radar data are interpolated with a Cressman weighting scheme [70] and a 1 km radius of influence in the horizontal and vertical.

#### 2.2.2. Dual-Doppler Analysis

_{r}measurements from the two radars and assumed hydrometeor fall speed relationships [31] are used to solve a set of linear equations to retrieve the horizontal (u, v) velocities within some prescribed error dictated by ARMOR and KTHX measurement error in V

_{r}and a beam-crossing angle requirement (30° as depicted in Figure 1) [72,73,74]. Vertical velocity is estimated by integrating the anelastic mass continuity equation using the retrieved horizontal winds (u, v) and assumed vertical velocity boundary conditions of 0 m s

^{−1}at the prescribed upper and lower boundaries. As in other studies [14,31,32,56,57], the variational integration method [75] is used to mitigate vertical velocity retrieval errors associated with errors in measured divergence and fall speeds [76,77].

^{−1}) and convective updraft volumes (km

^{3}) meeting specific minimum vertical velocity thresholds (>5 m s

^{−1}and >10 m s

^{−1}) are estimated in the mixed-phase region defined earlier (i.e., heights corresponding to environmental temperatures between −5 °C and −40 °C) because this is where the majority of charge separation and lightning initiation occurs, as discussed in Section 1. Environmental temperature data are obtained from the closest NOAA NWS representative atmospheric sounding to the dual-Doppler analysis domain in Figure 1, which is usually from Birmingham, Alabama (BMX; 33.16° N, 86.76° W), Nashville, Tennessee (BNA; 36.25° N, 86.57° W) or an average of both. Storm volumes are computed by counting the number of grid boxes that satisfy the convective updraft threshold in the mixed-phase region within the bounds of the two-dimensional footprint of the defined storm feature (Section 2.2.4) and multiplying that grid count by the grid volume, which is 1 km

^{3}in this study.

#### 2.2.3. Polarimetric Particle Identification

_{h}, Z

_{dr}, K

_{dp}, and ρ

_{hv}and environmental temperature profile data as inputs. The source of the temperature profile data is described in Section 2.2.2. Using REORDER, the NCAR PID output is interpolated to a Cartesian grid with 1 km horizontal and vertical resolution using a nearest-neighbor weighting scheme and 1 km radii of influence. For this study, the “graupel/small hail” category was used exclusively to represent graupel, which is a primary and necessary ingredient for significant cloud electrification and lightning production, as reviewed in Section 1. As discussed in earlier studies [13,31], including other PID categories with rimed precipitation ice, such as “graupel/rain”, “large hail” or “rain/hail” in the mixed-phase region does not improve the correlation between precipitation ice and lightning. Graupel/small hail also tends to dominate the rimed precipitation ice echo volume and mass in the mixed-phase region [13]. Since a grid box volume is 1 km

^{3}, the graupel volume is equal to the number of grid boxes with a PID of “graupel/small hail” in the mixed-phase region (−5 °C to −40 °C) within the boundaries of the two-dimensional storm footprint (Section 2.2.4). Following earlier studies [13,31,32], the graupel ice water content (IWC, g m

^{−3}) for each grid box is estimated from a radar reflectivity-IWC (z-IWC) relationship for graupel [79] (p. 3510, their Equation (5)) using the ARMOR Z

_{h}(after accounting for dielectric differences between z and Z

_{h}[80]) in each grid box characterized as “graupel/small hail” by the NCAR PID. The graupel mass used to generate a relationship with lightning flash rate in this study is calculated by summing the product of grid box graupel IWC (converted to kg km

^{−3}) and grid volume (1 km

^{3}) in all grid boxes in the mixed-phase region within the boundaries of the two-dimensional storm footprint. More succinctly, the graupel mass of interest is a mixed-phase region integrated storm quantity in kg or shown in 10

^{6}kg for convenience.

#### 2.2.4. Storm Identification and Tracking

^{−1}and >10 m s

^{−1}, graupel volume and graupel mass. One additional storm quantity computed is the 35 dBZ echo volume (km

^{3}) in the mixed-phase region, which is well correlated with lightning flash rates [25,48]. Because the grid box volume is 1 km

^{3}, the 35 dBZ echo volume is equivalent to the number of grid boxes characterized by Z

_{h}> 35 dBZ in the mixed-phase region within the two-dimensional storm boundary. As noted in prior studies [25,48,52,53], radar reflectivity in the mixed-phase region is sensitive to the presence of precipitation-sized particles, most likely, large rimed ice. Although not as accurate as the output from the polarimetric PID, the 35 dBZ echo volume is an approximate proxy for the presence of graupel and hail and can, therefore, be grouped with the PID-based microphysical parameters defined earlier.

#### 2.3. Linear Regression and Error Assessment

^{−1}. The normalized RMSE (NRMSE) and normalized MBE (NMBE) in percent (%) are also calculated. For direct comparison with a recent study [48], the NRMSE (NMBE) is defined as the RMSE (MBE) divided by the range of flash rates (i.e., maximum–minimum) in the sample being assessed. Normalizing error by the range is a standard statistical practice [85] and helps to account for the wide range of flash rates in the overall and individual storm samples. Note that in Section 3.2, the normalization is by the range of flash rate in the overall data set, including all 33 storms. In Section 3.3, the error of using relationships derived from the overall data set when applied to individual storms is being assessed as a function of storm average flash rate. In this case, the normalization in Section 3.3 is by the range of flash rate in individual storms, which varies. These individual storm errors provide insight into the likely lower bound of expected error when the relationships from this study are applied to other individual storms with similar flash rates. These errors are likely a lower bound as other factors such as regional variability of storm parameters and lightning behavior, radar estimation error, and model conceptual error may influence the outcomes of these relationships when applied to other situations, as discussed in Section 4.

## 3. Results

^{−1}and >10 m s

^{−1}, 35 dBZ echo volume, graupel mass and graupel volume) in the mixed-phase zone on the individual storm scale. Linear relationships between lightning and radar-inferred kinematic and microphysical parameters are then derived in Section 3.2 for the entire storm dataset (i.e., encompassing all Alabama storms in Table 1) using a zero y-intercept (Section 3.2.1) and a non-zero y-intercept (Section 3.2.2), assessed for overall performance and intercompared with each other and prior studies. Finally, error associated with the overall flash rate parameterization schemes derived in this study when applied to individual storms are evaluated as a function of the average storm flash rate in Section 3.3.

#### 3.1. Example of Lightning, Kinematic and Microphysical Properties in a Severe QLCS

^{−1}and the maximum flash rate was 78 min

^{−1}. With the exception of the maximum updraft, it can be seen that the flash rate in Figure 2 evolves in a similar manner as the other storm parameters, including rapid increases around 1458–1509 UTC, a steadier period of growth until 1533 UTC, then more rapid increases until a maximum in storm intensity around 1545–1556 UTC and the beginning of a weakening period from peak maturity before it moved out of the analysis domain.

^{−1}and >10 m s

^{−1}) and microphysical parameters (graupel volume, graupel mass and 35 dBZ echo volume) are provided in Figure 3. The lightning flash rate is highly correlated to most storm parameters, including especially the microphysical parameters (ρ = 0.96 to 0.98) and, to a slightly lesser extent, the updraft volumes (ρ = 0.87 to 0.90). Flash rate is least correlated to the maximum updraft (ρ = 0.50) in this particular severe storm. These correlations between flash rate and radar kinematic and microphysical parameters are generally similar to what has been found in prior studies of individual storms [8,9] and small samples of storms from similar regions [13,14,48]. The correlation between maximum flash rate and maximum updraft in this storm is lower than found in one study [13] but agrees more closely with other radar [48] and numerical cloud modeling [11,47] studies. It is worth noting that most kinematic (other than maximum updraft) and microphysical parameters are also well correlated to each other (ρ = 0.89 to 0.99), as noted in an earlier study [14]. As expected, the radar parameter with the highest correlation to maximum updraft is the updraft volume > 10 m s

^{−1}and the second highest is to updraft volume >5 m s

^{−1}, although all storm parameter correlations with maximum updraft are ≤0.66. The moderate-to-high correlations of most radar kinematic and microphysical parameters with each other and with flash rate in Figure 3 suggest that there should be little difference in the expected performance of various flash rate parameterization schemes with the possible exception of maximum updraft for this particular severe storm.

^{2}= 0.92 to 0.96) and very low implied flash rate retrieval error (NRMSE = 6% to 9%). Although their performance is not quite as good as the microphysical parameters, the updraft volume >5 m s

^{−1}and >10 m s

^{−1}both provide good results with low-to-moderate scatter, high explained variance (R

^{2}= 0.76 to 0.82) and low expected error (NRMSE = 14% to 16%). As expected, the scatter between flash rate and maximum updraft for this storm is large, the explained variance is fairly low (R

^{2}= 0.25) and the expected error is higher (NRMSE = 28%) than the other parameters. For this individual severe storm at least, the performance of the microphysical-based parameters is the best and they are all fairly similar, although a 35 dBZ echo volume is slightly better than graupel volume and graupel mass, which is consistent with a prior study [48]. These flash rate parameterization results are for one severe storm only and are customized to the storm of interest. As such, these outcomes should be considered ideal and not necessarily general in nature.

#### 3.2. Linear Relationships and Overall Performance

^{−1}to 101.7 min

^{−1}. The mode or most common storm has mean and maximum flash rates ≤10 min

^{−1}, which is not uncommon for non-severe storms or low-topped supercells in Alabama. In fact, half of the Alabama storm sample has a storm average flash rate <10 min

^{−1}and a storm maximum flash rate <21 min

^{−1}. This distribution of flash rates is typical for Alabama but lower than Colorado’s high mean flash rates [13,14], especially compared to a recent Colorado data sample dominated by high flash rate storms [48].

^{−1}, which are likely to be slightly worse. This suggestion will be explored in more detail in the following sub-sections along with an overall assessment of lightning–radar relationships and a comparison to prior studies.

#### 3.2.1. Overall Dataset with Zero Y-intercept

^{2}) is purposefully omitted from Table 2 since an R

^{2}for a regression solution forced through the origin does not have the same physical interpretation (i.e., % variance explained) and cannot be directly compared to an R

^{2}when the regression solution is not forced through the origin. Scatterplots of lightning flash rates versus radar parameters for all thunderstorms are shown in Figure 7 with the corresponding best fit lines from Table 2. The scatter between flash rate and radar-inferred microphysical and kinematic parameters is clearly much larger for the full sample of 33 Alabama storms (Figure 7), including low and high flash rate storms of varying types and severity, than for one high flash rate severe storm (Figure 4). In particular, the scatter in Figure 7 is much larger at low flash rates (i.e., <10 min

^{−1}) than it is at moderate-to-high rates. A similar trend of large scatter in lightning–radar relations at flash rates <10 min

^{−1}can be gleaned from the smaller samples of earlier studies [13,14,48], although the fraction of low flash rates is much larger in this study. As noted earlier, a little over half (17) of the 33 storms in this study have a storm average flash rate <10 min

^{−1}(Figure 5). In fact, the median (mean) lightning flash rate for all 515 analyzed samples in this study is 4.6 min

^{−1}(12.5 min

^{−1}) and fully 63% of the flash rates (i.e., during a radar sample volume time) are <10 min

^{−1}. These low flash rates occur in a variety of storm types, including non-severe multicellular storms and low-topped supercells, and are very common in Alabama. The amount of scatter qualitatively evident in the lightning flash rate versus radar parameters in Figure 7 is similar for each of the different microphysical and kinematic properties evaluated, including the increased scatter at low flash rates. Quantitative assessment of estimation error in Table 3 confirms that RMSE (NRMSE) is similar for the microphysical parameters and most kinematic parameters, ranging from 13–14 min

^{−1}(13–14%) for the linear equations in Table 2 and Figure 7. For updraft volume >10 m s

^{−1}, RMSE (NRMSE) is larger at 18 min

^{−1}(17%). Mean bias errors (Table 3) for the relations in Table 2 are small, ranging from −0.9% to 0.8%, which is to be expected since the lines are derived using WLS linear regression on the Alabama data in Figure 7.

^{−1}, and updraft volume >10 m s

^{−1}perform poorly on the Alabama dataset in this study, exhibiting much larger NMBE, NRMSE or both (Table 3).

^{−1}(NMBE = 28% and NRMSE = 55%).

^{−1}and >10 m s

^{−1}(Table 3) would make these relations unsuitable for use in Alabama storms. An additional challenge for flash rate parameterization equations based on updraft volume > 10 m s

^{−1}is that Alabama thunderstorms with non-zero flash rates often have little or no updrafts > 10 m s

^{−1}(Figure 9), which is why the relation from this study in Table 2 with a zero y-intercept predicts low or zero flash rates in these situations, resulting in a negative bias (Table 3). By comparison, the Colorado-only flash rate relation based on updraft volume > 10 m s

^{−1}[48] has a positive y-intercept (of 8.8 min

^{−1}), thus predicting a flash rate of 8.8 min

^{−1}even when the >10 m s

^{−1}updraft volume is zero (Figure 9).

^{−1}relation from the earlier study [14] is applied to the radar samples in this study (Table 4). Negative flash rates are even more frequently predicted (66–90%) when the graupel (or precipitation ice) mass relations are applied to the radar data in this study (Table 4). When negative flash rates are included in error estimation (Table 3), the bias errors for relations based on updraft volume > 5 m s

^{−1}and graupel mass from the earlier studies [13,14] as applied to radar data in this study are negative and accompanied by large RMSE, making them generally unsuitable. Similar results and conclusions regarding frequently predicted negative flash rates were also found by a recent study [48] that tested these earlier relations [13,14] on Colorado-only data. If the negative flash rates are replaced by zero before error estimation (Table 4), the bias errors for relations in the earlier studies [13,14] become less negative for graupel mass and even slightly positive for updraft volume > 5 m s

^{−1}when applied to radar samples in this study. The switch in bias from negative to positive when substituting zero for unphysical negative flash rates is related, in part, to the earlier updraft volume > 5 m s

^{−1}relation [14] over-estimating flash rate at moderate-to-large values of updraft volume, as seen in a comparison (Figure 10) of the flash rate parameterization equation for Alabama storms in [14] to those derived from the Alabama data in this study (Table 2).

^{−1}and maximum updraft velocity derived from Alabama and Colorado (or Alabama only) storms frequently predict unrealistic negative flash rates when applied to the Alabama storms in this study or other Colorado storms [48]. Differences in data and methodology seem less likely to be important since the same radar and lightning networks and most methods as used in the earlier studies [13,14,86] are used herein.

#### 3.2.2. Overall Dataset with Non-zero Y-intercept

^{2}for the regressed equations in Table 5 with non-zero y-intercepts suggest that parameterizations based on radar-inferred microphysical parameters can explain about 60% of the variance in flash rate, while radar-inferred kinematic parameters can explain about 50% based on updraft volume to about 40% based on maximum updraft velocity. The R

^{2}and explained variance of flash rates by radar parameters in this study are lower than in prior studies [13,14,48], although it is worth noting that the recent Colorado study [48] also found lower R

^{2}for relations based on kinematic quantities compared to microphysical ones. The higher R

^{2}in the prior studies is likely the result of smaller samples and larger mean flash rates, although there could be other possible explanations (e.g., differences in storm dynamics, microphysics, observational error, and conceptual model error).

^{−1}and maximum updraft velocity), the signs of the y-intercepts (i.e., positive for the former and negative for the latter) are the same as in this study. A consistent difference between the relations in Table 5 and all prior studies is that the y-intercept parameters tend to have smaller magnitudes in this study (0.3 to 4.2 min

^{−1}) compared to (8.8 to 16.7 min

^{−1}) in the Colorado relations [48] and (5.1 to 44.4 min

^{−1}) for the earlier studies of Alabama and Colorado (or Alabama only) storms [13,14,86]. As is also evident in Figure 10, the flash rate parameterization relations in this study with zero y-intercepts (Table 2) are very similar to those with non-zero y-intercepts (Table 5), which is expected since the y-intercepts in Table 5 are small (or not far from zero). As such, the error performance for the two sets of relations in this study are fairly similar with the non-zero y-intercept relations in Table 5 having slightly smaller magnitudes of MBE and RMSE (Table 3 and Table 4).

#### 3.3. Error as a Function of Flash Rate

^{−1}), while they are generally more similar at moderate-to-high flash rates (≥10 min

^{−1}). The magnitude of the |NMBE| and the NRMSE for all types of flash rate relations are generally a minimum at moderate flash rates, not coincidentally close to the median (10 min

^{−1}) and mean (15 min

^{−1}) of the 33 storm average flash rates, while the normalized errors tend to climb at the tails of the average storm flash rate, including high flash rates (> 30 min

^{−1}) and especially, low flash rates (<5 min

^{−1}). In fact, the error performance of most flash rate relations can only be characterized as undesirably high at these low flash rates, especially for the kinematic relations like maximum updraft. These errors are likely due to the high scatter in flash rates with radar-inferred kinematic parameters at low flash rate (Figure 7). The same is likely true at high flash rate, although small sample size at high flash rates may also play a role in how it affects the regression process, depending on the procedures implemented. In this study, a WLS linear regression is implemented, which has the effect of reducing the weight at and influence of the very-high-flash-rate samples due to increased error variance with flash rate. The small sample and WLS regression approach may explain, in part, the increased errors at high flash rates.

^{−1}), the error performance of maximum updraft is considerably worse for the zero y-intercept relations in Table 2 compared to the non-zero y-intercept relations in Table 5 (c.f., Figure 11 and Figure 12). To facilitate the comparison, the error difference (or delta error) between the two sets of relations is also shown in Figure 13. As others have argued [47], the presence of a negative y-intercept for kinematic quantities, like maximum updraft in this study (Table 5), may be interpreted as a threshold in maximum updraft that is physically necessary before lightning is possible. Of course, other factors (e.g., sample size, conceptual model error, observational error) may influence the regression resulting in a negative y-intercept. Conversely, at very low flash rates, the error performance for all the rest of the flash rate relations, especially updraft volume > 10 m s

^{−1}, is worse for the non-zero, and in these instances positive, y-intercept relations compared to the zero y-intercept relations (c.f., Figure 11 and Figure 12; Figure 13). As noted previously [47], a positive y-intercept in a flash rate parameterization equation is likely not physically driven as it implies a non-zero flash rate even when the radar-inferred kinematic or microphysical predictor is zero. Errors may affect the regression process, resulting in a physically unrealistic positive y-intercept in order to minimize the sum of the squares of the differences between the observed flash rates and those predicted by the linear function. In other words, it is a statistical outcome as influenced by observational error, not necessarily a physically realistic one. At the very high flash rate tail, there is some indication that the zero y-intercept relations perform worse, although it is only notable for a single storm (i.e., a tornadic supercell on 27 April 2011), which may be an outlier. So, although the choice of forcing the solution through the origin or not during regression may not affect the overall error performance much (Table 3 and Table 4) at most flash rates, it can affect it considerably in the tails, especially at low flash rates (Figure 13).

## 4. Discussion

- Conceptual model errors
- Observational errors
- Statistical errors

## 5. Conclusions

_{X}production. Conversely, lightning flash rate relations have been used to diagnose and nowcast high impact convective weather, such as aviation hazards and severe storms.

- When considering the entire data set, kinematic (i.e., updraft volume > 5 m s
^{−1}, updraft volume > 10 m s^{−1}, and maximum updraft velocity) and microphysical (i.e., graupel echo volume, graupel mass, and 35 dBZ echo volume) parameters are generally correlated to lightning flash rate (ρ = 0.60 to 0.76). However, these overall Pearson correlation coefficients are lower than found in individual high-flash-rate storms or in the smaller storm samples of past studies. Because the sample in this study is three- to five-times larger than past studies and contains a larger fraction of low-flash-rate storms for which various errors (e.g., conceptual, observational, statistical, as discussed in Section 4) may be larger, these lower correlations between flash rate and storm kinematic and microphysical parameters likely represent a more realistic assessment of typical storm and observational conditions, especially at low flash rate. - Maximum updraft velocity (ρ = 0.60) has the lowest overall correlation with flash rate, while graupel mass (ρ = 0.76) has the highest overall correlation with flash rate. With the exception of maximum updraft velocity, all of the other kinematic and microphysical parameters have similar overall correlations with flash rate, ranging from ρ = 0.69 to 0.76. In fact, the various radar-inferred microphysical parameters are even better correlated to each other (ρ = 0.91 to 0.97), as are the updraft volumes with each other (ρ = 0.92) or with the microphysical parameters (ρ = 0.81 to 0.92). From simple Pearson correlation coefficient analysis, it could be anticipated that the overall performance of the flash rate parameterizations based on all of the various microphysical and kinematic parameters tested herein would be fairly similar to each other, with the notable exception of maximum updraft velocity, which would be somewhat worse.
- Error analysis of the various flash rate parameterization relations developed and tested on the Alabama storms in the study find very low MBE or |NMBE| < 1% for all relations, as would be expected. When testing similar relations developed in other studies on the data in this study, the |NMBE| values are generally larger (>1%) and typically, much larger (>10%). All of the recently developed Colorado relations [48] overestimate flash rate when applied to the Alabama storms in this study, often with large positive NMBE, including 35 dBZ echo (11.2%), graupel mass (14.2%), updraft volume > 10 m s
^{−1}(20.4%), and updraft volume > 5 m s^{−1}(27.7%). Lower MBE’s for the recent Colorado relations include graupel volume (3.5%) and maximum updraft velocity (2.1%). When tested on the Alabama data in this study, nearly all earlier relations, except maximum updraft, based on Alabama-only or Alabama and Colorado combined storms [13,14] typically have negative bias errors (i.e., underestimate flash rate) overall, which were sometimes large in magnitude (|NMBE| = 1% to 23%). In fact, the negative y-intercepts of the earlier relations [13,14] often result in negative flash rates for 49% to 90% of the Alabama data sample in this study. On the other hand, the earlier relations [13,14] based on maximum updraft velocity exhibit large positive bias (NMBE = 21.6% and 10.5%) when applied to the data in this study. When considering MBE, the performance of most prior flash rate relations is not generally acceptable, resulting in significant over-estimation or under-estimation of flash rates, including frequent unphysical negative flash rates. - Error analysis of the various flash rate parameterization relations developed from and tested on the Alabama storms in the study find scatter error or NRMSE that are fairly low and similar to each other with no clear favorite, most values ranging from 12% to 14% and the largest NRMSE for updraft volume > 10 m s
^{−1}at 17%. The NRMSE values in this study are fairly similar to those NRMSE (12% to 19%) found by a recent Colorado study [48] when developing and testing the same flash rate parameterization relations exclusively on their Colorado data. The Colorado study also found larger NRMSE when estimating the flash rate from kinematic parameters. As might be expected, the NRMSE’s of flash rate relations derived in prior studies are larger than the relations derived herein when applied to the Alabama data in this study. However, sometimes the NRMSE’s associated with relations from prior studies are much larger. When the Colorado study relations [48] are applied to the Alabama data in this study, the NRMSE for estimating flash rate increase, sometimes notably to a range of 16% to 55%. The Colorado study [48] graupel volume (16%) and maximum updraft velocity (17%) relations have the lowest NRMSE while the NRMSE’s for the other Colorado study relations are much higher (34% to 55%) when applied to the Alabama data in this study. Similarly, the NRMSE’s for most flash rate parameterizations developed in the earlier studies of Alabama-only or Colorado and Alabama storms combined [13,14] when applied to the Alabama storms in this study are typically large (23% to 54%) with the exception of the graupel mass relation developed with Alabama-only data [13], which has an NRMSE of 16%. - When considering both bias error and RMSE, it is easy to come to the conclusion that most flash rate parameterization relations lack sufficient general applicability from one observational data set or study to the next, likely due to a combination of possible error sources discussed in Section 4 (e.g., conceptual model, observational and statistical errors). To improve the robustness and utility of these relations in both radar and model applications, future studies should attempt to isolate, quantify and mitigate the various error sources using a self-consistent radar-model analysis framework on a sufficiently large, diverse and representative sample of storms with similar assumptions and the ability to generate self-consistent kinematic and microphysical parameters between radar and cloud-resolving modeling systems.
- Prior studies and results presented herein have identified potential issues when applying linear flash rate parameterization equations with large-magnitude y-intercepts. As noted in item 3), prior flash rate relations with large-magnitude negative y-intercepts often result in a majority fraction of unphysical negative flash rates, significant negative bias errors and large RMSE. Similarly, flash rate relations from prior studies with large-magnitude positive y-intercepts have sometimes resulted in the over-estimation of flash rate, significant positive biases and large RMSE. Given these findings, a sensitivity test is conducted herein by deriving and testing two types of linear equations, (1) one in which the flash rate solution is forced through the origin (i.e., zero y-intercept) and (2) the other in which the y-intercept is allowed to vary (i.e., non-zero y-intercept) during WLS regression. Although the non-zero y-intercept equations perform slightly better (i.e., slightly lower MBE and RMSE) than the zero y-intercept equations in this study, the difference in overall outcomes is small. The magnitudes of the y-intercepts derived in the second set of relations in this study are small (0.3 to 4.2 min
^{−1}), which may be why there is little difference in error performance between the two sets overall. However, it should be noted that there are significant differences in the performance of the two types of relations at low flash rates, as will be discussed next in item 7. Large-magnitude y-intercepts in flash rate relations from prior studies are not applicable to the storms in this study and may not be generally applicable due to statistical issues (e.g., small sample sizes, heteroscedasticity), although the different effects of observational and conceptual model errors between studies cannot yet be ruled out. - To more completely assess expected performance, the flash rate relations derived (from the entire data set) herein are applied to each storm separately and the MBE and RMSE are evaluated for each storm individually as a function of storm mean flash rate, including for low-flash-rate storms. At low mean storm flash rates (<10 min
^{−1}), the errors associated with relations based on kinematic parameters, including updraft volume and especially maximum updraft velocity, are larger than for relations based on microphysical parameters (graupel volume, graupel mass, 35 dBZ echo volume), while all relations perform similarly at mean storm flash rates ≥10 min^{−1}. At low-mean-storm-flash rates, the errors associated with most relations are undesirably high, especially for the kinematic-based relations like maximum updraft. At very low flash rates (<1–3 min^{−1}), the error performance of maximum updraft is considerably worse for the zero y-intercept relations than the non-zero y-intercept relations in this study. Conversely, the error performance for all the rest of the flash rate relations, especially updraft volume > 10 m s^{−1}, is worse at very low flash rates for the non-zero, and in these instances positive, y-intercept relations compared to the zero y-intercept relations. The error structure of the relations at low flash rates is likely strongly influenced by observational error, although all error sources (including conceptual model and statistical errors) likely interact and contribute. - When considering all factors (e.g., low overall bias error, low overall RMSE, absence of negative flash rates, acceptable generality between studies, insensitivity to the choice of y-intercept, and relatively low bias error and low RMSE at the storm level for all flash rates, including low flash rates), the single-parameter flash rate parameterization relation with the best performance and most desirable overall characteristics found in this study is based on graupel volume. This conclusion may reflect (1) the primary, direct and causal role of graupel in the overall conceptual model of cloud electrification and lightning, (2) the relative robustness of fuzzy-logic-based polarimetric radar methods for identifying the bulk hydrometeor type (i.e., graupel) and (3) lower observational error associated with graupel echo volume compared to more complex radar products involving additional assumptions such as estimating precipitation ice mass from reflectivity or more complex methods such as dual-Doppler vertical motion retrievals. The finding of superior performance of the Z
_{h}> 35 dBZ echo volume relation in the Colorado study [48], even when applied to Alabama storms, could not be repeated herein, possibly due to the lack of non-isolated storms in the earlier study. Prior lightning studies of non-isolated Alabama storms [56] have found a high fraction of supercooled rain drops in the lower (i.e., warmer, −10 °C < T < −5 °C) portions of the mixed-phase zone (−40 °C < T < −5 °C) such that >35 dBZ echo at the warmer temperatures is often not associated with graupel. To confirm that graupel volume is superior to 35 dBZ echo volume (or vice versa, as found earlier [48]), this study should be repeated in a variety of regions that include frequent and diverse thunderstorm types, including low and high flash rate and non-severe and severe storms, with ample high-quality radar and lightning data such as in Cordoba, Argentina during the recent NSF RELAMPAGO (Remote Sensing of Electrification, Lightning, And Mesoscale/Microscale Processes with Adaptive Ground Observations) and DOE CACTI (Clouds, Aerosols, and Complex Terrain Interactions) joint field projects [97].

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Map of the radar and lightning networks centered over Northern Alabama and used in this study. The solid red (blue) circle represents the ARMOR (KHTX) radar at KHSV (Hytop, AL, USA). The green triangles depict the sensor locations of the NALMA. The black, dashed circles represent the ARMOR-KHTX dual-Doppler regions as defined by the 30° beam crossing angle. A distance scale is shown in the upper-right and latitude and longitude values are provided along the outside. Adapted from [56]. © Copyright [2015] AMS.

**Figure 2.**Temporal evolution of graupel volume (km

^{3}), graupel mass (10

^{6}kg), 35 dBZ echo volume (km

^{3}), convective updraft volume >5 m s

^{−1}and >10 m s

^{−1}(km

^{3}), maximum updraft velocity (m s

^{−1}) and lightning flash rate (min

^{−1}) for a severe storm embedded in a QLCS traversing the analysis domain over Northern Alabama (Figure 1) on 12 March 2010.

**Figure 3.**Pearson correlation coefficients between the time series of storm parameters shown in Figure 2 for a severe storm on 12 March 2010, including lightning flash rate (FPM, min

^{−1}) graupel volume (graupel, km

^{3}), graupel mass (10

^{6}kg), 35 dBZ echo volume (km

^{3}), convective updraft volume >5 m s

^{−1}(w5, km

^{3}), convective updraft volume >10 m s

^{−1}(w10, km

^{3}), and maximum updraft (m s

^{−1}).

**Figure 4.**Scatter plot of lightning flash rate versus radar-inferred storm parameters shown in Figure 2 for a severe storm on 12 March 2010, including (

**a**) graupel volume (km

^{3}), (

**b**) graupel mass (10

^{6}kg), (

**c**) 35 dBZ echo volume (km

^{3}), (

**d**) updraft volume >5 m s

^{−1}(km

^{3}), (

**e**) updraft volume >10 m s

^{−1}(km

^{3}) and (

**f**) maximum updraft (m s

^{−1}). The weighted least squares (WLS) linear regression is depicted as a solid red line and the resulting equation, where y = flash rate and x = radar parameter, is given for each scatterplot. The coefficient of determination (R

^{2}), the root mean square error (RMSE) and the normalized RMSE (NRMSE) are provided for each line.

**Figure 5.**Frequency and cumulative (%) histograms of storm (

**a**) mean and (

**b**) maximum flash rate (min

^{−1}) for all 33 storms in this study (Table 1).

**Figure 7.**Same as Figure 4 except the flash rate versus radar parameter scatterplots are for data encompassing all 33 storms in this study (Table 1). For each parameter, the linear equation (Table 2) that is forced through the origin (i.e., with zero y-intercept) is depicted as a solid red line. (

**a**) graupel volume (km

^{3}), (

**b**) graupel mass (10

^{6}kg), (

**c**) 35 dBZ echo volume (km

^{3}), (

**d**) updraft volume >5 m s

^{−1}(km

^{3}), (

**e**) updraft volume >10 m s

^{−1}(km

^{3}) and (

**f**) maximum updraft (m s

^{−1}).

**Figure 8.**Same as Figure 7 except flash rate parameterization equations derived in a recent study using lightning and radar data from Colorado thunderstorms [48] are also applied to the Alabama data in this study and plotted with red circles (B15) alongside the same results from this study using relations in Table 2 (blue triangles). Note that only (

**a**) graupel volume (km

^{3}), (

**b**) graupel mass (10

^{6}kg), (

**c**) 35 dBZ echo volume (km

^{3}), and (

**d**) updraft volume > 5 m s

^{−1}(km

^{3}) are shown here. The other two parameters are shown in Figure 9 in order to highlight aspects not possible in a log plot.

**Figure 9.**Similar to Figure 8 except predicted flash rates based on (

**a**) updraft volume > 10 m s

^{−1}(km

^{3}) and (

**b**) maximum updraft velocity (m s

^{−1}). Linear axes are utilized to emphasize the zero and negative flash rates predicted by the relations, which would not be possible on a log plot.

**Figure 10.**Flash rate parameterization equations based on radar-inferred microphysical or kinematic properties, including (

**a**) graupel volume (km

^{3}), (

**b**) graupel mass (10

^{6}kg), (

**c**) 35 dBZ echo volume (km

^{3}), (

**d**) updraft volume > 5 m s

^{−1}(km

^{3}), (

**e**) updraft volume > 10 m s

^{−1}(km

^{3}), and (

**f**) maximum updraft velocity (km

^{3}). Linear flash rate parameterization equations are shown for the Alabama storms in this study derived with a zero y-intercept (Table 2, red lines) and a non-zero y-intercept (Table 5, blue lines), the recent study using Colorado-only storms [48] (green lines) and the Alabama (AL) only storm relations from an earlier series of related studies [13,14,86] (black line).

**Figure 11.**Error performance when applying the various flash rate parameterization equations with zero y-intercept (Table 2) derived from the entire Alabama dataset to each of the 33 individual storms in this study (Table 1). Storm-level error statistics are plotted versus storm mean flash rate (min

^{−1}), including (

**a**) MBE (min

^{−1}), (

**b**) |NMBE| (%), (

**c**) RMSE (min

^{−1}), and (

**d**) NRMSE (%). Each type of relation has a different type and color marker, as shown in the legend in panel (

**a**).

**Figure 13.**Difference in percentage errors (%) between the various sets of linear equations with zero y-intercept (Table 2) and non-zero y-intercept (Table 5) versus storm mean flash rate (min

^{−1}) for (

**a**) Δ|NBE| = (|NBE| Table 2 Equation) − (|NBE| Table 5 Equation) and (

**b**) Δ NRMSE = (NRMSE Table 2 Equation) − (NRMSE Table 5 Equation). The type of flash rate parameterization equation is provided in the figure legend.

**Table 1.**Overview of Alabama storm properties, including date, cell number (#), analysis period (hh: mm UTC), storm type, severity type, storm life cycle phase(s) represented (G = growth, M = mature, D = decay), sample size (or number of radar volumes) and maximum lightning flash rate. The maximum Enhanced Fujita (EF) tornado damage scale is shown as part of the severity type.

Date (Cell #) | Period (UTC) | Storm Type (Severity) | Life Cycle | # Samples | Max. Flash Rate (min^{−1}) |
---|---|---|---|---|---|

20060719 (1) | 18:31–19:10 | Multicell (wind) | G, M | 8 | 37.6 |

20060719 (2) | 20:41–21:19 | Multicell (wind) | M, D | 8 | 58.1 |

20060719 (3) | 21:05–21:52 | Multicell | G. M | 9 | 20.5 |

20070403 (1) | 16:20–17:00 | Multicell (hail) | M, D | 11 | 7.0 |

20070403 (2) | 18:27–19:59 | Multicell (hail) | G, M | 23 | 45.6 |

20070403 (3) | 18:46–20:29 | Multicell (hail) | G, M | 26 | 22.4 |

20070404 | 02:52–03:22 | QLCS (EF1, hail, wind) | M | 7 | 76.0 |

20070601 (1) | 21:01–21:37 | Multicell | M, D | 9 | 2.1 |

20070601 (2) | 21:01–21:33 | Multicell | G, M | 8 | 3.9 |

20070706 (1) | 17:19–19:15 | Multicell | G, M, D | 24 | 47.2 |

20070706 (2) | 17:19–18:13 | Multicell | M, D | 12 | 16.8 |

20070817 (1) | 19:58–20:48 | Multicell (wind) | G, M, D | 12 | 26.5 |

20070817 (2) | 19:20–20:48 | Multicell (wind) | G, M, D | 21 | 32.6 |

20070817 (3) | 19:33–20:48 | Multicell | G, M, D | 18 | 25.7 |

20070817 (4) | 22:45–23:32 | Multicell | G, M, D | 9 | 15.4 |

20070914 | 16:44–17:07 | Multicell | G, M | 5 | 4.3 |

20080206 | 10:02–11:19 | Supercell (EF4) | M | 13 | 101.7 |

20090410 (1) | 17:16–17:48 | Supercell (hail) | G, M | 6 | 41.2 |

20090410 (2) | 18:17–18:56 | Supercell (hail, wind) | M, D | 9 | 69.4 |

20090410 (3) | 18:12–18:53 | Supercell (hail) | G, M | 9 | 35.9 |

20100121 (1) | 21:32–22:28 | Low-top supercell (EF0, hail) | G, M, D | 18 | 1.9 |

20100121 (2) | 21:29–23:33 | Low-top supercell (hail) | G, M, D | 39 | 9.3 |

20100312 | 14:52–16:08 | QLCS (hail, wind) | G, M | 13 | 78.0 |

20101026 (1) | 22:04–22:38 | Multicell | M | 7 | 1.7 |

20101026 (2) | 22:18–23:27 | Low-top supercell (EF0) | M | 16 | 4.7 |

20101026 (3) | 22:14–23:14 | Low-top supercell (EF1) | M | 12 | 8.6 |

20110427 | 19:55–20:38 | Supercell (EF4) | M | 9 | 71.8 |

20120518 | 22:03–23:59 | Multicell | G, M, D | 28 | 19.5 |

20120521 (1) | 19:35–20:32 | Multicell | G, M, D | 15 | 1.1 |

20120521 (2) | 19:35–21:23 | Multicell | G, M, D | 30 | 5.0 |

20120611 | 18:39–21:22 | Multicell | G, M, D | 33 | 6.6 |

20120614 (1) | 18:31–18:56 | Multicell | M | 8 | 1.7 |

20120614 (2) | 16:28–18:29 | Multicell | G, M, D | 40 | 22.9 |

**Table 2.**Linear equations for estimating lightning flash rate (f) from dual-polarization and dual-Doppler radar-inferred microphysical and kinematic properties in the mixed-phase zone (−5 °C to −40 °C) of a large sample of Alabama thunderstorms (i.e., 515 radar volumes of 33 thunderstorms on 17 different days, as summarized in Table 1). All regressed linear equations are forced to have a zero y-intercept (i.e., are forced through the origin).

Predictor Parameter Description (Radar Variable) | Radar Parameter Units | Flash Rate (f, min ^{−1}) Equation |
---|---|---|

Graupel Volume (GV) | km^{3} | f = (5.82 × 10^{−2}) × GV |

Graupel Mass (GM) | kg | f = (5.88 × 10^{−8}) × GM |

35 dBZ Echo Volume (V35) | km^{3} | f = (4.02 × 10^{−2}) × V35 |

Updraft Volume > 5 m s^{−1} (UV5) | km^{3} | f = (3.44 × 10^{−2}) × UV5 |

Updraft Volume > 10 m s^{−1} (UV10) | km^{3} | f = (1.00 × 10^{−1}) × UV10 |

Maximum Updraft Velocity (W_{max}) | m s^{−1} | f = (7.46 × 10^{−1}) × W_{max} |

**Table 3.**Summary of the estimation error associated with various lightning flash rate parameterization equations when applied to Alabama storms in Table 1. The source, including reference, and type of each flash rate parameterization equation are provided. Performance metrics for the estimation of flash rate from these equations include the mean bias error (MBE, min

^{−1}), the normalized MBE (NMBE, %), the root mean square error (RMSE, min

^{−1}) and the normalized RMSE (NRMSE, %). The NMBE and NRMSE are calculated by dividing the MBE and RMSE, respectively, by the range of observed flash rates in the Alabama thunderstorms summarized in Table 1. Note that some equations resulted in non-physical negative flash rates, which are included in error estimation.

Equation Source Reference | Equation Type | MBE (min^{−1}) | NMBE (%) | RMSE (min^{−1}) | NRMSE (%) |
---|---|---|---|---|---|

Graupel Volume | |||||

This study, Table 2 | 0.8 | 0.8 | 13.9 | 13.6 | |

This study, Table 5 | 0.6 | 0.6 | 13.3 | 13.1 | |

[48], Their Table 3 | 3.5 | 3.5 | 16.6 | 16.4 | |

Graupel Mass | |||||

This study, Table 2 | 0.7 | 0.7 | 13.9 | 13.7 | |

This study, Table 5 | 0.6 | 0.6 | 12.5 | 12.3 | |

[48], Their Table 3, Precipitation ice (PI) mass | 14.4 | 14.2 | 34.9 | 34.4 | |

[13], Their Table 6, All, PI mass | −23.0 | −22.6 | 25.8 | 25.3 | |

[13], Their Table 6, Alabama, PI mass | −11.5 | −11.3 | 16.7 | 16.4 | |

[13], Their Table 6, All, Graupel mass | −20.7 | −20.3 | 23.7 | 23.3 | |

35 dBZ Echo Volume | |||||

This study, Table 2 | 0.8 | 0.8 | 13.6 | 13.4 | |

This study, Table 5 | 0.6 | 0.6 | 13.2 | 13.0 | |

[48], Their Table 3 | 11.4 | 11.2 | 27.6 | 27.2 | |

Updraft Volume > 5 m s^{−1} | |||||

This study, Table 2 | 0.2 | 0.2 | 13.4 | 13.2 | |

This study, Table 5 | 0.4 | 0.4 | 12.8 | 12.6 | |

[48], Their Table 3 | 28.2 | 27.7 | 55.9 | 55.0 | |

[14], Their Table 3, All | −1.4 | −1.4 | 26.7 | 26.2 | |

[14], Their Table 3, Alabama | −1.9 | −1.9 | 26.6 | 26.2 | |

Updraft Volume > 10 m s^{−1} | |||||

This study, Table 2 | −0.9 | −0.9 | 17.6 | 17.3 | |

This study, Table 5 | 0.6 | 0.6 | 14.3 | 14.1 | |

[48], Their Table 3 | 20.7 | 20.4 | 45.6 | 44.8 | |

Maximum Updraft Velocity | |||||

This study, Table 2 | −0.2 | -0.2 | 14.3 | 14.1 | |

This study, Table 5 | −0.1 | -0.1 | 14.1 | 13.9 | |

[48], Their Table 3 | 2.1 | 2.1 | 17.5 | 17.2 | |

[14], Their Table 4, All | 10.7 | 10.5 | 54.8 | 53.9 | |

[86] ^{1}, Their Table 4.10, Alabama | 22.0 | 21.6 | 49.0 | 48.2 |

**Table 4.**Same as Table 3 except predicted non-physical negative flash rates are replaced by zero (0 min

^{−1}) for calculation of error statistics. Presented results are limited to those equations with negative y-intercepts that resulted in the prediction of negative flash rates. The percentage (%) of the total sample with predicted negative flash rates, which are set to zero for error estimation, is also shown.

Equation Source Reference | Equation Type | (% Negative) | MBE (min ^{−1}) | NMBE (%) | RMSE (min ^{−1}) | NRMSE (%) |
---|---|---|---|---|---|---|

Graupel Mass | ||||||

[13], Their Table 6, All, PI mass | 90% | −10.6 | −10.5 | 17.9 | 17.7 | |

[13], Their Table 6, Alabama, PI mass | 66% | −9.3 | −9.1 | 16.0 | 15.7 | |

[13], Their Table 6, All, Graupel mass | 87% | −10.2 | −10.0 | 17.3 | 17.0 | |

Updraft Volume > 5 m s^{−1} | ||||||

[14], Their Table 3, All | 49% | 3.1 | 3.0 | 25.2 | 24.8 | |

[14], Their Table 3, Alabama | 50% | 2.8 | 2.8 | 25.1 | 24.6 | |

Maximum Updraft Velocity | ||||||

This study, Table 5 | 2% | −0.1 | −0.1 | 14.1 | 13.9 | |

[48], Their Table 3 | 21% | 3.0 | 3.0 | 17.2 | 16.9 | |

[14], Their Table 4, All | 43% | 20.2 | 19.9 | 51.4 | 50.6 | |

[86] ^{1}, Their Table 4.10, Alabama | 24% | 24.9 | 24.5 | 48.4 | 47.6 |

**Table 5.**Same as Table 2 except that all regressed linear equations have a non-zero y-intercept (i.e., the solution is not forced to zero at the origin) and the Coefficient of Determination (R

^{2}) is provided.

Predictor Parameter Description (Radar Variable) | Parameter Units | Flash Rate (f, min^{−1}) Equation | Coefficient of Determination (R^{2}) |
---|---|---|---|

Graupel Volume (GV) | km^{3} | f = (5.50 × 10^{−2}) × GV + 0.5 | 0.54 |

Graupel Mass (GM) | kg | f = (5.08 × 10^{−8}) × GM + 1.7 | 0.57 |

35 dBZ Echo Volume (V35) | km^{3} | f = (3.88 × 10^{−2}) × V35 + 0.3 | 0.55 |

Updraft Volume > 5 m s^{−1} (UV5) | km^{3} | f = (3.13 × 10^{−2}) × UV5 + 1.4 | 0.53 |

Updraft Volume > 10 m s^{−1} (UV10) | km^{3} | f = (7.67 × 10^{−2}) × UV10 + 4.2 | 0.47 |

Maximum Updraft Velocity (W_{max}) | m s^{−1} | f = 1.00 × W_{max} − 4.2 | 0.36 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Carey, L.D.; Schultz, E.V.; Schultz, C.J.; Deierling, W.; Petersen, W.A.; Bain, A.L.; Pickering, K.E. An Evaluation of Relationships between Radar-Inferred Kinematic and Microphysical Parameters and Lightning Flash Rates in Alabama Storms. *Atmosphere* **2019**, *10*, 796.
https://doi.org/10.3390/atmos10120796

**AMA Style**

Carey LD, Schultz EV, Schultz CJ, Deierling W, Petersen WA, Bain AL, Pickering KE. An Evaluation of Relationships between Radar-Inferred Kinematic and Microphysical Parameters and Lightning Flash Rates in Alabama Storms. *Atmosphere*. 2019; 10(12):796.
https://doi.org/10.3390/atmos10120796

**Chicago/Turabian Style**

Carey, Lawrence D., Elise V. Schultz, Christopher J. Schultz, Wiebke Deierling, Walter A. Petersen, Anthony Lamont Bain, and Kenneth E. Pickering. 2019. "An Evaluation of Relationships between Radar-Inferred Kinematic and Microphysical Parameters and Lightning Flash Rates in Alabama Storms" *Atmosphere* 10, no. 12: 796.
https://doi.org/10.3390/atmos10120796