RADAR REFLECTIVITY-DERIVED THUNDERSTORM PARAMETERS
APPLIED TO STORM LONGEVITY FORECASTING
P. L. MacKeen1, H. E. Brooks, and K. L. Elmore1
NOAA, Environmental Research Labs, National Severe Storms Laboratory
Norman, OK 73069
1 - also affiliated with: Univ. Of Oklahoma, Cooperative Institute for Mesoscale Meteorological Studies
ABSTRACT
In order for the Federal Aviation Administration (FAA) to use air space more efficiently
during thunderstorm events, accurate storm longevity forecasts are needed. Relationships
between 16 radar reflectivity-derived storm characteristics and storm longevity are
examined to determine which, if any, of the storm characteristics are strongly related to
storm lifetime. Such relationships are potentially useful for the development of storm
longevity forecasts. The study includes 879 storms that formed over the Memphis, TN
area during fifteen late spring and summer convective days. Statistical analyses
comparing all 16 storm characteristics to the observed remaining lifetime show that these
storm characteristics are not good predictors for storm remaining lifetime.
I. Introduction
In the warm season, major air traffic delays often are caused by convective weather.
Although this seems straightforward, impacts on airport capacity by small areas of
convective weather can be profound. These impacts are discussed herein as they apply
to terminal areas within the United States. Impacts on airport capacity occur primarily
because arriving flight paths are more confined than departing flight paths; airport capacity
is more limited by the ability to accept arrivals than by the ability to discharge departures.
At major airports, arriving flights are constrained to fly over a point in space, or within
a narrow corridor around that point, called an arrival gate. In addition, these points must
be traversed at specific altitudes. Major airports typically maintain four arrival gates,
located roughly 60 km to the northeast, northwest, southwest and southeast of the airport;
departures are discharged in the cardinal directions. To maintain proper spatial separation
between arriving aircraft, all pass through the arrival gate in single-file at specified
intervals. At peak demand times, aircraft are expected to be able to pass through all four
arrival gates using minimal separation; any peak capacity reduction by either lost access
to an arrival gate or by an increase in interval spacing results in delays. Since safety
considerations prevent jet transport aircraft from penetrating active convection, convection
near or around an arrival gate causes that gate to become unavailable.
Currently, Air Traffic Control (ATC) responses to convective weather are reactive: either
holding distant departing flights until the weather is no longer effecting the arrival gate
or diverting flights to other, unaffected arrival gates, causing in-flight holds. Because in-
flight holds are expensive for air carriers and quickly fill up available airspace with
holding aircraft, they are avoided whenever possible. For aircraft already en route, flight
mileage (and fuel burn) may increase as they are redirected to available gates. At worst,
en route flights are forced to divert to alternate destinations at high cost to air carriers and
passengers.
Holding departing flights often leads to inefficient airspace use because ATC has no
reliable product which forecasts when adverse weather will dissipate or exit affected gates,
making them available again. As such, ATC holds flights at distant departure points until
the weather has moved out or dissipated, which means that there are significant periods
of available capacity that cannot be utilized since no flights are en route (Evans 1997).
Clearly, in order to use air space more efficiently during thunderstorm events, accurate
storm longevity forecasts are needed. Evans (1997) estimated spatial and temporal
accuracy required of convective weather forecasts. He found that lead times as short as
20 to 30 min would be operationally useful within about 74 km of the airport in a region
referred to as the airport terminal area (ATA). Location errors in the ATA must be within
5 to 10 km, whereas location errors within about 10 km of the runways require 5 to 10
min temporal and 2 km spatial accuracy. A Convective Forecasting Product Development
Team, comprised of scientists from Lincoln Laboratory (LL), the National Center for
Atmospheric Research (NCAR) and the National Severe Storms Laboratory (NSSL), is
collaborating to develop techniques for thunderstorm initiation, growth, and decay
forecasting that will meet the needs of the Federal Aviation Administration (FAA) as part
of the Integrated Terminal Weather System (ITWS) development (Wolfson et al. 1997).
This paper reports part of NSSL's findings related to this effort.
In this paper we analyze the utility of radar reflectivity-derived (Table 2) characteristics
as predictors of storm remaining lifetime. We investigated single and multicelled storms
that did not develop into well-organized convection. Since this study was designed to
examine predictors for convective events characterized by time scales on the order of tens
of minutes to a few hours, organized convective events with longer time scales (e.g. squall
lines) were excluded. Also, since there are no appropriate automated velocity products
applicable to non-supercell storms in the current National Weather Service (NWS)
operational suite of radar-based algorithms, none were used.
A number of previous radar-based convective longevity studies examined relationships
between storm longevity and storm characteristics such as size, intensity, and top height.
For example, Battan (1953) showed that for single-celled storms, storm duration increased
with increased storm horizontal extent for storms with horizontal diameters less than 3
miles. Battan (1952) also found that the storm top heights associated with "longer-lived"
(20 min) single-cell convection were higher than those associated with shorter-lived (~10
min) cells.
Whereas Battan's studies investigated relationships between storm duration and individual
storm characteristics, more recent studies (including the current study) have investigated
relationships between storm duration and multiple storm characteristics. For instance,
Henry (1993) analyzed the relationship between storm duration and both storm size,
measured by volume, and maximum reflectivity. The goal of Henry's study was very
similar to the goal of this study, but the data were somewhat different. In Henry's study,
storms were defined by a single reflectivity threshold, 35 dBZe, and a minimum volume
of 50 km3, using the Thunderstorm Identification, Tracking, Analysis and Nowcasting
(TITAN) algorithm (Dixon and Wiener 1993). Single and multicelled storms were
analyzed separately and storm characteristics were sampled every 30 minutes. Like the
current study, Henry's omitted supercells. The study showed that storms with volumes
greater than 400 km3 and maximum reflectivities of 53 dBZe or greater had a mean
remaining lifetime of 30 minutes or more. Also, simple (single-celled) storms had a much
greater probability of dissipating (83%) within 30 minutes than complex (merging) storms
(12%).
The relation of storm size and intensity to storm duration was also analyzed by Wilson
(1966). Wilson (1966) addressed the predictability of convection by determining how far
into the future various convective scales and intensities were likely to persist. In this
study, Wilson determined scale and intensity predictability by cross-correlating echo
patterns (5-40 mi wavelength) over several time intervals (5-180 min). Results showed
that forecast time scale decreased with decreasing echo scale, but increased with
increasing echo intensity (reflectivity). However, cross-correlations between reflectivity
patterns of different scale were quite variable; in some cases the same forecast time scale
appeared valid for different echo scales. Therefore, Wilson did not specify a general set
of rules for predicting echo longevity based on echo scale.
Others have addressed the relationship between the scale of weather disturbances, their
duration, and predictability. These studies are important to the current work because they
provide guidance for defining both the convective scale that falls within a radar's domain
and the time period over which the convective scale's duration may be forecast. Based
on Orlanski's (1975) scale definitions, convective scales that fall within the radar domain
used in this work (125 km) include thunderstorms, cloud clusters, and squall lines (scales
2 km - 200 km), with durations from 30 min to a day. These convective scales have
typical time periods of valid linear extrapolation which were estimated by Zipser (1983).
For instance, estimated valid linear extrapolation time scale for an individual thunderstorm
was 5-20 minutes, a severe thunderstorm 10 min-1 hr, and thunderstorms organized on the
mesoscale (e.g. squall lines, complexes) about 1-2 hrs. The current study addresses
whether reflectivity-derived characteristics (Table 2) associated with individual, non-
rotating thunderstorms can provide a convective duration forecast on the order of 30
minutes, a forecast time scale invalid for forecasts based on extrapolation alone.
In addition to the storm characteristics discussed previously, clear-air signatures, such as
outflow boundaries and other zones of localized convergence, have been applied to storm
longevity forecasting. Detection of these clear-air signatures can aid forecasting by
highlighting zones of enhanced lift that may help initiate or sustain convection. For
example, Wilson and Megenhardt (1997) showed that storms tend to be longer-lived when
boundary-relative storm motion is approximately zero. Although convergence boundaries
can be useful in storm longevity forecasts, their application is limited. For instance,
boundaries are only detectable close to the radar (~ 50 km) and are limited by terrain
blockage in mountainous areas. Furthermore, not all convergence boundaries are
associated with new convection. Stensrud and Maddox (1988) exemplified this scenario
through an analysis of colliding mesoscale outflows from two mesoscale convective
systems that moved into an area of potential instability without producing additional
convection. They concluded that the anvils associated with the two mesoscale convective
systems collided at approximately the same time as the outflows, producing an opposing
downward circulation that impeded convective initiation. In this case, the ability to
forecast convective initiation was limited by available observations.
Within the last decade, results from the above and other related studies have been used
to develop algorithms that forecast storm initiation, growth and decay. For instance,
Wolfson et al. (1994) developed the ITWS Microburst Prediction Algorithm using
machine-intelligent image processing and data-fusion techniques to detect regions of storm
growth and decay. The feasibility of using components of this algorithm to make short-
term convective forecasts for the FAA is currently under investigation. NCAR scientists
are also working toward short-term forecasts for the FAA. They continue to develop the
Auto-nowcaster (Wilson 1997), an automated system which utilizes radar, satellite, surface
and upper air weather observations, to make 0-60 min forecasts of thunderstorm initiation,
movement, and dissipation. In the United Kingdom, Hand and Conway (1995) developed
a rule-based model for convection that used radar-estimated rainfall rates and cloud top
temperatures to forecast the convective stage of a storm on 30 minute intervals out to 3
hrs. For a more thorough review of the history and the status of short-term convective
precipitation forecasts, see Wilson (1997).
For this study, we analyzed the statistical relationships between 16 radar reflectivity-
derived characteristics and observed remaining storm lifetime for our sample. The study
was based on radar reflectivity-derived characteristics that estimate storm height, size, and
intensity. These characteristics were analyzed to determine not only their general
relationships with storm remaining lifetime, but their practical value for storm longevity
prediction. This paper will show that simple treatment of these reflectivity-derived
parameters offers limited value for storm longevity prediction.
II. Data and Method
In this study, reflectivity-derived parameters were determined using Weather Surveillance
Radar-1988 Doppler (WSR-88D) Level II archived data that was collected using Volume
Coverage Pattern (VCP) 21, for fifteen days during the 1995-1996 late spring and summer
seasons in Memphis, TN (Table 1). Although radar data collected in VCP 111 would have
been ideal for this study, most of the available data had been collected using VCP 21.
Therefore, to avoid mixing data from two different VCPs, we limited the data set only to
that collected using VCP 21. In addition, the data were limited to the Memphis area
because Memphis, TN was a FAA ITWS test-bed during 1996 and the Convective
Weather PDT's efforts were focused on examining these data.
Soundings are not routinely taken at Memphis, TN. Therefore, to provide insight into the
environmental conditions within the Memphis radar domain for each day, the Convective
Available Potential Energy (CAPE) and the Bulk Richardson Number Shear (BRNSHR)
were determined for the most recent 00 UTC soundings from the two closest sounding
sites, Jackson, MS and Nashville, TN (Table 1). The calculation of CAPE was based on
lifting a well-mixed parcel from the lowest 500 m (mean temperature and mixing ratio),
and the BRNSHR is simply the denominator of the Bulk Richardson Number (BRN),
defined originally by Moncreiff and Green (1972) as
where and are the wind components of the difference between the density-weighted
mean winds over the lowest 6000 m and the lowest 500 m above ground level. Based on
these soundings, the environments associated with the examined storms likely contained
low-to-moderate BRNSHR (Stensrud et al. 1997) but a wide range in CAPE.
For each day, the WSR-88D reflectivity data were run through the Storm Cell
Identification and Tracking (SCIT) algorithm (Johnson et al. 1998) and the Hail Detection
Algorithm (HDA) (Witt et al. 1998). The SCIT algorithm identified the storms, calculated
storm characteristics (Table 2), and tracked storm movement. Johnson et al. (1998)
described explicitly the storm identification and tracking process. The HDA predicted the
probability of hail of any size, the probability of severe hail (as defined by the NWS), and
maximum expected hail size associated with each storm. Both the SCIT algorithm and
the HDA were developed to provide information about storm state and severity, and are
analyzed here to determine their value in statistical storm longevity forecasting.
Once the storms were identified by the (SCIT) algorithm, a radar meteorologist manually
verified the storm tracks. In order to manually verify the storm tracks, NSSL's Radar
Analysis and Detection System (RADS) (Sanger et al. 1995) was used to display both the
raw reflectivity data and the algorithm-derived storm tracks. Storm tracks were verified
to ensure that only accurate storm lifetimes were included in the data set. Next, storms
were selected that met the following criteria for the study's data set: 1) lifetime greater
than or equal to 12 minutes, 2) maximum reflectivity 40 dBZe or greater, and 3) storm
track within 30 to 125 km of the radar. These criteria were chosen to address FAA needs.
For instance, the 40 dBZe maximum reflectivity threshold was selected to define storm
lifetime because pilots tend to avoid convective areas containing reflectivity greater than
or equal 40 dBZe (Evans 1997). The range domain was chosen to 1) include airspace that
has greatly affected airport capacity and 2) minimize radar sampling errors. Finally,
reflectivity-based characteristic time series were created for each storm. Table 2 lists the
16 SCIT storm characteristics used in this study. Upon completion of this process, the
data set contained 879 storms and their 16 storm characteristic trends.
Statistical relationships between the 16 storm characteristic trends and storm remaining
lifetime were derived using linear univariate and multiple regression analysis. For the
linear univariate analysis, the magnitude of each storm characteristic for each volume scan
(every 6 minutes) throughout its lifetime was correlated with its remaining lifetime. This
analysis was also completed using the magnitude of a storm characteristic every other and
every third volume scan within a trend. However, this analysis failed to provide any
added benefit. Remaining lifetime was calculated by subtracting the current storm
duration from the total storm lifetime for each volume scan in a storm characteristic time
series. Also, remaining lifetime probability density functions (pdf) were determined for
each storm characteristic. For the linear multivariate analysis, the magnitudes of all the
storm characteristics were correlated with remaining lifetime.
In addition to analyzing the relationship between storm characteristic magnitude and
remaining lifetime, it was determined whether storm characteristic magnitude, in
conjunction with a simple measure of storm growth or decay, strengthened any relationship
with remaining lifetime. Storm growth or decay was measured by calculating storm
characteristic differences over one, two and three consecutive volume scans.
III. Results
Analysis methods showed that none of the storm characteristics used in this study were
good predictors for remaining storm lifetime. For example, both the univariate and
multivariate statistical analyses measured weak to moderately weak linear relationships
between radar-derived storm characteristics and remaining lifetime. Individually, the
strongest relationships (Pearson's linear product moment correlation coefficient (r) greater
than .3) were between variables that estimate storm intensity and height (Table 3).
Remaining lifetime was correlated slightly better with combinations of storm
characteristics than with any single characteristic. Combining all sixteen characteristics,
multiple linear regression analysis provided a multiple linear coefficient of r = .43.
To understand the relationship between these storm characteristics and remaining lifetime
better, discrete pdfs of remaining lifetime were constructed for each variable. Since
maximum reflectivity was most strongly correlated with remaining lifetime, maximum
reflectivity-based pdfs were used to illustrate the best example of storm characteristic time
series as a potential predictor of storm longevity (Fig. 1). These probability distributions
are interpreted as the probability of a storm having a certain remaining lifetime range,
given a maximum reflectivity value within one of the three maximum reflectivity
categories. For example, storms with a maximum reflectivity value between 30-50 dBZe
had the greatest probability (82%) of dissipating within 30 minutes, whereas storms with
a maximum reflectivity greater than 55 dBZe had only a 44% probability of dissipating
within 30 minutes. In contrast, the pdf associated with storm base height (see Fig. 2),
which was essentially uncorrelated with remaining lifetime, showed that base height fails
to discriminate remaining lifetime.
The remaining lifetime pdfs associated with storm trends and storm characteristic
magnitude were similar to those associated only with storm magnitude. Examples of these
pdfs are shown in Fig. 3. Comparison between Figs. 3a and 3b shows that storms with
mass less than or equal to 100 x 106 kg were more likely to dissipate within 30 minutes
than storms with mass greater than or equal to 200 x 106 kg. Also, regardless of mass
magnitude, "dissipating" storms were more likely to die within 30 minutes than growing
storms. However, adding a simple growth and decay measure to the data set did not
significantly improve storm remaining lifetime discrimination.
IV. Discussion
The general, qualitative relationships between storm duration and measures of storm
intensity, height, and size were similar to those determined by previous authors; namely
storm remaining lifetime increased with increasing storm top height, size, and intensity.
However, the purpose of this study was to determine the practical application of storm
characteristics in forecasts of storm remaining lifetime. In contrast to Henry's (1993)
study, storms were defined by SCIT using 7 reflectivity thresholds, single and multicelled
storms were analyzed jointly, and storms were sampled every volume scan (6 minutes).
As a result of storm definition differences, Henry's study calculated echo volume within
a 35 dBZe contour, whereas this study calculates only the storm core volume (Johnson et
al. 1998). The correlation coefficients calculated by Henry for both volume and maximum
reflectivity with respect to remaining storm lifetime, ranged from .39-.52. Henry's
correlation coefficients are somewhat larger than those found in this study, most likely
owing to computing storm characteristics based on a larger storm echo. However, both
studies show that relationships between storm characteristics and remaining lifetime are
not large enough to discriminate between short and long-lived storms.
Owing to the large size of the data set, all correlation coefficients are statistically
significant at the 99% confidence level. However, not all statistically significant results
have practical significance from an operational perspective. In the univariate portion of
this study, maximum reflectivity and remaining lifetime share the greatest coefficient of
determination (r2=.13). This means that only 13% of the variance in remaining lifetime
is explained by maximum reflectivity. The percentage of remaining lifetime variance
explained by the multivariate storm characteristics analysis is slightly greater (r2=.18). In
this light, these weak relationships make using this study's storm characteristics as storm
longevity predictors questionable at best, especially compared to the forecast criteria noted
by Evans (1997).
Given these results, a natural question is why radar reflectivity-derived storm characteristic
measurements, and especially their trends, are so poorly related to remaining lifetime.
There are several possible answers to this question. Since convective processes are
non-linear, linear relationships between storm characteristics and storm longevity
necessarily will be limited. In addition, relationships between these variables are also
limited by the WSR-88D's and algorithms' ability to observe, detect, and characterize
storms. Inherent radar sampling problems such as ground clutter contamination and
anomalous propagation all affect the final reflectivity fields ingested by storm detection
algorithms. Including these reflectivity features within an algorithm-defined storm results
in false detections.
Other radar sampling limitations like beam spreading and range-dependent beam height
can result in incorrect storm characteristic measurements, and especially their trends. As
a result, variations in the characteristics we calculate can be due to either actual changes
in storm state or variations in radar sampling. Howard et al. (1997) addressed this
problem by showing that the WSR-88D's inherent uncertainties (owing especially to radar-
range and the VCP) resulted in uncertainty in reflectivity-derived trends. Based on this
result, Howard et al. strongly suggested that the uncertainty associated with storm-based
parameters (e.g. storm top height, height of maximum reflectivity, etc.) needs to be
considered, especially when trying to develop relationships between thunderstorm
characteristics and rate of growth and decay. Whether consideration of height uncertainty
can improve the relationship between storm height parameters and storm growth and decay
is yet to be determined.
Two additional points to keep in mind are that 1) the WSR-88D reflectivity is a measure
of the scattering from hydrometeors produced by convection rather than the convective
process itself, and 2) reflectivity data are the result of convective processes temporally
integrated over an indeterminate time frame. Also, different storms may not enjoy
parallel evolution, which precludes the use of a single analysis method to capture storm
characteristic and remaining lifetime relationships to their fullest extent.
Although it is likely that radar reflectivity-derived storm characteristics fail to represent
adequately the convective processes that are needed for use in statistical forecasts of storm
longevity, reflectivity-derived characteristics combined with other data sources may
discriminate storm longevity better; especially for more organized convection. For
example, trends of velocity-derived characteristics associated with mesocyclones such as
maximum strength, depth, and mid-level vs low-level base may be correlated better with
remaining storm duration. Work addressing the use of both velocity and reflectivity-
derived characteristics associated with mesocyclones is currently underway and will be
reported subsequently. Additionally, environmental conditions measured by in situ,
satellite, and other data sources, or predicted by a model could provide information
concerning the environment's ability to support convection, or a specific type of
convection. One example of this type of analysis is examining mesoscale-model forecasts
of the environment ahead of an organized convective system, such as a squall line, to
determine the forecasts' relationship to the life cycle of the system. Cloud model results,
combined with both reflectivity and velocity radar observations also should be investigated
for their potential utility as storm duration guidance. For instance, convective evolution
forecasts could be provided from numerically simulated storm types and life stages shown
to match the storm structure (based on both reflectivity and velocity data) identified on
radar.
ACKNOWLEDGEMENTS
This research was completed in response to requirements and fundings by the
Federal Aviation Administration (FAA). The views expressed are those of the authors and
do not necessarily represent the official policy or position of the FAA. We thank
MIT/Lincoln Laboratory for supplying the radar data and John Cortinas for providing ascii-
conversion programs to expedite the sounding analysis. We also thank a number of our
colleagues for their comments and suggestions. In particular, we thank Dr. Rodger Brown,
Ken Howard, Dr. Bob Maddox, Barbara Brown, Mike Eilts, and the Convective Weather
Product Development Team. In addition, we thank Dr. C. A. Doswell and the two
anonymous reviewers for their reviews of the manuscript, which led to numerous
improvements. Thanks also to Mike Francis for his contribution to the storm tracking
verification.
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FIGURE LEGENDS
Figure 1. Remaining lifetime probability distribution functions for three maximum
reflectivity categories (30-50, 50-55, and 55 dBZe). N is the number of elements in each
distribution.
Figure 2. Remaining lifetime probability distribution functions for three base height
categories (0-2, 2-4, and 4 km). N is the number of elements in each distribution.
Figure 3. Remaining lifetime probability distribution functions for positive (solid line) and
negative (dashed line) changes in cell-based Vertically Integrated Liquid (VIL) time series
and storm mass a) greater than 0 but less than 100 x 106 kg and b) greater than or equal
to 200 x 106 kg. N is the number of elements in each distribution. Cell-based VIL
differences are denoted by VIL; the plus (negative) sign represents growth (dissipation).
Table 1. Dates and time periods of WSR-88D data and the CAPE and BRNSHR
(Nashville / Jackson) determined from the most recent 00 Z sounding for each day. M
denotes missing data.
Date Time Period (UTC) BRNSHR ( ),
6 June 1995 1615-1847, 1947-2202 5 / 1042 0 / 36
14 July 1995 1904-2345 M M
17 June 1995 2010-2330 88 / 12 10 / .12
17 August 1995 1723-0105 2765 / 2363 .18/ 6.88
19 August 1995 1946-0102 3329 / 1753 6.97 / .51
13 June 1996 1520-2122 1411 / 444 6.35 / 21.14
22 June 1996 1917-2315 1218 / M 11.6 / M
23 June 1996 2145-0020 0.0 / 1950 0.0 / 1.48
29 June 1996 1818-0020 11 / M .47 / M
08 July 1996 1926-1008 43 / 1935 43 / 25.12
16 July 1996 1857-0241 0.0 / 340 0.0 / .68
08 August 1996 1639-0516 522 / 467 18.64 / 2.28
12 August 1996 1700-0042 620 / 278 2.42 / 21.38
17 August 1996 1805-0105 766 / 162 8.41 / 7.71
30 August 1996 1642-0023 214 / 3.0 3.68 / 3.0
Table 2. Radar-derived storm characteristics from the SCIT and HDA algorithm
and their units.
Characteristics (Units)
Maximum Reflectivity (dBZ)
VIL (kg/m^2)
Volume (km^3)
Mass (kg x 10^6)
Area (km^2)
Storm Top Height (km)
Storm Base Height (km)
Top Height of 40 dBZe core (km)
Base Height of 40 dBZe core (km)
Probability of Hail (%)
Probability of Severe Hail (%)
Maximum Hail Size (inches)
Maximum Reflectivity Height (km)
Center of Mass Height (km)
Core Aspect Ratio (ratio of storm core depth to its width) (ratio)
Reflectivity Ratio (ratio of maximum reflectivity to reflectivity
at the lowest elevation angle) (ratio)
Table 3. Remaining lifetime and storm characteristic product-
moment correlation
coefficients at the 99% confidence level.
Characteristic Pearson's r
Maximum Reflectivity .36
Height of Maximum Reflectivity .34
Center of Mass Height .33
Top Height of 40 dBZ Core .33
Cell-based VIL .33
Storm Top Height .32
Core Aspect Ratio .29
Mass .27
Hail Probability .25
Reflectivity Ratio .23
Area .17
Volume .16
Base Height of 40 dBZe Core .13
Storm Base Height .05
Maximum Hail Size -.01
Severe Hail Probability -.01
Figure 1. Remaining lifetime probability distribution functions
for three maximum reflectivity categories (30-50, 50-55, and 55+ dBZ).
N = number of elements in each distribution.
Figure 2. Remaining lifetime probability distribution functions
for three base height categories (0-2, 2-4, and 4+ km). N = number of
elements in each distribution.
Figure 3. Remaining lifetime probability distribution functions
for positive (solid line) and negative (dashed line) changes in
VIL time series and storm mass: a) greater than 0 but less than 100 x 10^6 kg,
and b) greater than or equal to 200 x 10^6 kg. N is the number
of elements in each distribution. VIL differences are denoted by the triangle;
the plus (negative) sign represents growth (dissipation).
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