Quadratic Growth hypothesis , classification of cumulus Based on rate of Growth , concept of reaction time and cumulus motion speed

This paper is a new attempt to evaluate hail storm severity and hail suppression effects. In this paper a new concept of Quadratic Growth Hypothesis (QGH) has been proposed and examined in the prediction of hailstorm. Another new concept of Reaction Time (RT) has been presented which is useful for effective seeding in hail mitigation campaigns. Given complex nature of cumulus growth, rate of growth of cumulus (r) have been broadly categorized as Slow (r ≤ 0.2 dBZ/min), Moderate (0.2 < r < 0.8 dBZ/min) and Fast (r ≥ 0.8 dBZ/min). Often cumulus shows reverse growth too. It is found that QGH based predictions are 100% correct for Slow growing cumulus and 62.5% accurate for Moderate. However, QGH predictions are incorrect when the cumulus growth reverses or when it is Fast. Empirically a “QGH-Rectangle” has been identified wherein QGH is precisely valid. Prediction skill scores [= (Correct prediction/Total Predictions made)] of 0.79, 0.79 and 0.75 are obtained from scan intervals of 10, 12 and 19 minutes data, respectively. Amongst the three data sets, 10 minutes scan interval is operationally safer for RT computation during hail mitigation campaigns. In most of the cases RT may range from 17.3 to 29.6 minutes. Maximum RT of 43 minutes is also noted for slow growing cumulus. Linear Extrapolation (LE) has been used to predict the cumulus motion speed which has been observed from 5 m/s to as high as 19.3 m/s. It is noted that larger scan interval of DWR data would exhibit more consistent and reliable speed prediction by LE method. ABSTRACT Prabhat Kumar1 and deba Prasad Pati2

hypothesised that by artificially seeding of hail bearing cloud with CCN and IN aerosols, the condensed droplet sizes and concentrations or ice crystal concentrations increase within the cloud.They all compete to collect the available water vapour and grow larger altogether.As a result cloud water is distributed in to several small ice crystals or small hails above 0 °C isotherm within the cloud.These ice crystals or small hails would melt into water during their travel below zero degree isotherm and turn into rain or drizzle or limit the maximum size of hail particles that fall from the cloud.This hypothesis of hail control, however, does not specify the control of any specific size of hail.It is general strategy so that the sizes of newly born hailstones are small enough at the place of their origin itself.
Hypothesis was first applied by Georgia (part of erstwhile Soviet Union) and subsequently laid strong scientific foundation for the control of hailstorm world over.Between mid-1970s and beginning of 1980s two larger experiments were undertaken to evaluate the effectiveness of cloud seeding in Western Europe (Switzerland, France and Germany) known as The Great Experiment ~ SCIENTIFIC PAPERS ~VOLUME 51 JOURNAL OF WEATHER MODIFICATION (Federer et al., 1986) and another in U.S.A. known as National Hail Research Experiment (Knight et al., 1979).The results showed that statistically there was no significant difference in the occurrence of hail between seeded and not seeded hail bearing cloud.Albeit World Meteorological Organization-2007 acknowledged that evaluation of hail mitigation programs remains controversial, still several European countries continued with their Hail Suppression programs.Causes of unsatisfactory success in the hail control under several projects, worldwide, have been discussed by Kumar (2017).There are three preliminary steps in any effective hail mitigation operation.
i. Identify a growing cumulus in its early stage and forecast 'if it would turn into a hailstorm or not'.
ii.What is the reaction time (explained in 1.2) for the seeding operation, keeping in mind the operational and flight safety norms?
iii.What is the location and speed of motion of cloud?
First step is of great significance.In 1982 the "Thunderstorm Identification Tracking, Analysis and Nowcasting" (TITAN) developed software which could track the echo for forecasting of hail (Dixon and Wiener, 1993;Bally, 2004).Development of single and double "Polarized Doppler Weather Radar (DWR)" gave a shot in the arm to timely identify growing cumulus cloud (Wilson and Wilk 1982).In Oct 1996, Hail Detection Algorithm (HDA) by Witt et al., (1998) could provide hail indications.The centroid tracking algorithm of RAINBOW software defined cells based on user-defined single threshold level.But Johnson et al., (1998) showed that its procedure created false alarms or misses in the existence and vertical extent of cells, depending upon the settings of user.To overcome this problem, it was suggested that the storm cells are required to be detected with multiple thresholds of reflectivity ranging from 60 to 30 dBz to give more realistic results.The multiple threshold criterions for storm detection are implemented in the WDSS-II software -Warning Decision Support System Integrated Information.This software was developed by National Severe Storm Laboratory (USA) in collaboration with University of Oklahoma for real time analysis and visualization of remotely sensed weather radar data (Edwards and Thompson 1998;Fulton et al., 1998).In the WDSS-II, the Storm Cell Identification and Tracking (SCIT) algorithm uses centroid identification and tracking technique to identify and track individual storms and provide cell characteristic information (Roy et al., 2011).
The algorithm uses thresholds of reflectivity, length of segments, area of components (as well as other thresholds) and the presence in multiple levels to detect a storm.No velocity data are processed by this algorithm.This 3D depiction of storm is the input for HAD.Multi-Radar Severe Storm Analysis Program (MR-SSAP) - Stumpf et al., 2002 -combines the two-dimensional information from multiple radars and mosaics it into virtual volume scans (Lynn and Lakshmanan 2002), with the latest elevation scan of data replacing the one from a previous volume scan.HDA (Witt et al., 1998) was limited to showing one of the four possible indications for each storm identified e.g., positive, probable, none or insufficient data.Enhanced HDA was developed for hail detection which gave more information e.g., probability of Hail, probability of severe size of hail (diameter more than 1.9 cm), maximum expected hail size and Sever Hail index (SHI).Interactive Radar Information System (IRIS) is capable of generating products including 6 composite of all Doppler Weather Radar images.
For hail mitigation programs one not only needs spontaneous prediction of hail but also maximum possible reaction time (explained in Sec.1.2) for seeding.Hence a simple quadratic algorithm has been developed in the present paper as Prehail Detection Algorithm (PHDA) which not only identifies a growing cumulus in its early stage but also predicts the reaction time.For PHDA, IRIS is used to extract data from the PPI (Kumar and Debprasad, 2015).

Radar Reflectivity
Radar base reflectivity product is a display of echo intensity (reflectivity) measured in dBZ (decibels)."Reflectivity" is the amount of transmitted power returned to the radar receiver after hitting precipitation, compared to a reference power density at a distance of 1 meter from the radar antenna.

Reaction Time
It has been widely accepted that once hailstones have formed in the cloud it cannot be artificially reduced in size.Hence any exercise to control hailstone growth within the cloud must be attempted prior to its embryonic stage.Lower threshold of radar reflectivity of hailstone containing cloud has been well accepted as 45dBZ (Witt 1990;Witt et al., 1998;Singh et al., 2011;Srivastava et al., 2011) albeit it may be argued to be observed at 52 dBZ, too.Also in areas where there are heavy convective rains, 45 dBZ can also be achieved with no hail present.If the hails form at reflectivity higher than 45 dBZ then more reaction time would be available.But hails or no hails, operational safety would be relatively better if pre-planned hail-mitigation operation is for shorter period at lower threshold at 45 dBZ.Moreover for cloud at greater distances, even larger hydrometeors may exhibit reflectivity less than 45 dBZ because their effect is averaged out over the entire resolution volume, which could be of the order of cubic kilometres.Hence very long distance observational range may also affect this value.Radar measurements of storms at far ranges are generally less intense due to the larger volume of the radar beam at larger ranges.Moreover the radar does not measure particles at lower elevations at far distances/ranges.Therefore S-Band radar with operational range limit of ≈ 200 km is recommended for using the bench mark of 45 dBZ.
Advance prediction of time-period when a growing cumulus would attain 45 dBZ of reflectivity is, therefore, important for hail mitigation campaigns.Now following point: i. Total Reaction Time (TRT) may be defined as the time taken by any cumulus cloud with reflectivity 20 dBZ to grow till its reflectivity reaches 45 dBZ (Kumar and Pati 2015).
ii. Available Reaction Time (ART) is the time actually available within the TRT for action against the threatening cumulus cloud.
To restrict the formation of large hail size, seeding of the cloud must be done within the Available Reaction Time.Once the reflectivity of the cloud has grown more than 45 dBZ, this indicates that hails are already present within the cloud.These hails have to fall on the surface as hailstones after some melting.Hence seeding the cloud after the reaction time would be of little avail.Seeding within reaction time, therefore, will restrict large growth of any hydrometeor in the cloud.Consequently while falling through below 0 °C isotherm levels they will either partially melt down to very small size or completely meltdown into rain droplets.

Rate of Growth of Cumulus
Increase or decrease pattern of radar reflectivity is a complicated phenomenon which depends on dynamics and thermal conditions in the environmental background.Radar reflectivity of the cumulus clouds, therefore, could be termed as proxy of its growth or decay features.Hence before proposing the mathematical formulation of cumulus , where suffix n denotes the 'negative rate', has been computed from Tables 4, 5 and 6.The average r n = 0.3 dBz/min with standard deviation (σ) = 0.1.Therefore, in conformity with the definition given for r, the r n may also be termed as slow when r n ≤ 0.2 dBZ/min, moderate when 0.2 < r n < 0.4 dBZ/min and fast when r n ≥ 0.4 dBZ/min.These definitions would be used in the subsequent discussions in this paper.

Velocity of the Cloud and Cloud Seeding
Different types of seeding techniques have been described by Kumar (2017) The generated binary data file from DWR PPI data was interactively accessed through the inbuilt software feature of IRIS to arrive at the analysis data values rather than simple extrapolation from imageries.Console operator sitting in front of the scope ensured that same cluster is studied throughout its growth or decay.This ensured that all the clusters were correctly chased during the study.

Quadratic Growth hypothesis (QGh) alGorithm
Quadratic Growth Hypothesis (QGH) proposed in this paper is based on the assumption that slow to moderately growing cumulus reflectivity follows quadratic relation with time.To prove the validity of assumption if we assume that the hypothesis is incorrect then predictions made based on the assumptions should also be incorrect but on contrary we would see that predictions for hailstorm in slow growth category is 100% correct and in moderate growth category it is 62.3% correct (Fig. 36).Nevertheless it certainly needs further validation with larger data set in India and from other parts of the tropics and even extra-tropics, too, for examining its universal validity, under slow and moderate growth category in particular and all the categories in general.This would also clarify if the hypothesis is regional phenomena or global in nature.Mathematical details of prediction based on QGH, reaction time and speed of the cloud are categorically described below.Again validity of Quadratic Growth Hypothesis (QGH) is discussed in Section 7.0.

Hail Prediction and Reaction Time Based on QGH
It is generally expected that the Reaction Time is transient short span in a growing cumulus hence might range from a few minutes to an hour or so, depending on the rate of growth of convection.Further any two radar observations can be taken only after certain time interval 't'; where t is the scan time, hence if two observations are needed then it can be at the start time (T 0 ) and then again after t time i.e., at T 0 + t.Hence larger number of observations, for making prediction based on extrapolation scheme, would consume larger time and hence render shorter Available Reaction Time (ART).On the other hand if only two observations are taken at T 0 and T 0 + t then only straight line fit is possible.Straight line fit is not appropriate, since two consecutive observations exhibiting same reflectivity or decrease in reflectivity would always predict no hail.Skill score computed, based on this scheme is poor at 0.42.Hence better tradeoff between the reasonably good ART and least prediction time with high skill score is three observations at T 0 , T 0 + t and at T 0 + 2t.Hence a simple Quadratic Growth Hypothesis (QGH) has been adopted in the present paper.This hypothesis presumes that "Slow or Moderately Growing Cumulus" reflectivity (Z) in (dBZ) could be related with time (T) in minutes as Z = aT 2 + bT + c, where a, b and c are arbitrary constants.Let T i be the time of i th observation (i = 1, 2, 3, 4) and T 4 (i = 4) is the predicted time when 45 dBZ reflectivity would be achieved by the convective cloud.If (T i+1 -T i ) is the time interval between the two successive observations (in minutes), then (T 4 -T 3 ) is the Reaction Time.If the radar reflectivity at time T 1 , T 2 , T 3 in minutes be Z 1 , Z 2 , and Z 3 in dBZ respectively.With Quadratic Growth Hypothesis (QGH) the three may be presented as: Where a, b, c are arbitrary constants whose values can be obtained by solving equations ( 1), ( 2) and (3).Having obtained the values of constants a, b and c, the reaction time (T 4 -T 3 ) may be obtained by  16,19,25] and [11,16,23].
Radar operator has to include only these clusters in the computation of reaction time.If first scan is at 4 h 13 min UTC then T 1 = 13.Thereafter for 10 minutes scan interval T 2 = 23 and T 3 = 33 min.
Nevertheless radar operator has to be manually alert by closely identifying the shape and texture of the echoes during each scan and chase the specific echo keeping in mind the movement due to steering wind.New born echoes during the scans are to be astutely differentiated by their textures and shapes and are to be ignored while chasing any particular echo.

Speed Calculation
Significance of speed of the cloud has been discussed in Section 1.4 e.g., if cloud seeding has to done by vertical firing of rockets after landing of helicopter below the cloud then spontaneous speed would help the pilot to timely chase the suspicious cloud before landing below it.
If φ is the azimuth angle in degrees, measured clockwise from north and θ the angle made by the target with the positive x-axis (anticlockwise) then for first quadrant, as shown in Fig. 2, and for 2 nd , 3 rd and 4 th quadrants ; where where φ r is in radians and φ is in degrees.Conversion of r, θ into x and y components are made by taking x = r Cos θ and y = r Sin θ.
The speed is computed at the midpoint of two time observations by dividing the linear distance between the two points by the time interval.If locations of points A (r 1 , φ 1 ), B (r 2 , φ 2 ), C (r 3 , φ 3 ) are, as shown in Fig. 3.And if time associated with point A (r 1 , φ 1 ) is t 1 and the range and azimuth are r 1 and φ 1 respectively.Similarly at point B (r 2 , φ 2 ), time associated is t 2 and the range and azimuth are r 2 and φ 2 respectively and at point C (r 3 , φ 3 ), time associated is t 3 and the range and azimuth are r 3 and φ 3 respectively.

Then at time
, speed at AB is: and speed at BC at time .
Hence speed at time t 3 is given by: , where .

prehail detection alGorithm (phda) Based hail prediction and reaction time
As the available data from different regions of India had varying scan intervals of 10, 12 and 19 minutes hence corresponding data sets are named as D i j where i specifies the scan interval (e.g., 10, 12 or 19 minutes) and j specifies the number of hailstorms actually occurred in the data set.For example D 10 5 dataset means data with 10 minutes scan interval included 5 actually occurred hailstorms.Separate analyses of data sets helps to examine the role of scan time interval in influencing the skill scores of hail prediction and reaction time values.

PHDA Validation for Hail Prediction
Prediction skill score used in the present study is defined as the ratio of 'correct predictions' and the 'total predictions made' = (Correct prediction/Total Predictions made).

Prediction of Cloud Motion Speed by PHDA
Tables 7, 8 and 9 show the speed computation, based on the PHDA algorithm for 12 clusters each from D 10 5, D 12 12 and D 19 8 respectively.It may be noted that mean (standard Deviation) of the anomalies from the predicted speed in Tables 7, 8 and 9 are -1.16(3.0), 0.12 (2.5) and 0.7 (1.2) respectively.Although it indicates good accuracy in the predicted speed in all the three data sets but lowest value of standard deviation of anomalies in prediction is obtained in Table 9.This could indicate that accuracy of speed computation through PHDA algorithm could improve with larger scan-time data.
As the data in Tables 7, 8 and 9 has been collected from different places of India there is wide variation in cluster speed ranging from 5 m/s to as high as 19.3 m/s.This could be due to the regional and seasonal variability in the upper wind conditions in the Indian subcontinent.

example case on how to use phda
In this section a few example cases, each from D 10 5 and D 19 8, are presented as illustration for occurrence and non-occurrence cases of hailstorm correctly predicted by QGH.Graphical extrapolation of the quadratic curve will also guide readers on how the hypothesis works.

Occurrence of Hailstorm
Nagpur radar pictures of 16 th March 2013 taken from D 10 5 are shown in Figures 4, 5, 6 and 7 (cluster 1).Growth of cluster, marked inside rectangle, may be noted.

Non-occurrence of Hailstorm
Another cloud cluster of Nagpur radar picture of 16 h March 2012 is spotted and tracked in Fig. 8, 9 and 10 (cluster 3).This cluster did not show any significant growth in its reflectivity.From Fig. 8, 9 and 10, the PHDA finds complex roots; hence predicts for non-occurrence.It can also be graphically observed in Fig. 11 that although the                  Cloud cluster no.9 was spotted on 3 June 2013 in Mumbai radar as shown in Fig. 32, 33, 4 and 35.It may be observed that growth of reflectivity was fast (r = 0.58) during initial 19 minutes then it reversed back to fast decay (r n = 0.52) in next 19 minutes.PHDA predicted no hailstorm but hailstorm actually occurred at 10:36 UTC i.e., 19 minutes later.This was typical case of moderate r followed by fast r n .

QGH Based Algorithm
It has been observed in Section 5.0 that QGH algorithm presents correct forecast for a few moderately growing cumulus.Incorrect predictions in Section 6.0 were attributed to either 'slow followed by fast' or 'temporary reverse growth of cumuli'.To further analyze the relation between QGH and rate of growth of cumulus a scatter diagram was plotted for correct and incorrect predictions verses rate of growth of cumulus.In Fig. 36 the rate of growth of cumulus between first and second observation is plotted on x-axis and between second and third observation is on y-axis.Blue and Red dots correspond to correct and incorrect predictions respectively, based on QGH.Mostly growth of cumulus significantly slows down as it becomes taller.Therefore, all the cases but for one, lie on the right of 45° slant line.Large rectangle marked in Fig. 36    predicted hailstorm or no-hailstorm; albeit mostly QGH is not valid out of 'QGH-Rectangle' regime.

Reaction Time
RT computation is most consistent for data set D 12 12.Relatively lowest standard deviation in error indicates reliable range of reaction time 17.3 to 29.6 minutes.Maximum RT of 43 minutes is predicted by D 10 5 it is an indication that some (slower) growing cumulus can provide reasonably large reaction time of 43 minutes for conducting the seeding operation.
For fast growing cumulus it could be as low as 14 minutes only.Negative error differences are worrisome from operational safety point of view.
It ranges from 1 minute to 13 minute.In D 10 5 the maximum negative value is 8 minutes, which is lowest amongst the three data sets.Hence from operational safety point of view 10 minutes scan interval is safer for prediction of reaction time by PHDA.If the shorter than 10 minutes scan-time interval data will give larger and safer prediction by PHDA is scope for further research.

Speed of Cumulus
Cumulus data indicated that cumulus motion speed could range from 5 m/s to as high as 19.3 m/s.This could be due to the regional and seasonal variability in the upper wind conditions in the Indian subcontinent. A

Fig. 1 .
Fig.1.Example of PPI-display when easterly wind prevails.Black spots or shapes represent echo location during first scan.Blue spots or shapes are the locations of the same echoes during the second scan and red color echoes represent their locations during the third scan.The numbers close to echoes represent their reflectivity.Radar operator has to manually discard all those echoes whose reflectivity during the second scan are less than or equal to those during the first scan.It may be assumed that these echoes are not growing -although there may be exceptions of reverse growth.From north to south, the clusters are shown by cutting arrows which indicate growth e.g.,[07, 13, 19],[05, 14, 20],[08, 16, 23],[16, 19, 25] and [11, 16, 23].Radar operator has to include only these clusters in the computation of reaction time.

Fig. 3 .
Fig. 3. Cloud cluster locations A, B, C on PPI display at different time.
Fig.  4  shows the spotting of specific cluster having a very low reflectivity value (21 dBZ).The rectangular shed in Fig.5shows the tracking of same cluster with increased reflectivity of 23 dBZ.Fig.6shows the same cluster which has moved to different position with reflectivity of 27 dBZ.Fig.7shows occurrence of hailstorm at around 5:10 UTC.Note small speck of light yellow as indicated by arrow head within the rectangle.Prediction based on Pre Hail Detection Algorithm indicates hailstorm occurs at 4:57 UTC.Positive error of 13 minutes.Predicted reaction time was 27 minutes.It may be observed that Rate of growth (r) during first 10 minutes and next 10 minutes were moderate as r = 0.5 and 0.4 respectively.

Fig. 7 .
Fig. 7. Cluster inside the rectangular shed with Reflectivity: 45 dBZ, Range: 235 KM, Azimuth: 305°.Note small speck of light yellow color close to the head of arrow, within the rectangle.

Fig. 11 .
Fig. 11.Graphical representation of quadratic extrapolation.45 dBZ ordinates are not intersected.Quadratic Equation is shown on the figure.It predicts no hailstorm.x-axis is time interval in minutes and y-axis is the reflectivity in dBZ units.

Fig. 15 .Fig. 16 .
Fig. 15.Graphical representation of quadratic extrapolation.45 dBZ ordinates are intersected after certain interval of time.Quadratic Equation is shown on the figure.It predicts hailstorm.x-axis is time interval in minutes and y-axis is the reflectivity in dBZ units.

Fig. 19 .
Fig. 19.Graphical representation of quadratic extrapolation.45 dBZ ordinates are intersected after certain interval of time.Quadratic Equation is shown on the figure.It predicts no hailstorm.x-axis is time interval in minutes and y-axis is the reflectivity in dBZ units.
developed in the present work was separately applied to same time-interval data and then compared.The entire radar product, collected for the present study from India Meteorological Department, was without any spurious echoes, due to built-in software in the radar.
rd , 6 th and 8 th June time interval were even larger at time 19 minutes.This DWR at Mumbai was BEL Mk-II make S-Band (2700-2900 MHz).As the time intervals in different data were not same simple quadratic extrapolation algorithm

Table 7 .
Speed of Cumulus for D 10 5

Table 8 .
Speed of Cumulus for D 12 12

Table 9 .
Speed of Cumulus for D 19 8 comparative study of errors in speed prediction by linear extrapolation, of 12 clusters each from D 10 5, D 12 12 and D 19 8, shows higher degree of accuracy (i.e., mean error ≤ 1 m/s) in case of D 12 12 and D 19 8.However, relatively least standard deviation in errors in D 19 8 indicated larger scan interval would exhibit more consistent and reliable speed prediction with the presented extrapolation method.and satisfactory (62.5%) predictions for slow and moderately growing rate of cumulus cloud, respectively.Interestingly it is also observed from Fig. 36 that all growing cumuli falling within the rectangle 0.45 x 0.6 appear to empirically follow QGH.Hence it may be termed as 'QGH-Rectangle'.Although, to firmly authenticate the prediction, in future studies, it may verified for other regions, too, with larger data set.ii.QGH mostly fails to predict fast growing cumulus and also for cases when rate of growth temporarily reverses.It warrants different algorithm in future studies, to formulate fast and reverse growth, too.Albeit, it is noted that QGH predicted correctly, under these categories, too; twice in the present study.iii.As most cumulus growth fall in the category of slow or moderate range hence the skill score based on the QGH based algorithm is 0.79 for D 10 5, D 12 12.For D 19 8 it is bit lower (0 .75).iv.High 'cumulus reverse growth' of the order of -5.2 dBZ/min was observed in the present study.v.For hail mitigation campaigns, 10 minutes scan interval is operationally safe for RT computation.In most of the cases RT may range from 17.3 to 29.6 minutes.Maximum RT of 43 minutes was also noted for slow growing cumulus.If the shorter than 10 minutes scan-time interval data will give larger RT, is scope for further research.Hence efficacy of PHDA for shorter scan interval data (< 10 minutes) may be further examined in future researches to help compute better 'Reaction Time'.