Hailfall Data from a Fixed Network for the Evaluation of a Hail Modification Experiment

Richard A. Schleusenser Colorado State University

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John D. Marwitz Colorado State University

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William L. Cox Denver Public School System, Colo.

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Abstract

Haifall data collected from a fixed network in northeastern Colorado during three seasons (1960–62) included the estimated impact energy, duration of hailfall, most common stone size, maximum stone size, and number of stones per square inch. These basic data,X, along with the transformations, lnN, √X, 3X, and 1/X were analyzed by computer methods to determine which parameters could be used in a statistical analysis of hail suppression experiment. The gamma distribution function was fitted to the hailfall data by the method of maximum likelihood. A chi-square goodness of fit test was applied to the data, and one transformation was tested using a sequential analysis technique.

All parameters except impact energy and number of hailstones per square inch were eliminated from the statistical analysis because of bias, non-homogeneity, or sparsity of samples. Transformations which produced the minimum mean coefficient of variation were logarithm of impact energy (InE) and square root of the number of stones per square inch (√N1 − 6). It was determined that a target-control analysis was not feasible for the analysis of hail suppression experiment. A period of 3 to 5 years is believed necessary to detect changes of 10 to 25 per cent in the hail parameters. The gamma distribution function fitted only the √N1 − 6 data. From the results it was concluded that a sequential analysis test alone could not adequately evaluate the effectiveness of a hall modification experiment.

Abstract

Haifall data collected from a fixed network in northeastern Colorado during three seasons (1960–62) included the estimated impact energy, duration of hailfall, most common stone size, maximum stone size, and number of stones per square inch. These basic data,X, along with the transformations, lnN, √X, 3X, and 1/X were analyzed by computer methods to determine which parameters could be used in a statistical analysis of hail suppression experiment. The gamma distribution function was fitted to the hailfall data by the method of maximum likelihood. A chi-square goodness of fit test was applied to the data, and one transformation was tested using a sequential analysis technique.

All parameters except impact energy and number of hailstones per square inch were eliminated from the statistical analysis because of bias, non-homogeneity, or sparsity of samples. Transformations which produced the minimum mean coefficient of variation were logarithm of impact energy (InE) and square root of the number of stones per square inch (√N1 − 6). It was determined that a target-control analysis was not feasible for the analysis of hail suppression experiment. A period of 3 to 5 years is believed necessary to detect changes of 10 to 25 per cent in the hail parameters. The gamma distribution function fitted only the √N1 − 6 data. From the results it was concluded that a sequential analysis test alone could not adequately evaluate the effectiveness of a hall modification experiment.

F~BRU~tR-1965 SCHLEUSENER, MARWITZ AND COX 61I-Iailfall Data from a Fixed Network for the Evaluation of a Hail Modification Experiment~RIClIARI) A. SCHLErrSENEa, JOItN D. MARWITZColorado State University Am) WI~.n~A~ L. Cox Denver Public School System, Colo. (Manuscript received 2 October 1964)ABSTRACT Hailfall data collected from a fixed network in northeastern Colorado during three seasons (1960-62) included the estimated impact energy, duration of hailfall, most common stone size, maximum stone size, andnumber of stones per square inch. These basic data, X, along with the transformations; lnX, ~/~, '~, andI/X, were analyzed by computer methods to determine which parameters could be used in a statisticalanalysis of hail suppression experiment. The gamma distribution function was fitted to the hailfall data bythe method of maximum likelihood. A chi-square goodness of fit test was applied to the data, and one transformation was tested using a sequential analysis technique. All parameters except impact energy and number of hailstones per square inch were eliminated from thestatistical analysis because of bias, non-homogeneity, or sparsity of samples. Transformations which pro~duced the minimum mean coefficient of variation were logarithm of impact energy OnE) and square root ofthe number of stones per square inch (~_~). It was determined that a target-control analysis was notfeasible for the analysis of hail suppression experiment. A period of 3 to 5 years is believed necessary todetect changes of 10 to 25 per cent in the hail parameters. The gamma distribution function fitted only the~-0 data. From the results it was concluded that a sequential analysis test alone could not adequatelyevaluate the effectiveness of a hail modification experiment.1. Introduction Development of techniques for cloud modificationby seeding has led to a variety of attempts at weathermodification, including precipitation increase, hurricane modification, fog dispersal, and hail suppression.The natural variability of meteorological phenomena issuch that detection of any small change which mighthave been effected artificially is difficult. This difficultyis increased with the variability of the meteorologicalphenomenon being considered, and leads to frustrationfor cases in which high variance and low frequenciesof occurrence combine to require excessive periods oftime to draw valid conclusions concerning the effectsof modification attempts. These difficulties are further compounded for evaluation of attempts of hail suppression because of a lackof basic data on the nature and characteristics of hailfalls. For example, the only statistic concerning hailfalls which is readily available is "days with hail."While this parameter may serve to delimit regionaldifferences in the average annual frequency of hail,it leaves much to be desired in statistics which would beappropriate for detecting changes in hailfalls whichmight have been produced artificially. ~ Research supported by the Atmospheric Sciences Program,National Science Foundaton, NSF Grant G-23706. This paper presents a procedure for the evaluationof hail suppression using data on hailfalls for threeseasons (1960-1962) from the fixed hail network operated in northeastern Colorado by Colorado StateUniversity. Examination of these data provides aninsight into some of the physical properties of the HighPlains hailstorm, delimits the hailfall parameters whichmight be used in the design of a hail modification experiment, and points out the problems inherent in andrequirements of a statistical analysis of a hail modification experiment.2. Basic data For information on hailfalls obtained from theColorado State University network, see Schleusenerand Jennings (1960), Schleusener (1961), and Schleusener and Grant (1961). The following data werederived and prepared for computer analysis: From cooperative observers: Estimated impact energy, ft4b per ft~ E~ Duration of hailfall, minutes D~ Most common stone size (coded) MC Maximum stone size (coded) MX = D was removed from further analysis because of excessive bias.Most cooperators reported D in 5 to 10 minute increments.62 JOURNAL OF APPLIED METEOROLOGY VoLu~m4From the passive hM[ indicators: Impact energy, ft-lb per ft2 El Number of stones per square inch for (coded) diameters of hailstones 1-6 Ni-o 2-6 N2-~ 3-6 Ns_, 4-6 N~-0 These hail parameters were analyzed from the viewpoint of attempting to identify parameters which, onthe basis of physical reasoning, might be changed ifcloud seeding were effective in modification of hail.3. Transformations of basic data Transformations were applied to the basic data, toattempt to produce a minimum absolute dispersion ofthe individual coefficients of variation about the meancoefficient of variation (E I Cv~- Ovl ). In addition to thebasic observations, X, consideration was given to eachof the following transformations which were applied toeach of the observations: In X, 4~, ~ and 1IX. The selected transformations were In X for E, andEi and bx/-~ for MC, MX, Nx-o, N~-6, N~_6, and N4-6.The value of the mean, standard deviation, and thenumber of samples were determined for each transformed hail parameter set plus subsets by years, monthsand geographic locations. The transformations decreased the variance of the parameters considerablybut the coefficient of variation continued to be greaterthan un-ty in most cases.4. Tests of homogeneity of data The variables and transformed variables were testedby means of an F test for homogentity among years andamong months by a one way analysis of variance. From these analyses (Table 1) it was determinedthat the hypothesis of homogeneity among years forIn El 4~, N~-0, and ~/~2-~ must be rejected. Thehypothesis of homogeneity among months must alsobe rejected for 4~C and 4~_~. On this basis and since4~-~ (not listed in Table 1) was more variable thanM~-~-~, the parameters M~-~, 4~-~ and ~/~2_, wererejected from further analysis. The hypothesis ofhomogeneity among years for In E~ when using datafrom all of the hail network was rejected whereas whenusing only data from the west triangle of the networkit could not be rejected. The hypothesis of homogeneityamong years and months for In E,, and ~ couldnot be rejected. The parameters a/Na_6 and ~ wererejected from further analysis due to low frequenciesof occurrence. The remMning sets of parameters, consisting ofIn El, 4N--~_~ (retained for analysis despite homogeneityTABLE 1. F test results for significance of differences between parameter subsets.Parameter Among Among Between east and west Between north and south years months 60 61 62 60 61 62lnE~ F 0.90 0.34 4.00** 2.86* 1.72 3.94** 2.39 2.77**df (2X963) (2X1011) (4X324) (4X504) (4X283) (3X261) (3X348) (3,502)df (2X761) (2X726) (2X91) (2X275) (2X328) (2X49) (2X222) (2X423)~/~-~ F 51.00'* 19.60'*~f (2xuso) (2x1095)df (2X757) (2X717) (1Xlll) (1X196) (1X453)4N~4 F 18.55'* 2.69 4.0* (14.4' 2.73df (2X585) (2X571) (1X74) X152) (1X362)~/N~_~ F 1.18 0.48 0.24 5.70* 0.04dft (2X231) (2x228) (1x15) (1X59) (1x160)df (2X57)lnE~ F 1.28df (2X403)(west triangle)Notes: * Significant at 0.05 level, ** Significant at 0.01 level.F~u~'a-1965 SCHLEUSENER, MARWITZ AND COX 63considerations noted above), and In Ec, were arbitrarilydivided into subsets from'east to west and subsets fromnorth to south. The hypothesis of homogeneity amongeast-west subsets and north-south subsets could not berejected; consequently it was decided to add a 60-milenorth-south line of indicators through the west triangleof the network for the 1963 season.5. Effect of size of sampling area on variability of parameters The coefficient of variation was computed as afunction of sampling area for the 3 remaining hailparameters (Figs. 1, 2 and 3). From these computationsit was determined that the variance of the hail parameters would be changed very little if only half of thesampling area was used. For some of the hail parameters, a reduction in sampling area produced a decreasein variance. From these results and the results of thehomogeneity tests on In El it was concluded that thesize of the sampling area could be reduced, and theeast triangle of the indicator network was abandonedprior to the beginning of the 1963 season.6. Tests for normality of parameters The ~F~_6, In El and in E~ data sets and subsetsby years, were tested for normality. In nearly allsets and subsets the data were highly skewed right, withthe kurtosis ranging from leptokurtic to isokurtic toplatykurtic (Figs. 4, 5 and 6).7. Tests for independence of observationsCorrelation coefficients were computed betweencertain combinations of the hail parameters averagedover one region versus the parameters averaged overother regions approximately 25 miles away. Theresults indicate no significant correlation. 4.cCorrelation coefficients were also computed for thehail parameters In Ei and ~/N----~_~ between each dented ~ll~3'cindicator and its neighbor (and mean of neighbors) ~' 2.clocated approximately 2 and 4 miles away. Althoughthe correlation coefficients were all less than 0.50, thehypothesis of dependence could not be rejected ~'-(Table 2). 0TaBL~ 2. Number of significant and non-significant correlationsbetween a dented hail indicator and its neighbor (neighbors)located 2(4) miles aw~y. 2 miles 4 milesCorrelation Neighbors Neighbor Neighbors Neighbor NS 8 14 13 16 S 4 4 3 3 S* 12 6 8 5 Note: NS-~No significant correlation (0.05 level), S=Significant correlation (0.05 to 0.01 level), * S = Significant correlation(0.01 level). 1.0 0.8 ,.~ o.4 0.~ FRAGTION OF TOTAL LENGTH Fro. 1. Coefficient of variation, Cv, versus fraction of totallength of indicator lines for ~/N~-0. Total length is approximately200 miles.,.ot% 13.8 ~ ! , 06 -' ~ . : ~~ > 04 * ~ 0.~ 0~15 O~ 1.0 F~CTION OF' TOTAL LENOTH Fro. 2. Coefficient of variation, Cv, versus fraction of totallength of indicator llne for InEg. Total length is approximately200 miles. FRACTION OF AREA. Fro. 3. Coefficient of variation, Cv, versus fraction of totalarea containing cooperators for lnEo. Total area is approximately 6000 square miles.8. Tests to estimate the period of time required to detect scale changes in the hail parameters Computations were made to estimate the period oftime required to obtain significant differences in thehail parameters, assuming various scale changes in theparameters (Appendix A). These computations assume64 JOURNAL OF APPLIED METEOROLOGY VOL~J~tr. 4 , ~ i ~ ;-Ni'"~'-~"~_6 wherl Ni.8 is $fonas per squor, inchFro. 4. A histogram of the ~/~-G data.5C4C3CI0 7'[ ~77': '~,0~..i7 I III Kl"l~-~-~---~ o.o, . o.o, 0 50 I00 200 300 Ei, ft-lb per ft2IrIo. 5. A histogram of the El data.0.01 4~0~8 Ei, ft-lb pet' fl2FIG. 6. A histogram of the InEi data.that negative reports of hail occurrence can be obtained in those cases in which complete hail suppressionmight be attained, that all hail-producing storms wouldhave been subjected to a modification treatment duringthe'period of 15 May to 31 July, and that the averagenumber of hail samples will remain constant. It maybe noted that "success" in a hail modification experiment would increase difficulty of statistical analysisbecause of an increase in zero values. When these three (questionable) assumptions aremade, a period of 3 to 5 years is estimated to detectscale changes of 10 to 25 per cent in the hail parameters.(See Figs. 8, 9 and 10.)9. Gamma distribution parameters Hartley and Lewish (1959) have reported fitting atwo-parameter gamma distribution function to nonzero rainfall data. The first step in fitting the gammadistribution function to the data is to estimate thedistribution parameters. The most efficient method forestimating these distribution parameters is the methodof maximum likelihood. Maximum likelihood estimators are obtained by differentiating the likelihoodfunction and equating the derivative to zero. Greenwood and Durand (1960) have solved the complexequations in this procedure using polynominal approximations. Since this method involves a logarithm of thedata, it can be used only with data that are greaterthan zero. To eliminate zero values from the observedhail data, which are either positive or zero, 0.01 wasadded to each data sample. The gamma distribution function was fitted to thefollowing data: E6 lnE~, lnE~ for (~i>1), JYi-0,/~-0, and /N--~-0. Gamma distribution parameters forthese data are given in Table 3.T~ 3. Gamma distribution parameters for hailfall data. Gamma parameters Shape parameter, Scale parameter, Data T ~Ei 0.8211 16.1812lnE~ 0.9292 2.053lnEi (Ei> 1) 3.5488 0.6070/Y1-G 1.1587 2.7071~/~-0 4.2209 0.3727.95 ~/~_s 4.2210 0.3541.90 ~/~_~ 4.2209 0.3354.85 ~t~_o 4.2210 0.3168.75 ~/~-0 4.2210 0.27951.05 ~/~14 4.2210 0.39131.25 VN~4 4.2210 0.4659~/~_G 9.2925 0.142110. Test for goodness of fit A chi-square goodness of fit test was used to comparethe distribution of the hailfall data to a theoreticalgamma distribution. In applying this test, boundaries were selected suchthat the number of expected values exceeded 5 in eachcategory. It is permissible to set boundaries in anydesired manner, provided one is not influenced by theobserved frequencies. In applying this test an attemptwas made to obtain categories which contained approximately the same number of observations. It is believedF~BRUARV1965 SCHLEUSENER, MARWITZ AND COX 65TABLE 4. Cell boundaries and chi-square values for the goodness of fit test of hailfall data.Type of Cell boundaries Calculated data Cell 1 Cell 2 Cell 3 Cell 4 Cell 5 Cell 6 Cell 7 x~* RemarksEl 0.00-1.90"1.90-2.70 2.70-3.90 3.90-6.70 6.70-13.10 13.10-25.10 25.10-400.1 122.11,.'~ 0.00-1.30 1.30-2.70 2.70-3.90 3.90-7.50 7.50-14.10 14.10-25.1 25.t -330.1 56.1 2rid Run onlnEi 0.00-0.61 0.61-1.01 1.01-1.26 1.26-1.51 1.51- 1.86 1.86- 3.46 3.46- 6.01 208.8lnEi 0.00-2.41 2.41-2.91 2.91-3.21 3.2l-3.71 3.71- 3.91 3.91- 4.71 4.71- 5.91 96.9 2nd Run on lnEilnEi(E~>I) 0.00-0.79 0.79-1.01 1.01-1.26 1.26-1.51 1.51- 1.86 1.86- 3.06 3.06- 6.01 76.7N~-6 0.00-0.61 0.61-1.01 1.01-1.51 1.51-2.51 2.51- 4.51 4.51- 9.01 9.01- 41.51 28.0~/N--~_~ 0.00-0.77 0.77-1.01 1.01-1.26 1.26-1.51 1.51- 1.86 1.86- 3.01 3.01- 6.01 18.5~/~r~_~ 0.00-0.79 0.79-1.01 1.01-1.26 1.26-1.51 1.51- 1.86 1.86- 3.06 3.06- 6.01 6.5 2nd Run on * For 4 degrees of freedom and a 0.05 probability level, the tabulated x~ value is 9.5.** Cell boundaries are inclusive on the larger value (i.e., 0.00<cell 1 =< 1.90).that in using this method it was not possible to completely eliminate the influence of the observed frequencies. This factor produced a slight effect on thecalculated values of chi-square. Judicious selection ofboundaries reduced the chi-square value in all cases,but only for the ~/~_~ data did it affect the decision asto the goodness of fit. The boundaries used and theresults obtained with this test are presented in Table 4. If the gamma distribution function adequately fittedthe observed data, then the probability of obtaining achi-square value less than 9.49 is 0.95 with 4 degrees offreedom. Since only the ~/~_~ data produced a chisquare value less than 9.49, it was concluded that thegamma distribution did not adequately fit Ei, In Eland In ~i(s~i> 1), but that this distribution functiondid provide a marginal fit for ~/N--~_~ data.11. Sequential analysis testing Statistical tests such as Student's t, F, and xa set ana level for a null hypotheses (type I error) and n numberof samples and lets the ~ level for a specific alternativehypotheses (type II error) fall where it may. In thesequential analysis test both the a and/~ levels are setand the observations are tested sequentially. With eachnew observation one of the following decisions isreached: (1) accept the hypothesis, (2) reject the hypothesis, and (3) continue the experiment by taking anadditional observation. The experiment continues untilthe hypothesis is either accepted or rejected. In the sequential analysis test, if the distributionfunction under consideration has two parameters, it isnecessary to either test a composite hypothesis, or toreduce it to a simple hypothesis by assuming one of theparameters constant. It may be shown both theoretically(Appendix B) and experimentally that scale changeddata maintains a constant shape parameter (Table 3).Therefore, a simple hypothesis can be used when ascale changed data is considered (see Thorn, 1957,and Wald, 1947). Sequential analysis testing involves plotting a cumulative function of the data against the number of observations as shown diagrammatically in Fig. 7. In this test maximum and minimum values (from therejection and acceptance lines) corresponding to eachobservation are compared with the accumulated valueof the data. If the cumulative value remains betweenthese limits the test continues. When the cumulativevalue exceeds the maximum value the hypothesis isrejected with probability a of a type I error. If thecumulative value becomes less than the minimum valuethe hypothesis is accepted with probability ~ of a type IIerror. The slope of the rejection and acceptance linesand the difference between maximum and minimumvalues are functions of the hypothesis being tested andthe a and/~ levels selected. In selecting values for a and ~, economic considerations (cost of seeding versus potential benefit) suggestthe establishment of a low probability of error for typeI errors (rejecting a true hypothesis) with a higherprobability of error for the type II errors (accepting afalse hypothesis). Results of the sequential analysis test indicate that NUMBER OF OBSERVATIONS, nFro. 7. A hypothetical case illustrating the sequential analysis method for testing a hypothesis.66 JOURNAL OF APPLIED METEOROLOGY VoLu~u4 TABLE 5. Number of observations required in a hypotheticalcase to reach an accept or reject decision using a sequentialanalysis test and ~ hailfall data. These results were obtainedexperimentally by assuming certain indicated changes whichwere applied to all of the data. No. ofType of obs. redata Type of change Alpha Beta quired DecisionX/N~_0 5% scale decrease 0.01 0.05 405 Accept~/~_~ 10% scale decrease 0.01 0.05 135 Accept~/~-0 25% scale decrease 0.01 0.05 36 Accept~/~-0 5% scale decrease 0.01 0.01 732 Accept~/N-~-0 10% scale decrease 0.01 0.01 141 Acceptif the gamma distribution function fits the ~tN~_e data,a 5 per cent scale change with a and/5 both 0.01 wouldrequire 732 observations to obtain an accept or rejectdecision. The results calculated for other scale changeswith various a and ~ values are presented in Table 5.12. Conclusions 1) From the 9 hailfall parameters derived from data onhailfalls collected by the Colorado State Universityhail network, 6 were eliminated for use in any statisticalanalysis of hail modification because of bias, nonhomogeneity between years, or sparcity of samples.The remaining parameters were Ec, El, and N~_o. 2) The transformations which produce the minimum2:ICy,- 0, are In Ec, in E,, and 3) A north-south extension of the hail indicatornetwork can be made, and the east half of the networkcan be abandoned without significantly affecting thestatistical properties of the indicator data. 4) The hypothesis of dependence between .adjacentindicators spaced 2 to 4 miles apart cannot be rejected,even though the correlation coefficients are less than0.50. 5) A period of 3 to 5 years is estimated to be requiredto detect scale changes of 10 to 25 per cent in the hailparameters that might be accomplished by modification attempts. However, there are practical difficultiesinvolved in attaining the conditions assumed in theanalysis, one of the most difficult being the problem ofhandling zero values if complete hail suppression were tobe attained. 6) Lack of significant correlation between adjacentareas indicates that a target-control analysis is notfeasible for attempting to detect significant changesthat might result from a hail modification experiment. 7) Of the data collected and the transformationsstudied, only the ~/N-~-0 data can be fitted by a gammadistribution function and it provides only a marginalfit. 8) The sequential analysis test alone could notadequately evaluate the effectiveness of this hailmodification experiment. 9) Further work is presently being done to developprocedures not dependent on a fixed network foranalysis of effects which might be produced in a hailmodification experiment.APPENDIX A The following equation estimates the number ofreplications required to obtain significant results at aparticular probability level.r_> 2 (t~,q-l,)2 r = replications required.,= population standard deviation.m = population mean. a = scale change.t~= value of students' t at significance level p.t~=value of students' t at significance level 2(l-p) for a two-tailed test. Using the above equation and assuming variousscale changes and significance levels, the number ofreplications required to obtain significant results forIn Ei, In E.,. and ~/~-0 were computed and are presentedin Figs. 8, 9 and 10. m =1.92 ~=1.18 ~=1% 5% I0% 25% Sicj. lev.~----~_~'-- ~~3 ~ I % 50, 2/0~ 81 /io~o -,~oq 3, 3O % ~8,1~ 13 (~ % ~o,~ oo ~8op =0.90~5 % 79,300 93 27 /I 0 % 84 03 ~30% 40 ~6 65 p 135,000 IT '9 7, 800 8 I, 200 ~8 I 2 54,2( / 54? Sig. C~. ~~ % ~5 %P=095 /I0 % t30 %YEARS I O01960 = I 13 )1961 196~, 255 overage number of1962 455,1 observations per yearFro. 8. Estimated number of replications of lnEi required to obtainsignificant results at particular probability levels.FESRUAR-1965 SCHLEUSENER, MARWITZ AND COX 67 m - 2.888 ,1% 5% Sig. Lev. O/2~6( (1% 64,90~,.o. so/5 % 37, so~ /,0% 26,~0//0,' $ig. Lev. 104,000 / 40~ Sig. ~v. 52,~~=-'*~,o',. ,?.oo2 ~YEARS 50 I0o' - 2.0110% 25%/~ 104 60 42 17 i%- 1~3s ~0~ / ~64 2~2 / ~6)-)' ,~ / ~o~ ~o/ ~8/ zoZ ~,~ I m = 1.5050% $- t% 5% Sig. Lev. [,"'% 64,000 2~"'~ . 050)5 % 3,?,00026,5 '' )10% 25,90/~/A'1~I I [3,0% I0,4 S1~1. Lev. (, % ~4:5.000 5740\ /5 %125,000 4040157 P'0'90~10% 82,000 32~ /4133 '30' 51,700~070/ ~ L30 SIg. Lev. 000 685"~0~'''' (I % 171, p 095~5 %125,000 5000 /705, '' po%1o4,ooo 4~4o.// [3,0% 69,,00 2~0/42 28 YEARS I0 5 1960 - 223'~ 1961 302~ 322 average number of 1962 441J observations per yearF~o. 9. Estimated number o- replications of lnE~ required to obtainsignificant results at p~rticular probability levels. APPENDIX B The following proof shows that for scale-changeddata, the shape parameter, 7, of the gamma distributionfunction remains constant. The gamma distributionhas a probability function given by the equation 1 X'r-l--'~dt~. (1)The maximum likelihood estimates of /~, the scaleparameter, and T, the shape parameter are T= [1+ (l+4A/3)~/4A (2)and ~=~/~, (~)where ~ is the mean of xi's and A = log2--- ~ logxi (4)for P(xi) = (xl) (x~) (xo)'" (x,~). (6) Now assume that a gamma distribution also fitsxi', where a'= x,~(l+A), (7)where d~ is some scale or non-scale change and dg> --1.50%~' = 1,04 10% 25% ,~O~l 0:5 27 59 15 42 II 04 17 4 ~. 162 40 131 3,3 ~ a3 2i :) 1~9~l/ 74 69 O0 50 166 41 III 28 I 1960= III '~ average number of 1961 - 196~ 25:5 observolions per year 1962 - 453,)Fro. 10. Estimated number of replications of ~-~ required toobtain significant results at particular probability levels. Substituting (7) into (5) 1 ~ 1 A'=log- Z C~(1-Fd~)~-- logP[x,(lq-dg)~, (8)whereand since= log~,q- lo Xigi(lO)and log_~[a(l+A)]=log P(~0+log P(I+A), (12)consequently substituting (10), (11) and (12) into 1A':log.~q-log(lq-~.ixid,)'Let 1 1---logP(x~)---logP(lq-d~). (13) 1a=log(1--Z ~.~ xidl --~ logP(lZr-d/),(14)then substituting (4) and (14) into (13) gives A'=AJr-a05)68 JOURNAL OF APPLIED METEOROLOGY VOLUME4Now substituting (15) into (2) and (3) gives ~,'= [lq- (lq-4(Aq-a)/3)~/4(A+a)andand(16) lg'=.~'/v'=(:~q-l~ ~_~_~xd,)/3,'. (17)For d~=d, a constant l-g(1-bi ~xid)=l-g(lq-d)x n:~ ~=~ (18) 1 ~ - Y~ log(l-t-d) = log(lq-d), (19) ~/, ~~1therefore, substituting (18) and (19) into (14) a=0. (20) And finally substituting (20) into (16) yields theresult: '~'=~' (21) /~'= ~(l+d)/v. (22)Experimental tests (see Table 3) verified these resultsfor ~ data. The shape parameter, % remained constant for the scale changed data while the scale parameter,/~, varied as suggested in Eq. (22) above. REFERENCESGreenwood, J. Arthur, and David Durand, 1960: Aids for fitting the gamma distribution by maximum likelihood. Techno metrics, 2, 55-65.Hartley, H. O., and W. T. Lewish, 1959: Fitting of data to the two parameter gamma distributibn with special reference to rainfall data. Iowa State University of Science and Technology, Statistical Laboratory, Ames, Iowa.Schleusener, Richard A., 1961: Hailstorm damage to crops in Northeastern Colorado and an analysis 'of precipitation anomalies with a cloud-seeding program in 1959. J. Amer. Soc. Sugar Beet Tech., 11,413-428. , 1962: The 1959 hail suppression effort in Colorado, and evidence of its effectiveness. Nubila, 5, 31-59.-- and Lewis O. Grant, 1961: Characteristics of hailstorms in the Colorado State University Network, 1960-61. Proc. 9th Wea. Radar Conf., 140-145. , and Paul C. Jennings, 1960: An energy method for relative estimates of hail intensity. Bull. Amer. Meteor. Soc., 41, 372-376.Thorn, H. C. S., 1957: The frequency of hail occurrence, Technical Report No. 3; A method for the evaluation of hail suppression, Technical Report No. 4. Both in Final Report of the Advisory Committee on Weather Control, Vol. II.Wald, Abraham, 1947: Sequential Analysis. New York, John Wiley and Sons, Inc., 212 pp.

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