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Edwin L. Crow

Abstract

A graphical presentation of precipitation data has been used for some years in which the cumulative percentage of the total mass falling on various days of a sample of precipitation days, ordered from the largest to the smallest mass, is plotted against the cumulative percentage of days. This type of graph is called a “double cumulative curve” (DCC), but it is essentially the same as the Lorenz curve in economics. The paper reviews the literature, summarizes the properties, shows the DCC's for uniform, log-normal, exponential, gamma and degenerate distributions, studies the average effect of sample size, and presents a formula for testing the significance of the difference between two DCC's. This formula is applied to compare hail data of various types. It is concluded that a sample DCC is a biased estimate of the population DCC, but the bias becomes negligible as the sample size increases beyond ∼30.

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Edwin L. Crow

Abstract

A recurring problem in single-area randomized seeding experiments has been the assessment of statistical significance over all experimental units, such as days, when some of the units receive no precipitation. The present paper solves the problem in two ways: 1) the likelihood ratio test of the complete model including both zero and positive observations and 2) an approximate confidence interval for the ratio of mean values over all seeded and non-seeded experimental units. The results are obtained for both log-normal and gamma distributions. They are illustrated by numerical examples from the 1972–74 National Hail Research Experiment randomized seeding experiment.

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Alexis B. Long
,
Richard J. Matson
, and
Edwin L. Crow

Abstract

This paper reports on work carried out in the National Hail Research Experiment (NHRE) on hailpad materials, on procedures for reducing hailpad data, and on hailpad calibration. A recommendation is made for a pad constructed of 2.5 cm thick type-SI Styrofoam (manufactured by Dow Chemical USA) and sprayed with a 25–50 μm coating of white latex paint for protection from the deteriorating effects of sun-light. Calibration of the hailpad provides a relation between the minor axis of a dent in the pad and the dimensions of the stone producing the dent. It is recommended that measurements of the minor axis be categorized in size intervals no wider than 4 mm.

The NHRE laboratory technique for calibrating hailpads involves simulating a hailstone impact by dropping a steel sphere onto a pad from a height such that the impact kinetic energy achieved by the sphere equals that of a hailstone of equal diameter falling onto the pad in an environment with known horizontal wind. The pad is tilted to preserve the stone impact angle found in nature. A second-degree polynomial in sphere diameter D satisfactorily describes the calibration relation between D and the dent minor axis. Application of the calibration relation developed for the particular case of no wind to hailpads which have been hit by hail falling in a wind leads to an overestimate of hailstone diameter of approximately 0.5–1% per meter per second of wind speed. This effect of the wind is about twice as large as that found by others.

A theoretical expression is developed that explicitly relates the minor axis of a dent produced by a sphere to the diameter of the sphere. Two controlling parameters in this expression are the impact kinetic energy of the sphere and a factor p, with dimensions of pressure, which quantitatively embodies the response of a pad to a sphere impact. The effect of variations in p on the sphere diameter derived from dent minor axis and information supplied by Dow Chemical USA on possible variability in the compressive modulus of Styrofoam between manufacturing batches together suggest that the user of hailpads obtains a one time all the foam he may need for his work.

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G. Brant Foote
,
Ronald E. Rinehart
, and
Edwin L. Crow

Abstract

Radar data collected during the seeding experiment of the National Hail Research Experiment are used in a search for possible effects of seeding. Two types of variables, denoted by P and Q, are defined as daily integrals of reflectivity and areas of reflectivity above a given threshold. These and other radar variables are examined for correlation with hailfall at the ground and for seeding effect. Though several variables are closely associated with the occurrence of hail in the network, according to the present sample, none is highly correlated with the amount of hail. A method for measuring hailfall by radar recently used in Switzerland with apparently good results was not successful when applied to the Colorado area.

Ten radar variables were tested for seeding effect by comparing their values on seed and control days. Both the Student's t-test and the Wilcoxon-Mann-Whitney test were employed and gave comparable results. No variables tested showed a difference between seed and control days that was significant at the 10% level. An examination of regressions developed between two adjacent areas (one of which was expected to be much more strongly affected by seeding than the other) also failed to detect a statistically significant difference between seed and control days.

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Cleon J. Biter
,
Theodore W. Cannon
,
Edwin L. Crow
,
Charles A. Knight
, and
Philip M. Roskowski

Abstract

An airborne photographic system, in which the cameras are coupled with an inertial navigation system, was developed and used in a 1978 convective cloud study, Photogrammetric analysis from such a system is enhanced: cloud-feature positions can be determined without external references such as the earth's horizon or cloud base in the photographs, and the data reduction process can be considerably automated.

This paper describes the instrumentation, the photogrammetric theory, and the procedures for obtaining cloud measurements from the photographs. An empirical error analysis based on photographs of terrestrial targets is also presented. Cloud top heights determined without any reference height in the photographs are considered to be accurate to within 440 m at a range of 60 km. The largest source of error in determining cloud top height using the 1978 measurements is the uncertainty in determining the aircraft-to-cloud distance rather than inaccuracy in the photographic system. This error can be reduced in future programs by flying as closely in altitude as possible to the cloud features of interest.

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Edwin L. Crow
,
Alexis B. Long
,
James E. Dye
, and
Carlton W. Ulbrich

Abstract

The hailstone size (diameter) distributions measured by hailpads during the 1972-74 randomized seeding experiment of the National Hail Research Experiment are analyzed statistically for evidence of seeding effects and differences from year to year. Two approaches are taken, one comparing the entire empirical size distributions on seed days and on control days and the other comparing the mean diameters. The latter is based on the consistency with the exponential distribution (truncated at a prescribed minimum diameter), since the exponential distribution can be characterized completely by the difference between the mean diameter and the minimum diameter. Both approaches yield statistically significant results (10% level) only for 1974, when the hailstones were larger on seed days than on control days on the average. This may have resulted from the addition of seeding by rockets in 1974 or from differences in the hailpads used in that year. However, the physical hypothesis for the experiment predicted smaller stones on seed days; that tendency did appear in 1973 (though not significantly) and the difference was negligible in 1972.

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Edwin L. Crow
,
Alexis B. Long
,
James E. Dye
,
Andrew J. Heymsfield
, and
Paul W. Mielke Jr.

Abstract

An extensive statistical analysis is made of the precipitation data collected during the randomized seeding experiment conducted by the National Hail Research Experiment during 1972-74, aimed at testing the feasibility of diminishing hail by seeding with silver iodide. The major conclusion is that no effect of seeding is detected at the 10% significance level. This is true regardless of whether hail or rainfall response variables are considered, which of two methods of obtaining daily values for the response variables over the target area is used, or what distribution, if any, is assumed for the variables. Even though the ratios of hailfall or rainfall on seed days to those on control days are generally greater than 1, the confidence intervals attached to these ratios are so large, because of the large natural variance in each response variable and the small sample sizes, that the true underlying seeding effects could in every case have ranged from substantial decreases to large increases. The large confidence intervals emphasize the necessity of large sample sizes, large experimental areas or effective covariates for obtaining definitive results in precipitation modification experiments.

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G. Brant Foote
,
Charles G. Wade
,
James C. Fankhauser
,
Peter W. Summers
,
Edwin L. Crow
, and
Mark E. Solak

Abstract

An analysis of the seeding operations during the National Hail Research Experiment 1972–74 randomized seeding program is carried out for the purpose of critiquing the seeding procedures and establishing the actual rates at which seeding material was dispensed as opposed to the prescribed rates. The seeding coverage, a parameter defined in the paper, is found to be only about 50% on the average. The reasons for the low seeding coverage are discussed in terms of seeding logistics and storm evolution, and three case studies are presented to illustrate the problems that can arise. Some results on the rate at which storm cells can develop and on the duration of convective activity over a fixed target area are presented. It is concluded that seeding convective clouds using aircraft flying near cloud base is more difficult than is widely acknowledged.

Since the seeding operations were more thorough on some days than on others, one might reasonably expect that seeding effects, if they exist, would be more marked on the days with the higher coverage. Post hoc analyses that stratify the surface hail and rain data according to seeding coverage are presented. The results do not allow one to reject the hypothesis that seeding had no effect on surface precipitation.

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