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On Estimating Hail Frequency and Hailfall Area

Alexis B. LongNational Center for Atmospheric Research, Boulder CO 80307

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Abstract

Probabilistic and statistical concepts are used to examine how the number of hail observing sites within a region affects the accuracy of estimates of 1) the mean point frequency of hail within the region, 2) the overall regional frequency of hail, and 3) the area covered by individual hailfalls. A practically useful relationship ν̄ = /A is first derived to connect the mean point frequency ν̄ of hail within a region to the regional hail frequency N, to the mean area ā of the individual hailfalls and to the area A of the region. The error in estimating the mean frequency ν̄ is found to increase as n−1/2, where n is the number of sites placed within a region. If within a region there are proportionally more large hailstorms or if most of the area covered by hail is commonly due to a few large hailstorms, then fewer sites will be needed to estimate the mean point hail frequency. Of the 16 hailfalls detected by the 660 km2 1976 National Hail Research Experiment (NHRE) network of 603 hailpad sites, it is found, using a simple probabilistic expression, that 12 of the hailfalls still would have been detected using only 50 sites. The smaller hailfalls would have been the first to go undetected. There are diminishing returns in fielding sufficient instruments to detect all, or almost all, of the hailfalls in a region. For hailfalls with the lognormal distribution of areas observed by NHRE, the number of instruments needed increases exponentially with the number of hailfalls to be detected. A formula is derived for a correction to be made to the observed regional frequency of hailfalls of a given size so as to obtain the true frequency. For random networks the coefficient of variation of an estimate of the area of a hailfall is proportional to n−1/2. For a hailfall less than one-fifth as large as the instrumented region in which it lies, the coefficient of variation of an estimate of its area approximately equals nh−1/2, where nh is the expected number of sites within the hailfall.

Abstract

Probabilistic and statistical concepts are used to examine how the number of hail observing sites within a region affects the accuracy of estimates of 1) the mean point frequency of hail within the region, 2) the overall regional frequency of hail, and 3) the area covered by individual hailfalls. A practically useful relationship ν̄ = /A is first derived to connect the mean point frequency ν̄ of hail within a region to the regional hail frequency N, to the mean area ā of the individual hailfalls and to the area A of the region. The error in estimating the mean frequency ν̄ is found to increase as n−1/2, where n is the number of sites placed within a region. If within a region there are proportionally more large hailstorms or if most of the area covered by hail is commonly due to a few large hailstorms, then fewer sites will be needed to estimate the mean point hail frequency. Of the 16 hailfalls detected by the 660 km2 1976 National Hail Research Experiment (NHRE) network of 603 hailpad sites, it is found, using a simple probabilistic expression, that 12 of the hailfalls still would have been detected using only 50 sites. The smaller hailfalls would have been the first to go undetected. There are diminishing returns in fielding sufficient instruments to detect all, or almost all, of the hailfalls in a region. For hailfalls with the lognormal distribution of areas observed by NHRE, the number of instruments needed increases exponentially with the number of hailfalls to be detected. A formula is derived for a correction to be made to the observed regional frequency of hailfalls of a given size so as to obtain the true frequency. For random networks the coefficient of variation of an estimate of the area of a hailfall is proportional to n−1/2. For a hailfall less than one-fifth as large as the instrumented region in which it lies, the coefficient of variation of an estimate of its area approximately equals nh−1/2, where nh is the expected number of sites within the hailfall.

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