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Harold L. Crutcher

Route equivalent winds are an important aid to long range planning by aviation interests. The recent publication of point wind statistics by the U. S. Navy, NAVAER 50-1C-526, provides a considerable amount of material which can be used to probe the use of one route against another. Where many routes are to be examined, then recourse to punched card data and electronic computing machines is desirable. Such cards and machines are available at the National Weather Records Center where the material for the Navy publication was prepared.

In many cases, though, one may wish to examine only a few routes. This paper shows how the point wind statistics as published in the NAVAER 50-1C-526 may be readily converted to route wind statistics.

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Harold L. Crutcher

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Harold L. Crutcher

Abstract

For univariate or one-dimensional distributions the standard tables used for the Kolmogorov-Smirnov test are valid when testing whether a set of observations are from a completely specified distribution. If one or more parameters must be estimated from the sample, then the standard tables are no longer valid. Other tables must be used.

Lilliefors indicates the conservative extent of the tests when parameters are estimated for the sample and used with the standard tables. He provides valid tables for the univariate normal, exponential, gamma, and extreme value distributions when one or more parameters must be estimated from the sample. For the multivariate normal distributions the only tables known to the author are those of Malkovich and Afifi.

This note brings the problem to the attention of scientists who rely on the tools developed by statisticians to help them to use the appropriate tools correctly.

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Harold L. Crutcher

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Wind-data presentations are varied. Some serve one purpose better than another. A relatively new type of wind rose is presented in order that its potential may be exploited. This is the standard vector-deviation wind rose. It is an easily reproduced rose and may be constructed from values of the vector mean wind, standard deviations of the latitudinal and meridional wind components, and the correlation between the components. If the distribution is circular, rather than elliptical, only the vector mean and the standard vector deviation are needed. Methods to test for ellipticity and to construct the roses are given.

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Hans Panofsky and Harold L. Crutcher

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Harold L. Crutcher and Ledoph Baer

Abstract

A knowledge of winds and their distribution may be used to our advantage, for winds play an important part in all phases of our life. Winds are three dimensional but for most purposes they are considered only in their two dimensional from along horizontal surfaces.

Although exceptional are noted, winds in the free atmosphere may be considered to be homogeneous and normally distributed. They may be treated statistically. This requires the evaluation of the integral of the elliptical bivariate normal distribution over offset circles or polygons.

Various other problems in the fields of physical science also require the same treatment. Continued effort on the part of many individuals and organizations has resulted in the preparation of tabular material which facilitates the required evaluation of the integral.

The estimation of the frequency of winds from any given point, sector or area is now possible with the use of estimates of the statistical parameters of a wind distribution. The procedures discussed are useful particularly in regions for which original wind data are not available but where estimates of statistical parameters can he obtained.

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Harold L. Crutcher and Raymond L. Joiner

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In any study, the collection, processing and storage of data are important. Whether the data are clean, biased or contaminated is also important. Pollution or adulteration of data confuse the investigator.

Data do not necessarily fall into neatly packaged boxes or groups. Usually the data sets are mixtures of several types of phenomena. Some of these are basically deterministic in nature while others are not.

This paper illustrates the use of a clustering technique to separate mixed data sets into subsets which exhibit group characteristics. The investigator then assesses the relative importance of the subsets, the nature of the subsets, and perhaps makes an assumption as to whether a particular subset is biased, contaminated or adulterated, i.e., an assessment of the quality of the data may be made.

The technique is applicable to any data set which is multivariate normal. Here it is applied to the climatological set composed of the winds, temperatures and heights at the Canton Island 30 mb level with particular application to the quasi-biennial oscillation of the tropical equatorial stratosphere.

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Lee W. Falls and Harold L. Crutcher

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Transformation of statistics from a dimensional set to another dimensional set involves linear functions of the original set of statistics. Similarly, linear functions will transform statistics within a dimensional set such that the new statistics are relevant to a new set of coordinate axes.

A restricted case of the latter is the rotation of axes in a coordinate system involving any two correlated random variables. A special case is the transformation for horizontal wind distributions. Wind statistics are usually provided in terms of wind speed and direction (measured clockwise from north) or in east-west and north-south components. A direct application of this technique allows the determination of appropriate wind statistics parallel and normal to any preselected flight path of a space vehicle.

Among the constraints for launching space vehicles are critical values selected from the distribution of the expected winds parallel to and normal to the flight path. These procedures are applied to space vehicle launches at Cape Kennedy, Florida.

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William H. Haggard, Thaddeus H. Bilton, and Harold L. Crutcher

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The results of a statistical study of maximum measured rainfall amounts of tropical cyclone systems whose paths cross the Appalachian Mountains an presented. The study includes storms from 1900 through 1969 and considers primarily the maximum recorded rainfall in mountainous terrain. The criteria are that the tropical cyclone had to pass the 1000-ft contour and that the precipitation was measured inside the outer-most 1000-ft contour. In some cases the maxima were recorded at altitudes below the 1000-ft contour; these locations were in valleys surrounded by higher elevations. Supplemental information is presented for maximum rainfall anywhere after storm landfall, although passage of the system over the Appalachian region remained a criterion.

The data are modelled by the gamma distribution. Probabilities of exceeding a specified rainfall amount and rainfall amounts for specified probability levels are presented in tabular and graphical form.

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Harold L. Crutcher, Charles J. Neumann, and Joseph M. Pelissier

Abstract

This study focuses on the use of the bivariate normal distribution model to describe spatial distributions of tropical cyclone forecast errors. In this connection, it is found that forecast errors from the entire Atlantic tropical cyclone basin (Gulf of Mexico, Caribbean and North Atlantic) are multimodal and the fitting of these collective data to the usual unimodal bivariate normal distribution will be judged invalid by the usual statistical goodness-of-fit tests. While this is a recognized pitfall in classical statistics, it is often overlooked in meteorological application. The isolation of the clusters (components) and their statistical characteristics permit the issuance of forecast positions accompanied by more representative error ellipses.

The study continues with a bivariate clustering analysis of a set of 979 tropical cyclone 24 h forecast errors for the Gulf of Mexico, Caribbean and North Atlantic. These errors were collected from the entire tropical cyclone basin without regard to season or geography. The analysis shows that these errors could be drawn from two or possibly three parent bivariate normal distributions. A further analysis of the two clusters was made and it is shown that group membership is essentially a function of forecast “difficulty.” One group (essentially storms located in the Caribbean and Gulf of Mexico) has about one-half the component standard errors of the other group (the more northerly storms). A physical interpretation of the more complex three-mode clustering was not accomplished.

The study has application with regard to the future development of statistical prediction models and in connection with a recently inaugurated tropical cyclone “strike” probability concept.

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