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D. S. Crosby, H. E. Fleming, and D. Q. Wark

Abstract

The statistical minimum-rms inversion method used to obtain temperature profiles, requires estimates of covariance matrices and means for Planck function profiles of the atmosphere. In order to obtain these estimates over the pressure range of IWO to 0.01 mb, it was necessary to combine data from temperature measurements by radiosondes, rocketsondes and grenadesondes. Radiosonde data reaching the 10-mb level were extended to higher levels by means of a modified regression technique. Matrices and means have been obtained by this method for seasonal and geographical groupings in the Northern Hemisphere and the tropics. Details of the geographical and time changes in the matrices and the means are presented.

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Henry E. Fleming, David S. Crosby, and Mitchell D. Goldberg

Abstract

Layer-mean virtual temperatures retrieved from satellite measurements are more accurate than retrievals at specific pressure not only because an averaging process is involved, but also because of advantages in the retrieval process. In this note, a “retrieval efficiency” is derived to express this advantage over simple averaging as a function of layer thickness. The efficiency is examined for two common cases of retrieval initial guess: a statistical sample mean and a forecast profile obtained from a numerical prediction model. The advantage of the layer-mean retrieval clearly is demonstrated in both cases.

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L. J. Crone, L. M. Mcmillin, and D. S. Crosby

Abstract

Least squares or regression techniques have been used for many problems in satellite meteorology. Because of the large number of variables and the linear dependence among these variables, colinearity causes significant problems in the application of standard regression techniques. In some of the applications there is prior knowledge about the values of the regression parameters. Since there are errors in the predictor variables as well as the predictand variables, the standard assumptions for ordinary least squares are not valid. In this paper the authors examine several techniques that have been developed to ameliorate the effects of colinearity or to make use of prior information. These include ridge regression, shrinkage estimators, rotated regression, and orthogonal regression. In order to illustrate the techniques and their properties, the authors apply them to two simple examples. These techniques are then applied to a real problem in satellite meteorology: that of estimating theoretical computed brightness temperatures from measured brightness temperatures. It is found that the rotated and the shrinkage estimators make good use of the prior information and help solve the colinearity problem. Ordinary least squares, ridge regression, and orthogonal regression give unsatisfactory results. Theoretical results for the various techniques are given in an appendix.

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D. S. Crosby, L. C. Breaker, and W. H. Gemmill

Abstract

A universally accepted definition for vector correlation in oceanography and meteorology does not presently exist. To address this need, a generalized correlation coefficient, originally proposed by Hooper and later expanded upon by Jupp and Mardia, is explored. A short history of previous definitions is presented. Then the definition originally proposed by Hooper is presented together with supporting theory and associated properties. The most significant properties of this vector correlation coefficient are that it is a generalization of the square of the simple one-dimensional correlation coefficient, and when the vectors are independent, its asymptotic distribution is known; hence, it can be used for hypothesis testing. Because the asymptotic results hold only for large samples, and in practical situations only small samples are often available, modified sampling distributions are derived using simulation techniques for samples as small as eight. It is symmetric with respect to its arguments and has a simple interpretation in terms of canonical correlation. It is invariant under transformations of the coordinate axes, including rotations and changes of scale.

Finally, to assist in interpreting this vector correlation coefficient, several cases that lead to perfect correlation and zero correlation are examined, and the technique is applied to surface marine winds at two locations in the northwest Atlantic.

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L. C. Breaker, W. H. Gemmill, and D. S. Crosby

Abstract

In a recent study, Crosby et al. proposed a definition for vector correlation that has not been commonly used in meteorology or oceanography. This definition has both a firm theoretical basis and a rather complete set of desirable statistical properties. In this study, the authors apply the definition to practical problems arising in meteorology and oceanography. In the first of two case studies, vector correlations were calculated between subsurface currents for five locations along the southeastern shore of Lake Erie. Vector correlations for one sample size were calculated for all current meter combinations, first including the seiche frequency and then with the seiche frequency removed. Removal of the seiche frequency, which was easily detected in the current spectra, had only a small effect on the vector correlations. Under reasonable assumptions, the vector correlations were in most cases statistically significant and revealed considerable fine structure in the vector correlation sequences. In some cases, major variations in vector correlation coincided with changes in surface wind. The vector correlations for the various current meter combinations decreased rapidly with increasing spatial separation. For one current meter combination, canonical correlations were also calculated; the first canonical correlation tended to retain the underlying trend, whereas the second canonical correlation retained the peaks in the vector correlations.

In the second case study, vector correlations were calculated between marine surface winds derived from the National Meteorological Center's Global Data Assimilation System and observed winds acquired from the network of National Data Buoy Center buoys that are located off the continental United States and in the Gulf of Alaska. Results of this comparison indicated that 1) there was a significant decrease in correlation between the predicted and observed winds with increasing forecast interval out to 72 h, 2) the technique provides a sensitive indicator for detecting bad buoy reports, and 3) there was no obvious seasonal cycle in the monthly vector correlations for the period of observation.

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Larry M. McMillin, David S. Crosby, and Mitchell D. Goldberg

Abstract

A method for deriving a water vapor index is presented. An important feature of the index is the fact that it does not rely on radiosondes. Thus, it is not influenced by problems associated with radiosondes and the extent to which the horizontal variability of moisture invalidates the extrapolations from radiosonde measurements to satellite measurements. The index is derived by using channels that are insensitive to changes in moisture to predict a brightness temperature for one of the moisture channels and then by subtracting this predicted value from the observation. The predicted value represents the moisture value expected for the given temperature profile, and the difference between the predicted and measured values is the index. The subtraction removes the variability due to changes in atmospheric temperature from the moisture signal. This separation greatly enhances the ability to monitor atmospheric moisture patterns, especially near the ground and at high latitudes where some alternative methods have difficulties. The ability of the indices to display moisture patterns at all levels and latitudes is demonstrated.

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Mitchell D. Goldberg, David S. Crosby, and Lihang Zhou

Abstract

The Advanced Microwave Sounding Unit-A (AMSU-A) is the first of a new generation of polar-orbiting cross-track microwave sounders operated by the National Oceanic and Atmospheric Administration. A feature of a cross-track sounder is that the measurements vary with scan angle because of the change in the optical pathlength between the earth and the satellite. This feature is called the limb effect and can be as much as 30 K. One approach to this problem is to limb adjust the measurements to a fixed view angle. This approach was used for the older series of Microwave Sounding Units. Limb adjusting is important for climate applications and regression retrieval algorithms. This paper describes and evaluates several limb adjustment procedures. The recommended procedure uses a combined physical and statistical technique. The limb adjusted measurements were compared with computed radiances from radiosondes and National Centers for Environmental Prediction models. The model error was found to be less than the instrument noise for most of the temperature sounding channels. The error in the window channels was small relative to the observed range of these channels. Limb adjusted fields appear to be smooth. Statistical tests of the distributions of the adjusted measurements at each scan angle show them to be very similar.

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