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Isidore Halberstam
Shu-Lin Tung


Objective analysis can he performed on irregularly spaced observation points by fitting selected functions to the observations. A simple least-squares fit is found to be both impractical and ill-conditioned, the latter because of gaps in the data exceeding the smallest wavelengths of the fitting functions. Alternative fits can be made practical either by breaking down the least-squares fit into a step-wise fitting of several subsets or by imposing a “finality” condition that yields errors greater than least-squares, but are nevertheless bounded by a number related to the distribution of the data points. The method of finality involves sequential subtraction of each vector's successive contribution. Ordering of the vectors efficiently then becomes rather crucial. The ill-conditioned behavior of thew fits can be suppressed by adding first-guess information in areas that lack data. Tests of the methods using Hough functions evaluated at observation sites as the basis vectors and data from FGGE Ila and Illa as well as from a forecast field from the Air Force Geophysics Laboratory's global, spectral model show that the finality method is a viable alternative analysis procedure worth exploring. The step-wise least-square method is also a practical method if sufficient realistic data are distributed throughout the domain. Using residuals (forecast minus observation values) proves to be a more accurate procedure than using observed and forecast values themselves.

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