Some New Mathematical Methods for Variational Objective Analysis Using Splines and Cross Validation

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Abstract
Let Φ(x,y,p,t) be a meteorological field of interest, say, height, temperature, a component of the wind field, etc. We suppose that data concerning the field of the form ΦI = LiΦ + ϵi are where each Li is an arbitrary continuous linear functional and ϵi is a measurement error. The data Φi may be the result of theory, direct measurements, remote soundings or a combination of these. We develop a new mathematical formalism exploiting the method of Generalized Cross Validation (GCV), and some recently developed optimization results, for analyzing this data. The analyzed field ΦN,m,λ is the solution to the minimization problem: Find Φ in a suitable space of functions to minimize where Functions of d=1, 2 or 3 of the four variables x, y, p, t are also considered. The approach can he used to analyze temperature fields from radiosonde-measured temperatures and satellite radiance measurements simultaneously, to incorporate the geostrophic wind approximation and other information. In a test of the method (... Abstract Let Φ(x,y,p,t) be a meteorological field of interest, say, height, temperature, a component of the wind field, etc. We suppose that data concerning the field of the form ΦI = LiΦ + ϵi are where each Li is an arbitrary continuous linear functional and ϵi is a measurement error. The data Φi may be the result of theory, direct measurements, remote soundings or a combination of these. We develop a new mathematical formalism exploiting the method of Generalized Cross Validation (GCV), and some recently developed optimization results, for analyzing this data. The analyzed field ΦN,m,λ is the solution to the minimization problem: Find Φ in a suitable space of functions to minimize where Functions of d=1, 2 or 3 of the four variables x, y, p, t are also considered. The approach can he used to analyze temperature fields from radiosonde-measured temperatures and satellite radiance measurements simultaneously, to incorporate the geostrophic wind approximation and other information. In a test of the method (...

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