Analytic Approximation of Discrete Field Samples with Weighted Sums and the Gridless Computation of Field Derivatives

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  • 1 National Oceanic and Atmospheric Administration, Weather Research Program, Boulder, CO 80303
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Abstract

Objective analysis by weighted sums of discrete observations is equivalent to the approximation of the distribution of an observed parameter by a function which is also analytic, provided that the weighting function is both analytic and positive definite. When objective analysis is expressed as such an analytic approximation, then at any desired point of the analysis domain the approximating function and derivatives may be evaluated entirely as weighted sums of the observations and their corresponding coordinates. The discussion focuses on the Gaussian weighting scheme which is not only equivalent to an analytic approximation but which also yields expressions for derivatives that are formally very simple. This discussion includes the effects of recursive applications of the analytic approximation to arrive at a succession of analytic functions that progressively better approximate the observations. The recursive scheme itself is found to be expressible in a simple equation that effectively yields a multiple-pass result in one matrix computation. One and two dimensional examples of thisscheme and its recursive application are presented. The results show that the fidelity of the analytic approximations increases with 1) the number of passes used with 2) the increase in density of discrete samples and 3) the increase in the area sampled relative to that of the analysis domain.

Abstract

Objective analysis by weighted sums of discrete observations is equivalent to the approximation of the distribution of an observed parameter by a function which is also analytic, provided that the weighting function is both analytic and positive definite. When objective analysis is expressed as such an analytic approximation, then at any desired point of the analysis domain the approximating function and derivatives may be evaluated entirely as weighted sums of the observations and their corresponding coordinates. The discussion focuses on the Gaussian weighting scheme which is not only equivalent to an analytic approximation but which also yields expressions for derivatives that are formally very simple. This discussion includes the effects of recursive applications of the analytic approximation to arrive at a succession of analytic functions that progressively better approximate the observations. The recursive scheme itself is found to be expressible in a simple equation that effectively yields a multiple-pass result in one matrix computation. One and two dimensional examples of thisscheme and its recursive application are presented. The results show that the fidelity of the analytic approximations increases with 1) the number of passes used with 2) the increase in density of discrete samples and 3) the increase in the area sampled relative to that of the analysis domain.

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