A THEORETICAL INTERPRETATION OF ANISOTROPICALLY WEIGHTED SMOOTHING ON THE BASIS OF NUMERICAL VARIATIONAL ANALYSIS

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  • 1 The University of Oklahoma, Norman, Okla.
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Abstract

The weighting factors used in conventional objective analysis methods are reviewed on the basis of numerical variational analysis. Special emphasis is placed on anisotropy (ellipticity) of the factors. The weighting factors of the objective analysis methods were empirically determined and are two dimensional in a horizontal plane (x, y). Most of these weighting factors are isotropic. However, anisotropic weighting factors have recently been used to give greater weight to the upstream and downstream observations as compared to those of the crosswind direction.

A simple advection equation is used as a dynamical constraint in the numerical variational analysis in order to take into account quantitatively the effect of wind direction and speed on the anisotropy. A simple low-pass filter is also included in the variational formalism. A Green's function, derived for the Eular equation, is used to discuss the theoretical basis of the isotropic and anisotropic weighting factors.

The results obtained from the numerical variational analysis scheme suggest that the weights for the upstream and downstream observations should be of the same magnitude and as much as three times larger than the respective weights for the crosswind direction. These results were obtained by taking time t as a constant and considering a reasonable range of wind speeds. These suggestions seem to support the empirical anisotropic weighting factors proposed by Endlich and Mancuso. Additional discussion concerns weighting along the time coordinate simultaneously with the two space coordinates.

Abstract

The weighting factors used in conventional objective analysis methods are reviewed on the basis of numerical variational analysis. Special emphasis is placed on anisotropy (ellipticity) of the factors. The weighting factors of the objective analysis methods were empirically determined and are two dimensional in a horizontal plane (x, y). Most of these weighting factors are isotropic. However, anisotropic weighting factors have recently been used to give greater weight to the upstream and downstream observations as compared to those of the crosswind direction.

A simple advection equation is used as a dynamical constraint in the numerical variational analysis in order to take into account quantitatively the effect of wind direction and speed on the anisotropy. A simple low-pass filter is also included in the variational formalism. A Green's function, derived for the Eular equation, is used to discuss the theoretical basis of the isotropic and anisotropic weighting factors.

The results obtained from the numerical variational analysis scheme suggest that the weights for the upstream and downstream observations should be of the same magnitude and as much as three times larger than the respective weights for the crosswind direction. These results were obtained by taking time t as a constant and considering a reasonable range of wind speeds. These suggestions seem to support the empirical anisotropic weighting factors proposed by Endlich and Mancuso. Additional discussion concerns weighting along the time coordinate simultaneously with the two space coordinates.

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