On the Evaluation of Boundary Errors in the Barnes Objective Analysis Scheme

Patricia M. Pauley Department of Meteorology, University of Wisconsin-Madison, Madison, Wisconsin

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Abstract

The spectral response of the Barnes objective analysis scheme near data boundaries is the focus of this note. First of all, a modification of the results presented by Achtemeier is described. In order for the weighted sum (or integral) defining the Barnes scheme to provide an unbiased estimate of the field at grid points, the sum (or integral) of the normalized weights must equal one. The normalizing factor is therefore written as an integral whose limits of integration are kept identical to those for the weighted integral of observations defining the scheme, even as the integral is truncated near a boundary. This modification serves to phrase the theoretical form of the Barnes scheme in a manner that is more consistent with the commonly used discrete form of the scheme. The amplitude and phase-shifted responses using the proposed normalization at an interpolation point on a boundary differ from Achtemeier's results by a factor of two.

The amplitude and phase-shifted responses for a discrete application of the scheme are also examined using the Barnes scheme cast in rectangular coordinates. The amplitude and phase-shifted responses are integrated both using a small sampling interval to approximate the continuous case and using larger sampling intervals representative of typical observation spacings. These discrete results show that the phase shift near boundaries can be reduced by using a larger nondimensional sampling interval (or equivalently, a smaller smoothing scale length). However, this is at the expense of increasing the amplitude response of aliased unresolvable wavelengths. An estimate of the response at the boundary made from gridded values at the boundary confirms the discrete estimate of the response and the proposed modification of Achtemeier's results.

Abstract

The spectral response of the Barnes objective analysis scheme near data boundaries is the focus of this note. First of all, a modification of the results presented by Achtemeier is described. In order for the weighted sum (or integral) defining the Barnes scheme to provide an unbiased estimate of the field at grid points, the sum (or integral) of the normalized weights must equal one. The normalizing factor is therefore written as an integral whose limits of integration are kept identical to those for the weighted integral of observations defining the scheme, even as the integral is truncated near a boundary. This modification serves to phrase the theoretical form of the Barnes scheme in a manner that is more consistent with the commonly used discrete form of the scheme. The amplitude and phase-shifted responses using the proposed normalization at an interpolation point on a boundary differ from Achtemeier's results by a factor of two.

The amplitude and phase-shifted responses for a discrete application of the scheme are also examined using the Barnes scheme cast in rectangular coordinates. The amplitude and phase-shifted responses are integrated both using a small sampling interval to approximate the continuous case and using larger sampling intervals representative of typical observation spacings. These discrete results show that the phase shift near boundaries can be reduced by using a larger nondimensional sampling interval (or equivalently, a smaller smoothing scale length). However, this is at the expense of increasing the amplitude response of aliased unresolvable wavelengths. An estimate of the response at the boundary made from gridded values at the boundary confirms the discrete estimate of the response and the proposed modification of Achtemeier's results.

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