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## Abstract

This paper examines the response of the Barnes objective analysis scheme as a function of wavenumber or wavelength and extends previous work in two primary areas. First, the first- and second-pass theoretical response functions for continuous two-dimensional (2-D) fields are derived using Fourier transforms and compared with Barnes' (1973) responses for one-dimensional (1-D) waves. All responses are nondimensionalized with respect to a smoothing scale length, such that the first-pass responses are a function only of nondimensional wavelength. The 2-D response is of the same functional form as the 1-D response, with the 2-D wavenumber substituted for the 1-D wavenumber. The 2-D response departs significantly from the 1-D value (for the same *x*-component of the wavelength) when the *y*-component of the wavelength is less than approximately ten scale lengths, a condition applying to most fields with closed centers as well as open waves with a significant latitudinal variation.

Second, the continuous theoretical response for 1-D and 2-D waves is compared with the response for discrete applications of the scheme using uniformly spaced observations. This response is evaluated two different ways. The discrete theoretical response is found by discretizing the Barnes scheme with a 2-D “comb” function for cases in which interpolation points are either coincident with or midway between observation points. The response can then be evaluated with Fourier transforms in much the same way as for the continuous case. The discrete response evaluated in this manner attains a minimum at the Nyquist wavelength, with enhanced values for smaller unresolvable wavelengths resulting from aliasing. The discrete response is approximately equal to the sum of the responses for the original wavelength and for a primary aliased wavelength. At larger resolvable wavelengths, the discrete response approaches the continuous value as aliasing becomes negligible.

The second means of examining the response for discrete applications is through a Fourier series analysis of fields interpolated by the Barnes scheme. The “observations” in this context are given by analytic functions on a uniform mesh which may or may not differ from the analysis grid. A given input wavelength leads to an analysis which contains waves at one or more aliased wavelengths in addition to the original wavelength. Components of the response corresponding to each of these wavelengths can be estimated as the Fourier amplitude of the interpolated field divided by the amplitude of the input wave. The actual response from a discrete application of the Barnes scheme confirms the results of the analysis of the discrete theoretical response; aliasing to longer wavelengths is seen for nonresolvable wavelengths in the “observations,” while the actual response is close to the theoretical value for well resolved wavelengths for both 1-D and 2-D fields.

This analysis of the discrete and actual response supports the recommendations of Caracena et al. and Koch et al. for the relationship between the smoothing scale length and the observation spacing. Setting the smoothing scale length to approximately four-thirds of the observation spacing, the upper bound recommended by Caracena et al., yields a response close to the continuous value for resolvable wavelengths, with a reasonably small degree of aliasing. However, the case in which the smoothing scale length is equal to the observation spacing, the lower bound recommended by Caracena et al., retains an unacceptably high degree of aliasing in a typical two-pass application of the Barnes scheme.

## Abstract

This paper examines the response of the Barnes objective analysis scheme as a function of wavenumber or wavelength and extends previous work in two primary areas. First, the first- and second-pass theoretical response functions for continuous two-dimensional (2-D) fields are derived using Fourier transforms and compared with Barnes' (1973) responses for one-dimensional (1-D) waves. All responses are nondimensionalized with respect to a smoothing scale length, such that the first-pass responses are a function only of nondimensional wavelength. The 2-D response is of the same functional form as the 1-D response, with the 2-D wavenumber substituted for the 1-D wavenumber. The 2-D response departs significantly from the 1-D value (for the same *x*-component of the wavelength) when the *y*-component of the wavelength is less than approximately ten scale lengths, a condition applying to most fields with closed centers as well as open waves with a significant latitudinal variation.

Second, the continuous theoretical response for 1-D and 2-D waves is compared with the response for discrete applications of the scheme using uniformly spaced observations. This response is evaluated two different ways. The discrete theoretical response is found by discretizing the Barnes scheme with a 2-D “comb” function for cases in which interpolation points are either coincident with or midway between observation points. The response can then be evaluated with Fourier transforms in much the same way as for the continuous case. The discrete response evaluated in this manner attains a minimum at the Nyquist wavelength, with enhanced values for smaller unresolvable wavelengths resulting from aliasing. The discrete response is approximately equal to the sum of the responses for the original wavelength and for a primary aliased wavelength. At larger resolvable wavelengths, the discrete response approaches the continuous value as aliasing becomes negligible.

The second means of examining the response for discrete applications is through a Fourier series analysis of fields interpolated by the Barnes scheme. The “observations” in this context are given by analytic functions on a uniform mesh which may or may not differ from the analysis grid. A given input wavelength leads to an analysis which contains waves at one or more aliased wavelengths in addition to the original wavelength. Components of the response corresponding to each of these wavelengths can be estimated as the Fourier amplitude of the interpolated field divided by the amplitude of the input wave. The actual response from a discrete application of the Barnes scheme confirms the results of the analysis of the discrete theoretical response; aliasing to longer wavelengths is seen for nonresolvable wavelengths in the “observations,” while the actual response is close to the theoretical value for well resolved wavelengths for both 1-D and 2-D fields.

This analysis of the discrete and actual response supports the recommendations of Caracena et al. and Koch et al. for the relationship between the smoothing scale length and the observation spacing. Setting the smoothing scale length to approximately four-thirds of the observation spacing, the upper bound recommended by Caracena et al., yields a response close to the continuous value for resolvable wavelengths, with a reasonably small degree of aliasing. However, the case in which the smoothing scale length is equal to the observation spacing, the lower bound recommended by Caracena et al., retains an unacceptably high degree of aliasing in a typical two-pass application of the Barnes scheme.