On Interpolation and Evaluation of Derivatives from a Finite Number of Equally-Spaced Data Points

Ian Simmonds Atmospheric Environment Service, Montreal, Canada

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Abstract

Analytic fields, with several spectral variance power laws, are prescribed and evaluated at a finite number of equally-spaced points. For a given accuracy of interpolation, an unaliased truncated Fourier series is found to require less degrees of freedom than both cubic spline and two-point interpolation. With the input truncation chosen here, cubic spline is superior to linear interpolation, except for the roughest field. Very similar results hold for the accuracy of the first derivatives implied by these interpolation schemes.

When the errors in the first derivatives are examined only at the data points, however, the derivative of the aliased series is more accurate than that of the cubic spline. An even more accurate series of the same length can be obtained by analyzing the cubic spline passed through the points. The two finite-difference schemes tested have the largest errors.

Abstract

Analytic fields, with several spectral variance power laws, are prescribed and evaluated at a finite number of equally-spaced points. For a given accuracy of interpolation, an unaliased truncated Fourier series is found to require less degrees of freedom than both cubic spline and two-point interpolation. With the input truncation chosen here, cubic spline is superior to linear interpolation, except for the roughest field. Very similar results hold for the accuracy of the first derivatives implied by these interpolation schemes.

When the errors in the first derivatives are examined only at the data points, however, the derivative of the aliased series is more accurate than that of the cubic spline. An even more accurate series of the same length can be obtained by analyzing the cubic spline passed through the points. The two finite-difference schemes tested have the largest errors.

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