An automated method for recovering piecewise smooth functions on spheres free from Gibbs oscillations

Spectral methods using spherical harmonic coefficients are widely used in applications for geophysics and atmospheric sciences. The major draw-back in using spherical harmonic spectral methods occurs when the under-lying function is piecewise smooth. In this case, the well-known Gibbs phenomenon reduces the order of accuracy to first order and produces spurious oscillations, particularly in regions near the discontinuities. The frequently studied Gegenbauer reconstruction method has been shown to alleviate the effects of the Gibbs phenomenon while restoring the exponential accuracy of the spectral approximation. Since each reconstruction must be implemented only within smooth regions, the jump discontinuities of the piece-wise smooth function must first be located by an edge detection method. This study combines the recent developments in both edge detection and Gegenbauer reconstruction methods to construct an automated procedure for recovering horizontal or two-dimensional geophysical global fields free from Gibbs oscillations and without compromising the high order convergence properties of spectral methods. Numerical efliciency and robustness are discussed.