Isotropic windows and filters on the sphere

The study of geophysical data on global scale requires the modification of conventional spectral analysis and signal processing techniques from the real line to the sphere. These techniques often depend on the use of window functions (e.g., to improve the detection of periodic constituents and to suppress ripple effects in the impulse response of digital filters). Normalized window functions are also utilized as averaging filters. In this work, we present a comprehensive list of window functions for the analysis of signals defined on a spherical surface. We focus on classic tapering windows that are systematically used in signal processing applications, such as the Bartlett, Welch, generalized cosine, Bartlett-Hann and Tukey windows. The approach followed for their adaptation to the sphere results in isotropic (i.e., rotationally symmetric) window functions. We also examine their related filter kernels and provide expressions for their representation in the spatial and spectral domains. The spectral representation corresponds to a set of spherical harmonic coefficients, the calculation of which is entirely based on recurrence relations. We utilize these newly-developed relations to evaluate the spectrum of all filter kernels and compare their main characteristics, such as the main lobe width, first side lobe level and side lobe decay rate. Since the majority of these windows and filters have not been examined before, the present work extends significantly the current methods for localizing and filtering geophysical signals on the sphere.
The 28th IUGG General Assembly (IUGG2023) (Berlin 2023)