Polynomial families of isotropic windows and filters for geophysical signal analysis on the sphere

The study of global-scale geophysical signals 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., for localized spectral analysis and to improve the detection of periodic constituents). Normalized window functions are also utilized as averaging filters.In this work, we only focus on polynomial window functions. We present some families of polynomial windows that have been used in conventional signal processing, such as the B-spline, Singla-Singh, Kulkarni-type and generalized adaptive polynomial windows. We also demonstrate the possibility of approximating more sophisticated non-polynomial windows, such as the Kaiser, Lanczos and hyperbolic cosine windows, using their Taylor series expansion. 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 domain.Recent advances on the evaluation of spherical harmonic coefficients of polynomial functions also enable us to assess the spectral characteristics of all window functions and filter kernels examined. We compare their main spectral characteristics, such as the main lobe width, first side lobe level and side lobe decay rate. Since all of these windows and filters have not been examined on the sphere before, the present work extends the current methods for localizing and filtering geophysical signals on the sphere.