Multiscale Representation and Analysis of Spherical Data by Spherical Wavelets

Classic wavelet methods were developed in the Euclidean spaces for multiscale representation and analysis of regularly sampled signals (time series) and images. This paper introduces a method of representing scattered spherical data by multiscale spherical wavelets. The method extends the recent pioneering work of Narcowich and Ward [Appl. Comput. Harmon. Anal., 3 (1996), pp. 324–336] by employing multiscale rather than single-scale spherical basis functions and by introducing a bottom-up procedure for network design and bandwidth selection. Decomposition and reconstruction algorithms are proposed for efficient computation. An analytical investigation confirms the localization property of the resulting spherical wavelets. The proposed method is illustrated by numerical examples. It is also employed to analyze and compress a real-data set consisting of the surface air temperatures observed on a global network of weather stations.