Spherical wavelets and their statistical analysis with applications to meteorological data

Wavelets defined in the Euclidean spaces are known as a useful tool for representing signals and images of multi-scale components. This research introduces a method of constructing multi-scale spherical wavelets for representation and analysis of spherical fields such as the surface air temperatures observed over the globe. Narcowich and Ward (1996) proposed a construction method of spherical wavelets for scattered data on the unit sphere. However, their spherical wavelets always have the same spatial scale that only depends on the pre-determined bandwidth parameter regardless of the resolution level or the distribution of the station. We call this problem the single-scale problem. We propose a new method with a multiscale structure that can overcome the single-scale problem as a generalization of the work by Narcowich and Ward (1996). We discuss the multi-scale spherical basis representation, multiresolution analysis and define multi-scale spherical wavelets. Decomposition and reconstruction algorithms are also proposed for an efficient computation. Furthermore, we give mathematical results for the localization properties and stability properties of new spherical wavelet functions. Spatial adaptation methods based on multi-scale spherical wavelet coefficients are proposed and the statistical properties of spherical wavelet coefficients are discussed. Finally, multi-scale spherical wavelet representations for global surface air temperature data are described and through the experimental study of the data, we show that spherical wavelets are very powerful for detecting local activities from global trends and for the smoothing of spherical fields. Some important issues in practice such as the design of networks, the choice of spherical basis function, and the estimate of coefficients $b1$ are discussed.