Abstract

We investigate spatial prediction on the sphere in the presence of measurement errors, where data may be irregularly located. We first develop a special class of basis functions in the thin-plate-spline (TPS) function space on the sphere. These basis functions are ordered according to their level of smoothness from large-scale features to small-scale details, providing a multi-resolution representation and an orthogonal transformation of the data. Theoretically, we show that the number of basis functions selected by conditional Akaike information criterion is small, and the resulting reduced-rank estimate achieves a good convergence rate to the target function. In addition, we develop a multi-resolution mixed-effects spatial model on the sphere, by including a Gaussian spatial process to capture fine-scale information. Since large-scale features are captured by leading basis functions, the small-scale spatial process tends to have a short spatial dependence range, leading to a universal kriging estimate that allows rapid computations. A simulation experiment is performed, and an application to global sea-surface-temperature data observed from a satellite is given to demonstrate the effectiveness of the proposed method.