Compared with the traditional spherical harmonics, the spherical needlets are a new generation of spherical wavelets that possess several attractive properties. Their double localization in both spatial and frequency domains empowers them to easily and sparsely represent functions with small spatial scale features. This paper is divided into two parts. First, it reviews the spherical harmonics and discusses their limitations in representing functions with small spatial scale features. To overcome the limitations, it introduces the spherical needlets and their attractive properties. In the second part of the paper, a Matlab package for the spherical needlets is presented. The properties of the spherical needlets are demonstrated by several examples using the package.

Gaussian random fields have been one of the most popular tools for analyzing spatial data. However, many geophysical and environmental processes often display non-Gaussian characteristics. In this paper, we propose a new class of spatial models for non-Gaussian random fields on a sphere based on a multi-resolution analysis. Using a special wavelet frame, named spherical needlets, as building blocks, the proposed model is constructed in the form of a sparse random effects model. The spatial localization of needlets, together with carefully chosen random coefficients, ensure the model to be non-Gaussian and isotropic. The model can also be expanded to include a spatially varying variance profile. The special formulation of the model enables us to develop efficient estimation and prediction procedures, in which an adaptive MCMC algorithm is used. We investigate the accuracy of parameter estimation of the proposed model, and compare its predictive performance with that of two Gaussian models by extensive numerical experiments. Practical utility of the proposed model is demonstrated through an application of the methodology to a data set of high-latitude ionospheric electrostatic potentials, generated from the LFM-MIX model of the magnetosphere-ionosphere system.

The most dynamic electromagnetic coupling between the magnetosphere and ionosphere occurs in the polar upper atmosphere. It is critical to quantify the electromagnetic energy and momentum input associated with this coupling as its impacts on the ionosphere and thermosphere system are global and major, often leading to considerable disturbances in near-Earth space environments. The current general circulation models of the upper atmosphere exhibit systematic biases that can be attributed to an inadequate representation of the Joule heating rate resulting from unaccounted stochastic fluctuations of electric fields associated with the magnetosphere-ionosphere coupling. These biases exist regardless of geomagnetic activity levels. To overcome this limitation, a new multiresolution random field modeling approach is developed, and the efficacy of the approach is demonstrated using Super Dual Auroral Radar Network (SuperDARN) data carefully curated for the study during a largely quiet 4-hour period on February 29, 2012. Regional small-scale electrostatic fields sampled at different resolutions from a probabilistic distribution of electric field variability conditioned on actual SuperDARN LOS observations exhibit considerably more localized fine-scale features in comparison to global large-scale fields modeled using the SuperDARN Assimilative Mapping procedure. The overall hemispherically integrated Joule heating rate is increased by a factor of about 1.5 due to the effect of random regional small-scale electric fields, which is close to the lower end of arbitrarily adjusted Joule heating multiplicative factor of 1.5 and 2.5 typically used in upper atmosphere general circulation models. The study represents an important step toward a data-driven ensemble modeling of magnetosphere-ionosphere-atmosphere coupling processes.