This study explores the use of nonuniform fast spherical Fourier transforms on meteorological data that are arbitrarily distributed on the sphere. The applicability of this methodology in the atmospheric sciences is demonstrated by estimating spectral coefficients for nontrivial subsets of reanalysis data on a uniformly spaced latitude-longitude grid, a global cloud resolving model on an icosahedral mesh with 3-km horizontal grid spacing, and for temperature anomalies from arbitrarily distributed weather stations over the United States. A spectral correction technique is developed that can be used in conjunction with the inverse transform to yield data interpolated onto a uniformly spaced grid, with optional triangular truncation, at reduced computational cost compared to other variance conserving interpolation methods, such as kriging or natural spline interpolation. The spectral correction yields information that can be used to deduce gridded observational biases not directly available from other methods.

Numerous facets of scientific research implicitly or explicitly call for the estimation of probability densities. Histograms and kernel density estimates (KDEs) are two commonly used techniques for estimating such information, with the KDE generally providing a higher fidelity representation of the probability density function (PDF). Both methods require specification of either a bin width or a kernel bandwidth. While techniques exist for choosing the kernel bandwidth optimally and objectively, they are computationally intensive, since they require repeated calculation of the KDE. A solution for objectively and optimally choosing both the kernel shape and width has recently been developed by Bernacchia and Pigolotti (2011). While this solution theoretically applies to multidimensional KDEs, it has not been clear how to practically do so. A method for practically extending the Bernacchia-Pigolotti KDE to multidimensions is introduced. This multidimensional extension is combined with a recently-developed computational improvement to their method that makes it computationally efficient: a 2D KDE on 105 samples only takes 1 s on a modern workstation. This fast and objective KDE method, called the fastKDE method, retains the excellent statistical convergence properties that have been demonstrated for univariate samples. The fastKDE method exhibits statistical accuracy that is comparable to state-of-the-science KDE methods publicly available in R, and it produces kernel density estimates several orders of magnitude faster. The fastKDE method does an excellent job of encoding covariance information for bivariate samples. This property allows for direct calculation of conditional PDFs with fastKDE. It is demonstrated how this capability might be leveraged for detecting non-trivial relationships between quantities in physical systems, such as transitional behavior.