Localized spectral analysis on the sphere

Abstract

S U M M A R YIt is often advantageous to investigate the relationship between two geophysical data sets in the spectral domain by calculating admittance and coherence functions. While there exist powerful Cartesian windowing techniques to estimate spatially localized (cross-)spectral properties, the inherent sphericity of planetary bodies sometimes necessitates an approach based in spherical coordinates. Direct localized spectral estimates on the sphere can be obtained by tapering, or multiplying the data by a suitable windowing function, and expanding the resultant field in spherical harmonics. The localization of a window in space and its spectral bandlimitation jointly determine the quality of the spatiospectral estimation. Two kinds of axisymmetric windows are here constructed that are ideally suited to this purpose: bandlimited functions that maximize their spatial energy within a cap of angular radius θ 0 , and spacelimited functions that maximize their spectral power within a spherical harmonic bandwidth L. Both concentration criteria yield an eigenvalue problem that is solved by an orthogonal family of data tapers, and the properties of these windows depend almost entirely upon the space-bandwidth product N 0 = (L + 1) θ 0 /π . The first N 0 − 1 windows are near perfectly concentrated, and the bestconcentrated window approaches a lower bound imposed by a spherical uncertainty principle. In order to make robust localized estimates of the admittance and coherence spectra between two fields on the sphere, we propose a method analogous to Cartesian multitaper spectral analysis that uses our optimally concentrated data tapers. We show that the expectation of localized (cross-)power spectra calculated using our data tapers is nearly unbiased for stochastic processes when the input spectrum is white and when averages are made over all possible realizations of the random variables. In physical situations, only one realization of such a process will be available, but in this case, a weighted average of the spectra obtained using multiple data tapers well approximates the expected spectrum. While developed primarily to solve problems in planetary science, our method has applications in all areas of science that investigate spatiospectral relationships between data fields defined on a sphere.

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