Localized spectral analysis on the sphere

Mark A. Wieczorek

Département de Géophysique Spatiale et Planétaire
Institut de Physique du Globe de Paris
94701 St. Maur-des-Fossés, France

Frederik J Simons

Geosciences Department
Princeton University
Princeton NJ 08544, USA

Abstract

It 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, or concentration, of a window in space and its spectral bandlimitation jointly determine the quality of the spatiospectral estimation. We construct two kinds of axisymmetric windows: bandlimited functions that maximize their spatial energy within a cap of angular radius, and spacelimited functions maximizing their spectral power within a spherical harmonic bandwidth L. Both concentration criteria yield an eigenvalue problem solved by an orthogonal family of data tapers. The properties of our new windows depend almost entirely upon the space-bandwidth product N0=(L+1)θ0/π, with the first N0-1 windows nearly perfectly concentrated. The concentration of the best window approaches a lower bound imposed by a quantum-mechanical uncertainty principle. In order to make robust localized estimates of the admittance and coherence between two fields on the sphere, we propose a method that uses the optimally concentrated data tapers calculated with extreme computational efficiency. We show that the expectation of localized (cross-)power spectra calculated using our data tapers is nearly unbiased when the input spectrum is white and 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 will approach the expected spectrum; the approximation improves with the number of tapers used. 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.

Figures

1. Figure 01 The spectral response of (truncated) boxcar windows.
   
2. Figure 02 Eigenvalue behavior of the optimally concentrated axisymmetric windows or tapers.
   
3. Figure 03 Axisymmetric tapers in the spatial and spectral domains.
   
4. Figure 04 Scaled tapers with identical Shannon number in the spatial domain.
   
5. Figure 05 The spherical harmonic uncertainty product of the first four space-concentrated windows.
   
6. Figure 06 Comparison of the concentration factors and uncertainty products of the optimal tapers with those of truncated boxcars.
   
7. Figure 07 Performance comparison of the tapers with those of truncated boxcars on a geophysical data set.
   
8. Figure 08 Cumulative energy, weighted by the eigenvalue, of the Slepian functions illustrating their even coverage.
   
9. Figure 09 Theoretical single- and multitaper estimates for white and colored processes of white and red spectra.
   
10. Figure 10 Variance and covariance of multitaper spectral estimates.
    
11. Figure 11 Observed single- and multitaper estimates for white and colored processes of white and red spectra.