# Minimum-Variance Multitaper Spectral Estimation on the Sphere

@article{Wieczorek2007MinimumVarianceMS, title={Minimum-Variance Multitaper Spectral Estimation on the Sphere}, author={Mark A. Wieczorek and Frederik J. Simons}, journal={Journal of Fourier Analysis and Applications}, year={2007}, volume={13}, pages={665-692} }

We develop a method to estimate the power spectrum of a stochastic process on the sphere from data of limited geographical coverage. Our approach can be interpreted either as estimating the global power spectrum of a stationary process when only a portion of the data are available for analysis, or estimating the power spectrum from local data under the assumption that the data are locally stationary in a specified region. Restricting a global function to a spatial subdomain—whether by necessity…

## 108 Citations

Spectral estimation on a sphere in geophysics and cosmology

- Mathematics, Physics
- 2008

We address the problem of estimating the spherical-harmonic power spectrum Sl of a statistically isotropic scalar signal s(r) from noise-contaminated data d(r) = s(r) + n(r) on a region R of the unit…

CMB power spectrum estimation using wavelets

- Computer Science
- 2008

This work uses needlets (wavelets) on the sphere to construct natural and efficient spectral estimators for partially observed and beamed CMB with nonstationary noise and compares very favorably to the best pseudo-C{sub l} estimators, over the whole multipole range.

Construction of Overcomplete Multiscale Dictionary of Slepian Functions on the Sphere

- Computer ScienceICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
- 2019

The hierarchical equal area iso-latitude iso-longitude pixelization (HEALLPix) scheme for hierarchical partitioning of the sphere into equal area sub-regions called pixels is developed and its quaternary tree structure is presented.

Slepian functions and their use in signal estimation and spectral analysis

- Computer Science
- 2009

A theoretical overview of one particular approach to this "concentration" problem, as originally proposed for time series by Slepian and coworkers, in the 1960s, and how this framework leads to practical algorithms and statistically performant methods for the analysis of signals and their power spectra in one and two dimensions, and on the surface of a sphere.

Scalar and vector Slepian functions, spherical signal estimation and spectral analysis

- Computer Science
- 2013

A theoretical overview of one particular approach to this "concentration" problem, as originally proposed for time series by Slepian and coworkers in the 1960s, and how this framework leads to practical algorithms and statistically performant methods for the analysis of signals and their power spectra in one and two dimensions, and, particularly for applications in the geosciences.

Slepian Spatial-Spectral Concentration Problem on the Sphere: Analytical Formulation for Limited Colatitude–Longitude Spatial Region

- MathematicsIEEE Transactions on Signal Processing
- 2017

This paper exploits the expansion of spherical harmonics in the complex exponential basis to develop an analytical formulation for the Slepian concentration problem for a limited colatitude–longitude spatial region and designs a computationally efficient algorithm for the implementation of the proposed analytical formulation.

Revisiting Slepian concentration problem on the sphere for azimuthally non-symmetric regions

- Mathematics2011 5th International Conference on Signal Processing and Communication Systems (ICSPCS)
- 2011

The approach is different in a sense that the family of spatially concentrated bandlimited mutually orthogonal functions is obtained by maximizing the contribution of spherical harmonics components of all degrees and orders within the spectral bandwidth.

Iterative method to compute the maximal concentration Slepian band-limited eigenfunction on the sphere

- Mathematics2014 8th International Conference on Signal Processing and Communication Systems (ICSPCS)
- 2014

The Slepian concentration problem on the sphere to maximize the energy concentration of a band-limited (in spherical harmonic degree) function is formulated as an eigenvalue problem, the solution of…

Optimal-dimensionality sampling on the sphere: Improvements and variations

- Computer Science2017 International Conference on Sampling Theory and Applications (SampTA)
- 2017

This work has developed a method to place iso-latitude rings of samples with the objective of improving the well-conditioning of the linear systems involved in the computation of the spherical harmonic transform (SHT).

## References

SHOWING 1-10 OF 42 REFERENCES

Spectral estimation on a sphere in geophysics and cosmology

- Mathematics, Physics
- 2008

We address the problem of estimating the spherical-harmonic power spectrum Sl of a statistically isotropic scalar signal s(r) from noise-contaminated data d(r) = s(r) + n(r) on a region R of the unit…

The variance of multitaper spectrum estimates for real Gaussian processes

- MathematicsIEEE Trans. Signal Process.
- 1994

The authors show that near zero or Nyquist frequency this approximation is poor even for white noise and derive the exact expression of the variance in the general case of a stationary real-valued time series.

Multitaper spectral estimation of power law processes

- PhysicsIEEE Trans. Signal Process.
- 1998

It is shown that multitapering, or using sine or Slepian tapers, produces much better results than using the periodogram and is attractive compared with other competing methods when the technique is applied to a geophysical estimation problem.

Localized spectral analysis on the sphere

- Physics
- 2005

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…

Spectrum estimation and harmonic analysis

- MathematicsProceedings of the IEEE
- 1982

In the choice of an estimator for the spectrum of a stationary time series from a finite sample of the process, the problems of bias control and consistency, or "smoothing," are dominant. In this…

Spatiospectral Concentration on a Sphere

- MathematicsSIAM Rev.
- 2006

In the asymptotic limit of a small spatial region and a large spherical harmonic bandwidth, the spherical concentration problem reduces to its planar equivalent, which exhibits self-similarity when the Shannon number is kept invariant.

Gabor transforms on the sphere with applications to CMB power spectrum estimation

- Physics
- 2002

The Fourier transform of a data set apodized with a window function is known as the Gabor transform. In this paper we extend the Gabor transform formalism to the sphere with the intention of applying…

Spherical Slepian functions and the polar gap in geodesy

- Mathematics
- 2005

The estimation of potential fields such as the gravitational or magnetic potential at the surface of a spherical planet from noisy observations taken at an altitude over an incomplete portion of the…

Computing Fourier Transforms and Convolutions on the 2-Sphere

- Computer Science, Mathematics
- 1994

Convolution theorems generalizing well known and useful results from the abelian case are used to develop a sampling theorem on the sphere, which reduces the calculation of Fourier transforms and convolutions of band-limited functions to discrete computations.

Spectral analysis for physical applications : multitaper and conventional univariate techniques

- Mathematics
- 1996

Glossary of symbols 1. Introduction to spectral analysis 2. Stationary stochastic processes 3. Deterministic spectral analysis 4. Foundations for stochastic spectral theory 5. Linear time-invariant…