55 research outputs found

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

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    It is a well-known fact that mathematical functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access to, or are only interested in, a study area that is temporally or spatially bounded. In the geosciences we may be interested in spectrally modeling a time series defined only on a certain interval, or we may want to characterize a specific geographical area observed using an effectively bandlimited measurement device. It is clear that analyzing and representing scientific data of this kind will be facilitated if a basis of functions can be found that are "spatiospectrally" concentrated, i.e. "localized" in both domains at the same time. Here, we give a theoretical overview of one particular approach to this "concentration" problem, as originally proposed for time series by Slepian and coworkers, in the 1960s. We show 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, for scalar and vectorial signals defined on the surface of a unit sphere.Comment: Submitted to the 2nd Edition of the Handbook of Geomathematics, edited by Willi Freeden, Zuhair M. Nashed and Thomas Sonar, and to be published by Springer Verlag. This is a slightly modified but expanded version of the paper arxiv:0909.5368 that appeared in the 1st Edition of the Handbook, when it was called: Slepian functions and their use in signal estimation and spectral analysi

    An accelerating high-latitude jet in Earth's core

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    Observations of the change in Earth's magnetic field, the secular variation, provide information on the motion of liquid metal within the core that is responsible for its generation. The very latest high-resolution observations from ESA's Swarm satellite mission show intense field change at high-latitude localised in a distinctive circular daisy-chain configuration centred on the north geographic pole. Here we explain this feature with a localised, non-axisymmetric, westwards jet of 420 km width on the tangent cylinder, the cylinder of fluid within the core that is aligned with the rotation axis and tangent to the solid inner core. We find that the jet has increased in magnitude by a factor of three over the period 2000--2016 to about 40 km/yr, and is now much stronger than typical large-scale flows inferred for the core. The current accelerating phase may be a part of a longer term fluctuation of the jet causing both eastwards and westwards movement of magnetic features over historical periods, and may contribute to recent changes in torsional wave activity and the rotation direction of the inner core
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