Special Functions in Mathematical Geosciences: An Attempt at a Categorization

Freeden, Willi; Schreiner, Michael

doi:10.1007/978-3-642-01546-5_31

Cited by 8 publications

(2 citation statements)

References 19 publications

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“…The variation of the symbolsˆĴ .n/ for different scales J represents the trade-off between spatial and frequency localization (see illustration below). For a more precise quantitative categorization in terms of the uncertainty principle, the reader is referred to Freeden (1998) and the contribution Freeden and Schreiner (2010a) …”

Section: Wavelets As Frequency Packagesmentioning

confidence: 99%

Multiscale Modeling of the Geomagnetic Field and Ionospheric Currents

Gerhards¹

2014

Handbook of Geomathematics

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This chapter gives a brief overview on the application of multiscale techniques to the modeling of geomagnetic problems. Two approaches are presented: one focusing on the construction of scaling and wavelet kernels in frequency domain and the other one focusing on a spatially oriented construction resulting in locally supported wavelets. Both approaches are applied exemplarily to the modeling of the crustal field, the reconstruction of radial current systems, and the definition of a multiscale power spectrum.

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Section: Wavelets As Frequency Packagesmentioning

confidence: 99%

Multiscale Modeling of the Geomagnetic Field and Ionospheric Currents

Gerhards¹

2014

Handbook of Geomathematics

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“…geographical, domain, where they are either optimally concentrated or within which they are exactly limited, and which at the same time are exactly confined within a certain bandwidth, or maximally concentrated therein. The measure of concentration is invariably a quadratic energy ratio, which, though only one choice out of many (Donoho & Stark 1989;Freeden & Windheuser 1997;Riedel & Sidorenko 1995;Freeden & Schreiner 2010;Michel 2010) is perfectly suited to the nature of the problems we are attempting to address. These are, for example: How do we make estimates of signals that are noisily and incompletely observed?…”

Section: Introductionmentioning

confidence: 99%

Scalar and Vector Slepian Functions, Spherical Signal Estimation and Spectral Analysis

Simons¹,

Plattner²

2015

Handbook of Geomathematics

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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.

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Comparison of Different Smoothing Methods for Initial Data of the DSN-PC Sensor Network

Vas

Tóth

2020

Communications in Computer and Information Science

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Special Functions in Mathematical Geosciences: An Attempt at a Categorization

Cited by 8 publications

References 19 publications

Multiscale Modeling of the Geomagnetic Field and Ionospheric Currents

Multiscale Modeling of the Geomagnetic Field and Ionospheric Currents

Scalar and Vector Slepian Functions, Spherical Signal Estimation and Spectral Analysis

Comparison of Different Smoothing Methods for Initial Data of the DSN-PC Sensor Network

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