Biorthogonal Locally Supported Wavelets on the Sphere Based on Zonal Kernel Functions

Freeden, Willi; Schreiner, Michael

doi:10.1007/s00041-006-6905-0

Cited by 14 publications

(7 citation statements)

References 21 publications

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“…Freeden and Schreiner (2007) ;Freeden and Gutting (2013) are interested in a compromise connecting zonal function expressions and structured grids on the sphere to obtain fast algorithms.…”

Section: Resultsmentioning

confidence: 99%

Special Functions in Mathematical Geosciences: An Attemptat a Categorization

Freeden

Schreiner

2013

Handbook of Geomathematics

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This chapter reports on the current activities and recent progress in the field of special functions of mathematical geosciences. The chapter focuses on two major topics of interest, namely, trial systems of polynomial (i.e., spherical harmonics) and polynomially based (i.e., zonal kernel) type. A fundamental tool is an uncertainty principle, which gives appropriate bounds for both the qualification and quantification of space and frequency (momentum) localization of the special (kernel) function under consideration. The essential outcome is a better understanding of constructive approximation in terms of zonal kernel functions such as splines and wavelets.

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“…Freeden and Schreiner (2007) ;Freeden and Gutting (2013) are interested in a compromise connecting zonal function expressions and structured grids on the sphere to obtain fast algorithms.…”

Section: Resultsmentioning

confidence: 99%

Special Functions in Mathematical Geosciences: An Attemptat a Categorization

Freeden

Schreiner

2013

Handbook of Geomathematics

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“…The power of wavelets is based on a multiresolution analysis which is well localized both in the space and in the frequency domains. Basically, this is done by convolving the function against rotated and dilated versions of the mother wavelet [5]. Here, Abel-Poisson scaling function and its corresponding wavelets are used as the mother wavelet and 16 GRACE L2 Release 04 monthly gravity data from GeoForschungsZentrum (GFZ), spanning the period from January 2010 to May 2011, are processed.…”

Section: Wavelet Transform and Resultsmentioning

confidence: 99%

Coseismic geoid height changes of the 2011 Tohoku-Oki earthquake in GRACE monthly gravity data

Park¹,

Hong

2013

Current Research on Material, Architecture and Civil Engineering

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Abstract. A Mw 9.0 earthquake occurred off the coast of Japan in the Pacific Ocean on March 11, 2011. At shorter time and spatial scales, mass redistribution caused by an earthquake produces local variations of the geoid reaching a few centimeters. In this paper, we apply the bilinear wavelets analysis based on the Abel-Poisson scale and its corresponding wavelet functions to GRACE monthly gravity data in order to detect coseismic changes of the geoid heights due to the Tohoku-Oki earthquake. As a result, we can clearly distinguish that multiscale geoid heights are decreased temporarily in the order about -4 meter on average throughout the region between February 2011 and March 2011 when compared with monthly geoid heights observed by GRACE satellite.

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“…For an overview, see [8]. Within these different methods, two essential approaches can be identified: The first one uses certain grids on the sphere and tries to transfer concepts of the one-dimensional wavelet analysis to the sphere.…”

Section: Main Ideamentioning

confidence: 99%

Locally supported spherical wavelets based on block grids

Schreiner

2007

Proc Appl Math and Mech

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This paper presents a method for approximating spherical functions from discrete data of a block-wise grid structure. The essential ingredients of the approach are scaling and wavelet functions within a biorthogonalisation process generated by locally supported zonal kernel functions. In consequence, geophysically and geodetically relevant problems involving rotationinvariant pseudodifferential operators become attackable. A multiresolution analysis is formulated enabling a fast wavelet transform similar to the algorithms known from classical tensor product wavelet theory. Main IdeaSeveral different concepts for the construction of wavelets are known in the literature. For an overview, see [8]. Within these different methods, two essential approaches can be identified: The first one uses certain grids on the sphere and tries to transfer concepts of the one-dimensional wavelet analysis to the sphere. The approaches with longitude/latitude grids or triangulations fall into this category. The second approach put the emphasis on the solution of rotation-invariant pseudodifferential equations on the sphere (as proposed, e.g., in [3]). For this concept, zonal functions are an appropriate tool. In this case, there is -at first glance -no need for a particular point distribution on the sphere.In this paper we are interested in a new compromise for the construction of spherical wavelets. In fact, our wavelets are based on zonal functions but also relate to structured grids in order to obtain fast algorithms that are easy to implement.The key points of our paper can be characterized as follows:• The wavelets and the scaling function are based on zonal kernel functions so that our approach is well-suited for the solution and regularization of rotation-invariant pseudodifferential equations.• The wavelets are locally supported so that the integration can be made efficient.• The construction is based on a biorthogonal system of zonal functions, which gives us almost all advantages of an orthogonal approach.• A new block-wise grid structure is used on the sphere: The points lie on circles of constant latitude to ensure that the construction of a biorthogonal system is not expensive. But the grid gets sparser in the polar regions so that there are not too large differences in the distances between neighboring points.• The grid is organized in blocks of size 2 p by 2 p−1 so that the algorithms can be formulated easily.• Finally, a scaling equation is established with only a few coefficients. In fact, we end up with a fast wavelet transform which is completely similar to the algorithms known from tensor product approaches of Euclidean wavelet theory.What is the prize that we have to pay for all these advantages? It is by no means dramatic: First, we do not develop a multiresolution analysis of the whole space of square-integrable functions but on finite dimensional subspaces. In fact, we have -at the beginning of the analysis -to choose a finest level of resolution. From numerical point of view, this is what has to be done for p...

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Biorthogonal Locally Supported Wavelets on the Sphere Based on Zonal Kernel Functions

Cited by 14 publications

References 21 publications

Special Functions in Mathematical Geosciences: An Attemptat a Categorization

Special Functions in Mathematical Geosciences: An Attemptat a Categorization

Coseismic geoid height changes of the 2011 Tohoku-Oki earthquake in GRACE monthly gravity data

Locally supported spherical wavelets based on block grids

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