2008
DOI: 10.1111/j.1365-246x.2007.03700.x
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Numerical aspects of a spline-based multiresolution recovery of the harmonic mass density out of gravity functionals

Abstract: S U M M A R YWe show the numerical applicability of a multiresolution method based on harmonic splines on the 3-D ball which allows the regularized recovery of the harmonic part of the Earth's mass density distribution out of different types of gravity data, for example, different radial derivatives of the potential, at various positions which need not be located on a common sphere. This approximated harmonic density can be combined with its orthogonal anharmonic complement, for example, determined out of the … Show more

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Cited by 35 publications
(12 citation statements)
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References 22 publications
(31 reference statements)
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“…eqn [53]). A multitude of wavelet families have been constructed, including those that are zonal (Chambodut et al, 2005;Fengler et al, 2007;Freeden and Windheuser, 1997;Guilloux et al, 2009;Holschneider et al, 2003;Holschneider and Iglewska-Nowak, 2007;Li, 1999;Michel and Wolf, 2008;Panet et al, 2011;Schmidt et al, 2007;Wiaux et al, 2007) and directional (Audet, 2011(Audet, , 2013McEwen et al, 2007). Once the family of wavelets has been constructed for each scale s, it is straightforward to construct the wavelet transform.…”
Section: Wavelet Analysismentioning
confidence: 99%
“…eqn [53]). A multitude of wavelet families have been constructed, including those that are zonal (Chambodut et al, 2005;Fengler et al, 2007;Freeden and Windheuser, 1997;Guilloux et al, 2009;Holschneider et al, 2003;Holschneider and Iglewska-Nowak, 2007;Li, 1999;Michel and Wolf, 2008;Panet et al, 2011;Schmidt et al, 2007;Wiaux et al, 2007) and directional (Audet, 2011(Audet, , 2013McEwen et al, 2007). Once the family of wavelets has been constructed for each scale s, it is straightforward to construct the wavelet transform.…”
Section: Wavelet Analysismentioning
confidence: 99%
“…The basis functions of the (scalar) spline S(x) = N k=1 a k F k K H (·, x), x ∈ B, are chosen according to the data structure. This is advantageous if one wants to locally adapt the resolution of the spline to the density of the data grid (see Michel and Wolf (2008)). Using the idea of a matching pursuit in our new approach, we select the basis functions to best match the structure of the solution, which provides us with alternative advantages.…”
Section: Outlook: Application Of a Matching Pursuit Techniquementioning
confidence: 99%
“…At each step, we add more data and reduce the "hat-width" of the spline basis functions in order to obtain a better resolution. Our approach is based on the spline-wavelet method in Fengler et al (2006) and Michel and Wolf (2008). The proofs are partially analogous.…”
Section: Spline Multiresolutionmentioning
confidence: 99%
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“…Localised functions for the inverse gravimetry problems have been developed and successfully applied for the last ten years. In this context, wavelet-based multiscale methods (see [Freeden and Michel, 2004, Michel, 1998, 2002a,b, 2005a, Michel and Fokas, 2008) were constructed as well as spline methods (see Wolf, 2008]). The latter use reproducing kernel Hilbert spaces and are more flexible with respect to the use of different types of data and are a further development of the theory of spherical splines [Freeden, 1981a[Freeden, ,b, 1999.…”
Section: Introductionmentioning
confidence: 99%