Translated origin spherical cap harmonic analysis

Santis, A. De

doi:10.1111/j.1365-246x.1991.tb04615.x

Cited by 41 publications

(20 citation statements)

References 18 publications

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“…Certainly, using both sets of Legendre functions ( n − k even and n − k odd), which are not orthogonal to each other, might destabilize the determination of the harmonic coefficient by the least‐squares procedure (De Santis 1991). Further, in the case of overdetermined systems, if the experimental errors are neglected, the accuracy of the solution suffers mainly from the condition of the coefficient matrix of the redundant system of equations, the distribution of the data, and, of course, their resolution.…”

Section: Asha Expansion Of a Load Functionmentioning

confidence: 99%

Modelling of mantle postglacial relaxation in axisymmetric geometry with a composite rheology and a glacial load interpolated by adjusted spherical harmonics analysis

Forno

Gasperini

2007

Geophysical Journal International

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S U M M A R YAlthough studies on glacial isostatic adjustment usually assume a purely linear rheology, we have previously shown that mantle relaxation after the melting of Laurentide ice sheet is better described by a composite rheology including a non-linear term. This modelling is, however, based on axially symmetric geometry and glacial forcing derived from ICE-3G and suffers from a certain amount of arbitrariness in the definition of the ice load. In this work we apply adjusted spherical harmonics analysis to interpolate the ice thicknesses of ICE-3G and ICE-1 glaciological models. This filters out the non-axisymmetric components of the ice load by considering only the zonal terms in the spherical harmonics expansion. The resulting load function is used in finite-element simulation of postglacial rebound to compare composite versus purely linear rheology. Our results confirm that composite rheology can explain relative sea level (RSL) data in North America significantly better than a purely linear rheology. The performance of composite rheology suggests that in future investigations, it may be better to use this more physically realistic creep law for modelling mantle deformation induced by glacial forcing.The modelling of mantle deformation induced by mass redistribution between oceans and ice sheets during glacial cycles (Haskell 1935;Cathles 1975;Peltier & Andrews 1976) is one of the most important tools for inferring the rheological behaviour of the Earth. The majority of postglacial rebound (PGR) studies are conducted assuming a linear viscoelastic rheology for the Earth's mantle, even though there is strong evidence of non-linearity from mineral microphysics (Ranalli 1995). The point here is that if the flow law is linear and the structure of the Earth quite uniform, the problem can be treated by rigorous spectral methods taking advantage of the superposition principle. When some terms of the governing equation are non-linear or the rheological structure is complex (e.g. laterally varying), the superposition of the harmonic responses is no longer applicable. These are the reasons why the first models with a non-Newtonian mantle (Post & Griggs 1973;Brennen 1974;Crough 1977) were based on unrealistic assumptions and their results need to be interpreted with caution. More rigorous solutions of the non-linear case have recently been obtained through finite elements (FE), but only models with non-linear zones in either the upper mantle or the lower mantle performed better than a purely linear reference model (Wu 1999(Wu , 2002, while a uniform non-Newtonian mantle yielded a close but still worse fit to relative sea level (RSL) data than a Newtonian model (Wu 2001). Therefore, a purely non-Newtonian rheology had never appeared to be preferable to a linear creep law. Nevertheless, the observations of traces of deformation due to both diffusion and dislocation creep in laboratory samples suggest that a realistic mantle rheology might be composite, with linear or non-linear creep mechanisms temporarily and local...

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Section: Asha Expansion Of a Load Functionmentioning

confidence: 99%

Modelling of mantle postglacial relaxation in axisymmetric geometry with a composite rheology and a glacial load interpolated by adjusted spherical harmonics analysis

Forno

Gasperini

2007

Geophysical Journal International

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“…(1). Given the observable v, we want to compute urY r x nYm u nm r na nm r or its functionals in the space surrounding g. Usually, we make a ®t by LS on g and this again produces a strong numerical instability (De Santis 1991).…”

Section: Problem (Cap)mentioning

confidence: 99%

“…A dierent approach to Problem 2 is represented by the so called ®nite cap solutions. This approach comes from a solution of the Laplace equation in a cut cone (Haines 1985;De Santis 1991;Hwang and Chen 1997) The separation of variables, in spherical coordinates, leads to the de®nition of the eigenfunctions mm of the Laplace±Beltrami operator on g, which are generalizations of Legendre's functions to non-integer values of m; however, the particular set of eigenfunctions of D r depends on the conditions imposed on the boundary of g.…”

Section: Problem (Cap)mentioning

confidence: 99%

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Band-limited functions on a bounded spherical domain: the Slepian problem on the sphere

1999

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The Slepian problem consists of determining a sequence of functions that constitute an orthonormal basis of a subset of R (or R 2 ) concentrating the maximum information in the subspace of square integrable functions with a band-limited spectrum. The same problem can be stated and solved on the sphere. The relation between the new basis and the ordinary spherical harmonic basis can be explicitly written and numerically studied. The new base functions are orthogonal on both the subspace and the whole sphere. Numerical tests show the applicability of the Slepian approach with regard to solvability and stability in the case of polar data gaps, even in the presence of aliasing. This tool turns out to be a natural solution to the polar gap problem in satellite geodesy. It enables capture of the maximum amount of information from non-polar gravity ®eld missions.

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“…However, it was Haines (1985a,b) who first made use of these functions in approximating the geomagnetic field with the introduction of spherical cap harmonics. Other applications of spherical cap harmonics are mostly found in geomagnetic literature; for example in De Santis (1991) and Haines & Torta (1994). A simple approximated alternative based on ordinary spherical harmonics properly adjusted to the cap is given by De Santis ( 1992).…”

Section: Introductionmentioning

confidence: 99%

Fully normalized spherical cap harmonics: application to the analysis of sea-level data from TOPEX/POSEIDON and ERS-1

Hwang

Chen

1997

Geophysical Journal International

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We introduce two sets of fully normalized harmonics for the spectral analysis of functions defined on a spherical cap. The harmonics are the products of Fourier functions and the fully normalized associated Legendre functions of non-integer degree. Using Sturm-Liouville theory for boundary-value problems, we present two convenient and stable formulae for computing the zeros of the associated Legendre functions that form two sets of orthogonal functions. Formulae for the stable numerical evaluation of the fully normalized associated Legendre functions of non-integer degree that avoid the gamma function are also derived. The result from the expansions of sealevel anomaly from altimetry into Set 2 fully normalized cap harmonics shows fast convergence of the series, and the degree variances decay rapidly without aliasing effects. The zero-degree coefficients (Set 2) of sea-level anomaly from TOPEX/ POSEIDON (T/P) and ERS-1 indicate an El Nifio event during 1993 January-1993 July, and a La Niiia event during 1993 November-1994 July, although the ERS-1 result is less obvious. Ocean circulations over the South China Sea and the Kuroshio area are clearly identified with the low-degree expansions of sea-surface topography (SST) from T/P and ERS-1. A cold-core eddy of 4" in diameter centred at 17.5"N, 118"E was detected with the expansion of SST from T/P cycle 47, and a property of the cap harmonics is used to compute this eddy's kinetic energy. The kinetic energy is at a low in winter and high in summer, and its variation seems to be periodic with an amplitude of 0.4 m2 s-'.

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Translated origin spherical cap harmonic analysis

Cited by 41 publications

References 18 publications

Modelling of mantle postglacial relaxation in axisymmetric geometry with a composite rheology and a glacial load interpolated by adjusted spherical harmonics analysis

Modelling of mantle postglacial relaxation in axisymmetric geometry with a composite rheology and a glacial load interpolated by adjusted spherical harmonics analysis

Band-limited functions on a bounded spherical domain: the Slepian problem on the sphere

Fully normalized spherical cap harmonics: application to the analysis of sea-level data from TOPEX/POSEIDON and ERS-1

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