Triangulated spherical splines for geopotential reconstruction

Lai, Ming-Jun; Shum, C. K.; Baramidze, Victoria; Wenston, Paul

doi:10.1007/s00190-008-0283-0

Cited by 22 publications

(7 citation statements)

References 24 publications

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“…The theory and computation of bivariate splines become matured and has had continued growth; see the monograph by Lai and Schumaker (2007) for basic theories of bivariate, trivariate, and spherical splines, and see Awanou, Lai, and Wenston (2006) and Baramidze, Lai, and Shum (2006) for some numerical implementations of multivariate splines for data fitting and their application to geopotential reconstruction in Lai et al (2009). We refer to Lai (2008) for a survey of multivariate splines for scattered data fitting and some numerical examples.…”

Section: Introductionmentioning

confidence: 99%

Bivariate Penalized Splines for Regression

Lai¹,

Wang²

2013

STAT SINICA

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In this paper, the asymptotic behavior of penalized spline estimators is studied using bivariate splines over triangulations and an energy functional as the penalty. A convergence rate for the penalized spline estimators is derived that achieves the optimal nonparametric convergence rate established by Stone (1982). The asymptotic normality of the proposed estimators is established and shown to hold uniformly over the points where the regression function is estimated. The size of the asymptotic conditional variance is evaluated, and a simple expression for the asymptotic variance is given. Simulation experiments have provided strong evidence that corroborates the asymptotic theory. A comparison with thin-plate splines is provided to illustrate some advantages of this spline smoothing approach.

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Section: Introductionmentioning

confidence: 99%

Bivariate Penalized Splines for Regression

Lai¹,

Wang²

2013

STAT SINICA

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“…This distinguishes the Slepian functions philosophically from the eigenfunctions of full-phase-space localization operators (Daubechies 1988;Simons et al 2003) or wavelets (Daubechies and Paul 1988;Olhede and Walden 2002), with which they nevertheless share strong connections (Lilly and Park 1995;Shepp and Zhang 2000;Shen 2004, 2005). Strict localization of this type remained the driving force behind the development of Slepian functions over fixed geographical domains on the surface of the sphere (Albertella et al 1999;Miranian 2004;) -which have numerous applications in geodesy (Albertella and Sacerdote 2001;Han et al 2008a;, geomagnetism (Simons et al 2009;Schott and Thébault 2011), geophysics (Han and Ditmar 2007;Han et al 2008b;Han and Simons 2008;Harig et al 2010), biomedical (Maniar and Mitra 2005;Mitra and Maniar 2006) and planetary (Evans et al 2010;Han 2008;Han et al 2009;Wieczorek and Simons 2005) science, and cosmology (Dahlen and Simons 2008;Wieczorek and Simons 2007) -as opposed to approaches using spherical wavelets (Chambodut et al 2005;Faÿ et al 2008;Fengler et al 2007;Freeden and Windheuser 1997;Holschneider et al 2003;Kido et al 2003;McEwen et al 2007;Panet et al 2006;Schmidt et al 2006), needlets (Marinucci et al 2008), splines (Amirbekyan et al 2008;…”

Section: Introductionmentioning

confidence: 99%

Spatiospectral concentration in the Cartesian plane

Simons

Wang

2011

Int J Geomath

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We pose and solve the analogue of Slepian's time-frequency concentration problem in the two-dimensional plane, for applications in the natural sciences. We determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the plane, or, alternatively, of strictly spacelimited functions that are optimally concentrated in the Fourier domain. The Cartesian Slepian functions can be found by solving a Fredholm integral equation whose associated eigenvalues are a measure of the spatiospectral concentration. Both the spatial and spectral regions of concentration can, in principle, have arbitrary geometry. However, for practical applications of signal representation or spectral analysis such as exist in geophysics or astronomy, in physical space irregular shapes, and in spectral space symmetric domains will usually be preferred. When the concentration domains are circularly symmetric in both spaces, the Slepian functions are also eigenfunctions of a Sturm-Liouville operator, leading to special algorithms for this case, as is well known. Much like their one-dimensional and spherical counterparts with which we discuss them in a common framework, a basis of functions that are simultaneously spatially and spectrally localized on arbitrary Cartesian domains will be of great utility in many scientific disciplines, but especially in the geosciences.Comment: 34 pages, 7 figures. In the press, International Journal on Geomathematics, April 14th, 201

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“…Moreover, the methods proposed by Lindgren et al (2011) and by Niu et al (2019) can deal with more general forms of non-planar domains. Regularised least-square smoothing methods able to fit data scattered on spheres include Wahba (1981), Baramidze et al (2006), & Lai et al (2009), while Duchamp & Stuetzle (2003) and SR-PDE (Ettinger et al, 2016;Lila et al, 2016; work on general non-planar domains. Finally, other smoothing methods over general nonplanar domains include nearest-neighborhood techniques (see, e.g., Hagler et al, 2006) and heat-kernel smoothing (see, e.g., Chung et al, 2005Chung et al, , 2017, and references therein).…”

Section: Data Over Complex Domainsmentioning

confidence: 99%

Spatial Regression With Partial Differential Equation Regularisation

Sangalli

2021

Int Statistical Rev

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This work gives an overview of an innovative class of methods for the analysis of spatial and of functional data observed over complicated two-dimensional domains. This class is based on regression with regularising terms involving partial differential equations. The associated estimation problems are solved resorting to advanced numerical analysis techniques. The synergical interplay of approaches from statistics, applied mathematics and engineering endows the methods with important advantages with respect to the available techniques, and makes them able to accurately deal with data structures for which the classical techniques are unfit. Spatial regression with differential regularisation is illustrated via applications to the analysis of eco-colour doppler measurements of blood-flow velocity, and to functional magnetic resonance imaging signals associated with neural connectivity in the cerebral cortex.

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Triangulated spherical splines for geopotential reconstruction

Cited by 22 publications

References 24 publications

Bivariate Penalized Splines for Regression

Bivariate Penalized Splines for Regression

Spatiospectral concentration in the Cartesian plane

Spatial Regression With Partial Differential Equation Regularisation

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