Spatially restricted data distributions on the sphere: the method of orthonormalized functions and applications

Pail, Roland; Plank, Gernot; Schuh, Wolf-Dieter

doi:10.1007/s001900000153

Cited by 28 publications

(23 citation statements)

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“…Since timelimited functions cannot be simultaneously bandlimited in the frequency domain, nor vice versa, the optimally concentrated signal is considered to be the one with the least energy outside the interval of interest. The concentration problem has been extended and generalized for the purpose of signal estimation, representation and analysis on geographical domains by Albertella et al (1999), Pail et al (2001) and in geodesy, and by Wieczorek and Simons (2007) and Dahlen and Simons (2008) in more general settings. The quadratic maximization of the spatial energy of bandlimited functions is one way to achieve localization in one domain while curbing leakage in the other.…”

Section: The Spherical Slepian Basismentioning

confidence: 99%

The spherical Slepian basis as a means to obtain spectral consistency between mean sea level and the geoid

Slobbe

Simons

Klees

2012

J Geod

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The mean dynamic topography (MDT) can be computed as the difference between the mean sea level (MSL) and a gravimetric geoid. This requires that both data sets are spectrally consistent. In practice, it is quite common that the resolution of the geoid data is less than the resolution of the MSL data, hence, the latter need to be low-pass filtered before the MDT is computed. For this purpose conventional low-pass filters are inadequate, failing in coastal regions where they run into the undefined MSL signal on the continents. In this paper, we consider the use of a bandlimited, spatially concentrated Slepian basis to obtain a low-resolution approximation of the MSL signal. We compute Slepian functions for the oceans and parts of the oceans and compare the performance of calculating the MDT via this approach with other methods, in particular the iterative spherical harmonic approach in combination with Gaussian low-pass filtering, and various modifications. Based on the numerical experiments, we conclude that none of these methods provide a lowresolution MSL approximation at the sub-decimetre level. In particular, we show that Slepian functions are not appropriate basis functions for this problem, and a Slepian representation of the low-resolution MSL signal suffers from broadband leakage. We also show that a meaningful definition of a lowresolution MSL over incomplete spherical domains involves orthogonal basis functions with additional properties that Slepian functions do not possess. A low-resolution MSL signal, spectrally consistent with a given geoid model, is obtained by a suitable truncation of the expansions of the MSL signal in terms of these orthogonal basis functions. We compute one of these sets of orthogonal basis functions using the Gram-Schmidt orthogonalization for spherical harmonics. For the oceans, we could construct an orthogonal basis only for resolutions equivalent to a spherical harmonic degree 36. The computation of a basis with a higher resolution fails due to inherent instabilities. Regularization reduces the instabilities but destroys the orthogonality and, therefore, provides unrealistic low-resolution MSL approximations. More research is needed to solve the instability problem, perhaps by finding a different orthogonal basis that avoids it altogether.

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Section: The Spherical Slepian Basismentioning

confidence: 99%

The spherical Slepian basis as a means to obtain spectral consistency between mean sea level and the geoid

Slobbe

Simons

Klees

2012

J Geod

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“…These approaches can be interpreted as introducing additional artificial (prior) information to the normal equation systems, for the purpose of stabilizing the solution either globally (spectral domain techniques) or regionally in those regions causing the problem (spatial techniques). Another approach to cope with the problem of such an inhomogeneous data distribution is the introduction of a new set of (orthonormalized) base functions on the sphere (Albertella et al, 1999;Pail et al, 2001), which is dedicated to the specific data coverage.…”

Section: Simulations and Resultsmentioning

confidence: 99%

On the combination of global and local data in collocation theory

Pail¹,

Reguzzoni

Sansò

et al. 2010

Stud Geophys Geod

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Due to the successful operation of dedicated satellite gravity missions, nowadays highaccuracy global gravity field models have become available. This triggers the challenge to optimally combine this long to medium wavelength gravity field information derived from space-borne data with high-resolution terrestrial gravity data. In this paper, the least squares collocation concept is revised with the attempt to consistently unify the combination procedure in such a way that the full information contained in both data sets is merged. For example, in local or regional geoid determination the remove-restore method is usually applied only partially taking into account the accuracy of the global model coefficients used for the long-wavelength reduction. The key advantage of the extended formulation is the fact that it automatically accounts for the error covariance of all data types involved. The applicability, feasibility and performance of the proposed method is investigated in the frame of numerical closed-loop simulations. The two main fields of application, i.e., the improvement of a global gravity field model by terrestrial gravity field data, and, vice versa, the support to a regional geoid solution by the incorporation of a global gravity field model, have been analyzed and assessed. Although applied under simplified conditions, it could be shown that the method works and is practically applicable

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“…This method would allow to generalize the scheme also to non-spherical data gaps, as it was done, e.g., in Pail et al (2001).…”

Section: T I K H O N O V a N D K A U L A R E G U L A R I Z A T I O Nmentioning

confidence: 96%

“…where a represents the area of the bounded region σ (Hwang, 1991;Pail et al, 2001). By defining σ as northern and southern spherical cap, and by applying Eq.…”

Section: T I K H O N O V a N D K A U L A R E G U L A R I Z A T I O Nmentioning

confidence: 99%

GOCE Data Processing: The Spherical Cap Regularization Approach

Metzler

Pail

2005

Stud Geophys Geod

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Due to the sun-synchronous orbit of the satellite gravity gradiometry mission GOCE, the measurements will not be globally available. As a consequence, using a set of base functions with global support such as spherical harmonics, the matrix of normal equations tends to be ill-conditioned, leading to weakly determined low-order spherical harmonic coefficients. The corresponding geopotential strongly oscillates at the poles.Considering the special configuration of the GOCE mission, in order to stabilize the normal equations matrix, the Spherical Cap Regularization Approach (SCRA) has been developed. In this approach the geopotential function at the poles is predescribed by an analytical continuous function, which is defined solely in the spatially restricted polar regions. This function could either be based on an existing gravity field model or, alternatively, a low-degree gravity field solution which is adjusted from GOCE observations. Consequently the inversion process is stabilized.The feasibility of the SCRA is evaluated based on a numerical closed-loop simulation, using a realistic GOCE mission scenario. Compared with standard methods such as Kaula and Tikhonov regularization, the SCRA shows a considerably improved performance.

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Spatially restricted data distributions on the sphere: the method of orthonormalized functions and applications

Cited by 28 publications

References 0 publications

The spherical Slepian basis as a means to obtain spectral consistency between mean sea level and the geoid

The spherical Slepian basis as a means to obtain spectral consistency between mean sea level and the geoid

On the combination of global and local data in collocation theory

GOCE Data Processing: The Spherical Cap Regularization Approach

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