Regional gravity modeling in terms of spherical base functions

Schmidt, Michaël; Fengler, Martin; Mayer‐Gürr, Torsten; Eicker, Annette; Kusche, Jürgen; Sánchez, Laura; Han, Shin‐Chan

doi:10.1007/s00190-006-0101-5

Cited by 136 publications

(119 citation statements)

References 48 publications

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“…Heikkinen 1981; Barthelmes 1986Barthelmes , 1988Vermeer 1982Vermeer , 1983Vermeer , 1984Vermeer , 1995, the radial multipoles (e.g. Marchenko 1998;Marchenko et al 2001), the Blackman functions of Blackman and Tukey (1958) (Schmidt et al 2004(Schmidt et al , 2005a(Schmidt et al , 2007, and the Poisson wavelets of Holschneider et al (2003) (e.g. Chambodut et al 2005;Panet et al 2006;Klees and Wittwer 2007b), see also Holschneider and Iglewska-Nowak (2007).…”

Section: Introductionmentioning

confidence: 99%

A data-driven approach to local gravity field modelling using spherical radial basis functions

Klees

Tenzer

Prutkin

et al. 2008

J Geod

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We propose a methodology for local gravity field modelling from gravity data using spherical radial basis functions. The methodology comprises two steps: in step 1, gravity data (gravity anomalies and/or gravity disturbances) are used to estimate the disturbing potential using least-squares techniques. The latter is represented as a linear combination of spherical radial basis functions (SRBFs). A data-adaptive strategy is used to select the optimal number, location, and depths of the SRBFs using generalized cross validation. Variance component estimation is used to determine the optimal regularization parameter and to properly weight the different data sets. In the second step, the gravimetric height anomalies are combined with observed differences between global positioning system (GPS) ellipsoidal heights and normal heights. The data combination is written as the solution of a Cauchy boundary-value problem for the Laplace equation. This allows removal of the non-uniqueness of the problem of local gravity field modelling from terrestrial gravity data. At the same time, existing systematic distortions in the gravimetric and geometric height anomalies are also absorbed into the combination. The approach is used to compute a height reference surface for the Netherlands. The solution is compared with NLGEO2004, the official Dutch height reference surface, which has been computed using the same data but a Stokes-based approach with kernel modification and a geometric six-parameter "corrector surface" to fit the gravimetric solution to the GPS-levelling points. A direct comparison of both height reference surfaces shows an RMS difference of 0.6 cm; the maximum difference is 2.1 cm. A test at R. Klees (B) · R. Tenzer · I. Prutkin · T. Wittwer

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Section: Introductionmentioning

confidence: 99%

A data-driven approach to local gravity field modelling using spherical radial basis functions

Klees

Tenzer

Prutkin

et al. 2008

J Geod

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“…They are versatile in that their approximation characteristics and spatial distribution can be adjusted, making it possible to use them for all kinds of data sets and for combining different types of observations (e.g., Lieb et al 2016). Regional gravity field modeling with RBFs can be done using numerical integration (e.g., Freeden and Schneider 1998;Schmidt et al 2002;Liu and Sideris 2003;Roland and Denker 2005) or least-squares estimation approaches (e.g., Schmidt et al 2007;Lieb et al 2016). In this work, we use the latter, which is the common geodetic approach, facilitating error analysis and propagation.…”

Section: Introductionmentioning

confidence: 99%

On the equivalence of spherical splines with least-squares collocation and Stokes’s formula for regional geoid computation

Ophaug

Gerlach

2017

J Geod

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This work is an investigation of three methods for regional geoid computation: Stokes's formula, leastsquares collocation (LSC), and spherical radial base functions (RBFs) using the spline kernel (SK). It is a first attempt to compare the three methods theoretically and numerically in a unified framework. While Stokes integration and LSC may be regarded as classic methods for regional geoid computation, RBFs may still be regarded as a modern approach. All methods are theoretically equal when applied globally, and we therefore expect them to give comparable results in regional applications. However, it has been shown by de Min (Bull Géod 69:223-232, 1995. doi:10.1007/BF00806734) that the equivalence of Stokes's formula and LSC does not hold in regional applications without modifying the cross-covariance function. In order to make all methods comparable in regional applications, the corresponding modification has been introduced also in the SK. Ultimately, we present numerical examples comparing Stokes's formula, LSC, and SKs in a closed-loop environment using synthetic noise-free data, to verify their equivalence. All agree on the millimeter level.

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“…However, for 0 θ ≠ π the function V in 0 θ as well as its derivative in θ must be arbitrary [Schmidt M., 2007]:…”

Section: Initial Datamentioning

confidence: 99%

Geodesy, Cartography and Aerial Photography

Yankiv-Vitkovska¹,

Dzhuman²

2017

ГЕОДЕЗІЯ, КАРТОГРАФІЯ І АЕРОФОТОЗНІМАННЯ

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Aim. The widespread use of global navigation satellite systems (GNSS) has led to the development of new methods designed to determine and accumulate the index of ionosphere ionization (VTEC). Using these data it is possible to significantly improve the accuracy and reliability of determining the coordinates of the observation point. Therefore, the task of constructing a model of ionization index is relevant. Method. To construct a spatial model, we used the spherical Legendre functions of the first kind of real order with integer degree as a basic system of functions. We found the magnitude of order using the Sturm-Liouville theory since it depended on the size of the investigated region. Such system of functions form two orthogonal systems of functions in the region under study (the sphere segment), but does not have recurrence relations between functions, therefore, it is necessary to use function expansion in a hypergeometric series to find them. Also in order to find unknown coefficients of the model it is necessary to use the Tikhonov regularization parameter, since the matrix of normal equations will not be stable. For calculating the time model of the ionosphere the coefficients of different spatial models were expanded in series of power polynomials. Results. Based on the data of the ionization parameter values obtained of 19 permanent stations of the ZAKPOS network using the Trimble Pivot Platform software, the spatial-temporal model of this parameter was constructed using the Legendre spherical functions up to the 3rd order as well as with power polynomials up to 3rd order. The standard deviation between the measured and model values of the VTEC parameter does not exceed 1TECU. The scientific novelty and practical significance. We developed algorithm for construction of the spacetime model of the ionosphere parameter. A ionosphere model of high resolution is obtained, which can be used to solve geodetic tasks in order to provide the necessary accuracy in determining the coordinates of the point, as well as to study and forecast the space weather.

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Regional gravity modeling in terms of spherical base functions

Cited by 136 publications

References 48 publications

A data-driven approach to local gravity field modelling using spherical radial basis functions

A data-driven approach to local gravity field modelling using spherical radial basis functions

On the equivalence of spherical splines with least-squares collocation and Stokes’s formula for regional geoid computation

Geodesy, Cartography and Aerial Photography

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