An evaluation of accuracy of stokes' series and of some attempts to improve his theory

Molodensky, M. S.; Yeremeyev, V. F.; Yurkina, M. I.

doi:10.1007/bf02528170

Cited by 8 publications

(6 citation statements)

References 5 publications

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“…Equations ( 60), (61) extend similar equations (11), (12) of classic geodesy to the realm of general relativity. The main difference is that the covariant Laplace operator in (60), (61) is taken in curved space with the metric ḡαβ .…”

Section: The Master Equation For the Anomalous Gravity Potentialmentioning

confidence: 85%

Towards an exact relativistic theory of Earth's geoid undulation

2015

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The present paper extends the Newtonian concept of the geoid in classic geodesy towards the realm of general relativity by utilizing the covariant geometric methods of the perturbation theory of curved manifolds. It yields a covariant definition of the anomalous (disturbing) gravity potential and formulate differential equation for it in the form of a covariant Laplace equation. The paper also derives the Bruns equation for calculation of geoid's height with full account for relativistic effects beyond the Newtonian approximation. A brief discussion of the relativistic Bruns formula is provided. (Sergei M. Kopeikin), e_mazurova@mail.ru (Elena M. Mazurova), rector@ssga.ru (Alexander P. Karpik) 1 It is the same J. B. Listing who introduced in 1847 the term topology in mathematics.

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Section: The Master Equation For the Anomalous Gravity Potentialmentioning

confidence: 85%

Towards an exact relativistic theory of Earth's geoid undulation

2015

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“…The first scientific studies of coordinate transformation procedures between different terrestrial reference frames appeared in the 1960s [2,5,6], so scientists like M. S. Molodensky (1909Molodensky ( -1991, M. Bursa (1929Bursa ( -2019, and H. Wolf (1910Wolf ( -1994) can be considered as pioneers of coordinate transformation. Development and usage of global navigation satellite systems (GNSS), primarly NAVSTAR GPS in the 1980s [23,24], for solving everyday geodetic tasks required linking national coordinate reference frames to global ones [3,7] and pointed out that, often, inherited (historical) coordinate reference systems are inappropriate for the application of new technologies.…”

Section: Review Of Previous Studiesmentioning

confidence: 99%

“…Coordinate transformation procedures are also distinguished by the number of transformation parameters. In geodesy and geoinformatics, often used coordinate transformation procedures, due to the number of parameters, are: 3-parameter [2,3], 5-parameter [2,4], 7-parameter [2,[5][6][7], 8-parameter [8], 9-parameter [8], and 12-parameter [9]. Transformation parameters for all mentioned transformation procedures can be obtained from identical points in two different coordinate reference frames.…”

Section: Introductionmentioning

confidence: 99%

Detail Explanation of Coordinate Transformation Procedures Used in Geodesy and Geoinformatics on the Example of the Territory of the Republic of Croatia

Banković,

Dubrović,

Banko

et al. 2024

Applied Sciences

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With strong technology improvement, especially GNSS, coordinate transformation has become an essential tool in everyday practice. For this reason, it is beneficial to have a good understanding of the mathematical process of coordinate transformation, as well as the method for calculating transformation parameters. The purpose of this article is to provide a detailed explanation of the coordinate transformation procedure and the calculation of transformation parameters for seven transformation models: 3-parameter, 5-parameter standard and abridged, 7-parameter, 8-parameter, 9-parameter, and 12-parameter. For each transformation model, a basic transformation equation, a procedure for calculating transformation parameters, inverse expression for reversible transformation, as well as characteristics of each model are provided. As an addition, for each model, a numerical example of calculating transformation parameters for the territory of the Republic of Croatia is provided. Additionally, for the example of the 7-parameter transformation, importance of reversible transformation, rotation convention, as well as the form of the rotation matrix used during transformation are demonstrated.

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“…The above principle is even more visible when we model a quasi-geoid or a figure of Earth according to the theory of Molodensky (Molodensky, 1958;Molodensky et al, 1962). In this case, surface measurements are not necessary, but only precise measurements in a finite number of points.…”

Section: Introductionmentioning

confidence: 99%

The Best Robust Estimation Method to Determine Local Surface

Borowski¹,

Banas²

2019

BJMC

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Geoid/quasi-geoid modelling in a local (small) area is often made with polynomials. The optimal version of a polynomial is found by testing its successive versions using different statistical parameters, i.e. with different degree and number of coefficients. In this publication, the authors presented 3 approaches to search for an optimal version of the polynomial. Two of them were based on one of the robust estimation methods, i.e. the Danish method. Therefore, the question arises whether this method is suitable for this type of work. The publication analyses 6 different methods of robust estimation and the method of least squares. The control parameters of the methods were selected in such a way as to obtain a comparable accuracy of fitting the polynomial into the points each time. Then, the actual accuracy obtained for each of them was compared. The result of the study is a ranking of methods in terms of the most accurate fitting results.

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An evaluation of accuracy of stokes' series and of some attempts to improve his theory

Cited by 8 publications

References 5 publications

Towards an exact relativistic theory of Earth's geoid undulation

Towards an exact relativistic theory of Earth's geoid undulation

Detail Explanation of Coordinate Transformation Procedures Used in Geodesy and Geoinformatics on the Example of the Territory of the Republic of Croatia

The Best Robust Estimation Method to Determine Local Surface

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