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Fitting a Triaxial Ellipsoid to a Geoid Model

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Fitting a Triaxial Ellipsoid to a Geoid Model J. Geod. Sci. 2020; 10:69–82 Research Article Open Access G. Panou*, R. Korakitis, and G. Pantazis Fitting a triaxial ellipsoid to a geoid model DOI: https://doi.org/10.1515/jogs-2020-0105 comprehensive list by Heiskanen (1962). Also, recent de- Received March 10, 2020; accepted July 8, 2020 terminations of the parameters of triaxiality (semi-axes ax, Abstract: The aim of this work is the determination of the ay and b, longitude of major semi-axis λ0) are presented parameters of Earth’s triaxiality through a geometric t- in Table 1. However, the dierence between the equatorial ting of a triaxial ellipsoid to a set of given points in space, semi-axes ax and ay, as determined in the satellite era, is as they are derived from a geoid model. Starting from a smaller than the one previously proposed. Some of the rea- Cartesian equation of an ellipsoid in an arbitrary refer- sons have been explained in Burša (1972). ence system, we develop a transformation of its coe- Generally, there are three methods for the determina- cients into the coordinates of the ellipsoid center, of the tion of the parameters of triaxiality: i) the astrogeodetic, three rotation angles and the three ellipsoid semi-axes. ii) the gravimetric and iii) the satellite method. The gravi- Furthermore, we present dierent mathematical models metric method is given in detail by Heiskanen (1962). In for some special and degenerate cases of the triaxial el- this method, the parameters of triaxiality are determined lipsoid. We also present the required mathematical back- using the least-squares method by minimizing the sum ground of the theory of least-squares, under the condition of squares of gravity anomalies ∆g. However, in the pre- of minimization of the sum of squares of geoid heights. satellite era, as mentioned by Heiskanen (1962), the in- Also, we describe a method for the determination of the ability to determine the triaxiality parameters with the re- foot points of the set of given space points. Then, we pre- quired accuracy is due to the lack of enough data. The pare suitable data sets and we derive results for various satellite method is described in several papers, e.g. Burša geoid models, which were proposed in the last fty years. (1970, 1971, 1972) and Burša and Šíma (1980). In this Among the results, we report the semi-axes of the triax- method, the requested parameters are determined using ial ellipsoid of geometric tting with four unknowns to be the least-squares method by minimizing the radial dis- 6378171.92 m, 6378102.06 m and 6356752.17 m and the equa- tance between a given geoidal point and the correspond- torial longitude of the major semi-axis –14.9367 degrees. ing point on the surface of the ellipsoid. On the other hand, Also, the parameters of Earth’s triaxiality are directly esti- once some gravity eld model is adopted, from the second- mated from the spherical harmonic coecients of degree degree coecients of spherical harmonics, the orienta- and order two. Finally, the results indicate that the geoid tions of the principal axes and the magnitudes of the prin- heights with reference to the triaxial ellipsoid are smaller cipal moments may be determined by a well-known theory than those with reference to the oblate spheroid and the (Chen and Shen, 2010). improvement in the corresponding rms value is about 20 Nowadays, we can obtain all the data required for per cent. applying any of the aforementioned methods. Indeed, there are several satellite-only or combined global grav- Keywords: algebraic tting, coordinates conversion, geo- ity eld models available by the International Cen- metric tting, least squares method, principal axes tre for Global Earth Models (ICGEM) (http://icgem.gfz- potsdam.de/home.html). Also, according to Barthelmes 1 Introduction (2014), the satellite-only models are computed from satel- lite measurements alone, whereas ground and altimetry The problem of determining the parameters of Earth’s tri- data are additionally used for the combined models. Fur- axiality has been recognized since 1859 and various treat- thermore, the spatial resolution of the satellite-only mod- ments by dierent scientists have been reported in the els is lower, but the accuracy is nearly uniform over the Earth. In this work, we determine the parameters of Earth’s *Corresponding Author: G. Panou: Department of Surveying En- triaxiality through a least-squares tting of a triaxial el- gineering, National Technical University of Athens, 15780 Athens, Greece, E-mail: [email protected] lipsoid to a set of ‘almost equally spaced’ geoid points, R. Korakitis, G. Pantazis: Department of Surveying Engineering, whose coordinates are of equal precision and are derived National Technical University of Athens, 15780 Athens, Greece Open Access. © 2020 G. Panou et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. 70 Ë G. Panou, R. Korakitis, and G. Pantazis, Fitting a triaxial ellipsoid to a geoid model Table 1. Parameters of Earth’s triaxiality in the satellite era ◦ Reference ax (m) ay (m) b (m) λ0 ( ) ax − ay (m) Burša and Fialová (1993) 6378171.36 ±0.30 6378101.61 ±0.30 6356751.84 ±0.30 –14.93 ±0.05 69.75 ±0.42 Burša and Šíma (1980) 6378172 6378102.7 6356752.6 –14.9 69.3 Eitschberger (1978) 6378173.435 6378103.9 6356754.4 –14.8950 69.5 Burša (1972) 6378173 6378104 6356754 –14.8 69 Burša (1970) 6378173 ±10 6378105 ±18 6356754 ±10 –14.8 ±5 68 ±21 from a global gravity eld model. The condition for the of the triaxial ellipsoid, including the oblate spheroid tting is the minimization of the sum of squares of geoid and the sphere. heights N, known as geometric tting (non-linear), using – the problem of geometric tting is solved following approximate values for the unknowns derived from an al- an iterative approach, as a least-squares problem with gebraic tting (linear). Theoretically, the results of geo- constraints involving additional parameters and us- metric tting are dierent from those of algebraic tting ing a very large data set. In contrast, Bektas (2015) and some of the advantages of the geometric tting are re- uses the Levenberg-Marquardt algorithm for solving ferred to in Bektas (2015). The required mathematical back- the non-linear least-squares problem of geometric t- ground of the theory of least-squares is included in Mikhail ting, in order to determine its parameters. In addition, (1976) and Dermanis (2017). Furthermore, in order to illu- he uses a general algorithm for the determination of minate the dierences between the triaxial ellipsoid and the distance between a data point and its closest point the oblate spheroid, which has been adopted in geodesy, on the ellipsoid, as presented in Bektas (2014). Finally, we consider special and degenerate cases of the triaxial el- as an application of his method, he presents a numer- lipsoid in the tting. For this scope, we start from a Carte- ical example for a virtual body using only 12 points. sian equation of an ellipsoid in an arbitrary reference sys- – the presented procedure can be also used for a geomet- tem, as given in Hirvonen (1971), and we develop a trans- ric tting of the physical surface of the Earth, as men- formation of the unknowns of the tting into parameters, tioned in Tserklevych, Zaiats and Shylo (2016), as well following the algorithm of determining an error ellipsoid as of the physical surfaces of other celestial bodies, from a 3 × 3 variance-covariance matrix (for the details see as presented in Karimi, Ardalan and Farahani (2016, Hirvonen (1971)). Taking into account the improvement of 2017), who use an algebraic tting in their work. This the global gravity eld models in the last fty years, we capability is of special importance for other celestial also derive results for several past geoid models. Finally, bodies, where the lack of gravity eld models does not we compare the results with those directly derived from the permit the estimation of their triaxial parameters. coecients of a global gravity eld model. – it vividly demonstrates and veries the numerical sta- The main contributions of the present work can be bility of the values of the triaxial parameters of the summarized as follows: Earth. As pointed out in Heiskanen (1962), the triaxial- – it oers updated values for the triaxial parameters of ity is geophysically interesting and important. For ex- the Earth, derived from a large volume of contempo- ample, it is important in modifying the Earth rotation rary data, lling the gap in the relevant geodetic lit- theory, since it is closely related to the principal mo- erature since the seventies, which was mostly due to ments of inertia of the Earth (Chen and Shen, 2010). the use of the oblate spheroid as the adopted refer- Hence, this task should be a challenge for the follow- ence surface. Nowadays, however, the accuracy re- ing years. quirements are higher and the computing capabilities are greatly improved. – starting from a Cartesian equation of an ellipsoid in an arbitrary reference system, we develop a transfor- mation of its polynomial coecients into geometrical parameters, like the coordinates of the ellipsoid cen- ter, the three rotation angles and the three ellipsoid semi-axes. Furthermore, we present dierent mathe- matical models for some special and degenerate cases G. Panou, R. Korakitis, and G. Pantazis, Fitting a triaxial ellipsoid to a geoid model Ë 71 2 Cartesian equation of an ellipsoid 2.1 Triaxial ellipsoid and special cases in an arbitrary reference system 2.1.1 General case T1 The Cartesian
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