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Ellipsoidal orthogonal curvilinear coordinates referred to a triaxial reference ellipsoid are not used in geodesy perhaps because of their theoretical complexity. However, many surveying and geodetic problems could be resolved by simply transforming variables between local frames. Herein it is shown that the computations of local displacements and variance-covariance matrices referred to currently used local curvilinear geodetic coordinate frames gives approximately the same answers than when referred to local ellipsoidal (triaxial) curvilinear frames. The theoretical procedures to obtain the required transformations between local geodetic and local ellipsoidal (triaxial) frames are introduced and examples explained. [ABSTRACT FROM AUTHOR]

By introducing the auxiliary variable with respect to the reduced latitude, a new closed-form method for transforming Cartesian to geodetic coordinates has been proposed based on the solution of a special constructed unary quartic equation. The algorithm comes with rigorous and concise procedure of root-finding. Moreover, through theoretical analysis, different approaches with respective pros and cons to determine the geodetic latitude and height have been explored. Besides fast computation, numerical experiments covering the magnitude of the geodetic height from - 6.33 × 10 6 m to 10 10 m have also shown that the new method can be operational with high precision at almost any point including the region near or at the pole, the equator and the center of the reference ellipsoid. Considering the accuracy, efficiency and adaptability simultaneously, it is prospective to be applied into computation and inspection on critical occasions in comparison to existing methods. [ABSTRACT FROM AUTHOR]

Through transformation from geocentric to topocentric space rectangular coordinate system and taking advantage of the plain geometrical relationships from the chord between two points on the ellipsoidal surface, the problems of the distances and azimuths on the reference ellipsoid can be converted to several elementary equations and corresponding corrective terms. Based on this, an improved chord method has been proposed in the article. Numeric experiments have validated that the new algorithm can be effective for arbitrary distances. Moreover, the solution can also be operational without singularity in the polar or equatorial regions, in the situations where the difference in longitude is 180°, and in other special cases. And with suitable corrections, the improved method can be prospectively applied to other problems of distances and azimuths with corresponding precision. [ABSTRACT FROM AUTHOR]

We present a new, physically motivated triaxial reference ellipsoid for the Earth. It is an equipotential surface in the gravity field and closely approximates the geoid, akin to the conventional reference ellipsoid of revolution. According to Burša and Fialová (Studia Geophysica et Geodaetica 37(1):1–13, 1993), the triaxial reference ellipsoid is uniquely, but not exclusively, specified by the body's total mass, the dynamic form factors of polar and equatorial flattening, the longitude of the equatorial major axis, the rotation rate, and the designated surface potential. We model the gravity field using triaxial ellipsoidal harmonics. While they are rarely considered practical for near-spherical planets, we leverage an intrinsic property that ellipsoidal harmonics yield an exact expression for the constant potential on a triaxial ellipsoid. A practical procedure is proposed to solve for the ellipsoidal parameters that converge iteratively to fulfill the exact condition of equipotentiality. We present the solution for the Earth Gravitational Model 2008. [ABSTRACT FROM AUTHOR]

The geoid, according to the classical Gauss–Listing definition, is, among infinite equipotential surfaces of the Earth's gravity field, the equipotential surface that in a least squares sense best fits the undisturbed mean sea level. This equipotential surface, except for its zero-degree harmonic, can be characterized using the Earth's global gravity models (GGM). Although, nowadays, satellite altimetry technique provides the absolute geoid height over oceans that can be used to calibrate the unknown zero-degree harmonic of the gravimetric geoid models, this technique cannot be utilized to estimate the geometric parameters of the mean Earth ellipsoid (MEE). The main objective of this study is to perform a joint estimation of W0, which defines the zero datum of vertical coordinates, and the MEE parameters relying on a new approach and on the newest gravity field, mean sea surface and mean dynamic topography models. As our approach utilizes both satellite altimetry observations and a GGM model, we consider different aspects of the input data to evaluate the sensitivity of our estimations to the input data. Unlike previous studies, our results show that it is not sufficient to use only the satellite-component of a quasi-stationary GGM to estimate W0. In addition, our results confirm a high sensitivity of the applied approach to the altimetry-based geoid heights, i.e., mean sea surface and mean dynamic topography models. Moreover, as W0 should be considered a quasi-stationary parameter, we quantify the effect of time-dependent Earth's gravity field changes as well as the time-dependent sea level changes on the estimation of W0. Our computations resulted in the geoid potential W0 = 62636848.102 ± 0.004 m2 s−2 and the semi-major and minor axes of the MEE, a = 6378137.678 ± 0.0003 m and b = 6356752.964 ± 0.0005 m, which are 0.678 and 0.650 m larger than those axes of GRS80 reference ellipsoid, respectively. Moreover, a new estimation for the geocentric gravitational constant was obtained as GM = (398600460.55 ± 0.03) × 106 m3 s−2. [ABSTRACT FROM AUTHOR]

The author continues to investigate additional planetary-level stresses that occur in the crust due to distributed tangential mass forces. Such forces may be related to the daily rotation of the Earth and movements of the relatively solid core relative to the geocenter. In [Rebetskii, 2016], he discusses how the tangential mass forces in the continental crust are influencing additional meridional and latitudinal stresses and attempted to explain regularities of planetary fracturing. In this paper, he considers the role of the tangential mass forces in the occurrence of lateral movement of the lithospheric plates.The author proposes to estimate amplitudes of the tangential mass forces from the difference between the two global ellipsoids of rotation. The reference ellipsoid averages the level surface of the gravity potential, and the second ellipsoid averages the physical surface of the Earth, separately considering continents and oceans. The Earth’s dynamic compression factor estimated from satellite data is 1/305.5. This value corresponds well to the average polar compression of the two rotation ellipsoids, which approximately describes the shape of the Earth’s physical surface. Thus, in the first approximation, the polar compression of the Earth’s physical surface is less than that of the reference ellipsoid (1/298.25) that approximately describes the shape of the level surface of gravity (i.e. geoid).Gravity vectors deviate from the normal to the physical surface of the Earth by relatively small angles, according to calculations from the data on rotation ellipsoids (a maximum value of 16.4 at the 45° latitude). Tangential mass forces are thus small (2.15×10–4 G/cm3 at the 45° latitude). Due to small tangential forces, shear stresses about 0.3 MPa may occur at the base of the continental lithosphere (depths of 120–150 km). In their turn, such stresses can cause a shear flow in the asthenosphere, which provides for movements of the lithospheric plates at velocities of a few centimeters per year. The estimates in this study suggest that the tangential mass forces can be viewed as a possible source of the movements of the lithospheric plates.Regional rotation ellipsoids, that average the physical surface of the continental and oceanic parts of the Earth, were estimated separately for the northern and southern hemispheres. The largest deviations of the ellipsoids from the reference ellipsoid were revealed for the oceanic parts of both hemispheres of the Earth. The regional ellipsoids for the oceanic parts show smaller polar compression (1/313.1 in the northern hemisphere, and 1/306.9 in the southern hemisphere) than that of the reference ellipsoid, and this predetermines the north-south orientation of the tangential mass forces from the poles to the equator. Compared to the reference ellipsoid, polar compression values estimated for the regional ellipsoids of the continental crust are larger (1/296.2) in the northern hemisphere and smaller (1/303.2) in the southern hemisphere. According to the calculations, the oceanic lithosphere makes the major contribution to submeridional movements of the continental plates.

The aim of this paper is to study the size, shape, and orientation of the asteroid Vesta based on the Dawn spacecraft observations. In this line, three main reference surfaces are defined; the geoid, reference ellipsoid, and best-fit ellipsoid. To be consistent with International Astronomical Union (IAU) standards, all the computations are done in the Claudia Double–Prime coordinate system. The geoid and its potential value W 0 are computed by fitting an equipotential surface to the shape of Vesta in a least-squares sense regarding the topographic bias effect. We find that the topographic bias effect on the geoid potential value as well as the geoidal and topographic heights is significant. The geoid potential value W 0 is estimated equal to 68709 ± 24 m 2 /s 2 . The reference ellipsoid is defined as an equipotential ellipsoid which best fits the geoid. The reference ellipsoid is computed based on the fundamental geodetic constants which define a geodetic reference system (GRS). The semi-axes of the triaxial reference ellipsoid are found to be equal to 280413 ± 104 m, 274572 ± 102 m, and 231253 ± 86 m with the equatorial semi-major axis longitude 8.29°E, while the semi-axes of the biaxial reference ellipsoid are 278556 ± 117 m and 229921 ± 76 m. The results show that the geoidal heights with respect to the triaxial reference ellipsoid are significantly different from the geoidal heights with respect to the biaxial reference ellipsoid. The parameters of the best-fit ellipsoid are estimated by fitting geometrical ellipsoids with various degrees of freedom to Vesta's shape in a least-squares sense. We report the semi-axes of the general best-fit ellipsoid with 9 degrees of freedom equal to 284562 ± 75 m, 277248 ± 73 m, and 226405 ± 57 m. Regarding the orientation of Vesta, we find that the angle between the equatorial semi-major axis of the general best-fit ellipsoid and the X-axis of the coordinate system is 9.17 ∘ ± 0.47 , and the angle between its polar axis and the Z-axis is equal to 0.63 ∘ ± 0.05 . Furthermore, the spherical harmonic coefficients of the shape model of Vesta up to degree 180 are computed, giving the ability to estimate the parameters of the general best-fit ellipsoid. The significant differences observed between the parameters of the general best-fit ellipsoid derived from the spherical harmonic coefficients of the shape model, and those derived from the fitting are due to the large polar and equatorial flattenings of Vesta. We estimate that the offset of the center of figure from the center of mass is 1350 ± 53 m, and the mean radius of Vesta is 261403 ± 39 m. [ABSTRACT FROM AUTHOR]

Earth Gravitational Models (EGMs) describe the Earth’s gravity field including the geoid, except for its zero-degree harmonic, which is a scaling parameter that needs a known geometric distance for its calibration. Today this scale can be provided by the absolute geoid height as estimated from satellite altimetry at sea. On the contrary, the above technique cannot be used to determine the geometric parameters of the Mean Earth Ellipsoidal (MEE), as this problem needs global data of both satellite altimetry and gravimetric geoid models, and the standard technique used today leads to a bias for the unknown zero-degree harmonic of the gravimetric geoid height model. Here we present a new method that eliminates this problem and simultaneously determines the potential of the geoid (W0) and the MEE axes. As the resulting equations are non-linear, the linearized observation equations are also presented.

Chronometric geodesy applies general relativity to study the problem of the shape of celestial bodies including the earth, and their gravitational field. The present paper discusses the relativistic problem of construction of a background geometric manifold that is used for describing a reference ellipsoid, geoid, the normal gravity field of the earth and for calculating geoid's undulation (height). We choose the perfect fluid with an ellipsoidal mass distribution uniformly rotating around a fixed axis as a source of matter generating the geometry of the background manifold through the Einstein equations. We formulate the post-Newtonian hydrodynamic equations of the rotating fluid to find out the set of algebraic equations defining the equipotential surface of the gravity field. In order to solve these equations we explicitly perform all integrals characterizing the interior gravitational potentials in terms of elementary functions depending on the parameters defining the shape of the body and the mass distribution. We employ the coordinate freedom of the equations to choose these parameters to make the shape of the rotating fluid configuration to be an ellipsoid of rotation. We derive expressions of the post-Newtonian mass and angular momentum of the rotating fluid as functions of the rotational velocity and the parameters of the ellipsoid including its bare density, eccentricity and semi-major axes. We formulate the post-Newtonian Pizzetti and Clairaut theorems that are used in geodesy to connect the parameters of the reference ellipsoid to the polar and equatorial values of force of gravity. We expand the post-Newtonian geodetic equations characterizing the reference ellipsoid into the Taylor series with respect to the eccentricity of the ellipsoid, and discuss the small-eccentricity approximation. Finally, we introduce the concept of relativistic geoid and its undulation with respect to the reference ellipsoid, and discuss how to calculate it in chronometric geodesy by making use of the anomalous gravity potential.