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A geometric technique is introduced to estimate the height of volcanic eruption columns using the generally discarded near-limb portion of geostationary imagery. Such oblique observations facilitate a height-by-angle estimation method by offering close to orthogonal side views of eruption columns protruding from the Earth ellipsoid. Coverage is restricted to daytime point estimates in the immediate vicinity of the vent, which nevertheless can provide complementary constraints on source conditions for the modelling of near-field plume evolution. The technique is best suited to strong eruption columns with minimal tilting in the radial direction. For weak eruptions with severely bent plumes or eruptions with expanded umbrella clouds the radial tilt/expansion has to be corrected for either visually or using ancillary wind profiles. Validation on a large set of mountain peaks indicates a typical height uncertainty of ±500 m for near-vertical eruption columns, which compares favourably with the accuracy of the common temperature method.

In this paper, we consider the double-satellite localization under the earth ellipsoid model of the Wideband Geodetic System (WGS-84) using the Time Difference of Arrival (TDOA) and the Angle-of-Arrival (AOA). Several closed-form solution algorithms via the pseudo-linearization of the measurement equations are presented to efficiently estimate the location. These algorithms include the Weighted Least Squares (WLS), the Constrained Total Least Squares (CTLS), and the Taylor-Series Iteration (TSI). Performance comparison of the proposed methods with the Cramér-Rao Lower Bound (CRLB) in the simulation is shown to demonstrate that the proposed algorithms are feasible and have stable performance. Keywords: Angle-of-Arrival (AOA), Constrained Total Least Squares (CTLS), Cramér-Rao Lower Bound (CRLB), Double-satellite positioning system, Taylor-Series Iteration (TSI), Time Difference of Arrival (TDOA), Weighted Least Squares (WLS), Wideband Geodetic System (WGS-84)

With the increase in satellite communication interference, the tri-satellite time difference of arrival (TDOA) localization technique, which is an effective method to determine the location of the interference using sensors or antennas, has been developed rapidly. The location of the interference source is determined through the intersection of the TDOA lines of position (LOP). However, when the two TDOA LOP have two mirrored intersection points, it is theoretically difficult to determine the real location. Aiming at this problem, a method for eliminating mirrored location based on multiple moment TDOA is proposed in this paper. First, the TDOA results are measured at multiple moments using the cross-ambiguity function (CAF), and the localization equation set is established based on the World Geodetic System (WGS)-84 earth ellipsoid model. Then, the initial location result can be obtained by solving the equation set through the Newton iteration method. Finally, the high-precision location result after eliminating the mirrored location is obtained after the single moment localization based on the initial location. Simulation experiments and real measured data processing results verify the effectiveness of the proposed method. It still has good robustness under the condition of large measurement errors and deviations from the prior initial values.

The datum for sounding reduction is a permanently fixed surface, to which the depths displayed on the nautical charts and the tide tables heights refer. The International Hydrographic Organization recommends adopting the lowest astronomical tide as a chart datum, although its calculation can be complex because it varies both spatially and temporally. The ever increasing accuracy of 3D positioning with Global Navigation Satellite Systems requires that the chart datum is referenced to the ellipsoid WGS84. The aims of this paper are to calculate the lowest astronomical tide and to develop an empirical model to determine the distance between the lowest astronomical tide and the WGS84 ellipsoid, for a region of the Southwest Atlantic Ocean between latitudes 36°S and 54°S and longitude 54°W, on a 5km x 5km grid. Harmonic constants from the Centre for Topographic studies of the Oceans and Hydrosphere are used to calculate the lowest astronomical tide. To refer it to the WGS84 ellipsoid, results from mean sea level models and along-track sea level heights provided by Archiving, Validation and Interpretation of Satellite Oceanographic data are utilized. The final product has been designed for open waters and will be useful both for the development of relevant marine activities in the area, as well as to increase the efficiency of hydrographic surveys while contributing to more precise navigation in critical areas.

This investigation implements a least-squares methodology to fit a triaxial ellipsoid to a set of three-dimensional Cartesian coordinates obtained from present-day geospatial techniques, materializing the terrestrial frame ITRF2014. To approximate, as much as possible previous research on this topic, the original spatial values of the station coordinates were “reduced” to the surface of the EGM2008 geoid model by introducing a simple and straightforward procedure. The mathematical model adopted in all LS solutions is the standard quadric surface polynomial equation parameterizing a triaxial ellipsoid. Functionally related to these polynomial coefficients are nine geometric parameters: the three ellipsoid semi-axes, its origin location with respect to the current conventional geocentric terrestrial frame, and the three rotations defining its spatial orientation. The final results are compatible with the pioneering work started by Bursa in 1970 and, lately, by a recent publication by Panou and colleagues in that incorporates updated geoid models.

The term navigation is commonly used to describe several different but related concepts. Some use the term for the process of determining the position (in some reference frame) and the orientation of the vehicle only, whereas others also include the problem of how to get from a point A to a point B in the navigation concept. In this chapter, we focus solely on the former meaning of the term. In other words, we define navigation to mean estimation of the six degrees of freedom of a rigid body (any vehicle or device), i.e., the position and orientation of the body. The position can be expressed in various reference frames (local or global), and the orientation is defined as the spatial orientation of the body frame relative to a local reference tangential to the Earth ellipsoid. This is more precisely defined in Section 5.2. It is often customary to include the estimation of linear and angular velocities in the navigation problem. The uncertainties of the estimates are often also part of the navigation output, and the navigation can be performed in real time, or in post -processing.

According to the classical Gauss–Listing definition, the geoid is the equipotential surface of the Earth’s gravity field 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, the 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). In this study, we perform 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 m2s-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 the GRS80 reference ellipsoid, respectively. Moreover, a new estimation for the geocentric gravitational constant was obtained as GM = (398600460.55 ± 0.03) × 106 m3s-2.

El presente artículo define la ruta metodológica usada en el proyecto “Red Hidrográfica de Referencia Vertical para los principales puertos marítimos Colombianos”, durante la generación del marco vertical de referencia denominado “Superficie Hidrográfica de Referencia Vertical (SHRV2016)” para las bahías de Buenaventura y Málaga en el Pacífico Colombiano, a partir de tres componentes principales: la determinación matemática de datums verticales asociados a registros históricos de nivel de agua, el establecimiento de una red de vértices geodésicos de primer orden, y la medición de alturas en la superficie del mar con referencia al elipsoide WGS84 (World Geodetic System), empleando receptores del sistema GNSS (Global Navigation Satellite System) en modo diferencial RTK (Real Time kinematic). Los puntos de referencia en tierra fueron utilizados para dar el control vertical de precisión hacia los datums en el agua, y a su vez para la instalación de receptores base GNSS encargados de enviar las correcciones diferenciales en tiempo real durante la medición de alturas en la superficie del mar; Estas mediciones junto a la densificación de datums verticales calculados para la zona, constituyeron la base en la elaboración de la SHRV2016, modelo que con una resolución espacial de 500 metros, refiere al elipsoide WGS84 los datums verticales de nivel de agua, y que además de estandarizar las mediciones que realizan los diferentes usuarios en las aguas jurisdiccionales relativas a un datum vertical; permite aprovechar las ventajas de la tecnología GNSS para realizar la medición y corrección de nivel de agua en tiempo real con el menor grado de incertidumbre posible durante la ejecución de levantamientos hidrográficos, optimizando tiempo, calidad y recursos, especialmente en lugares como las Bahías de Buenaventura y Málaga, donde el régimen de marea hace compleja la actividad de corrección debido a la variabilidad en rango y fase propia del sector, lo cual obliga a instalar múltiples estaciones mareográficas, realizar tareas de topografía y geodesia en lugares de difícil acceso, y esperar al termino del levantamiento hidrográfico para incluir los registros de las estaciones en la corrección final de las profundidades. En un mediano plazo, las SHRV permitirán la generación de un modelo completo de referencia de los datums de marea hacia el elipsoide, que abarque completamente los litorales y regiones insulares del país, hasta los límites del mar territorial, gracias a la combinación con datos de altimetría satelital e instrumentos de observación mar adentro como boyas GPS, que permitan la correcta fusión con los modelos generados a nivel local en cada una de las bahías y puertos.

Accurate determination of the specular point is important for simulating and processing reflected global navigation satellite system signals for remote sensing applications. Existing methods for determining the specular point are based on both spherical and ellipsoidal approximations of the Earth and employ either Snell's law or Fermat's principle to formulate the problem. By analysis and simulation, it is shown that these methods produce significant errors at intermediate latitudes. In this paper, a novel formulation for the solution of the specular point is proposed that satisfies Snell's Law on the WGS84 ellipsoid. The proposed method is compared with the existing methods for various receiver orbit configurations and algorithm augmentations. It is shown that the proposed method is more accurate than the existing methods and more computationally efficient than the minimum path length (MPL) method. Additionally, the resultant grazing angles, MPL method errors, and specular point locations as a function of the receiver orbit are investigated, leading to the finding that the likelihood of an error is significant for geometries favored by reflectometry applications.

The paper focuses on the comparison of metrics of three most common reference surfaces of the Earth used in geodesy (excluding the plane which also belongs to reference surfaces used in geodesy when dealing with small areas): a sphere, an ellipsoid of revolution and a triaxial ellipsoid. The two latter surfaces are treated in a more detailed way. First, the mathematical form of the metric tensors using three types of coordinates is derived and the lengths of meridian and parallel arcs between the two types of ellipsoids are compared. Three kinds of parallels, according to the type of latitude, can be defined on a triaxial ellipsoid. We show that two types of parallels are spatial curves and one is represented by ellipses. The differences of curvature of both kinds of ellipsoid are analysed using the normal curvature radii. Priority of the chosen triaxial ellipsoid is documented by its better fit with respect to the high-degree geoid model EIGEN6c4 computed up to degree and order 2160.

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 m2/s2. 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.

Taking into account the heat exchange, the horizontal inhomogeneities of the gravity field were shown to lead to the origin of dynamic perturbations (flows) in the density stratified medium. The amplitude of the horizontal component of the originating flows is proportional to the product of a geoid deviation from a uniform Earth ellipsoid and buoyancy frequency.

In view of the problems existing in the current wellbore positioning method, a true three-dimensional wellbore positioning method was presented, and an example for analysis was given. The current positioning method uses the grid north as the reference datum to the north, positions in the horizontal plane based on the map projection coordinates, and positions in the vertical direction based on the elevation coordinates. It has inherent defects and errors, as the two positioning methods above are independent of each other, and only use the constant meridian convergence and constant magnetic declination at the wellhead to calculate the borehole trajectory for the whole well. Based on the earth ellipsoid and its calculating principle, the transformation relationship between the wellhead coordinate system, geodetic coordinate system and elevation coordinate system was revealed, and the true three-dimensional wellbore positioning method using the true north as the reference datum to the north was presented. Analysis results of an example show that the current positioning method yields a smaller vertical depth and a larger horizontal displacement for the target, and produces larger errors compared with the true three-dimensional positioning method. The true three-dimensional positioning method has fundamentally solved the problems existing in the current positioning method, accurately positioning the relative location between the target and the wellhead, and significantly improves the accuracy and reliability of borehole trajectory design. Key words: drilling theory, directional drilling, wellbore trajectory, wellbore positioning

The physical surface of the earth has irregular shape which is not mathematically defined, therefore the shape of the Earth is approximated with mathematically defined surfaces such as ellipsoid and sphere. The developing of a global positioning systems, thus and modern navigation systems, as effect produce large amounts of data which contain the problem of homogeneity. This problem could be exceed if all the data are store in the same coordinate system. Hence the need for data transformation from local coordinate systems to the global coordinate systems. Global level implies WGS84 ellipsoid and UTM projection while national coordinate system of Republic Serbia is Gauss-Kruger with Bessel ellipsoid. This coordinate system of Republic Serbia on a global level has a local character. Applying appropriate mathematical models and functions it is possible to transform coordinates from one system to another and vice versa. The paper describes coordinate transformations from Gauss-Kruger coordinate system ellipsoid Bessel to UTM projection WGS84 ellipsoid and vice versa, and also an application which provides transformation of its kind that is made using open source environment. Name of the application is TRANS7_GK_UTM_GK and it can be found and used on the web page of the faculty for Mining and Geology under the link http://gk2utm.rgf.bg.ac.rs with a user guide.

By convention the absolute bias in sea surface height (SSH) is the difference between the altimeter and the in-situ reference SSH heights above the Earth ellipsoid. Both the absolute and the relative bias of the CryoSat-2 and Sentinel-3A missions are derived in this study at four stations along the German coasts.

The method for calculation of geodetic problem on earth ellipsoid surface is the basic premise for the realization of high-precision marine delimitation technology. Based on the analysis of the construction principle of the existing solution of geodetic problem method, the algorithm for direct geodetic problem solution(DGPS) based on nested coefficient method and Bessel’s inverse geodetic problem solution(IGPS), which are suitable for various distance solutions, are optimized. Then, the experimental verification scheme of the solution of geodetic problem is designed. Finally, the crossing and reducing accuracy of different calculation examples are verified by different geodetic problem solution methods, and the experiment shows that: the verified accuracy of point-position plotting, distance calculation and azimuth calculation meet the requirements of the indexes of high-precision marine delimitation.

The series in ellipsoidal harmonics for derivatives of the Earth’s gravity potential are used only on the reference ellipsoid enveloping the Earth due to their very complex mathematical structure. In the current study, the series in ellipsoidal harmonics are constructed for first- and second-order derivatives of the potential at satellite altitudes; their structure is similar to the series on the reference ellipsoid. The point P is chosen at a random satellite altitude; then, the ellipsoid of revolution is described, which passes through this point and is confocal to the reference ellipsoid. An object-centered coordinate system with the origin at the point P is considered. Using a sequence of transformations, the nonsingular series in ellipsoidal harmonics is constructed for first and second derivatives of the potential in the object-centered coordinate system. These series can be applied to develop a model of the Earth’s potential, based on combined use of surface gravitational force measurements, data on the satellite orbital position, its acceleration, or measurements of the gravitational force gradients of the first and second order. The technique is applicable to any other planet of the Solar System.

Despite incontrovertible scientific evidence to support a round Earth, GIS software implementation typically models the Earth with flat map projections. This choice has consequences that vary from mild to severe. This paper explores solutions that range from moderate measures to correct for map projection errors to radical revisions of standard practice that place all calculations on the ellipsoid. On examination, the best practice requires a distinct projection choice for each geometric operation. Multiple reasons, from the technical to ethical, justify revised practices and increased vigilance for the errors induced by inappropriate projections.

The standard spheroidal reference surface also referred as the reference ellipsoid is a part of mapping basic infrastructures of a country. In Cameroon, the ellipsoid Clarke 1880 with unknown parameters used for a long time as reference system in basic mapping, has lead to the use of World Geodetic System 1984 (WGS 84) whose parameters are well known. Meanwhile, the latter is not accurate locally due to the fact that the best ellipsoid is the one that fits very well with the local geoid. In order to look for the parameters of the local ellipsoid that fits best to the local geoid (Cameroon Geoid Model 2011 (CGM11)), the Global Positionning System (GPS) data made of 525 geodetic ground control points of the new geodetic network of Cameroon set up in 2011 were used. These GPS measures provide for each point the values of the ellipsoid height and the elevation which are used to determine the parameters of the local ellipsoid model for Cameroon through the least square form of the Molodensky analytical method. The results are given as the difference in ellipsoidal height relative to the parameters of the WGS84 ellipsoid assuming the two ellipsoids in their parallel. These results show that the obtained ellipsoid fits better to the local geoid with 1.072 m as the standard deviation value, is improved considerably in comparison to the previous studies conducted in Cameroon whose standard deviation was fairly equal to 1.679 m.

Orthometric heights, useful for many engineering and geoscience applications, can be obtained by GPS (Global Positioning System) surveys only when an accurate geoid undulation model (that supplies the vertical separation between the geoid and WGS84 ellipsoid) is available for the considered topic area. Global geoid height models (i.e., EGM2008), deriving from satellite gravity measurements suitably integrated with other data are free available on web, but their accuracy is often not sufficient for the user’s purposes. More accurate local models can nevertheless be acquired, but often only for a fee. GPS/levelling surveys are suitable for determining a local, accurate geoid model, but may be too expensive. This paper aims to demonstrate that GNSS (Global Navigation Satellite System) Permanent Station documents (monographs), freely available on the web and supplying orthometric and ellipsoidal heights, permit to calculate precise geoidal undulations useful to perform global geoid modelling on a local area. In fact, in this study 25 GNSS Permanent Stations (GNSS PS), located in North-Western Italy are considered: the differences between GNSS PS geoidal heights and the corresponding EGM2008 1′ × 1′ ones are used as a starting dataset for Ordinary Kriging applications. The resulting model is summed to the EGM2008 1′ × 1′, obtaining a better-performed model of the interest area. The accuracy tests demonstrate that the resulting model is better than EGM2008 grids to produce contours from a GPS dataset for large-scale mapping.

The accurate computation of positions of points from DGPS/GNSS observations acquired on the WGS84 ellipsoid in each country requires the application of transformation parameters. This is because, positions are determined with respect to the local datum/ellipsoid adopted for geodetic computation in each country. Prior to the derivation of the new/unified transformation parameters which were recently published by the Office of the Surveyor-General of the Federation, OSGOF, various sets of transformation parameters were determined by different multinational oil companies for use in Nigeria. Since the positions of reasonable number of points had been computed using these old sets of transformation parameters before the new/unified ones were derived and promulgated, there is need to verify which of the positions computed with the old sets of transformation parameters agree with the positions computed with the new/unified ones. Consequently, this paper compares the positions computed from DGPS/GNSS observations using the various, old sets and the unified transformation parameters. GNSS observations of four different points were processed with Compass post processing software using the various, old sets and the unified transformation parameters to obtain the positions of the points. The coordinates of each point obtained from the processing of the observations using the old sets of transformation parameters were compared with those obtained using the unified ones. The comparison results show that the positions computed using three different (SPDC, AGIP and DMA) sets of transformation parameters agree with those computed using the unified ones.

In recent years, with the rapid development of information technology and the improvement of people’s quality of life, location service has become an indispensable part of people’s life. In the outdoor environment, only three satellites can be received in some locations. In this case, the traditional satellite positioning method is difficult to obtain more accurate positioning results. The existing satellite positioning technology uses differential barometric altimeter to construct the earth ellipsoid model, which can be located under the condition of visible three satellites. However, its positioning accuracy is poor. Therefore, this paper presents a virtual satellite positioning technology based on differential pressure altimetry and PDR to solve the above issues. The technology needs to use the coordinate of the previous positioning result as the reference point coordinate to establish virtual satellite pseudo-range observation equations. Newton iterative method is used to get the result of single-point positioning result. The Kalman filter is used to fuse the inertial sensor information and single-point positioning results to get more accurate positioning results. The experimental verification shows that the proposed technology can also locate in the visible three-satellite case with the accuracy of1.82 meters (1 σ), which is 27.63% more accurate than the existing three-satellite positioning technology based on terrestrial mobile communication base stations. When four satellite signals are received, the proposed technology has a positioning accuracy of about 1.44 meters (1 σ), which is 32.08% more accurate than the test of same environment with only four satellites for positioning.

To use the dynamics of GPS satellites to complement geometrical Earth rotation and orientation parameters from VLBI has always been a challenge. Geometric VLBI differs from other space geodesy techniques, such as GNSS, SLR and DORIS, in that it does not rely on satellite dynamics to estimate terrestrial reference frame parameters. Here we present a geometrical approach that combines dynamic and geometric variations in the Earth’s rotation with temporal gravity field variations. We demonstrate that this novel approach provides a new insight into the Earth’s interior, especially into processes and dynamics associated with the Earth’s fluid outer core and the great earthquakes over the last 10 and 100 years. Firstly, we demonstrate that by combining two LAGEOS satellites in low MEO orbit we can remove errors in secular orbit perturbations stemming from low zonal harmonics (J2) and give new insights into the Earth’s rotation and nutation rates. Nutation rates were first estimated from GPS data including orbit determination of GPS satellites (Rothacher et al. 1999). Here we extend the theoretical model of nutation rates and show how, with the nodal separation close to 180° of the two LAGEOS satellites, common orbital errors in terms of nodal and apsidal orbit precession are eliminated. This approach based on celestial mechanics opens up the possibility of using satellite dynamics to determine rates of nutation and variations in length-of-day (LOD) very accurately and correlate them against the variations in the temporal gravity field (errors in J2 are eliminated). This then leads us to the unexplained rate of variations in dynamical LOD estimated from GPS/LAGEOS (orbits driven by Earth’s gravity) and from geometrical LOD from VLBI (external measure of Earth’s orientation). We show how the rotation of spherical harmonics can explain this unresolved effect since rotation of the tri-axial Earth ellipsoid is the real physical phenomenon measured by gravity field missions as well as by SLR to LAGEOS satellites. We show how the geometrical rotation of spherical harmonics is equivalent to temporal gravity field variations and in the case of second degree harmonics is directly proportional to the rate of variations in LOD. This was presented for the first time in Svehla (2008). The conventional IERS mean pole model is in very good agreement with the terrestrial pole of the GRACE monthly gravity field models (derived from \( \bar{C}_{21} \) and \( \bar{S}_{21} \) gravity field coefficients). We show that temporal variations in the orientation of the tri-axial Earth ellipsoid (sectorials) are taking place along the equatorial plane, i.e., sharing the same axis of rotation within

Due to the complicated structure of their expressions, the ellipsoidal harmonic series for the derivatives of the Earth’s gravitational potential are commonly applied only on a reference ellipsoid. They depend on the first- and second-order derivatives of the associated Legendre functions of both kinds and contain a few singular terms. We construct ellipsoidal harmonic expansions in the exterior space for the first and second potential derivatives, which are similar to the series on the reference ellipsoid enveloping the Earth. We take a point P at an arbitrary altitude above the reference ellipsoid and construct the ellipsoid of revolution confocal to it, which passes through this point. The conventional complicated singular expressions for the first and second potential derivatives in the local north-oriented ellipsoidal reference frame, with the origin at the point P, are transformed into non-singular ellipsoidal harmonic series, which do not contain the first- and second-order derivatives of the associated Legendre functions. The resulting series have an accuracy of the squared eccentricity. These series can be applied for constructing a geopotential model, which is based, simultaneously, on the surface gravity data and the data of satellite missions, which provide measurements of the accelerations and/or the gravitational gradients. When the eccentricity of the considered external ellipsoid is equated to zero, the ellipsoid becomes an external sphere passing through the point P and the constructed ellipsoidal harmonic expansions are converted into non-singular spherical harmonic series for the first and second potential derivatives in the local north-oriented spherical reference frame.

Many small solar system bodies such as asteroids or small satellites have irregular shapes, often approximated by the reference surface of a triaxial ellipsoid. Map projections for the triaxial ellipsoid are needed to present the incoming data in the form of maps. In this paper the formulae of equal-area cylindrical and azimuthal projections of the triaxial ellipsoid were derived and practically implemented for the first time using as an example the asteroid 253 Mathilde. This paper is the final in a series of papers devoted to all main classes of projections of the triaxial ellipsoid. Before this, the authors obtained equidistant along meridians projection and Jacobi conformal projection for the triaxial ellipsoid.

Voronoi diagram on the earth surface is a powerful tool to study spatial proximity at continental or global scale. However, its computation remains challenging because geospatial features have complex shapes. This paper presents a raster-based algorithm to generate Voronoi diagrams on earth's surface. The algorithm approximates the exact point-to-point geographical distances using the cell-to-cell geographical distances calculated by a geographical distance transform. The result is a distance image on which Voronoi diagram is delineated. Compared to existing methods, the proposed algorithm calculates geographical distances based on an earth ellipsoid and allows Voronoi generators to take complex shapes. Most importantly, its approximation error is bounded thus enabling users to control the accuracy of the Voronoi diagram through grid resolution. We developed a raster-based method to generate Voronoi diagrams on the earth.The method is based on geographical distance transform.Voronoi generators are allowed to have any arbitrary shape.The accuracy of the Voronoi diagram can be controllable through grid resolution.

It is a general trend to adopt the geocentric coordinate system as a geodetic datum for the international measurement community. The definition and realization of Chinese geocentric three-dimensional coordinate system (CGCS2000) which has been employed since July 1st, 2008 were introduced in detail. The defining parameters and derived constants of the reference ellipsoid used were given. The comparison between CGCS2000 and WGS84 was carried out. The differences of geodetic coordinates of a point between the two coordinate systems, normal gravity and vertical gradient of normal gravity on the two ellipsoids caused by the change of the flattening of the ellipsoid were analyzed. The results show that these differences could be neglected in view of present measurement accuracies.

This paper presents an application of a new adaptive filtering algorithm based on a standard earth ellipsoid model in tracking ballistic missile in free flight phase. The traditional Extended Kalman Filtering algorithm (EKF) has defect in two aspects: 1) observation values applied in observation equation mainly include azimuth, pitch angle and radial distance (or radial velocity), without Doppler frequency; 2) Jacobi matrix of the system needs to be calculated, and high nonlinearization could result in degradation of calculation precision. In this paper, an algorithm used for tracking ballistic missile in free flight phase which is based on UKF observation equation with Doppler frequency is presented. The algorithm has been applied in hybrid coordinate system of the standard earth ellipsoid model (the earth is assumed as an ellipsoid planet). It has higher tracking precision compared with EKF algorithm. Monte Carlo simulation conducted to model ballistic missile in free flight phase by using MATLAB software justified the efficacy and the reasonableness of the new algorithm.

The scroll imaging of a space camera can improve its temporal resolution. However, the space camera shows different image motion speeds and drift angles at different positions of field of view in the scroll imaging because of the earth ellipsoid and varied scroll angles. To overcome the problem, the formulas of image motion speeds and drift angles at different positions of field of view were deduced for the scroll imaging based on earth ellipsoid. By taking a high resolution space camera carried on a stereoscopic mapping satellite for an example, the influence of row transfer periods and drift angles adjusting uniformly and individually on Modulation Transfer Function(MTF) was analyzed. Analysis results and on-orbit testing experiments under a constraint of 5% drop for the MTF indicate that the row transfer periods should be adjusted individually if the integral stages are more than 22 in scroll imaging with scroll angle of 10. In comparison to adjusting row transfer periods uniformly, the drop of MTF along the track direction would be cut down from 10.28% to 0.11% by adjusting row transfer periods individually. If integral stages are fixed to 16, the scroll angle by adjusting row transfer periods uniformly should not be over 13.2.

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.

Since the continuation of an external gravity field inside topographic masses by a harmonic function results in topographic bias, geoid computation by means of global gravity models (GGMs) in terms of external-type series of spherical harmonics, at locations where the GGMs are evaluated inside the topographic masses, will be biased. Consequently, if the reference ellipsoid is defined based on the geoid, it will also be biased. In this paper, the effects of topographic bias on the geoid and reference ellipsoid of Venus, Mars, and the Moon are studied. Moreover, a thorough error analysis in the geoid and reference ellipsoid computation is presented, and it is shown that the estimated standard deviation (STD) of the geoid potential value, the geoidal heights, and the semimajor and semiminor axes of the reference ellipsoid are independent of the topographic bias. According to the results, the effects of topographic bias on the geoid potential value and the semimajor and semiminor axes of the reference ellipsoid in comparison with their estimated STDs are insignificant for Venus, Mars, and the Moon. Moreover, the effect of topographic bias on the geoidal heights of Venus as compared with the estimated STD of its geoidal heights is insignificant. However, the effects of topographic bias on the geoidal heights of Mars and the Moon can be significant, especially in high mountains such as the Tharsis volcanic region on Mars.

Summary: In recent years, with the development of the virtual reality technology and data acquisition technology, people increase the demand of GIS visualization. Especially accounting for occupying 70 percent of global area, and as a based environmental visualization, global ocean visualization is particularly important in some applications. This paper studies the global ocean visualization and modeling techniques under the framework of the WGS84 ellipsoid and achieves a method of rapid global ocean photorealistic rendering. The main research works are as follow: 1. In the height field modeling, with the law of statistical and spectral marine and Phillip wave spectrum, we can produce a single height map which considers the wind farm on the ocean wave magnitude of impact; 2. With ellipsoid rectangular grid mapping relationship, the single height map produced above will be mapped to the ellipsoid repeatedly, and achieve a goal of global ocean height field modeling; 3. With the conversion of screen space coordinate system and the rectangular spatial coordinate system, sampling points can be acquired by the view-dependent ellipsoid; 4. With the introduction of global bathymetric data, and came through the GPU for rapid sampling, so that we can get sampling points related to transparency and depth values to achieve a global ocean and land border processing.

Double-star navigation and positioning system use Double-star positioning system which basic principles are the intersection of three balls. Two measurements relate to the distance of the two satellites and the user, and the third measurements which relate to the user elevation are the critical factor to determine the positioning precision. For this reason, based on the Earth ellipsoid geometry and geodetic latitude knowledge, this paper derives the accurate model of the third measurement, and establishes a user location algorithm. The algorithm is simple and easy to understand, and simulation tests show that the algorithm has fast convergence and accurate positioning.

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.

Gravimetry entails measuring the acceleration of gravity because of the attraction experienced by a mass that is part of the entire mass of the Earth system. By providing information on the distribution of mass within the Earth, gravimetry is a geodesy discipline: a form of science with the goal of determining the shape of the Earth, its gravitational field and its rotation. The shape of the Earth is not only characterized by its surface topography, but also by its gravitational field, particularly the geoid, which is the equipotential gravity materialized by the ocean surface ideally at rest, meaning a surface where water is in gravitational equilibrium. The geoid extends under the continents, thus defining level zero of elevation. With respect to the ellipsoid that is the best approximation of the shape of the Earth, or reference ellipsoid, the geoid deviations up to ± 100 m.

The large ellipsoidal aluminium plate system for missile in lab simulation[2], reaching a length of nearly 100 meters, is an important component of the laboratory. How to make it well-designed to avoid repetition and reduce the loss is the key to the project. This paper proposes the calculation methods andmodels for the geometrical parameters such as longitude and latitude line length, radius of curvature , keel curve direction on ellipsoid and deflecting ellipsoid. The practical problem is resolved successfully by designing the calculation models.

The InSAR (Interferometric SAR) processing steps for DEM generation consist of the coregistration of two SAR data, interferogram generation, phase filtering, phase unwrapping, phase to height conversion, and geocoding, etc. In this study, we developed the precise algorithm for phase to height conversion, including the ambiguity method taking into account Earth ellipsoid, Schw?visch method, and the refined ambiguity method suitable for the interferometric pair with non-parallel obit. From the testing with JERS-1 orbit we found that the height error by traditional ambiguity method reaches to about 40 m during phase to height conversion. The proposed methods are very useful in generating precise InSAR DEM, especially in the case of using non-parallel InSAR pair due to unstable orbit control such as JERS-1 or intentional orbit control such as Cross-InSAR pair between ERS2 and ENVISAT satellite.

The paper presents modified Lin and Wang’s (1995) and Hedgley’s (1976) algorithms to tackle the problem of transforming Cartesian to geodetic coordinates on a triaxial ellipsoid. Originally, the methods were developed for an ellipsoid of revolution but due to their universality, they may be adapted to the more complicated problem stated on a triaxial ellipsoid what is in fact done in this work. Two modified methods are compared to the vector method recently introduced by Feltens. The modified methods turn out to be more accurate and faster than the algorithm presented by Feltens.

High-resolution earth-observing satellites acquire substantial amount of geospatial images. In addition to high image quality, high-resolution satellite images (HRSI) provide unprecedented direct georegistration accuracy, which have been enabled by accurate orbit determination technology. Direct georegistration is carried out by relating the determined position and attitude of camera to the ground target, i.e., projecting an image point to the earth ellipsoid using the collinearity equation. However, the apparent position of ground target is displaced due to the atmosphere and satellite velocity causing significant georegistration bias. In other words, optic ray from the earth surface to satellite cameras at 400~900km altitude refracts due to the thick atmosphere which is called atmospheric refraction. Velocity aberration is caused by high traveling speed of earth-observing satellites, approximately 7.7 km/s, relative to the earth surface. These effects should be compensated for accurate direct georegistration of HRSI. Therefore, this study presents the equation and the compensation procedure of atmospheric refraction and velocity aberration. Then, the effects are simulated at different image acquisition geometry to present how much bias is introduced. Finally, these effects are evaluated for Quickbird and WorldView-1 based on the physical sensor model.

Field observations refer to the vertical of the geoid, because the geodetic instruments are put in the working position in respect to the vertical. It is not appropriate for the calculations on the ellipsoid surface, which request measured elements to be referred to the normal of the reference ellipsoid. This means that the field observations must be reduced from the vertical to the normal, i.e. they must be somehow reduced from the Earth's physical surface to the surface of the reference ellipsoid. There are two methods to do this: Pizzetti's method and Helmert's method. Pizzeti's method projects the measured elements with the aid of the vertical onto the geoid surface and then, with the aid of the normal to the ellipsoid, projects them further onto the surface of the reference ellipsoid. Helmert's (or Bruns-Helmert's) method projects the measured elements directly onto the surface of the reference ellipsoid with the aid of the normal to the ellipsoid. But, until the middle of the 20th century, none of them were used. This means that the field observations were not properly reduced onto the ellipsoid's surface; they were reduced only to the geoid's surface and then used as if they were reduced to the ellipsoid. The reasons fot that are simple. On the one hand, it was almost impossible to determine the relation between the geoid and the ellipsoid at each desired point on the Earth's physical surface (first of all the geoid undulations N and the components of the deflection of the vertical i and c), so it was very difficult to calculate the reductions. On the other hand, the consequences of neglecting the reduction from the geoid to the ellipsoid were too small, compared with the accuracy of the observations in that time.

The imaging principles of a three-line stereo mapping camera was analyzed.To overcome the differences of image motion speeds and drift angles of the forward-view camera,nadir-view camera and backward-view camera,the calculation formulas of image motion speeds and drift angles for forward-view,nadir-view and backward-view cameras based on an ellipsoid model were deduced.By taking a three-line stereo mapping for an example and using the WGS84 earth ellipsoid model,the influences of image motion speed matching remains and drift angle remains on the image quality of a three-line stereo mapping camera were analyzed when the forward-view and backward-view cameras both adjust the row transfer periods and drift angles same as the nadir-view camera.Results indicate that with 5% reduction of MTF as a constraint,if more than 2 TDI stages are used nadir-view camera,the forward-view and backward-view cameras should adjust their row transfer periods respectively;However,if drift angles of forward-view and backward-view cameras are adjusted according to nadir-view camera,TDI stages should be no more than 40.

Lunar geodetic parameters, which play an important role in lunar exploration, can be calculated from the gravity and topography data. With the CE-1 altimetry data and LP gravity model, we calculate such geodetic parameters as the principle moment of inertia, the principle inertia axes, equatorial radius, polar radius, mean radius, flattening and offset between center of mass and center of figure (DCOM-COF). According to the CE-1 altimetry data and the above geodetic parameters, a tri-axial ellipsoid (CE-1-LAM-GEO) and a tri-axial level ellipsoid (CE-1-LAM-LEVEL) are calculated individually, providing mass center and figure center offset (DCOM-COF) and parameters more reliable in direction and magnitude.