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Cosmic ray cut-off rigidity tables and maps over the world concerning the epochs 2010, 2015 and the current one 2020 have been constructed. These maps display the effective cut-off rigidity in every five degrees in latitude and in longitude at the altitude of 20 km above the surface of the international reference ellipsoid. The values of the geomagnetic cut-off rigidity were calculated in every 5° in latitude and in every 15° in longitude applying the well-known method of particle trajectory calculations resulted from the theory of the particle motion in the Earth's magnetic field. The applied software employed the 12th Generation of the International Geomagnetic Reference Field (IGRF 12) and trajectories were calculated at 0.01 GV intervals in order to determine the vertical cut-off rigidity for each location. Beyond the use of the calculated cut-off rigidity values as a basic reference of charged particle access to different geographical locations during quiet and/or more intense geomagnetic periods, these results can be used for a long- term forecasting of the geomagnetic conditions variations.

The advent of Global Navigation Satellite System (GNSS) and Earth Gravitational Model (EGM)particularly Global Positioning System (GPS) has modernized geodetic surveying in providing horizontal and vertical positions of points with a sub-meter level of accuracy over the reference ellipsoid. The GPS gives ellipsoidal heights which makes the conversion of the heights to orthometric heights possible by incorporating a geoid model. The conventional method of determining orthometric height is tedious, time-consuming, and labour intensive. This study entails the determination of orthometric height using GNSS and EGM data. A total of forty-nine (49) stations selected within the study area were occupied for GPS observation using South DGPS instrument in static mode for the position and ellipsoidal height determination. The geoidal height values of the GPS derived data were computed using GeoidEval utility software with reference to three different EGMs (EGM2008, EGM96 and EGM84). In order to determine the orthometric heights of the selected stations, the difference between the EGM geoidal height values (NEGM) and the ellipsoidal heights were computed. The results show that the orthometric height obtained with respect to EGM2008 gives better results with a standard deviation of 9.530m and a standard error of 1.361m. The study reveals that the use of GNSS and EGM data for orthometric height determination is less expensive, less tedious, accurate and time-saving compared to the conventional approach of geodetic and spirit levelling.

SUMMARY An algorithm and software are developed for fast and accurate evaluation of the elements of the geomagnetic field represented in high-degree (>720) solid spherical harmonics at many scattered points in the space above the surface of the Earth. The algorithm is based on representation of the geomagnetic field elements in solid ellipsoidal harmonics and application of tensor product needlets. Open source FORTRAN and MATLAB realizations of this algorithm that rely on data from the Enhanced Magnetic Models 2015, 2017 (EMM2015, EMM2017) have been developed and extensively tested. The capabilities of the software are demonstrated on the example of the north, east and down components of the geomagnetic field as well as the derived horizontal intensity, total intensity, inclination and declination. For the range from −417 m under the Earth reference ellipsoid up to 1000 km above it the FORTRAN and MATLAB versions of the software run 465 and 189 times faster than the respective FORTRAN and MATLAB versions of the software using the standard spherical harmonic series method, while the accuracy is less than 1 nT and the memory (RAM) usage is 9 GB.

A transversal coordinate frame based on a reference ellipsoid is proposed to solve problems associated with strapdown inertial navigation systems (SINSs) in polar regions. However, a commercial-grade SINS can still suffer from cumulative errors even if the transversal coordinate frame is used. To solve this problem, complementary sensors can be used to constitute an integrated navigation system. Because SINS navigational quantities are transformed to the transversal coordinate frame in polar regions, we analyze the performance of a global positioning system and a celestial navigation system in this paper when conducting the coordinate transformation. The derivation and evaluation results show how we reconcile the benefits in SINSs and the negative effects in attitude sensors when adopting the transversal coordinate frame.

The orthometric and normal heights are the two altitude systems most used in the practical realization of the local and national vertical geodesic data. The main difference between these two concepts is that the actual mean gravity along the plumb line between the geoid and the topographic surface theoretically defines the orthometric heights, as well as the normal mean gravity along the plumb line between the reference ellipsoid and teluroid is used in the definition of normal heights. In the Brazilian geodetic system, only a correction of non-parallelism is applied, resulting in the normal-orthometric heights. The objective of this work is to calculate the separation between geoid and quasi-geoid obtained through the systems of normal-orthometric and rigorous orthometric heights. By means of numerical tests, it was observed that the separation is greater when using the brazilian system with values between -0.5 and 0.5 meters. On the other hand, by using the new rigorous heights, the average gravity along the long plumb line is evaluated, taking into account the variations of topographic densities and masses above the geoid, the values are smaller, being in the range of -0.05 to 0.2 meters. It can be said that the rigorous orthometric heights system is more accurate, since it has the smallest difference between the geoid/quasi-geoid separation.

The airborne Operation IceBridge and spaceborne CryoSat-2 missions observe polar sea ice at different spatial and temporal scales as well as with different sensor suites. Comparison of data products from IceBridge and CryoSat-2 is complicated by the fact that they use different geophysical corrections: reference ellipsoid, geoid model, tide model, and atmospheric corrections to derive surface elevation and sea-ice freeboard. In this paper, we compare sea-ice surface elevation and freeboard using eight coincident CryoSat-2, Airborne Topographic Mapper (ATM), and Land, Vegetation, and Ice Sensor (LVIS) observations with direct IceBridge underflights of CryoSat-2 ground tracks. We apply identical geophysical corrections to CryoSat-2 and IceBridge data to eliminate elevation biases due to these differences and focus on differences due to retracker performance. The IceBridge ATM and LVIS elevation and freeboard and Snow Radar snow depth data sets are averaged to each CryoSat-2 footprint for comparison. With snow depth measurements, we are able to compare elevations and freeboards at the snow/ice interface for five different CryoSat-2 retrackers (ESA, GSFCv1, AWI, JPL, and GSFCv2) and IceBridge. The overall mean of freeboard differences between GSFCv2, ESA, AWI, JPL retrackers, and ATM are in agreement within 0.05 m. However, the five different CryoSat-2 retrackers show distinct differences in mean elevation over leads and over floes. This suggests that the physical interpretation of the different retrackers needs to be considered depending on usage, for example, elevations from CryoSat-2 retrackers need to be carefully calibrated before comparing with elevation from other satellites for long-term surface elevation trends.

Global Navigation Satellite System (GNSS) low-cost multi-frequency receivers are argued as an alternative to geodetic receivers for many applications. Calibrated low-cost antennas recently became available on the market making low-cost instruments more comparable with geodetic ones. The main goal of this research was to evaluate the noise of low-cost GNSS receivers, to compare the positioning quality from different types of low-cost antennas, and to analyze the positioning differences between low-cost and geodetic instruments. The results from a zero baseline test indicated that the u-blox multi-frequency receiver, namely, ZED-F9P, had low noise that was at the sub-millimeter level. To analyze the impact of the antennas in the obtained coordinates, a short baseline test was applied. Both tested uncalibrated antennas (Tallysman TW3882 and Survey) demonstrated satisfactory positioning performance. The Tallysman antenna was more accurate in the horizontal position determination, and the difference from the true value was only 0.1 mm, while, for the Survey antenna, the difference was 1.0 mm. For the ellipsoid height, the differences were 0.3 and 0.6 mm for the Survey and Tallysman antennas, respectively. The comparison of low-cost receivers with calibrated low-cost antennas (Survey Calibrated) and geodetic instruments proved better performance for the latter. The geodetic GNSS instruments were more accurate than the low-cost instruments, and the precision of the estimated coordinates from the geodetic network was also greater. Low-cost GNSS instruments were not at the same level as the geodetic ones, however, considering their cost, they demonstrated excellent performance that is sufficiently appropriate for various geodetic applications.

In this paper, a combination of satellite altimetry (SA) and tide-gauge (TG) observations is used to determine the pattern of vertical deformation at southern coasts of Caspian Sea. Satellite altimetry measures the variations of sea water surface with a respect to a reference ellipsoid, while tide gauge data gives the sea level relative to a particular ground station; consequently, the difference between these two observations will give the vertical motion at the ground station. However, altimeter satellites still have problems at shallow waters or coastal areas which leads to inaccurate and corrupted sea surface height at tide gauge stations. To minimize this imprecision, two strategies are applied for the selection of the suitable SA points near the TG stations. The First one is to choose those close passes to the TG stations, provided that the SA data is not corrupted (range observation is available). The second strategy is to extrapolate the SA points at TG stations using Spatio- Temporal Kriging method. In both point of views, use is made of re-tracking methods to improve the ranges of SA observations near the coast. Amplitudes of tidal components are removed from both SA and TG observations to derive the residual tide free signals. Afterward, a linear trend is fitted to the residual signals in the sense of least square; the difference between these two, supplies the vertical deformation of the ground station. The results illustrate that the average vertical crustal motion at Anzali, Noshahr and Neka are -1.387, -1.903 and -3.14 mm/year, respectively.

Advanced Topographic Laser System (ATLAS) instrument on-board NASA's Ice, Cloud, and Land Elevation Satellite-2 (ICESat-2) enabled the capability of detecting certain properties of the Earth's surface at a photon level. Primary applications of ICESat-2 are for cryosphere but extendable to measure and determine the changes in elevation for land terrain and canopy. ICESat-2 ATL08 data product is a level 3A product that contains the best fit height above a reference ellipsoid processed in fixed 100 m data segments, which typically uses more than 100 signal photons from level 2 product namely ATL03. In this article, ICESat-2 ATL08 has been evaluated for its accuracy. Accuracy assessment of a point ellipsoidal height can be done using known benchmarks or survey points. In the absence of known points, the performance of these points can be assessed by hydro-flatness test as well as by comparing the satellite observed surface water level with gauged water table height. Both these tests have been done on 5 large reservoirs in the Indian sub-continent. Hydro-flatness test on ICESat-2’s ATL08 data product determined the permissible variations that occur on the surface water body due to ripples, wavelets, and streaks at a centimeter-level. Water level from ICESat-2 data will be in ellipsoidal height and cannot be compared with gauged data as it gives the height of the water surface above an established altitude where the stage is zero, because of this we have compared the change of surface water level from different dates. The results indicate that the surface water levels from ICESat-2’s ATL08 data product are in line with gauged data at centimeter-level accuracy.

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.

Islamic studies, including fiqh, using an integration-interconnection approach with other sciences are important to do to get a more complete perspective. One of them is the interconnection of “hisab-rukyat” fiqh with geodesy. In general, the study of the Qibla direction, the beginning of prayer times, the beginning of the qamariah month, and eclipses in “hisab-rukyat” fiqh, uses the general formula of spherical trigonometry with a round earth reference. Round earth references in the study of the region differ from the real condition of the earth in the form of an ellipsoid as described in geodesy. To get more valid results, it is important to convert latitude and place height. The latitude of the place obtained by reference or GPS is the geodetic latitude with the earth reference in the form of an ellipsoid, whereas the spherical trigonometry formula refers to the shape of a round Earth. The place height obtained through GPS is the ellipsoid height, not the desired orthometric height (above the average surface of sea water). By converting these data, studies of the “hisab-rukyat” fiqh can be more valid. By increasing the validity of the calculation results, the interconnection between “hisab-rukyat” fiqh and geodesy appears in the form of complementation, meaning that data and findings of geodesy can complement data and analysis in “hisab-rukyat” fiqh, so that it is possible to draw more valid conclusions.

In geodesy three surfaces, the physical surface of the earth, the geoid and the reference ellipsoid are encountered giving rise to orthometric height (h), the ellipsoidal height (H) and the geoidal separation (N). The orthometric height and the ellipsoidal height are with reference to the geoid and the reference ellipsoid respectively. The vertical separation between the ellipsoid and the geoid is the geoidal separation. A mathematical relation depicting the surface of the geoid with regard to the reference ellipsoid is the geoid model. It relates the geoidal separation with the horizontal location.The Global Navigational Satellite System provides precise location of points on the surface of the earth. The vertical location provided is the ellipsoidal height which needs conversion to a more usable format, the orthometric height. This is done by integrating ellipsoidal heights with a geoid model. The accuracy of conversion depends on the accuracy of geoid model. Therefore, development of geoid model has become a current area of research in geodesy.Objective of this study is to develop a local geoid model by employing various polynomial models and thereafter to analyse the accuracy of these models. The test area is in Papua New Guinea. The geometric method is used for computation of the geoidal separation from ellipsoidal and orthometric heights and thereafter the horizontal coordinates and the geoidal separation are used to develop the geoid surface using second, third and fourth degree polynomials. The study shows that the third degree polynomial provided an accuracy of ±20 cm. Keywords: Geodesy, Orthometric height, Ellipsoidal height, Geoidal separation, Geoid model

Rising sea levels pose one of the greatest threats to coastal zones. However, sea-level changes near the coast, particularly absolute sea-level changes, have been less well monitored than those in the open ocean. In this study, we aim to investigate the potential of Global Navigation Satellite Systems Interferometric Reflectometry (GNSS-IR) to measure coastal absolute sea-level changes and tie on-land (coastal GNSS) and offshore (satellite altimetry) observations into the same framework. We choose three coastal GNSS stations, one each in regions of subsidence, uplift and stable vertical land motions, to derive both relative sea levels and sea surface heights (SSH) above the satellite altimetry reference ellipsoid from 2008 to 2020. Our results show that the accuracy of daily mean sea levels from GNSS-IR is

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.

Alturas niveladas pertenecientes a las Redes de Nivelación Fundamental (RNF), sirven de base para el establecimiento de la referencia vertical fundamental de cada país. Cuando estas alturas no están asociadas a observaciones del campo de gravedad terrestre, teniendo por tanto características puramente geométricas, pierden utilidad en la solución de problemas relacionados con fenómenos de carácter regional y global vinculados con la dinámica del Sistema Tierra. Adicionalmente, diferentes Data Verticales Locales (DVL) no materializan la misma referencia, debido a que el NMM determinado en cada DVL presenta discrepancias en relación a un Sistema de Referencia Vertical Global (SRVG). Alturas geométricas provenientes de posicionamiento satelital (GNSS), podrían ser usadas para la determinación de alturas físicas si fuese conocida la relación entre el elipsoide de nivel (superficie de referencia para las alturas geométricas) y una superficie de referencia basada en el campo de la gravedad terrestre. Una aproximación a dicha relación es obtenida de la diferencia entre las alturas elipsoidales y niveladas (h-Hn). Los resultados obtenidos en el presente trabajo muestran que las Redes Neuronales Artificiales (RNA) son un método promisorio en la predicción de la relación geométrica h-Hn. El entrenamiento de una RNA multicapa ha sido realizado, la información de base proviene de registros de la RNF de Ecuador (RNFE) asociados a observaciones GPS (GPS/nivelación), los resultados muestran que la predicción de h-Hn puede ser realizada registrando errores máximos de 7 cm.

This paper presents two new direct symbolic-numerical algorithms for the transformation of Cartesian coordinates into geodetic coordinates considering the general case of a triaxial reference ellipsoid. The problem in both algorithms is reduced to finding a real positive root of a sixth degree polynomial. The first approach consists of algebraic manipulations of the equations describing the geometry of the problem and the second one uses Gr\"obner bases. In order to perform numerical tests and accurately compare efficiency and reliability, our algorithms together with the iterative methods presented by M. Ligas (2012) and J. Feltens (2009) have been implemented in C++. The numerical tests have been accomplished by considering 10 celestial bodies, referenced in the available literature. The obtained results clearly show that our algorithms improve the aforementioned iterative methods, in terms of both efficiency and accuracy., Comment: 17 pages

In the past, geoid was computed from gravity anomaly data using Stokes or Molodensky approaches. Obtaining gravity anomaly data is difficult because it needs some reductions of gravity from surface of the earth to the geoid using orthometric height from spirit level measurement. In the modern era, gravity anomaly data may be replaced by gravity disturbance data. It only required gravity and GNSS (Global Navigation Satellite System) measurement. This research aimed to determine geoid using Hotine’s approach. Disturbance data were generated from archived free air anomaly of airborne gravimetry in Sulawesi area. South East Sulawesi province was selected as a case study area. In this study, gravity observation was calculated at an altitude of 4000 m above the reference ellipsoid. Gravity estimation at the same height aims to increase the precision of the downward continuation process to the geoid. Hotine integral is calculated above the geoid, so that the gravity disturbance data is downwarded to the geoid. The geoid undulation is graded from north to south. Geoid from airborne gravity around Pegunugan Mekongga in the northern part of Southeast Sulawesi Province has the largest geoid undulation which reaches 63 m, while the geoid in the southern part of Buton Island reaches 52 m. Geoid validation of airborne gravity at 13 test points produces a standard deviation of ± 0.050 m. The standard deviation is much smaller than the results of geoid testing from airborne data in North Sulawesi, Central Sulawesi and Southeast Sulawesi. This fact indicates that the Hotine approach has the potential to produce a precise geoid if used in geoid-based airborne gravity calculations.

The adjustment problem of the so-called combined (hybrid, integrated) network created with GNSS vectors and terrestrial observations has been the subject of many theoretical and applied works. The network adjustment in various mathematical spaces was considered: in the Cartesian geocentric system on a reference ellipsoid and on a mapping plane. For practical reasons, it often takes a geodetic coordinate system associated with the reference ellipsoid. In this case, the Cartesian GNSS vectors are converted, for example, into geodesic parameters (azimuth and length) on the ellipsoid, but the simple form of converted pseudo-observations are the direct differences of the geodetic coordinates. Unfortunately, such an approach may be essentially distorted by a systematic error resulting from the position error of the GNSS vector, before its projection on the ellipsoid surface. In this paper, an analysis of the impact of this error on the determined measures of geometric ellipsoid elements, including the differences of geodetic coordinates or geodesic parameters is presented. Assuming that the adjustment of a combined network on the ellipsoid shows that the optimal functional approach in relation to the satellite observation, is to create the observational equations directly for the original GNSS Cartesian vector components, writing them directly as a function of the geodetic coordinates (in numerical applications, we use the linearized forms of observational equations with explicitly specified coefficients). While retaining the original character of the Cartesian vector, one avoids any systematic errors that may occur in the conversion of the original GNSS vectors to ellipsoid elements, for example the vector of the geodesic parameters. The problem is theoretically developed and numerically tested. An example of the adjustment of a subnet loaded from the database of reference stations of the ASG-EUPOS system was considered for the preferred functional model of the GNSS observations.

The transversal coordinate system has been proposed to solve the problem of navigation in the Polar Region for a strapdown inertial navigation system (INS). However, it is assumed that the Earth is a perfect sphere in the transversal coordinate system, which is not accurate for a high-precision INS. To solve this problem, we consider the Earth as a reference ellipsoid and then deduct its radii of curvature in our transversal coordinate system. In addition, with the aid of the Bessel geodetic projection method, transformations of navigation quantities from a local-level coordinate system to our transversal coordinate system are derived based on the reference ellipsoid. Numerical results indicate that our method based on the reference ellipsoid performs better than the method based on a reference sphere.

Time-varying Stokes coefficients estimated from GRACE satellite data are routinely converted into mass anomalies at the Earth’s surface with the expression proposed for that purpose by Wahr et al. (J Geophys Res 103(B12):30,205–30,229, 1998). However, the results obtained with it represent mass transport at the spherical surface of 6378 km radius. We show that the accuracy of such conversion may be insufficient, especially if the target area is located in a polar region and the signal-to-noise ratio is high. For instance, the peak values of mean linear trends in 2003–2015 estimated over Greenland and Amundsen Sea embayment of West Antarctica may be underestimated in this way by about 15%. As a solution, we propose an updated expression for the conversion of Stokes coefficients into mass anomalies. This expression is based on the assumptions that: (i) mass transport takes place at the reference ellipsoid and (ii) at each point of interest, the ellipsoidal surface is approximated by the sphere with a radius equal to the current radial distance from the Earth’s center (“locally spherical approximation”). The updated expression is nearly as simple as the traditionally used one but reduces the inaccuracies of the conversion procedure by an order of magnitude. In addition, we remind the reader that the conversion expressions are defined in spherical (geocentric) coordinates. We demonstrate that the difference between mass anomalies computed in spherical and ellipsoidal (geodetic) coordinates may not be negligible, so that a conversion of geodetic colatitudes into geocentric ones should not be omitted.

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.

Terrain visualization is an important aspect of many applications, especially of 3D geographic information systems, which try to preserve the accuracy of the applied terrain model. A discrepancy of data representation, efficient streaming and the application to a chosen mathematical model of the whole planet, still imposes a challenge to a number of large terrain rendering engines. In this paper, we present ellipsoidal clipmaps - a new way to apply geometry clipmaps to an ellipsoid. The ellipsoid is divided into three partitions, seamlessly stitched together. Unlike previous geometry clipmap approaches, based on preloaded grid upon which the terrain is elevated, the underlying ellipsoidal grid is generated on the fly in the vertex shader. The proposed method guarantees sub-pixel precision of the planet Earth reference ellipsoid surface calculation, exploiting GPU single precision floating-point arithmetic. A tight coupling with the implementation of clipmaps enables consistent caching and fast data streaming, while the usage of equidistant cylindrical projection for the source data minimizes preprocessing and preserves low-to-moderate texture distortion. Graphical abstractDisplay Omitted HighlightsA real-time visualization of the planet Earth.The surface is divided into three partitions seamlessly stitched in the globe.A sub-pixel precise WGS84 reference ellipsoid generation on the fly.Consistent caching and fast data streaming through geometry clipmap application.Minimizing texture distortion through the usage of geographic projection.

A surface spherical harmonic expansion of gravity anomalies with respect to a geodetic reference ellipsoid can be used to model the global gravity field and reveal its spectral properties. In this paper, a direct and rigorous transformation between solid spherical harmonic coefficients of the Earth’s disturbing potential and surface spherical harmonic coefficients of gravity anomalies in ellipsoidal approximation with respect to a reference ellipsoid is derived. This transformation cannot rigorously be achieved by the Hotine–Jekeli transformation between spherical and ellipsoidal harmonic coefficients. The method derived here is used to create a surface spherical harmonic model of gravity anomalies with respect to the GRS80 ellipsoid from the EGM2008 global gravity model. Internal validation of the model shows a global RMS precision of $$

Integrated navigation algorithms under the grid frame have been proposed based on the Kalman filter (KF) to solve the problem of navigation in some special regions. However, in the existing study of grid strapdown inertial navigation system (SINS)/Doppler velocity log (DVL) integrated navigation algorithms, the Earth models of the filter dynamic model and the SINS mechanization are not unified. Besides, traditional integrated systems with the KF based correction scheme are susceptible to measurement errors, which would decrease the accuracy and robustness of the system. In this paper, an adaptive robust Kalman filter (ARKF) based hybrid-correction grid SINS/DVL integrated navigation algorithm is designed with the unified reference ellipsoid Earth model to improve the navigation accuracy in middle-high latitude regions for marine application. Firstly, to unify the Earth models, the mechanization of grid SINS is introduced and the error equations are derived based on the same reference ellipsoid Earth model. Then, a more accurate grid SINS/DVL filter model is designed according to the new error equations. Finally, a hybrid-correction scheme based on the ARKF is proposed to resist the effect of measurement errors. Simulation and experiment results show that, compared with the traditional algorithms, the proposed navigation algorithm can effectively improve the navigation performance in middle-high latitude regions by the unified Earth models and the ARKF based hybrid-correction scheme.

All geodetic measured quantities from the physical surface of the Earth are reduced to the geoid and surface of adopted reference ellipsoid through applying corrections on the measured values, as well projected to the map projection. Before calculating the Cartesian coordinates of points in a formal state geodetic coordinate system, the horizontal lengths between points on the topography surface first should to be reduced from the topography to the mean see level (geoid), then from the geoid to the reference ellipsoid, and after that projected from the ellipsoid into the map projection. Up to now, within the research analyses have been performed for defining of most appropriate State Plane coordinate system, the choice of most suitable national map projection is based on the requirements of accurate representation of the earth surface on the plane with minimum distortions. This choice is usually based on the research analyses performed for defining of most appropriate State Plane coordinate system. During the literature review, analyses of the length changes between points within successive projections from the earth surface to the chosen state map projection were not found. Recognition of the length differences for whole country areas mapped on the State Plane coordinate system are very important data for many geodetic and cartographic applications. This research aims to develop and provide national GIS datasets of lengths' differences on a grid with cells 1x1 km covering national territories of seven Balkan countries – Albania, Bosnia and Herzegovina, Bulgaria, Croatia, Kosova, FYRo Macedonia and Montenegro. Assessment and comparison of successive length differences between four surfaces (from the Earth surface, through geoid and ellipsoid, to the map projection) have been realized through calculation of mean linear deformations of 1km lengths, dispersion, maximum and minimum values, as well as their positive, negative or zero values. The differences between the real length on earth surface and on a map projection due to elevation factor were evaluated, that gave an impression on the quality of state geodetic coordinate systems, of the seven analyzed countries.

Physical heights in geodesy are expressed through the theory of potential, and their definition is based on the concept of geoid and equipotential sur faces. When defining the conceptually simpler ellip soidal heights, only the geometric aspects of the Earth's body are considered. In recent years it becomes rel atively easy to determine the ellipsoidal heights by using modern technologies of global positioning. However, these heights are measured from the geocentric ellipsoid, while, for the determination of the ellipsoidal he ight related to the reference ellipsoid, it is necessary to know the geoid undulation relative to the consi dered ellipsoid. The results of the global navigation sat ellite system measurements must be transformed into the state coordinate system. This procedure is called t he datum transformation. To carry out the transform ation it is necessary to know the ellipsoidal heights in bot h systems. As these heights are often not known in the state coordinate system, they are replaced by known value s of the physical heights. The datum transformation based on the indicated approximation certainly chan ges the values of the determined transformation parameters, as well as the values of the transforme d coordinates. The intensity of these changes depen ds on several factors, but the change of the transformati on results itself affects geodetic computations in different ways.

The determination of the geoid in South Korea is a national imperative for the modernization of height datums, specifically the orthometric height and the dynamic height, that are used to monitor hydrological systems and environments with accuracy and easy revision, if necessary. The geometric heights above a reference ellipsoid, routinely obtained by GPS, lead immediately to vertical control with respect to the geoid for hydrological purposes if the geoid height above the ellipsoid is known accurately. The geoid height is determined from gravimetric data, traditionally ground data, but in recent times also from airborne data. This paper illustrates the basic concepts for combining these two types of data and gives a preliminary performance assessment of either set or their combination for the determination of the geoid in South Korea. It is shown that the most critical aspect of the combination is the gravitational effect of the topographic masses above the geoid, which, if not properly taken into account, introduces a significant bias of about 8 mgal in the gravity anomalies, and which can lead to geoid height bias errors of up to 10 cm. It is further confirmed and concluded that achieving better than 5 cm precision in geoid heights from gravimetry remains a challenge that can be surmounted only with the proper combination of terrestrial and airborne data, thus realizing higher data resolution over most of South Korea than currently available solely from the airborne data.

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.

Forward gravity modeling in the spectral domain traditionally relies on spherical approximation. However, this level of approximation is insufficient for some present day high-accuracy applications. Here we present two solutions that avoid the traditional spherical approximation in spectral forward gravity modeling. The first solution (the extended integration method) applies integration over masses from a reference sphere to the topography and applies a correction for the masses between ellipsoid and sphere. The second solution (the harmonic combination method) computes topographic potential coefficients from a combination of surface spherical harmonic coefficients of topographic heights above the ellipsoid, based on a relation among spherical harmonic functions introduced by Claessens (2005). Using a degree 2160 spherical harmonic model of the topographic masses, both methods are applied to derive the Earth's ellipsoidal topographic potential in spherical harmonics. The harmonic combination method converges fastest and—akin to the EGM2008 geopotential model—generates additional spherical harmonic coefficients in spectral band 2161 to 2190 which are found crucial for accurate evaluation of the ellipsoidal topographic potential at high degrees. Therefore, we recommend use of the harmonic combination method to model ellipticity in spectral-domain forward modeling. The method yields ellipsoidal topographic potential coefficients which are “compatible” with global Earth geopotential models constructed in ellipsoidal approximation, such as EGM2008. It shows that the spherical approximation significantly underestimates degree correlation coefficients among geopotential and topographic potential. The topographic potential model is, for example, of immediate value for the calculation of Bouguer gravity anomalies in fully ellipsoidal approximation.

The paper presents a solution to one of the basic problems of computational geodesy - conversion between Cartesian and geodetic coordinates on a biaxial ellipsoid. The solution is based on what is known in the literature as “latitude equation”. The equation is presented in three different parameterizations commonly used in geodesy - geodetic, parametric (reduced) and geocentric latitudes. Although the resulting equations may be derived in many ways, here, we present a very elegant one based on vectors orthogonality. As the “original latitude equations” are trigonometric ones, their representation has been changed into an irrational form after Fukushima (1999, 2006). Furthermore, in order to avoid division operations we have followed Fukushima’s strategy again and rewritten the equations in a fractional form (a pair of iterative formulas). The resulting formulas involving parametric latitude are essentially the same as those introduced by Fukushima (2006) (considered the most efficient today). All the resulting variants are solved with Newton’s second-order and Halley’s third-order formulas. It turns out that all parameterizations of the “latitude equation” show a comparable level of performance.

Civil engineering projects require the orthometric height of points (i.e., the height of points above geoid). Thus, to make use of the Global Navigation Satellite System (GNSS) in engineering surveying, geoid undulation (i.e., the separation between an ellipsoid and geoid surface) must be known. In this study, an attempt has been made to determine the geoid undulation using the astro-geodetic method. Observations were taken in a network of seven stations spread over 300 km2 around Dehra Dun, India. Astronomical observations of seven stars belonging to different constellations were taken with a theodolite having a least count of 1.0 s. Global positioning system (GPS) observations were taken using dual-frequency geodetic GPS receivers with a 0.1-mm-accurate digital level. Values of astro-geodetic geoid undulations (of stations) have been found close to that computed from field levelling. Further, the values have been compared with those obtained from global geoid models such as EIGEN6C3stat and EGM2...

In geopotential space, the fundamental geodetic parameter W 0 defines the Gauss–Listing geoid which can be used to best represent the Earth’s mean sea level (MSL) and hence specifies a conventional zero height level to unify vertical datums employed by mapping agencies throughout the world. Further, W 0 cannot be considered invariant as the parameter varies temporally as a direct response to sea level change and mass redistributions. This study determines W 0 and its rate, dW 0/dt, by utilizing altimetric MSL models and an independent mean dynamic topography (MDT) model to define points on the geoid. W 0 and dW 0/dt are estimated by two approaches: (i) by means of a global gravity field model (GGM) and (ii) within normal gravity field space as the geopotential value of the best fitting reference ellipsoid. The study shows that uncertainty in W 0 is mainly influenced by MDT while the choice of methodology, GGM and MSL data coverage are not significant within reason. Our estimate W 0 = 62636854.2 ± 0.2 m2 s−2 at epoch 2005.0 differs by 1.8 m2s−2 from the International Astronomical Union reference value. This study shows that, at a sub-decadal time scale, the time variation dW 0/dt stems mainly from sea level change with negligible effect from gravity field variations. dW 0/dt = (−2.70 ± 0.03) × 10−2 m2 s−2 year−1, corresponding to a MSL rise of 2.9 mm year−1, is evaluated from sea level change based on 16 years of TOPEX and Jason-1 data.

The availability of high-resolution global digital elevation data sets has raised a growing interest in the feasibility of obtaining their spherical harmonic representation at matching resolution, and from there in the modelling of induced gravity perturbations. We have therefore estimated spherical Bouguer and Airy isostatic anomalies whose spherical harmonic models are derived from the Earth's topography harmonic expansion. These spherical anomalies differ from the classical planar ones and may be used in the context of new applications. We succeeded in meeting a number of challenges to build spherical harmonic models with no theoretical limitation on the resolution. A specific algorithm was developed to enable the computation of associated Legendre functions to any degree and order. It was successfully tested up to degree 32,400. All analyses and syntheses were performed, in 64 bits arithmetic and with semi-empirical control of the significant terms to prevent from calculus underflows and overflows, according to IEEE limitations, also in preserving the speed of a specific regular grid processing scheme. Finally, the continuation from the reference ellipsoid's surface to the Earth's surface was performed by high-order Taylor expansion with all grids of required partial derivatives being computed in parallel. The main application was the production of a 1' x 1' equiangular global Bouguer anomaly grid which was computed by spherical harmonic analysis of the Earth's topography-bathymetry ETOPO1 data set up to degree and order 10,800, taking into account the precise boundaries and densities of major lakes and inner seas, with their own altitude, polar caps with bedrock information, and land areas below sea level. The harmonic coefficients for each entity were derived by analyzing the corresponding ETOPO1 part, and free surface data when required, at one arc minute resolution. The following approximations were made: the land, ocean and ice cap gravity spherical harmonic coefficients were computed up to the third degree of the altitude, and the harmonics of the other, smaller parts up to the second degree. Their sum constitutes what we call ETOPG1, the Earth's TOPography derived Gravity model at 1' resolution (half-wavelength). The EGM2008 gravity field model and ETOPG1 were then used to rigorously compute 1' x 1' point values of surface gravity anomalies and disturbances, respectively, worldwide, at the real Earth's surface, i.e. at the lower limit of the atmosphere. The disturbance grid is the most interesting product of this study and can be used in various contexts. The surface gravity anomaly grid is an accurate product associated with EGM2008 and ETOPO1, but its gravity information contents are those of EGM2008. Our method was validated by comparison with a direct numerical integration approach applied to a test area in Morocco-South of Spain (Kuhn, private communication 2011) and the agreement was satisfactory. Finally isostatic corrections according to the Airy model, but in spherical geometry, with harmonic coefficients derived from the sets of the ETOPO1 different parts, were computed with a uniform depth of compensation of 30 km. The new world Bouguer and isostatic gravity maps and grids here produced will be made available through the Commission for the Geological Map of the World. Since gravity values are those of the EGM2008 model, geophysical interpretation from these products should not be done for spatial scales below 5 arc minutes (half-wavelength).

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.

In this study, we suggested the period of tide observations is proper to calculate the mean sea level(MSL) precisely on Incheon tide station using wavelet analysis, and newly determined then the vertical reference surface of Korea using the calculated MSL. In order to calculate the height difference between the calculated MSL and specific ground station (ICGP) near the Incheon tide stations, we performed the laser measurements directly to the sea surface where located below ICGP. The orthometric-height of ICGP was determined that corrected the height difference to the calculated MSL using linear interpolation method. Finally, we connected the orthometric-height of ICGP with the original benchmark (ORBM) using GPS leveling methods for determining the new orthometric-height of ORBM. As the results, there is a variation amount of 0.026m between the new MSL was calculated in this study and old MSL was calculated in 1910's. Also, there is a difference of 0.035m between the new and old orthometric-heights of ORBM. The connection (or leveling) error of 0.009m was revealed in new orthometric height of ORBM with consideration of MSL variation which may caused by the error of GPS ellipsoid height and/or geoid model. In this study, we could be determined precisely the orthometric-height of ORBM based on the new MSL of Incheon Bay using only GPS leveling method, not a spirit leveling method. Therefore, it is necessary to determine the vertical datum strictly using long-term and continuously tide observations more than 19 years and to use the GPS leveling method widely in the height leveling work for the effective changeover from the orthonormal to the orthometric in national height system.

The object of this paper is to compare the accuracy and the characteristic of various methods of solving the geodetic inverse problem for the geodesic lines which be in the standard case and special cases(antipodal, near antipodal, equatorial, and near equatorial situation) on the WGS84 reference ellipsoid. For this, the various algorithms (classical and recent solutions) to deal with the geodetic inverse problem are examined, and are programmed in order to evaluate the calculation ability of each method for the precise geodesic determination. The main factors of geodetic inverse problem, the distance and the forward azimuths between two points on the sphere(or ellipsoid) are determined by the 18 kinds of methods for the geodetic inverse solutions. After then, the results from the 17 kinds of methods in the both standard and special cases are compared with those from the Karney method as a reference. When judging these comparison, in case of the standard geodesics whose length do not exceed 100km, all of the methods show the almost same ability to Karney method. Whereas to the geodesics is longer than 4,000km, only two methods (Vincenty and Pittman) show the similar ability to the Karney method. In the cases of special geodesics, all methods except the Modified Vincenty method was not proper to solve the geodetic inverse problem through the comparison with Karney method. Therefore, it is needed to modify and compensate the algorithm of each methods by examining the various behaviors of geodesics on the special regions.

Transformation of coordinates usually invokes level-wised processes wherein several sets of complicated equations are calculated. Unfortunately, the accuracy may be corrupted due to the accumulation of inevitable errors between the transformation processes. This paper presents a genetic-based method for generating regressive models for direct transformation from global positioning system (GPS) signals to 2-D coordinates. Since target coordinates for a GPS application can be obtained by using simpler transformation formulas, the computational costs and inaccuracy can be reduced. The proposed method, though does not exclude systematic errors due to the imperfection on defining the reference ellipsoid and the reliability of GPS receivers, effectively reduces the statistical errors when the accurate Cartesian coordinates are known from the independent sources. From the experimental results where the target datums TWD67 is investigated, it seems that the proposed method can serve as a direct and feasible solution to the transformation of GPS coordinates.

A review is given of the geodetic concepts necessary for oceanographers to make use of satellite gravity data to define the geoid, and to interpret the resulting product. The geoid is defined, with particular attention to subtleties related to the representation of the permanent tide, and the way in which the geoid is represented in ocean models. The usual spherical harmonic description of the gravitational field is described, together with the concepts required to calculate a geoid from the spherical harmonic coefficients. A brief description is given of the measurement system in the GOCE satellite mission, scheduled for launch shortly. Finally, a recipe is given for calculation of the ocean dynamic topography, given a map of sea surface height above a reference ellipsoid, a set of spherical harmonic coefficients for the gravitational field, and defining constants.

Transforming height information that refers to an ellipsoidal Earth reference model, such as the geometric heights determined from GPS measurements or the geoid undulations obtained by a gravimetric geoid solution, from one geodetic reference frame (GRF) to another is an important task whose proper implementation is crucial for many geodetic, surveying and mapping applications. This paper presents the required methodology to deal with the above problem when we are given the Helmert transformation parameters that link the underlying Cartesian coordinate systems to which an Earth reference ellipsoid is attached. The main emphasis is on the effect of GRF spatial scale differences in coordinate transformations involving reference ellipsoids, for the particular case of heights. Since every three-dimensional Cartesian coordinate system ‘gauges’ an attached ellipsoid according to its own accessible scale, there will exist a supplementary contribution from the scale variation between the involved GRFs on the relative size of their attached reference ellipsoids. Neglecting such a scale-induced indirect effect corrupts the values for the curvilinear geodetic coordinates obtained from a similarity transformation model, and meter-level apparent offsets can be introduced in the transformed heights. The paper explains the above issues in detail and presents the necessary mathematical framework for their treatment.

The purpose of the research was to compare measuring orthometric heights using differential leveling, static global positioning observations, and real-time kinematic (RTK) Global Positioning System (GPS) observations in the vicinity of the University of Conn. campus. Height observations were recorded at temporary and permanent monuments using differential leveling, static GPS, and RTK. Heights at permanent monuments with published elevations were observed using each method and served as control and to check the quality of all measurements. Three-dimensional reference ellipsoid coordinates derived from GPS observations and published orthometric heights were used to create local geoid models, which were then used to convert ellipsoid heights observed with the static GPS and RTK to orthometric heights. The resulting values were used to compare between the measuring methods. In general, static GPS performed better than RTK. Several polynomial surface models of different orders were fitted to the geoid heights...

We apply general relativity to construct the post-Newtonian background manifold that serves as a reference spacetime in relativistic geodesy for conducting relativistic calculation of the geoid's undulation and the deflection of the plumb line from the vertical. We chose an axisymmetric ellipsoidal body made up of perfect homogeneous fluid uniformly rotating around a fixed axis, as a source generating the reference geometry. We, then, reformulate and extend hydrodynamic calculations of rotating fluids done by previous researchers to the realm of relativistic geodesy to set up algebraic equations defining the shape of the post-Newtonian reference ellipsoid. To complete this task, we explicitly perform all integrals characterizing gravitational field inside the fluid body and represent them in terms of the elementary functions depending on its eccentricity. We fully explore the coordinate freedom of the equations describing the post-Newtonian ellipsoid and demonstrate that the fractional deviation of the post-Newtonian level surface from the Maclaurin ellipsoid can be made much smaller than the previously anticipated estimate based on the coordinate gauge advocated by Bardeen and Chandrasekhar. We also derive the gauge-invariant relations of the post-Newtonian mass and the angular velocity of the rotating fluid with the parameters characterizing the shape of the post-Newtonian ellipsoid. We formulate the post-Newtonian theorems of Pizzetti and Clairaut that are used in geodesy to connect the geometric parameters of the reference ellipsoid to the physically measurable force of gravity at the pole and equator. Finally, we expand the post-Newtonian geodetic equations to the Taylor series with respect to the eccentricity of the ellipsoid and discuss their practical applications., 32 pages, 2 figures, 71 references. Text improved and extended, one more figure included

Global positioning system (GPS) technology has been frequently used to monitor geological hazards associated with ground deformations, such as long-term landslides and subsidence. When GPS data are processed, they yield ellipsoid heights, which are the distances above a smooth ellipsoid surface. However, orthometric heights are often used in practical surveying and engineering applications. Orthometric height is a physical quantity that refers to the surface of the geoid. In this study, a more practical alternative definition of orthometric height is used. This approximation is the one commonly implemented in practical surveying and engineering applications to compute relative orthometric height values. This well-known procedure computes orthometric heights by combining GPS-measured ellipsoid height and a geoid model. Any type of orthometric height is a physically based quantity. GPS alone, which is a geometric technique, cannot directly measure orthometric heights. This study investigates the ver...

In an absolute sense and over long (>100 km) baselines, the AUSGeoid98 gravimetric-only geoid model does not always allow the accurate transformation of Global Positioning System (GPS)-derived ellipsoidal heights to Australian Height Datum (AHD) heights in all regions of Australia. This is due predominantly to the well-known biases and distortions in the AHD, but long-wavelength errors in the gravimetric geoid model or GPS errors cannot be ruled out. Until the AHD is rigorously redefined, an interim solution is sought where co-located GPS and AHD heights are used to distort AUSGeoid98 such that it provides a better model of the separation between the base of the AHD and the GRS80 reference ellipsoid. This data combination was implemented using least squares collocation (LSC) gridding. Importantly, GPS-AHD data not used in the LSC combination were used to assess the improvement using a cross-validation technique. Using this cross-validation, RMS noise of 14 mm and correlation length of 2500 km for the LSC covariance function were optimised empirically. The standard deviation of the differences between the final combined model and the unused GPS-AHD data is ±156 mm, compared to ±282 mm for AUSGeoid98 alone. It is anticipated that the same technique will be used to produce a new Australian “geoid” model.

In this paper, we present techniques for ellipsoid fitting which are based on minimizing the sum of the squares of the geometric distances between the data and the ellipsoid. The literature often uses "orthogonal fitting" in place of "geometric fitting" or "best-fit". For many different purposes, the best-fit ellipsoid fitting to a set of points is required. The problem offitting ellipsoid is encounteredfrequently intheimage processing, face recognition, computer games, geodesy etc. Today, increasing GPS and satellite measurementsprecisionwill allow usto determine amore realistic Earth ellipsoid. Several studies have shown that the Earth, other planets, natural satellites, asteroids and comets can be modeled as triaxial ellipsoids Burša and Šima (1980), Iz et all (2011). Determining the reference ellipsoid for the Earth is an important ellipsoid fitting application, because all geodetic calculations are performed on the reference ellipsoid. Algebraic fitting methods solve the linear least squares (LS) problem, and are relatively straightforward and fast. Fitting orthogonal ellipsoid is a difficult issue. Usually, it is impossible to reach a solution with classic LS algorithms. Because they are often faced with the problem of convergence. Therefore, it is necessary to use special algorithms e.g. nonlinear least square algorithms. We propose to use geometric fitting as opposed to algebraic fitting. This is computationally more intensive, but it provides scope for placing visually apparent constraints on ellipsoid parameter estimation and is free from curvature bias Ray and Srivastava (2008). Neste trabalho, apresentamos a adequação do elipsóide com base na soma dos mínimos quadrados das distâncias geométricas entre os dados e o elipsóide. A literatura muitas vezes aborda a "adequação ortogonal" em lugar da "adequação geométrica" na "melhor adequação". Para muitos propósitos diferentes, a melhor adequação do elipsóide para o conjunto de pontos é necessária. O problema da adequação do elipsóide é encontrado frequentemente no processamento de imagens, reconhecimento de superfícies, jogos de computador, geodésia, etc. Hoje, o aumento da precisão das medidas GPS e de satélites permite que se determine um elipsóide da Terra mais realístico. Diversos estudos, tem demonstrado que a Terra, outros planetas, satélites naturais, asteróides e cometas podem ser modelados com o elipsóides triaxiais, Bursa e Sima (1980), Iz et al (2011). Determinar o elipsóide adequado para a Terra é importante porque se aplica a todos os cálculos geodésicos que são executados sobre o elipsóide de referência. Métodos adequados algebricamente resolvem o problema de adequação por mínimos quadrados lineares, e são relativamente rápaidos e diretos. A adequação do elipsóide ortogonal é um assunto difícil. Normalmente, é impossível encontrar uma solução com algorítimos clássicos de mínimos quadrados. A razão é que eles são frequentemente encontrados com problemas de convergência. Portanto, é necessário usar algorítimos especiais, por exemplo, algorítimos de mínimos quadrados com não lineares. Nós propomos o uso da adequação geométrica em contraste com a adequação algébrica. Isto é computacionalmente mais intensivo, mas dá margem ao uso de injunções relativas na estimação a dos parâmetros do elipsóide e elimina as tendências da curvatura, Ray e Srivastava (2008).