On the geoid and orthometric height vs. quasigeoid and normal height

Abstract The geoid, but not the quasigeoid, is an equipotential surface in the Earth’s gravity field that can serve both as a geodetic datum and a reference surface in geophysics. It is also a natural zero-level surface, as it agrees with the undisturbed mean sea level. Orthometric heights are physical heights above the geoid,while normal heights are geometric heights (of the telluroid) above the reference ellipsoid. Normal heights and the quasigeoid can be determined without any information on the Earth’s topographic density distribution, which is not the case for orthometric heights and geoid. We show from various derivations that the difference between the geoid and the quasigeoid heights, being of the order of 5 m, can be expressed by the simple Bouguer gravity anomaly as the only term that includes the topographic density distribution. This implies that recent formulas, including the refined Bouguer anomaly and a difference between topographic gravity potentials, do not necessarily improve the result. Intuitively one may assume that the quasigeoid, closely related with the Earth’s surface, is rougher than the geoid. For numerical studies the topography is usually divided into blocks of mean elevations, excluding the problem with a non-star shaped Earth. In this case the smoothness of both types of geoid models are affected by the slope of the terrain,which shows that even at high resolutions with ultra-small blocks the geoid model is likely as rough as the quasigeoid model. In case of the real Earth there are areas where the quasigeoid, but not the geoid, is ambiguous, and this problem increases with the numerical resolution of the requested solution. These ambiguities affect also normal and orthometric heights. However, this problem can be solved by using the mean quasigeoid model defined by using average topographic heights at any requested resolution. An exact solution of the ambiguity for the normal height/quasigeoid can be provided by GNSS-levelling.

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On the advantage of normal heights

Geodesy and Cartography ◽

10.22389/0016-7126-2018-939-9-2-9 ◽

2018 ◽

Vol 939 (9) ◽

pp. 2-9

Author(s):

V.V. Popadyev

Keyword(s):

Gravitational Field ◽

Tide Gauge ◽

General Term ◽

Reference Ellipsoid ◽

Height Anomaly ◽

Normal Height ◽

Anomalous Field ◽

Normal Heights ◽

Basic Concepts ◽

Orthometric Heights

The author analyzes the arguments in the report by Robert Kingdon, Petr Vanicek and Marcelo Santos “The shape of the quasigeoid” (IX Hotin-Marussi Symposium on Theoretical Geodesy, Italy, Rome, June 18 June 22, 2018), which presents the criticisms for the basic concepts of Molodensky’s theory, the normal height and height anomaly of the point on the earth’s surface, plotted on the reference ellipsoid surface and forming the surface of a quasigeoid. The main advantages of the system of normal heights, closely related to the theory of determining the external gravitational field and the Earth’s surface, are presented. Despite the fact that the main advantage of Molodensky’s theory is the rigorous determining the anomalous potential on the Earth’s surface, the use of the system of normal heights can be shown and proved separately. To do this, a simple example is given, where the change of marks along the floor of a strictly horizontal tunnel in the mountain massif is a criterion for the convenience of the system. In this example, the orthometric heights show a change of 3 cm per 1.5 km, which will require corrections to the measured elevations due the transition to a system of orthometric heights. The knowledge of the inner structure of the rock mass is also necessary. It should be noted that the normal heights are constant along the tunnel and behave as dynamic ones and there is no need to introduce corrections. Neither the ellipsoid nor the quasi-geoid is a reference for normal heights, because so far the heights are referenced to initial tide gauge. The points of the earth’s surface are assigned a height value; this is similar to the ideas of prof. L. V. Ogorodova about the excessive emphasis on the concept of quasigeoid. A more general term is the height anomaly that exists both for points on the Earth’s surface and at a distance from it and decreases together with an attenuation of the anomalous field.

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Strategy for Connecting to the IHRF: Case Study for the Tide Gauge of Cananeia-SP

Anuário do Instituto de Geociências - UFRJ ◽

10.11137/1982-3908_2021_44_41104 ◽

2021 ◽

Vol 44 ◽

Author(s):

Fabio Luiz Albarici ◽

Gabriel Do Nascimento Guimarães ◽

Marcelo Carvalho Santos ◽

Jorge Luiz Alves Trabanco

Keyword(s):

Standard Deviation ◽

Tide Gauge ◽

Tide Gauges ◽

Normal Height ◽

Numerical Tests ◽

Lower Accuracy ◽

Normal Heights ◽

Congruence Analysis ◽

Orthometric Heights

In July 2018, IBGE launched the new heights of the Brazilian Geodetic System (BGS), the normal height, which has associated gravity. These new heights are replacing the old normal-orthometric ones, in which there was only the non-parallelism correction. The IBGE informs that the values farther from the origin, have less accuracy. This lower accuracy may interfere in the future, the connection of the local tide gauges to IHRF (International Reference Frame Height). Thus, this paper proposes the integration of the local tide gauge of Cananeia-SP to the IHRF. In order to validate the methodology, the normal, Helmert, and rigorous orthometric heights using two distinct references: the Imbituba-SC tide gauge, as the origin of the BGS and the Cananeia-SP tide gauge, as a local tide gauge to be integrated into the IHRF. Calculating the three heights through these two origins, we analyzed the discrepancies in comparison to the heights calculated by IBGE. Numerical tests indicate that there was an improvement in terms of a mean and standard deviation when using the Cananeia gauge as origin in the calculation of normal, Helmert, and rigorous heights. In the congruence analysis, the calculations indicate that the highest standard deviation is presented when using IBGE normal heights. Thus, we have a new origin that is reliable and functional, can be integrated with the IHRF, where the Helmert and rigorous orthometric heights have the best statistical results.

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On the uncertainty of height anomaly differences predicted by least-squares collocation

Journal of Geodetic Science ◽

10.1515/jogs-2020-0111 ◽

2020 ◽

Vol 10 (1) ◽

pp. 53-61

Author(s):

E. Mysen

Keyword(s):

Least Squares ◽

Spatial Separation ◽

Height Anomaly ◽

Surface Model ◽

Least Squares Collocation ◽

Normal Heights ◽

Gravimetric Quasigeoid ◽

The Difference ◽

Grid Interpolation ◽

Gps Observations

AbstractA network of pointwise available height anomalies, derived from levelling and GPS observations, can be densified by adjusting a gravimetric quasigeoid using least-squares collocation. The resulting type of Corrector Surface Model (CSM) is applied by Norwegian surveyors to convert ellipsoidal heights to normal heights expressed in the official height system NN2000. In this work, the uncertainty related to the use of a CSM to predict differences in height anomaly was sought. As previously, the application of variograms to determine the local statistical properties of the adopted collocation model led to predictions that were consistent with their computed uncertainties. For the purpose of predicting height anomaly differences, the effect of collocation was seen to be moderate in general for the small spatial separations considered (< 10 km). However, the relative impact of collocation could be appreciable, and increasing with distance, near the network. At last, it was argued that conservative uncertainties of height anomaly differences may be obtained by rescaling output of a grid interpolation by \sqrt \Delta, where Δ is the spatial separation of the two locations for which the difference is sought.

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Exploring the Limits of Floating-Point Resolution for Hardware-In-the-Loop Implemented with FPGAs

Electronics ◽

10.3390/electronics7100219 ◽

2018 ◽

Vol 7 (10) ◽

pp. 219 ◽

Cited By ~ 9

Author(s):

Alberto Sanchez ◽

Elías Todorovich ◽

Angel de Castro

Keyword(s):

The State ◽

Floating Point ◽

Hardware In The Loop ◽

State Variables ◽

Numerical Resolution ◽

Digital Devices ◽

State Variable ◽

Simulation Step ◽

Field Programmable ◽

The Difference

As the performance of digital devices is improving, Hardware-In-the-Loop (HIL) techniques are being increasingly used. HIL systems are frequently implemented using FPGAs (Field Programmable Gate Array) as they allow faster calculations and therefore smaller simulation steps. As the simulation step is reduced, the incremental values for the state variables are reduced proportionally, increasing the difference between the current value of the state variable and its increments. This difference can lead to numerical resolution issues when both magnitudes cannot be stored simultaneously in the state variable. FPGA-based HIL systems generally use 32-bit floating-point due to hardware and timing restrictions but they may suffer from these resolution problems. This paper explores the limits of 32-bit floating-point arithmetics in the context of hardware-in-the-loop systems, and how a larger format can be used to avoid resolution problems. The consequences in terms of hardware resources and running frequency are also explored. Although the conclusions reached in this work can be applied to any digital device, they can be directly used in the field of FPGAs, where the designer can easily use custom floating-point arithmetics.

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Impact of different centroid means on the accuracy of orthometric height modelling by geometric geoid method

International Journal of Scientific Reports ◽

10.18203/issn.2454-2156.intjscirep20201267 ◽

2020 ◽

Vol 6 (4) ◽

pp. 124 ◽

Cited By ~ 1

Author(s):

Oluyori P. Dare ◽

Eteje S. Okiemute

Keyword(s):

Root Mean Square ◽

Arithmetic Mean ◽

Orthometric Height ◽

Harmonic Mean ◽

Geometric Method ◽

Mean Square ◽

Geoid Model ◽

Microsoft Excel ◽

Orthometric Heights ◽

The Impact

Background: Orthometric height, as well as geoid modelling using the geometric method, requires centroid computation. And this can be obtained using various models, as well as methods. These methods of centroid mean computation have impacts on the accuracy of the geoid model since the basis of the development of the theory of each centroid mean type is different. This paper presents the impact of different centroid means on the accuracy of orthometric height modelling by geometric geoid method.Methods: DGPS observation was carried out to obtain the coordinates and ellipsoidal heights of selected points. The centroid means were computed with the coordinates using three different centroid means models (arithmetic mean, root mean square and harmonic mean). The computed centroid means were entered accordingly into a Microsoft Excel program developed using the Multiquadratic surface to obtain the model orthometric heights at various centroid means. The root means square error (RMSE) index was applied to obtain the accuracy of the model using the known and the model orthometric heights obtained at various centroid means. Results: The computed accuracy shows that the arithmetic mean method is the best among the three centroid means types.Conclusions: It is concluded that the arithmetic mean method should be adopted for centroid computation, as well as orthometric height modelling using the geometric method.

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On the determination of a locally optimized Ellipsoidal model of the Geoid surface in sea areas

IOP Conference Series Earth and Environmental Science ◽

10.1088/1755-1315/906/1/012036 ◽

2021 ◽

Vol 906 (1) ◽

pp. 012036

Author(s):

Persephone Galani ◽

Sotiris Lycourghiotis ◽

Foteini Kariotou

Keyword(s):

Data Acquisition ◽

Sea Level ◽

Reference Ellipsoid ◽

Triaxial Ellipsoid ◽

Geoid Model ◽

Orthometric Heights ◽

The One ◽

Best Fit ◽

Local Geoid

Abstract Deriving a local geoid model has drawn much research interest in the last decade, in an endeavour to minimize the errors in orthometric heights calculations, inherited by the use of global geoid reference models. In most parts of the earth, the local geoid surface may be tens of meters away from the Global Reference biaxial Ellipsoid (WGS84), which create numerus problems in topographic, environmental and navigational applications. Several methods have been developed for optimizing the precision of the calculation of the geoid heights undulations and the accuracy of the corresponding orthometric heights calculations. The optimization refers either to the method used for data acquisition, or to the geometrical method used for the determination of the best fit local geoid model. In the present work, we focus on the reference ellipsoid used for the geometric and geoid heights determination and develop a method to provide the one that fits best to the local geoid surface. Moreover, we consider relatively small sea regions and near to coast areas, where the usual methods for data acquisition fail more or less, and we pay attention in two directions: To obtain accurate measured data and to have the best possible reference ellipsoid for the area at hand. In this due, we use the “GNSS-on-boat” methodology to obtain direct sea level data, which we induce in a Moore Penrose pseudoinverse procedure to calculate the best fit triaxial ellipsoid. This locally optimized reference ellipsoid minimizes the geometric heights in the region at hand. The method is applied in two closed sea areas in Greece, namely Corinthian and Patra’s gulf and also in four regions in the Ionian Sea, which exhibit significant geoid alterations. Taking into account all factors of uncertainty, the precision of the mean sea level surface, produced by the “GNSS on boat” methodology, had been estimated at 5.43 cm for the gulf of Patras, at 3.76 cm for the Corinthian gulf and at 3.31 for the Ionian and Adriatic Sea areas. The average difference of this surface and the local triaxial reference ellipsoid, calculated in this work, is found to be less than 15 cm, whereas the corresponding difference with respect to WGS84 is of the order of 30m.

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A regional Stokes-Helmert geoid determination for Costa Rica (GCR-RSH-2020): computation and evaluation

Contributions to Geophysics and Geodesy ◽

10.31577/congeo.2020.50.2.3 ◽

2020 ◽

Vol 50 (2) ◽

pp. 223-247

Author(s):

Jaime GARBANZO-LEÓN ◽

Alonso VEGA FERNÁNDEZ ◽

Mauricio VARELA SÁNCHEZ ◽

Juan Picado SALVATIERRA ◽

Robert W. KINGDON ◽

...

Keyword(s):

Costa Rica ◽

Gravity Data ◽

Accurate Determination ◽

Costa Rican ◽

Geopotential Model ◽

Geoid Model ◽

Satellite Gravity ◽

Common Solution ◽

The Difference ◽

The University

GNSS observations are a common solution for outdoor positioning around the world for coarse and precise applications. However, GNSS produces geodetic heights, which are not physically meaningful, limiting their functionality in many engineering applications. In Costa Rica, there is no regional model of the geoid, so geodetic heights (h) cannot be converted to physically meaningful orthometric heights (H). This paper describes the computation of a geoid model using the Stokes-Helmert approach developed by the University of New Brunswick. We combined available land, marine and satellite gravity data to accurately represent Earth's high frequency gravity field over Costa Rica. We chose the GOCO05s satellite-only global geopotential model as a reference field for our computation. With this combination of input data, we computed the 2020 Regional Stokes-Helmert Costa Rican Geoid (GCR-RSH-2020). To validate this model, we compared it with 4 global combined geopotential models (GCGM): EGM2008, Eigen6C-4, GECO and SGG-UM-1 finding an average difference of 5 cm. GECO and SGG-UM-1 are more similar to the GCR-RSH-2020 based on the statistics of the difference between models and the shape of the histogram of differences. The computed geoid also showed a shift of 7 cm when compared to the old Costa Rican height system but presented a slightly better fit with that system than the other models when looking at the residuals. In conclusion, GCR-RSH-2020 presents a consistent behaviour with the global models and the Costa Rican height systems. Also, the lowest variance suggests a more accurate determination when the bias is removed.

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RELATIONSHIP OF MAXIMUM DEFLECTIONS AND NATURAL FREQUENCIES VIBRATIONS OF ISOTROPIC CIRCULAR PLATES WITH INHOMOGENEOUS RESISTANCE CONDITIONS ON THE OUTER AND INNER CONTOUR

Building and reconstruction ◽

10.33979/2073-7416-2021-98-6-36-42 ◽

2021 ◽

Vol 98 (6) ◽

pp. 36-42

Author(s):

A.V. TURKOV ◽

◽

S.I. POLESHKO ◽

E.A. FINADEEVA ◽

K.V. MARFIN ◽

...

Keyword(s):

Variable Thickness ◽

Constant Thickness ◽

Circular Plates ◽

Outer Contour ◽

Numerical Studies ◽

The Difference ◽

Relationship Of ◽

The Relationship ◽

Hinged Support ◽

Natural Transverse

The relationship between the maximum deflections from a static uniformly distributed load W0 and the fundamental frequency of natural transverse vibrations of a round isotropic plate of linearly variable thickness with thickening to the edge under homogeneous conditions of support along the outer contour, depending on the ratio of the thickness of the plate in the center to the thickness along the edge, is considered. According to the results of the study, graphs of the dependence of the maximum deflection and the frequency of natural vibrations of the plate on the ratio t1 / t2 are constructed. It is shown that for round plates of linearly variable thickness at t1/t2<1.1 coefficient K with an accuracy of 5.9% coincides with the analytical coefficient for round plates of constant thickness. Numerical studies shows that when the ratio of the thicknesses on the contour and in the center is equal to two, the difference in the coefficient K, which depends on the relationship between the static and dynamic characteristics of the platinum, is about 25% for hinged support along the contour and up to 37% for rigid support. This indicates a more significant effect of uneven mass distribution for such boundary conditions.

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On the topographic effects by Stokes’ formula

Journal of Geodetic Science ◽

10.2478/jogs-2014-0014 ◽

2014 ◽

Vol 4 (1) ◽

Cited By ~ 1

Author(s):

L.E. Sjöberg

Keyword(s):

Gravity Anomaly ◽

Bouguer Anomaly ◽

Gravity Anomalies ◽

Geoid Height ◽

Topographic Effects ◽

Gravimetric Geoid ◽

Bouguer Gravity ◽

The Third ◽

Stokes Formula ◽

The Difference

AbstractTraditional gravimetric geoid determination relies on Stokes’ formula with removal and restoration of the topographic effects. It is shown that this solution is in error of the order of the quasigeoid-to-geoid difference, which is mainly due to incomplete downward continuation (dwc) of gravity from the Earth’s surface to the geoid. A slightly improved estimator, based on the surface Bouguer gravity anomaly, is also biased due to the imperfect harmonic dwc the Bouguer anomaly. Only the third estimator,which uses the (harmonic) surface no-topography gravity anomaly, is consistent with the boundary condition and Stokes’ formula, providing a theoretically correct geoid height. The difference between the Bouguer and no-topography gravity anomalies (on the geoid or in space) is the “secondary indirect topographic effect”, which is a necessary correction in removing all topographic signals.

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Combined approach for the unification of levelling networks in New Zealand

Journal of Geodetic Science ◽

10.2478/v10156-011-0012-0 ◽

2011 ◽

Vol 1 (4) ◽

pp. 324-332 ◽

Cited By ~ 9

Author(s):

Robert Tenzer ◽

Viliam Vatrt ◽

Luzi Gan ◽

Ahmed Abdalla ◽

Nadim Dayoub

Keyword(s):

New Zealand ◽

Gravity Data ◽

Gravity Anomalies ◽

Combined Approach ◽

Network Adjustment ◽

The North ◽

Geoidal Geopotential ◽

Normal Heights ◽

Orthometric Heights ◽

Local Vertical Datum

Combined approach for the unification of levelling networks in New ZealandThe unification of levelling networks in New Zealand is done using a combined approach. It utilises the joint levelling network adjustment and the geopotential-value approach. The levelling and normal gravity data are used for a joint adjustment of the levelling networks at the South and North Islands of New Zealand while fixing the heights of tide gauges in Dunedin and Wellington. The results reveal a good quality of levelling data; the STD of residuals is 2 mm for the whole country. The comparison of the newly determined and original normal-orthometric heights confirms the presence of large local vertical datum offsets and systematic levelling errors. Since the geopotential-value approach is based on the Molodensky's theory, the newly adjusted normal-orthometric heights are converted to the normal heights. This conversion is based on applying the cumulative normal to normal-orthometric height correction computed from levelling and gravity anomaly data. In the absence of the observed gravity data the gravity anomalies along levelling lines are generated fromEGM2008. The GPS-levelling data and EGM2008 are used to estimate the average offsets of the jointly adjusted levelling networks at the North and South Islands with respect to World Height System defined by the adopted geoidal geopotential value of W0 = 62636856 ± 0.5 m2s-2; the estimated offsets are 10.6 cm and 27.5 cm.

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