World Height System Research Articles

Estimation of the zero-height geopotential value and datum offset for the Kenya vertical network using an optimized earth model

The vertical height system in Kenya is anchored on a single levelling-based tidal gauge, which is referred to as mean sea level. The fusion of existing height systems into the world height system is one of the primary objectives of the International Height Reference System (IHRS) implementation. Computing the datum offset with regard to the global IHRS datum and the zero-height geopotential value can help achieve this goal in part. This paper studies the approach of using a Global Geopotential model (GGM) with the GNSS-levelling as a convenient method for vertical datum offsets’ computation that may connect two or more vertical datums. To reduce the geoid omission error of the GGMs, an optimized GGM is developed and utilized up to a maximum of 3600° and order. For the first time, the numerical analyses of this work reveal the zero-height geopotential and its associated standard error of the Kenyan vertical datum, from which the datum offsets with the IHRS were estimated. Values of 62636850.996±0.104m2s−2 and −24.5±1.04cm were obtained as the geopotential of the zero-height and vertical datum offset with respect to the global value, respectively.

Simulation experiments on high-precision VGOS time transfer for future geopotential difference determination

Known for unique ultrahigh spatial resolution and timing accuracy, the VLBI global observing system (VGOS) time transfer of ultralong baselines can achieve better long-term stability compared to other techniques. Because of the 1 mm positioning accuracy of the VGOS, comparing high-performance atomic clocks with VGOS time transfer can be used to accurately determine the geopotential difference between two remote stations and unify the world height system in the framework of the theory of general relativity. To investigate its performance in determining geopotential differences, we schedule continuous VGOS sessions with a 7-station network, simulate and estimate the corresponding VGOS observables and determine the geopotential differences of six baselines. Furthermore, the coefficients of the tropospheric turbulence model and clock stability are varied to analyze their influences. The results show that when using atomic clocks with 1.0 × 10 - 16 @1 d stability, the accuracy and precision of the weighted averaged geopotential difference remain at the level of 8 m 2 s −2 and are sensitive to variations in the troposphere. The improvement by upgrading clock performance is limited by other error sources. In addition, connecting twin telescopes with identical atomic clocks and combining clock parameters in estimation can improve the precision of the estimated geopotential difference by approximately 4%.

Normal gravity field in relativistic geodesy

Modern geodesy is subject to a dramatic change from the Newtonian paradigm to Einstein's theory of general relativity. This is motivated by the ongoing advance in development of quantum sensors for applications in geodesy including quantum gravimeters and gradientometers, atomic clocks and fiber optics for making ultra-precise measurements of the geoid and multipolar structure of the Earth's gravitational field. At the same time, VLBI, SLR, and GNSS have achieved an unprecedented level of accuracy in measuring coordinates of the reference points of the ITRF and the world height system. The main geodetic reference standard is a normal gravity field represented in the Newtonian gravity by the field of a Maclaurin ellipsoid. The present paper extends the concept of the normal gravity field to the realm of general relativity. We focus our attention on the calculation of the first post-Newtonian approximation of the normal field that is sufficient for applications. We show that in general relativity the level surface of the uniformly rotating fluid is no longer described by the Maclaurin ellipsoid but is an axisymmetric spheroid of the forth order. We parametrize the mass density distribution and derive the post-Newtonian normal gravity field of the rotating spheroid which is given in a closed form by a finite number of the ellipsoidal harmonics. We employ transformation from the ellipsoidal to spherical coordinates to deduce the post-Newtonian multipolar expansion of the metric tensor given in terms of scalar and vector gravitational potentials of the rotating spheroid. We compare these expansions with that of the normal gravity field generated by the Kerr metric and demonstrate that the Kerr metric has a fairly limited application in relativistic geodesy. Finally, we derive the post-Newtonian generalization of the Somigliana formula for the gravity field on the reference ellipsoid.

Formulation of geopotential difference determination using optical-atomic clocks onboard satellites and on ground based on Doppler cancellation system

In this study, we propose an approach for determining the geopotential difference using high-frequency-stability microwave links between satellite and ground station based on Doppler cancellation system. Suppose a satellite and a ground station are equipped with precise optical-atomic clocks (OACs) and oscillators. The ground oscillator emits a signal with frequency f a towards the satellite and the satellite receiver (connected with the satellite oscillator) receives this signal with frequency f b which contains the gravitational frequency shift effect and other signals and noises. After receiving this signal, the satellite oscillator transmits and emits, respectively, two signals with frequencies f b and f c towards the ground station. Via Doppler cancellation technique, the geopotential difference between the satellite and the ground station can be determined based on gravitational frequency shift equation by a combination of these three frequencies. For arbitrary two stations on ground, based on similar procedures as described above, we may determine the geopotential difference between these two stations via a satellite. Our analysis shows that the accuracy can reach 1 m 2 s − 2 based on the clocks’ inaccuracy of about 10 −17 (s s −1 ) level. Since OACs with instability around 10 −18 in several hours and inaccuracy around 10 −18 level have been generated in laboratory, the proposed approach may have prospective applications in geoscience, and especially, based on this approach a unified world height system could be realized with one-centimetre level accuracy in the near future.

An attempt to link the Brazilian Height System to a World Height System

This paper deals with the geopotential approach to investigate the present Brazilian Height System (BHS). Geopotential numbers are derived from Global Positioning System (GPS) satellite surveying and disturbing potential on selected benchmarks. A model for the disturbing potential can be obtained by an existing set of spherical harmonic coefficients such as the Earth Gravity Model 2008 (EGM08). The approach provides absolute evaluation of local normal geopotential numbers (aka spheropotential numbers) related to a so-called World Height System (WHS). To test the validity of the proposed methodology, a numerical experiment was carried out related to a test region in Southern Brazil. The accuracy of the derived geopotential numbers was tested versus local normal geopotential numbers based on 262 GPS/leveling points. The root mean square error (RMSE) value for metric offset of BHS derived from geopotential numbers and the disturbing potential modeling in the test area was estimated to be near 0.224 meters in the absolute view. Therefore, since these spheropotential numbers are referred to a local datum, these results of comparisons may be an indicator of the mean bias of local network due to the effect of local Sea Surface Topography (SSTop) and possible offset between the unknown reference for the BHS and the quasigeoid model in the region.

Geopotential numbers from GPS satellite surveying and disturbing potential model: a case study of Parana, Brazil

This article deals with the approach to determining the geopotential numbers from Global Positioning System (GPS) satellite surveying and disturbing potential at a point on the Earth's surface. The disturbing potential was solved by the Brovar-type solution of Molodenskii's boundary value problem but in the context of the fixed problem. The solution is compatible with the techniques for smoothing of external field: removerestore techniques, residual terrain model, and use of high-resolution global geopotential models. This method provides an absolute geopotential numbers related to the so-called world height system. As an example, we calculate the disturbing potential model in Southern Brazil. The general condition of the sparse and absent gravity values is the main point to be solved. The absolute and relative accuracies of the quasigeoid model were tested vs. GPS/leveling data. Based on 99 GPS/leveling points, the mean value of fitting for the quasigeoid model was estimated near 0.032 m in the absolute view and 0.2 ppm in the relative view.

Combined approach for the unification of levelling networks in New Zealand

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.

Assessment of the LVD offsets for the normal-orthometric heights and different permanent tide systems—a case study of New Zealand

The geopotential-value approach is utilized in this study to estimate the average offsets of local vertical datums (LVDs) in New Zealand realized in the system of normal-orthometric heights. The LVD offsets are taken relative to the World Height System (WHS). We adopt the geoidal geopotential value W 0 = 62,636,856 m2 s−2 for a definition of WHS. The conversion of heights between different permanent tide systems is taken into consideration. The geopotential-value approach utilizes Molodensky’s theory of the normal heights. The normal-orthometric heights at global positioning system (GPS)-leveling points are thus first converted to the normal heights. The normal to normal-orthometric height correction is computed and applied along the leveling lines using the leveling data, and the gravity disturbances are computed approximately from the EGM08 global geopotential model. The numerical study is conducted for 18 LVDs in the North and South Islands of New Zealand. The LVD offsets are estimated from EGM08 to GPS-leveling data. The estimated average LVD offsets vary between 1 cm (Wellington 1953 LVD) and 37 cm (One Tree Point 1964 LVD).

Earth’s Dimension Specified by Geoidal Geopotential

The TOPEX/POSEIDON (T/P) satellite altimeter data from January 1,1993 to January 3, 2001 (cycles 11–305) was used for investigating the long-term variations of the geoidal geopotential W0 and the geopotential scale factor R 0 = GM/W 0 (GM is the adopted geocentric gravitational constant). The mean values over the whole period covered are W0 = (62 636 856.160 ±0.002) m2·s-2, R0 = (6 363 672.544 8 ±0.000 2) m. The actual accuracy is limited by the altimeter calibration error (2–3cm) and it is conservatively estimated to be about ±0.5m2·s-2 (±5cm). The differences between the yearly mean sea surface (MSS) levels, came out as follows: 1993–1994: -(1.2 ±0.7) mm, 1994–1995: (0.5 ±0.7) mm, 1995–1990: (0.5 ±0.7) mm, 1996–1997: (0.1 ±0.7) mm, 1997–1998: -(0.5 ±0.7) mm, 1998–1999: (0.0 ±0.7) mm and 1999–2000 (0.6 ±0.7) mm. The corresponding rate of change in the MSS level (or R 0) during the whole period of 1993–2000 is (0.02±0.07) mm/y. The value W 0 was found to be quite stable (see table), it depends only on the adopted GM, ω and the volume enclosed by surface W = W 0. W 0 can also uniquely define the reference (geoidal) surface that is required for a number of applications, including World Height System and General Relativity in precise time keeping and time definitions, that is why W 0 is considered to be suitable for adoption as a primary astrogeodetic parameter. Furthermore, W 0 provides a scale parameter for the Earth that is independent of the tidal reference system. After adopting a value for W 0, the semi-major axis a of the Earth’s general ellipsoid can easily be derived. However, an apriori condition should be posed first. Two conditions have been examined, namely an ellipsoid with the corresponding geopotential which fits best W 0 in the least squares sense and an ellipsoid which has the global geopotential average equal to W 0. It is demonstrated that both α-valwes are practically equal to the value obtained by the Pizzetti’s theory of the level ellipsoid: a = (6 378 136.7±0.05) m.

Determination of Geopotential Differences between Local Vertical Datums and Realization of a World Height System

The methodology developed for connecting Local Vertical Datums (LVD) was applied to the Australian Height Datum (AHD) and the North American Vertical Datum (NAVD88). The geopotential values at AHD and NAVD88 were computed and the corresponding vertical offset of 974 mm with rms 51 mm was obtained between the zero reference surfaces defined by AHD and NAVD88. The solution is based on the four primary geodetic parameters, the GPS/levelling sites and the geopotential model EGM96. The Global Height System (or the Major Vertical Datum) can be defined by a geoidal geopotential value used in the solution as the reference value, or by the geopotential value of the LVD, e.g. NAVD88.

GEOIDAL GEOPOTENTIAL AND WORLD HEIGHT SYSTEM

The geoidal geopotential value of W 0 = 62 636 856.0 ± 0.5m 2 s −2 , determined from the 1993 –1998 TOPEX/POSEIDON altimeter data, can be used to practically define and realize the World Height System. The W 0 -value can also uniquely define the geoidal surface and is required for a number of applications, including General Relativity in precise time keeping and time definitions. Furthermore, the W 0 -value provides a scale parameter for the Earth that is independent of the tidal reference system. All of the above qualities make the geoidal potential W 0 ideally suited for official adoption as one of the fundamental constants, replacing the currently adopted semi-major axis a of the mean Earth ellipsoid. Vertical shifts of the Local Vertical Datum (LVD) origins can easily be determined with respect to the World Height System (defined by W 0 ), in using the recent EGM96 gravity model and ellipsoidal height observations (e.g. GPS) at levelling points. Using this methodology the LVD vertical displacements for the NAVD88 (North American Vertical Datum 88), NAP (Normaal Amsterdams Peil), AMD (Australian Height Datum), KHD (Kronstadt Height Datum), and N60 (Finnish Height Datum) were determined with respect to the proposed World Height System as follows: −55.1 cm, −11.0 cm, +42.4 cm, −11.1 cm and +1.8 cm, respectively.

Separation between reference surfaces of selected vertical datums

This paper discusses the separation between the reference surface of several vertical datums and the geoid. The data used includes a set of Doppler positioned stations, transformation parameters to convert the Doppler positions to ITRF90, and a potential coefficient model composed of the JGM-2 (NASA model) from degree 2 to 70 plus the OSU91A model from degree 71 to 360. The basic method of analysis is the comparison of a geometric geoid undulation derived from an ellipsoidal height and an orthometric height with the undulation computed from the potential coefficient model The mean difference can imply a bias of the datum reference surface with respect to the geoid. Vertical datums in the following countries were considered: England, Germany, United States, and Australia. The following numbers represent the bias values of each datum after adopting an equatorial radius of 6378136.3m: England (-87 cm), Germany (4 cm), United States (NGVD29 (-26 cm)), NAVD88 (-72 cm), Australia AHD (mainland, -68 cm); AHD (Tasmania, -98 cm). A negative sign indicates the datum reference surface is below the geoid. The 91 cm difference between the datums in England and Germany has been independently estimated as 80 cm. The 30 cm difference between AHD (mainland) and AHD (Tasmania) has been independently estimated as 40 cm. These bias values have been estimated from data where the geometric/ gravimetric geoid undulation difference standard deviation, at one station, is typically ±100 cm, although the mean difference is determined more accurately. The results of this paper can be improved and expanded with more accurate geocentric station positions, more accurate and consistent heights with respect to the local vertical datum, and a more accurate gravity field for the Earth. The ideas developed here provide insight on the determination of a world height system.