The use of gravity data to determine orthometric heights at the Hong Kong territories

Albertini Nsiah Ababio; Robert Tenzer

doi:10.1515/jag-2022-0012

Artikel

The use of gravity data to determine orthometric heights at the Hong Kong territories

Albertini Nsiah Ababio und Robert Tenzer

Veröffentlicht/Copyright: 4. August 2022

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Manuskript einreichen Informationen für Autor*innen

Aus der Zeitschrift Journal of Applied Geodesy Band 16 Heft 4

Abstract

The Hong Kong Principal Datum (HKPD) is the currently adopted official geodetic vertical datum at the Hong Kong territories. The HKPD is practically realized by heights of levelling benchmarks. The HKPD heights are, however, neither normal nor orthometric. The reason is that heights of levelling benchmarks were determined from precise levelling measurements, but without involving gravity observations along levelling lines. To reduce systematic errors due to disregarding the gravity information along levelling lines, we used terrestrial and marine gravity data to interpolate gravity values at levelling benchmarks in order to compute and apply the orthometric correction to measured levelling height differences. Our results demonstrate the importance of incorporating the gravity information even for a relatively small region but characterized by a rough topography with heights of levelling benchmarks exceeding several hundreds of meters. According to our estimates, the orthometric correction reaches (and even slightly exceeds) ±2 cm, with maxima along levelling lines crossing mountain chains.

Keywords: gravity; levelling; orthometric height; vertical geodetic control

Funding source: Research Grants Council, University Grants Committee

Award Identifier / Grant number: 15218819

Funding statement: The work presented in this article was supported by the by the Hong Kong GRF RGC project 15218819: “The modernization of height datum in the Hong Kong territories”.

Appendix A The error analysis

The relation between the (actual) error of the orthometric correction ε O C and the errors ε Δ h and ε H of measured height differences and heights of levelling benchmarks can be derived from Eq. (10) in the following form

(A.1) ε O C = 1 g ‾ i + 1 g i + g i + 1 2 − g ‾ i + 1 ε Δ h + g ‾ i g ‾ i + 1 − 1 ε H .

Assuming that for relatively small distances between levelling benchmarks g ‾ i ≈ g ‾ i + 1 and g i ≈ g i + 1 , the expression in Eq. (A.1) simplifies to

(A.2) ε O C ≈ 1 g ‾ g − g ‾ ε Δ h .

Substituting from Eq. (3) for the mean gravity g ‾, the term g − g ‾ in Eq. (A.2) is further rearranged into the following form

(A.3) g − g ‾ ≅ g − g + 1 2 ∂ g ∂ h H ≅ 1 2 ∂ γ ∂ h H + 2 π G ρ H ,

where ∂ γ / ∂ h ≅ 3 × 10 − 6 m s − 2 .

Inserting from Eq. (A.3) back to Eq. (A.2), we arrive at

(A.4) ε O C ≈ H g ‾ 1 2 ∂ γ ∂ h + 2 π G ρ ε Δ h .

The mean actual gravity g ‾ in Eq. (A.4) can be replaced by the mean normal gravity γ ‾, so that

(A.5) ε O C ≈ H γ ‾ 1 2 ∂ γ ∂ h + 2 π G ρ ε Δ h .

For the values 2 − 1 ∂ γ / ∂ h ≅ 1.5 × 10 − 6 m s − 2 and 2 π G ρ ≈ 1.1 × 10 − 6 and maximum heights of levelling benchmarks in Hong Kong H ≈ 500 m, the uncertainties in height differences ε Δ h ≤ 10 cm cause errors in computed values of the orthometric correction that are completely negligible.

Similarly, the errors of orthometric correction ε O C due to errors ε Δ g F A of interpolated free-air gravity anomalies can be estimated. From Eq. (10), the relation between these errors is found to be

(A.6) ε O C ≈ Δ h g ‾ ε Δ g F A ≅ Δ h γ ‾ ε Δ g F A .

For maximum height differences Δ h ≈ ±100 m along levelling lines in Hong Kong (see Fig. 12), the error ε Δ g F A ≈ ± 10 mGal = ± 1 × 10 − 4 m s − 2 introduces the error in computed values of the orthometric correction of about ±1 mm.

Appendix B Comparison of different gravity interpolation techniques

We used the kriging, natural neighbour, least-squares collocation, nearest neighbour, and radial basis functions to interpolate the simple planar Bouguer gravity anomalies at levelling benchmarks and compared them with the interpolated values obtained by applying the inverse distance weighting for the weight w = s i , i + 1 − 2 . The statistics of differences is summarized in Table B.1.

Table B.1

Statistics of differences (in mGal) between the interpolation results obtained by applying the kriging, natural neighbour, least-squares collocation, nearest neighbour, and radial basis functions and the interpolated values of the simple planar Bouguer gravity anomalies at levelling benchmarks (Table 3) computed by applying the inverse distance weighting (for the weight w = s i , i + 1 − 2 ).

Method	Min	Max	Mean	STD
Kriging	−3.04	2.58	0.10	0.74
Natural Neighbor	−3.25	2.50	0.09	0.67
Nearest Neighbor	−4.04	4.12	0.27	0.82
Radial Basis function	−3.501	3.92	0.21	0.88
Least-squares collocation	−3.23	2.87	0.12	0.75

References

[1] Artemjev ME, Kaban MK, Kucherinenko VA, Demjanov GV, Taranov VA (1994) Subcrustal density inhomogeneities of the Northern Eurasia as derived from the gravity data and isostatic models of the lithosphere. Tectonoph 240: 248–280.10.1016/0040-1951(94)90275-5Suche in Google Scholar

[2] Davies S (2013) The principal datum: Some puzzles associated with the Rifleman’s bolt. Journal of the Royal Asiatic Society Hong Kong Branch 53: 109–133. https://www.jstor.org/stable/23891239.Suche in Google Scholar

[3] Electronic and Geophysical Services ltd (1991) Regional gravity survey of Hong Kong. Final Report, Job Number HK50190, Hong Kong.Suche in Google Scholar

[4] Filmer MS, Featherstone WE, Kuhn M (2010) The effect of EGM2008-based normal, normal-orthometric and Helmert orthometric height systems on the Australian levelling network. J Geod 84(8): 501–513.10.1007/s00190-010-0388-0Suche in Google Scholar

[5] Foroughi I, Tenzer R (2017) Comparison of different methods for estimating the geoid-to-quasigeoid separation. Geophys J Int 210: 1001–1020.10.1093/gji/ggx221Suche in Google Scholar

[6] Fotopoulos G, Kotsakis C, Sideris MG (2003) How accurately can we determine orthometric height differences from GPS and geoid data. J Surv Eng 1: 1–10.10.1061/(ASCE)0733-9453(2003)129:1(1)Suche in Google Scholar

[7] Fotopoulos G (2005) Calibration of geoid error models via a combined adjustment of ellipsoidal, orthometric and gravimetric geoid height data. J Geod 79: 111–123.10.1007/s00190-005-0449-ySuche in Google Scholar

[8] Grebenitcharsky RS, Rangelova EV, Sideris MG (2005) Transformation between gravimetric and GPS/levelling-derived geoids using additional gravity information. J Geodyn 39: 527–544.10.1016/j.jog.2005.04.008Suche in Google Scholar

[9] Guo D-M, Xu H-Z, Chen M (2017) Precise Geoid Determination over Hong Kong from Heterogeneous Data Sets using a Hybrid Method, Mar Geod 40(2-3): 160–171.10.1080/01490419.2017.1309330Suche in Google Scholar

[10] Heiskanen WA, Moritz H (1967) Physical Geodesy. WH Freeman and Co., New York, London and San Francisco.10.1007/BF02525647Suche in Google Scholar

[11] Helmert FR (1884) Die mathematischen und physikalischen Theorien der höheren Geodäsie, Vol 2, Teubner, Leipzig.Suche in Google Scholar

[12] Helmert FR (1890) Die Schwerkraft im Hochgebirge, insbesondere in den Tyroler Alpen. Veröff Königl Preuss Geod Inst, No. 1.Suche in Google Scholar

[13] Hinze WJ (2003) Bouguer reduction density, why 2.67? Geophysics 68(5): 1559–1560.10.1190/1.1620629Suche in Google Scholar

[14] Hwang C, Hsiao YS (2003) Orthometric corrections from leveling, gravity, density and elevation data: a case study in Taiwan. J Geod 77: 279–291.10.1007/s00190-003-0325-6Suche in Google Scholar

[15] Klees R, Prutkin I (2010) The combination of GNNS-levelling data and gravimetric (quasi-) geoid heights in the presence of noise. J Geod 84: 731–749.10.1007/s00190-010-0406-2Suche in Google Scholar

[16] Luo Z, Ning J, Chen Y, Yang Z (2005) High Precision Geoid Models HKGEOID-2000 for Hong Kong and SZGEOID-2000 for Shenzhen, China. Mar Geod 28(2): 191–200.10.1080/01490410590953758Suche in Google Scholar

[17] Pizzetti P (1911) Sopra il calcolo teorico delle deviazioni del geoide dall‘ellissoide. Atti R Accad Sci Torino 46: 331–350.Suche in Google Scholar

[18] Prutkin I, Klees R (2008) The non-uniqueness of local quasi-geoids computed from terrestrial gravity anomalies. J Geod 82(3): 147–156.10.1007/s00190-007-0161-1Suche in Google Scholar

[19] Rapp RH (1997) Use of potential coefficient models for geoid undulation determinations using a spherical harmonic representation of the height anomaly/geoid undulation difference. J Geod 71: 282–289.10.1007/s001900050096Suche in Google Scholar

[20] Sjöberg LE (1995) On the quasigeoid to geoid separation. Manuscr Geod 20(3): 182–192.Suche in Google Scholar

[21] Sjöberg LE (1999) On the downward continuation error at the Earth’s surface and the geoid of satellite derived geopotential models. Boll Geod Sci Affin 58(3): 215–229.Suche in Google Scholar

[22] Sjöberg LE (2006) A refined conversion from normal height to orthometric height. Stud Geophys Geod 50: 595–606.10.1007/s11200-006-0037-5Suche in Google Scholar

[23] Somigliana C (1929) Teoria Generale del Campo Gravitazionale dell’Ellisoide di Rotazione. Memoire della Societa Astronomica Italiana IV: 425, Milano.Suche in Google Scholar

[24] Tenzer R, Vaníček P, Santos M, Featherstone WE, Kuhn M (2005) The rigorous determination of orthometric heights. J Geod 79(1-3): 82–92.10.1007/s00190-005-0445-2Suche in Google Scholar

[25] Tenzer R, Moore P, Novák P, Kuhn M, Vaníček P (2006) Explicit formula for the geoid-to-quasigeoid separation. Stud Geoph Geod 50: 607–618.10.1007/s11200-006-0038-4Suche in Google Scholar

[26] Tenzer R, Vatrt V, Abdalla A, Dayoub N (2011) Assessment of the LVD offsets for the normal-orthometric heights and different permanent tide systems – a case study of New Zealand. Appl Geomat 3(1):1–8.10.1007/s12518-010-0038-5Suche in Google Scholar

[27] Tenzer R, Hirt CH, Claessens S, Novák P (2015) Spatial and spectral representations of the geoid-to-quasigeoid correction. Surv Geophys 36:627.10.1007/s10712-015-9337-zSuche in Google Scholar

[28] Tenzer R, Hirt Ch, Novák P, Pitoňák M, Šprlák M (2016) Contribution of mass density heterogeneities to the geoid-to-quasigeoid separation. J Geod 90(1): 65–80.10.1007/s00190-015-0858-5Suche in Google Scholar

[29] Tenzer R, Foroughi I, Pitoňák M, Šprlák M (2017) Effect of the Earth’s inner structure on the gravity in definitions of height systems. Geophys J Int 209 (1): 297–316.10.1093/gji/ggx024Suche in Google Scholar

[30] Tenzer R, Chen W, Rathnayake S, Pitoňák M (2021) The effect of anomalous global lateral topographic density on the geoid-to-quasigeoid separation. J Geod 95: 12.10.1007/s00190-020-01457-6Suche in Google Scholar

[31] Vaníček P, Tenzer R, Sjöberg LE, Martinec Z, Featherstone WE (2005) New views of the spherical Bouguer gravity anomaly. Geophys J Int 159: 460–472.10.1111/j.1365-246X.2004.02435.xSuche in Google Scholar

[32] Wu Y, Luo Z, Chen W, Chen Y (2017) High-resolution regional gravity field recovery from Poisson wavelets using heterogeneous observational techniques. Earth Planets Space 69:34.10.1186/s40623-017-0618-2Suche in Google Scholar

Received: 2022-04-18

Accepted: 2022-07-18

Published Online: 2022-08-04

Published in Print: 2022-10-26

Sie haben derzeit keinen Zugang zu diesem Inhalt.

Artikel in diesem Heft

https://doi.org/10.1515/jag-2022-0012

Schlagwörter für diesen Artikel

gravity; levelling; orthometric height; vertical geodetic control