Advertisement

Journal of Geodesy

, Volume 82, Issue 4–5, pp 231–248 | Cite as

Astronomical-topographic levelling using high-precision astrogeodetic vertical deflections and digital terrain model data

  • Christian HirtEmail author
  • Jakob Flury
Original Article

Abstract

At the beginning of the twenty-first century, a technological change took place in geodetic astronomy by the development of Digital Zenith Camera Systems (DZCS). Such instruments provide vertical deflection data at an angular accuracy level of 0.̋1 and better. Recently, DZCS have been employed for the collection of dense sets of astrogeodetic vertical deflection data in several test areas in Germany with high-resolution digital terrain model (DTM) data (10–50 m resolution) available. These considerable advancements motivate a new analysis of the method of astronomical-topographic levelling, which uses DTM data for the interpolation between the astrogeodetic stations. We present and analyse a least-squares collocation technique that uses DTM data for the accurate interpolation of vertical deflection data. The combination of both data sets allows a precise determination of the gravity field along profiles, even in regions with a rugged topography. The accuracy of the method is studied with particular attention on the density of astrogeodetic stations. The error propagation rule of astronomical levelling is empirically derived. It accounts for the signal omission that increases with the station spacing. In a test area located in the German Alps, the method was successfully applied to the determination of a quasigeoid profile of 23 km length. For a station spacing from a few 100 m to about 2 km, the accuracy of the quasigeoid was found to be about 1–2 mm, which corresponds to a relative accuracy of about 0.05−0.1 ppm. Application examples are given, such as the local and regional validation of gravity field models computed from gravimetric data and the economic gravity field determination in geodetically less covered regions.

Keywords

Astronomical levelling Vertical deflection Digital Zenith Camera System (DZCS) Digital Terrain Model (DTM) Least-squares collocation (LSC) 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Boedecker G (1976) Astrogravimetrisch-topographisches Nivellement. Wiss. Arb. Lehrst. Geod. Phot. u Kart. TU Hannover Nr. 64Google Scholar
  2. Bomford G (1980) Geodesy, Fourth edn. Clarendon Press, OxfordGoogle Scholar
  3. Bosch W, Wolf H (1974) Über die Wirkung von topographischen Lokal-Effekten bei profilweisen Lotabweichungs-Prädiktionen. Mitteilungen aus dem Institut für Theoretische Geodäsie der Universität Bonn Nr. 28Google Scholar
  4. Bürki B (1989) Integrale Schwerefeldbestimmung in der Ivrea-Zone und deren geophysikalische Interpretation. Geodätisch-geophysikalische Arbeiten in der Schweiz, Nr. 40. Schweizerische Geodätische KommissionGoogle Scholar
  5. Bürki B, Müller A, Kahle H-G (2004) DIADEM: the new digital astronomical deflection measuring system for high-precision measurements of deflections of the vertical at ETH Zurich. Electronic Proc. IAG GGSM2004 meeting in Porto, Portugal. Published also in: CHGeoid 2003, Report 03-33 A Marti U et al. (ed), Bundesamt für Landestopographie (swisstopo), Wabern, SchweizGoogle Scholar
  6. Campbell J (1971) Eine Erweiterung der Theorie des astronomischen Nivellements bei Einbeziehung von Schweremessungen. Wiss. Arb. Lehrst. Geod. Phot. u. Kart. TU Hannover Nr. 49Google Scholar
  7. Elmiger A (1969) Studien über Berechnung von Lotabweichungen aus Massen, Interpolation von Lotabweichungen und Geoidbestimmung in der Schweiz. Mitt. Inst. Geod. Phot. ETH Zürich Nr. 12Google Scholar
  8. Flury J (2002) Schwerefeldfunktionale im Gebirge—Modellierungsgenauigkeit, Messpunktdichte und Darstellungsfehler am Beispiel des Testnetzes Estergebirge. Deutsche Geodätische Kommission C 557Google Scholar
  9. Flury J (2006) Short wavelength spectral properties of gravity field quantities. J Geod 79(10-11):624–640. doi: 10.1007/s00190-005-0011-yCrossRefGoogle Scholar
  10. Flury J, Gerlach C, Hirt C, Schirmer U (2006) Heights in the Bavarian Alps: mutual validation of GPS, levelling, gravimetric and astrogeodetic quasigeoids. Proc. Geodetic Reference Frames 2006, submittedGoogle Scholar
  11. Forsberg R, Tscherning CC (1981) The use of height data in gravity field approximation by collocation. J Geophys Res 86(B9):7843–7854CrossRefGoogle Scholar
  12. Galle A (1914) Das Geoid im Harz. Königl. Preuß. Geod. Inst. Nr 61. BerlinGoogle Scholar
  13. GOCE GRAND (2005) GOCE-Graviationsfeldanalyse Deutschland II. Joint Proposal to BMBF research theme 2 Rummel R et al. (ed) Institut für Astronomische und Physikalische Geodäsie, TU MunichGoogle Scholar
  14. Gurtner W (1978) Das Geoid der Schweiz. Geodätisch-geophysikalische Arbeiten in der Schweiz, Nr. 32. Schweizerische Geodätische KommissionGoogle Scholar
  15. Heiskanen WA, Moritz H (1967) Physical geodesy. W.H. Freeman, San FranciscoGoogle Scholar
  16. Heitz S (1968) Geoidbestimmung durch Interpolation nach kleinsten Quadraten aufgrund gemessener und interpolierter Lotabweichungen. Deutsche Geodätische Kommission C 124Google Scholar
  17. Helmert FR (1880/1884) Die mathematischen und physikalischen Theorien der höheren Geodäsie. Teubner, Leibzig (reprint Minerva, Frankfurt a.M. 1961)Google Scholar
  18. Helmert FR (1901) Zur Bestimmung kleiner Flächenstücke des Geoids aus Lotabweichungen mit Rücksicht auf Lotkrümmung. Sitzungsberichte Königl. Preuß. Akad. der Wissenschaften zu Berlin. 2. Mitteilung:958–975Google Scholar
  19. Hirt C (2004) Entwicklung und Erprobung eines digitalen Zenitkamerasystems für die hochpräzise Lotabweichungsbestimmung. Wissenschaftliche Arbeiten der Fachrichtung Geodäsie und Geoinformatik an der Universität Hannover Nr 253. http://edok01. tib.uni-hannover.de/edoks/e01dh04/393223965.pdfGoogle Scholar
  20. Hirt C (2006) Monitoring and analysis of anomalous refraction using a digital zenith camera system. Astron Astrophys 459:283–290. doi: 10.1051/0004-6361:20065485CrossRefGoogle Scholar
  21. Hirt C, Bürki B (2002) The digital zenith camera—a new high-precision and economic astrogeodetic observation system for real-time measurement of deflections of the vertical. In: Tziavos I (ed) Proceedings of the 3rd meeting of the international gravity and geoid commission of the international association of geodesy, Thessaloniki 161–166Google Scholar
  22. Hirt C, Seeber G (2005) High-resolution local gravity field determination at the sub-millimeter level using a digital zenith camera system. Dynamic Planet, Cairns 2005, Tregoning P, Rizos C (eds) IAG Symposia 130:316–321Google Scholar
  23. Hirt C, Denker H, Flury J, Lindau A, Seeber G (2006) Astrogeodetic validation of gravimetric quasigeoid models in the German Alps—first results. Accepted paper presented at 1. Meeting of the International Gravity Field Service, IstanbulGoogle Scholar
  24. Høg E, Fabricius C, Makarov VV, Urban S, Corbin T, Wycoff G, Bastian U, Schwekendiek P, Wicenec A (2000) The Tycho-2 catalogue of the 2.5 million brightest stars. Astron Astrophys 355:L27–L30Google Scholar
  25. Jekeli C (1999) An analysis of vertical deflections derived from high-degree spherical harmonic models. J Geod 73(1):10–22 doi: 10.1007/s001900050213CrossRefGoogle Scholar
  26. Jekeli C, Li X (2006) INS/GPS vector gravimetry along roads in Western Montana. OSU Report 477Google Scholar
  27. Kobold F (1957) Die astronomischen Nivellemente in der Schweiz. ZfV 82(4):97–103Google Scholar
  28. Levallois JJ, Monge H (1978) Le geoid Européen, version 1978. Proceedings of the International Sympoium on the Geoid in Europe and the Mediterranean Area, Ancona, 1978:153–164Google Scholar
  29. Mader K (1951) Das Newtonsche Raumpotential prismatischer Körper und seine Ableitungen bis zur dritten Ordnung. Öst Z Vermess Sonderheft 11Google Scholar
  30. Marti U (1997) Geoid der Schweiz 1997. Geodätisch-geophysikalische Arbeiten in der Schweiz Nr. 56. Schweizerische Geodätische KommissionGoogle Scholar
  31. Meier HK (1956) Über die Berechnung von Lotabweichungen für Aufpunkte im Hochgebirge. Deutsche Geodätische Kommission C 16Google Scholar
  32. Molodenski MS, Eremeev VF, Yurkina MI (1962) Methods for study of the external gravitational field and figure of the Earth. Translated from Russian (1960) Israel Program for Scientific Translations Ltd, JerusalemGoogle Scholar
  33. Moritz H (1980) Advanced physical geodesy. Wichmann KarlsruheGoogle Scholar
  34. Moritz H (1983) Local geoid determination in mountain regions. OSU Report 352Google Scholar
  35. Nagy D, Papp G, Benedek J (2000) The gravitational potential and its derivatives for the prism. J Geod 74(7-8):552–560. doi: 10.1007/s001900000116CrossRefGoogle Scholar
  36. Nagy D, Papp G, Benedek J (2002) Erratum: corrections to “The gravitational potential and its derivatives for the prism”. J Geod 76(8):475–475. doi: 10.1007/s00190-002-0264-7CrossRefGoogle Scholar
  37. Niethammer T (1932) Nivellement und Schwere als Mittel zur Berechnung wahrer Meereshöhen. Astronomisch geodätische Arbeiten in der Schweiz, Nr 20. Schweizerische Geodätische KommissionGoogle Scholar
  38. Petrović, S. (1996) Determination of the potential of homogeneous polyhedral bodies using line integrals. J Geod 71 (1):44–52. doi: 10.1007/s001900050074Google Scholar
  39. Tenzer R, Vanícek P, Santos M, Featherstone WE, Kuhn M (2005) The rigorous determination of orthometric heights. J Geod 79(1–3):82–92. doi: 10.1007/s00190-005-0445-2CrossRefGoogle Scholar
  40. Torge W (2001) Geodesy, Third edn. W. de Gruyter, BerlinGoogle Scholar
  41. Tsoulis D (1999) Analytical and numerical methods in gravity field modelling of ideal and real masses. Deutsche Geodätische Kommission C 510Google Scholar
  42. Tsoulis D (2001) Terrain correction computations for a densely sampled DTM in the Bavarian Alps. J Geod 75(5-6):291–307. doi: 10.1007/s001900100176CrossRefGoogle Scholar
  43. Zacharias N, Zacharias, MI, Urban SE, Høg E (2000) Comparing Tycho-2 astrometry with UCAC1. Astron J 120:1148–1152CrossRefGoogle Scholar
  44. Zacharias N, Urban SE, Zacharias MI, Wycoff GL, Hall DM, Monet DG, Rafferty TJ (2004) The second US naval observatory CCD astrograph catalog (UCAC2). Astron J 127:3043–3059CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Institut für ErdmessungUniversität HannoverHannoverGermany
  2. 2.Department of GeomaticsHafenCity University HamburgHamburgGermany
  3. 3.Institut für Astronomische und Physikalische GeodäsieTechnische Universität MünchenMunichGermany
  4. 4.Center for Space ResearchAustinUSA

Personalised recommendations