Journal of Geodesy

, Volume 81, Issue 2, pp 121–136

# A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling

• B. Heck
• K. Seitz
Original Article

## Abstract

The calculation of topographic (and iso- static) reductions is one of the most time-consuming operations in gravity field modelling. For this calculation, the topographic surface of the Earth is often divided with respect to geographical or map-grid lines, and the topographic heights are averaged over the respective grid elements. The bodies bounded by surfaces of constant (ellipsoidal) heights and geographical grid lines are denoted as tesseroids. Usually these ellipsoidal (or spherical) tesseroids are replaced by “equivalent” vertical rectangular prisms of the same mass. This approximation is motivated by the fact that the volume integrals for the calculation of the potential and its derivatives can be exactly solved for rectangular prisms, but not for the tesseroids. In this paper, an approximate solution of the spherical tesseroid integrals is provided based on series expansions including third-order terms. By choosing the geometrical centre of the tesseroid as the Taylor expansion point, the number of non-vanishing series terms can be greatly reduced. The zero-order term is equivalent to the point-mass formula. Test computations show the high numerical efficiency of the tesseroid method versus the prism approach, both regarding computation time and accuracy. Since the approximation errors due to the truncation of the Taylor series decrease very quickly with increasing distance of the tesseroid from the computation point, only the elements in the direct vicinity of the computation point have to be separately evaluated, e.g. by the prism formulas. The results are also compared with the point-mass formula. Further potential refinements of the tesseroid approach, such as considering ellipsoidal tesseroids, are indicated.

## Keywords

Topographic reduction Newton’s integral Tesseroid Prism method Point-mass modelling

## References

1. Anderson EG (1976) The effect of topography on solutions of Stokes’ problem. Unisurv S-14, Rep, School of Surveying. University of New South Wales, KensingtonGoogle Scholar
2. Forsberg R (1984) A study of terrain reductions, density anomalies and geophysical inversion methods in gravity field modelling. Rep 355, Department of Geodetic Science. The Ohio State University, ColumbusGoogle Scholar
3. Forsberg R (1985) Gravity field terrain effect computations by FFT. Bull Géod 59:342–360
4. Forsberg R, Sideris MG (1989) On topographic effects in gravity field approximation. In: Kejlsø E, Poder K, Tscherning CC (eds) Festschrift to Torben Krarup. Geodaetisk Institut Meddelelse No 58, København, pp 129–148Google Scholar
5. Forsberg R, Tscherning CC (1997) Topographic effects in gravity field modelling for BVP. In: Sansò F, Rummel R (eds) Geodetic boundary value problems in view of the one centimetre geoid. Lecture Notes in Earth Sciences, vol 65. Springer, Berlin Heidelberg New York, pp 241–272Google Scholar
6. Grüninger W (1990) Zur topographisch-isostatischen Reduktion der Schwere. PhD Thesis, Universität KarlsruheGoogle Scholar
7. Hackney RI, Featherstone WE (2003) Geodetic versus geophysical perspective of the ‘gravity anomaly’. Geophys J Int 154(1):35–43, erratum in 154(2):596Google Scholar
8. Hammer S (1939) Terrain corrections for gravimeter stations. Geophysics 4:184–194
9. Harrison JC, Dickinson M (1989) Fourier transform methods in local gravity field modelling. Bull Géod 63:149–166
10. Heck B (2003a) On Helmert’s methods of condensation. J Geod 77(3–4):155–170, DOI: 10.1007/s00190-003-0318-5
11. Heck B (2003b) Rechenverfahren und Auswertemodelle der Landesvermessung. Klassische und moderne Methoden, 3rd edn. Wichmann, HeidelbergGoogle Scholar
12. Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman, San FranciscoGoogle Scholar
13. Heiskanen WA, Vening Meinesz FA (1958) The Earth and its gravity field. McGraw-Hill, New YorkGoogle Scholar
14. Jekeli C, Serpas JG (2003) Review and numerical assessment of the direct topographical reduction in geoid determination. J Geod 77(3–4):226–239, DOI: 10.1007/s00190-003-0320-y
15. Jung K (1961) Schwerkraftverfahren in der Angewandten Geophysik. Akademische Verlagsgesellschaft. Geest& Portig, LeipzigGoogle Scholar
16. Klose U, Ilk KH (1993) A solution to the singularity problem occurring in the terrain correction formula. Manuscr Geod 18:263–279Google Scholar
17. Koch KR (1965) Die topographische Schwere- und Lotabweichungsreduktion für Aufpunkte in geneigtem Gelände. Allg Vermess.-Nachr 11:438–441Google Scholar
18. Kuhn M (2000) Geoidbestimmung unter Verwendung verschiedener Dichtehypothesen. Reihe C, Heft Nr 520. Deutsche Geodätische Kommission, MünchenGoogle Scholar
19. Kuhn M, Seitz K (2005) Comparison of Newton’s integral in the space and frequency domains. In: Sansò F (ed) A window on the future of geodesy. Proceedings of the IAG General Assembly, Sapporo, Japan 2003. IAG Symposia, vol 128. Springer, Berlin Heidelberg New York, pp 386–391Google Scholar
20. LaFehr TR (1991a) Standardization in gravity reduction. Geophysics 56(8):1170–1178
21. LaFehr TR (1991b) An exact solution for the gravity curvature (Bullard B) corrections. Geophysics 56 (8):1179–1184
22. Lemoine FG, Kenyon SC, Factor JK, Trimmer RG, Pavlis NK, Chinn DS, Cox CM, Klosko SM, Luthcke SB, Torrence MH, Wang YM, Williamson RG, Pavlis EC, Rapp RH and Olson TR (1998) The development of the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) geopotential model EGM96. Report TP-1998–206861, NASA Goddard Space Flight Center, Greenbelt, 575 ppGoogle Scholar
23. Li YC, Sideris MG (1994) Improved gravimetric terrain corrections. Geophys J Int 119:740–752
24. MacMillan WD (1930) Theoretical Mechanics, vol 2: the Theory of the potential. McGraw-Hill, New York (reprinted by Dover Publications, New York 1958)Google Scholar
25. Mader K (1951) Das Newtonsche Raumpotential prismatischer Körper und seine Ableitungen bis zur dritten Ordnung. Österr Z Vermess Sonderheft, vol 11Google Scholar
26. Martinec Z (1998) Boundary value problems for gravimetric determination of a precise geoid. Lecture notes in Earth Sciences, vol 73. Springer, Berlin Heidelberg New YorkGoogle Scholar
27. Martinec Z, Vanicěk P, Mainville A and Véronneau M (1996) Evaluation of topographical effects in precise geoid computation from densely sampled heights. J Geod 70(11):746–754, DOI: 10.1007/BF00867153Google Scholar
28. Moritz H (1968) On the use of the terrain correction in solving Molodensky’s problem. Rep 108, Department of Geodetic Science. The Ohio State University, ColumbusGoogle Scholar
30. Nagy D (1966) The gravitational attraction of a right rectangular prism. Geophysics 31:362–371
31. 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/s001900000116
32. Nagy D, Papp G, Benedek J (2002) Corrections to “The gravitational potential and its derivatives for the prism”. J Geod 76(8):475, DOI: 10.1007/s00190-002-0264-7
33. Novák P, Grafarend EW (2005) Ellipsoidal representation of the topographical potential and its vertical gradient. J Geod 78(11–12):691–706, DOI: 10.1007/s00190-005-0435-4
34. Novák P, Vaniček P, Martinec Z, Véronneau M (2001) Effects of the spherical terrain on gravity and the geoid. J Geod 75(9–10):491–504, DOI: 10.1007/s001900100201Google Scholar
35. Nowell DAG (1999) Gravity terrain corrections – an overview. J Appl Geophys 42:117–134
36. Papp G, Wang ZT (1996) Truncation effects in using spherical harmonic expansions for forward local gravity field modelling. Acta Geod Geoph Hung 31(1–2):47–66Google Scholar
37. Paul MK (1974) The gravity effect of a homogeneous polyhedron for three-dimensional interpretation. Pure Appl Geophys 112:553–561
38. Petrović S (1996) Determination of the potential of homogeneous polyhedral bodies using line integrals. J Geod 71(1):44–52, DOI: 10.1007/s001900050074
39. Rodriguez E, Morris CS, Belz JE, Chapin EC, Martin JM, Daffer W, Hensley S (2005) An assessment of the SRTM topographic products. Technical Report JPL D-31639, Jet Propulsion Laboratory, Pasadena, 143 ppGoogle Scholar
40. Schwarz K-P, Sideris MG, Forsberg R (1990) The use of FFT techniques in physical geodesy. Geophys J Int 100:485–514
41. Seitz K, Heck B (2001) Tesseroids for the calculation of topographic reductions. Abstracts “Vistas for Geodesy in the New Millenium”, IAG 2001 Scientific Assembly 2–7 September 2001, Budapest, Hungary, 106Google Scholar
42. Sideris MG (1985) A fast Fourier transform method for computing terrain corrections. Manuscr Geod 10:66–73Google Scholar
43. Smith DA (2000) The gravitational attraction of any polygonally shaped vertical prism with inclined top and bottom faces. J Geod 74(5): 414–420, DOI: 10.1007/s001900000102
44. Smith DA (2002) Computing components of the gravity field induced by distant topographic masses and condensed masses over the entire Earth using the 1-D FFT approach. J Geod 76(3):150–168, DOI: 10.1007/s00190-001-0227-4
45. Smith DA, Robertson DS, Milbert DG (2001) Gravitational attraction of local crustal masses in spherical coordinates. J Geod 74(11–12):783–795, DOI: 10.1007/s001900000142
46. Talwani M, Ewing M (1960) Rapid computation of gravitational attraction of three-dimensional bodies of arbitrary shape. Geophysics 25:203–225
47. Tsoulis D (1999) Analytical and numerical methods in gravity field modelling of ideal and real masses. Reihe C, Heft Nr 510. Deutsche Geodätische Kommission, MünchenGoogle Scholar
48. Tsoulis D, Tziavos IN (2002) A comparison of some existing methods for the computation of terrain correction in local gravity field modelling. In: Tziavos IN (ed) Gravity and Geoid 2002. Proceedings of the 3rd meeting of the international gravity and geoid commission, Thessaloniki, pp 156–160Google Scholar
49. Tsoulis D, Wziontek H, Petrović S (2003) A bilinear approximation of the surface relief in terrain correction computations. J Geod 77(5–6):338–344, DOI: 10.1007/s00190-003-0332-7
50. Vaniček P, Novák P, Martinec Z (2001) Geoid, topography, and the Bouguer plate or shell. J Geod 75(4):210–215, DOI: 10.1007/s001900100165
51. Vaniček P, Tenzer R, Sjöberg LE, Martinec Z, Featherstone WE (2004) New views of the spherical Bouguer gravity anomaly. Geophys J Int 159:460–472