Journal of Geodesy

, Volume 92, Issue 5, pp 471–485 | Cite as

Using radial basis functions in airborne gravimetry for local geoid improvement

  • Xiaopeng Li
Original Article


Radial basis functions (RBFs) have been used extensively in satellite geodetic applications. However, to the author’s knowledge, their role in processing and modeling airborne gravity data has not yet been fully advocated or extensively investigated in detail. Compared with satellite missions, the airborne data are more suitable for these kinds of localized basis functions especially considering the following facts: (1) Unlike the satellite missions that can provide global or near global data coverage, airborne gravity data are usually geographically limited. (2) It is also band limited in the frequency domain. (3) It is straightforward to formulate the RBF observation equations from an airborne gravimetric system. In this study, a set of band-limited RBF is developed to model and downward continue the airborne gravity data for local geoid improvement. First, EIGEN6c4 coefficients are used to simulate a harmonic field to test the performances of RBF on various sampling, noise, and flight height levels, in order to gain certain guidelines for processing the real data. Here, the RBF method not only successfully recovers the harmonic field but also presents filtering properties due to its particular design in the frequency domain. Next, the software was tested for the GSVS14 (Geoid Slope Validation Survey 2014) area in Iowa as well as for the area around Puerto Rico and the US Virgin Islands by use of the real airborne gravity data from the Gravity for the Redefinition of the American Vertical Datum (GRAV-D) project. By fully utilizing the three-dimensional correlation information among the flight tracks, the RBF can also be used as a data cleaning tool for airborne gravity data adjustment and cleaning. This property is further extended to surface gravity data cleaning, where conventional approaches have various limitations. All the related numerical results clearly show the importance and contribution of the use of the RBF for high- resolution local gravity field modeling.


Radial basis functions Airborne gravimetry Data cleaning Regional gravity field modeling 



The author sincerely appreciates all of the useful comments and suggestions from the Editors and all the anonymous reviewers, especially reviewer 3 who gave very useful insights to further discuss the numerical results and very practical instructions to present them in a much better way. The author also thanks Mr. Heck, and Mr. Hilla at NGS for their help on improving the language usage and writing styles. Without all of these above mentioned help, the paper cannot be in the current shape that can be considered here.

Supplementary material


  1. Bates AP, Khalid A, Kennedy RA (2017) Efficient computation of Slepian functions for arbitrary regions on the sphere. IEEE Trans Signal Process 65(16):4379–4393. doi: 10.1109/TSP.2017.2712122 CrossRefGoogle Scholar
  2. Bentel K, Schmidt M, Denby CR (2013) Artifacts in regional gravity representations with spherical radial basis functions. J Geod Sci 3:173–187. doi: 10.2478/jogs-2013-0029 Google Scholar
  3. Bentel K, Schmidt M, Gerlach C (2013b) Different radial basis functions and their applicability for regional gravity field representation on sphere. GEM Int J Geomath 4:67–96. doi: 10.1007/s13137-012-0046-1 CrossRefGoogle Scholar
  4. Brozena JM, Peters MF (1988) An airborne gravity study of eastern North Carolina. Geop 53(2):245–253. doi: 10.1190/1.1442459 CrossRefGoogle Scholar
  5. Cheng MK, Shum CK, Tapley B (1997) Determination of longterm changes in the earth’s gravity field from satellite laser ranging observations. J Geophys Res 102(22):377–390. doi: 10.1029/97JB01740 Google Scholar
  6. Childers VA, Bell RE, Brozena JM (1999) Airborne gravimetry: an investigation of filtering. Geop 64:64–69. doi: 10.1190/1.1444530 Google Scholar
  7. Colombo O (1981) Global geopotential modeling from satellite-to-satellite tracking, Report No. 317, Ohio State UniversityGoogle Scholar
  8. Eicker A (2008) Gravity field refinement by radial basis functions from in-situ satellite data. Ph.D. Thesis, University of Bonn, Bonn, GermanyGoogle Scholar
  9. Eicker A, Schall J, Kusche J (2013) Regional gravity modelling from spaceborne data: case studies with GOCE. Geophys J Int 196:1431–1440. doi: 10.1093/gji/ggt485 CrossRefGoogle Scholar
  10. Forsberg R (1987) A new covariance model for inertial gravimetry and gradiometry. J Geophys Res 92(B2):1305–1310. doi: 10.1029/JB092iB02p01305 CrossRefGoogle Scholar
  11. Forsberg R, Olesen AV, Keller K, Møller M (2001) Airborne gravity and geoid surveys in the Arctic and Baltic seas. In: Proceedings of international symposium on kinematic systems in geodesy, geomatics and navigation, Banff, Canada, pp 586–593Google Scholar
  12. Forsberg R, Olesen AV (2010) Airborne gravimetry. In: Gouchang X (ed) Sciences of geodesy-I, monograph. Spinger, Berlin, pp 83–104CrossRefGoogle Scholar
  13. Förste C, Bruinsma SL, Abrikosov O, Lemoine JM, Marty JC, Flechtner F, Balmino G, Barthelmes F, Biancale R (2014) EIGEN-6C4 The latest combined global gravity field model including GOCE data up to degree and order 2190 of GFZ Potsdam and GRGS Toulouse. GFZ Data Services. doi: 10.5880/icgem.2015.1
  14. Gilardoni M, Reguzzoni M, Sampietro D (2016) GECO: a global gravity model by locally combining GOCE data and EGM2008. Stud Geophys Geod 60(2):228–247. doi: 10.1007/s11200-015-1114-4 CrossRefGoogle Scholar
  15. Hansen P (1990) Truncated singular value decomposition solutions to discrete ill-posed problems with ill-determined numerical rank. SIAM J Sci Stat Comput 11:503–518CrossRefGoogle Scholar
  16. Heiskanen WA, Mortiz H (1967) Physical geodesy. Freeman, San FranciscoGoogle Scholar
  17. Hubbert S, Le Gia QT, Morton TM (2015) Spherical radial basis functions, theory and applications, SpringerBriefs in mathematics. doi: 10.1007/978-3-319-17939-1_2
  18. Hwang C, Hsiao YS, Shih HC (2006) Data reduction in scalar airborne gravimetry: theory, software and case study in Taiwan. Comput Geosci 32:1573–1584. doi: 10.1016/j.cageo.2006.02.015 CrossRefGoogle Scholar
  19. Hwang C, Hsiao YS, Shih HC, Yang M, Chen KH, Forsberg R, Olesen AV (2007) Geodetic and geophysical results from a Taiwan airborne gravity survey: Data reduction and accuracy assessment. J Geophys Res 112(B04407). doi: 10.1029/2005JB004220
  20. Jekeli C (2000) Inertial navigation systems with geodetic applications. Walter de Gruyter, BerlinGoogle Scholar
  21. Jekeli C (2005) Spline representation of functions on a sphere for geopotential modeling, Technical report 475, Department of Geodetic Science, the Ohio State UniversityGoogle Scholar
  22. Keller W, Hirsch M (1993) Downward continuation versus free-air reduction in airborne gravimetry. In: Montag H, Reigber C (eds) Geodesy and physics of the earth. International association of geodesy symposia, vol 112. Springer, Berlin. doi: 10.1007/978-3-642-78149-0_62 Google Scholar
  23. Klees R, Wittwer T (2007) Local gravity field modelling with multipole wavelets. In: Tregoning P, Rizos C (eds) Dynamic planet monitoring and understanding a dynamic planet with geodetic and oceanographic tools. In: International association of geodesy symposia 130: 303–308. doi: 10.1007/978-3-540-49350-1_45
  24. Klees R, Tenzer R, Prutkin I, Wittwer T (2008) A data-driven approach to local gravity field modeling using spherical radial basis functions. J Geod 82:457–471. doi: 10.1007/s00190-007-0196-3 CrossRefGoogle Scholar
  25. Kwon JH, Jekeli C (2001) A new approach for airborne vector gravimetry using GPS/INS. J Geod 74(10):690–700. doi: 10.1007/s001900000130 CrossRefGoogle Scholar
  26. Li X (2007) Moving base INS/GPS vector gravimetry on a land vehicle. Ph.D. dissertation, OSU report 486, the Ohio State University, Columbus OHGoogle Scholar
  27. Li X (2009) Comparing the Kalman filter with a Monte Carlo-based artificial neural network in the INS/GPS vector gravimetric system. J Geod 83(9):797–804. doi: 10.1007/s00190-008-0293-y CrossRefGoogle Scholar
  28. Li X (2011a) Strapdown INS/DGPS airborne gravimetry tests in the Gulf of Mexico. J Geod 85(9):597–605. doi: 10.1007/s00190-011-0462-2 CrossRefGoogle Scholar
  29. Li X (2011b) An exact formula for the tilt correction in scalar airborne gravimetry. Journal of Applied Geodesy. 5(2): 81–85, ISSN (Online) 1862–9024, ISSN (Print) 1862–9016. doi: 10.1515/jag.2011.007
  30. Li X (2017) Modeling the North American Vertical Datum of 1988 Errors in the Conterminous United States, under reviewGoogle Scholar
  31. Li X, Crowley JW, Holmes SA, Wang YM (2016) The contribution of the GRAV-D airborne gravity to geoid determination in the Great Lakes region. Geophys Res Lett 43:4358–4365. doi: 10.1002/2016GL068374 CrossRefGoogle Scholar
  32. Lieb V, Schmidt M, Dettmering D, Börger K (2016) Combination of various observation techniques for regional modeling of the gravity field. J Geophys Res 121(5):3825–3845. doi: 10.1002/2015JB012586 CrossRefGoogle Scholar
  33. Lin M, Denker H, Müller J (2014) Regional gravity field modeling using free-positioned point masses. Stud Geophys Geod 58(2):207–226. doi: 10.1007/s11200-013-1145-7 CrossRefGoogle Scholar
  34. Muller PM, Sjogren WL (1968) Mascons: lunar mass concentrations. Science 161(3842):680–684. doi: 10.1126/science.161.3842.680 CrossRefGoogle Scholar
  35. Luyendyk APJ (1997) Processing of airborne magnetic data. Journal of Australian Geology & Geophysics 17(2):31–38. doi: fae9173a-71af-71e4-e044-00144fdd4fa6Google Scholar
  36. Naeimi M (2013) Inversion of satellite gravity data using spherical radial base functions. Ph.D. thesis. Munchen, ISBN 978-3-7696-5123-2Google Scholar
  37. Novák P, Heck B (2002) Downward continuation and geoid determination based on band-limited airborne gravity data. J Geod 76(5):269–278. doi: 10.1007/s00190-002-0252-y CrossRefGoogle Scholar
  38. Pail R, Bruinsma S, Migliaccio F, Förste C, Goiginger H, Schuh W, Höck E, Reguzzoni M, Brockmann JM, Abrikosov O, Veicherts M, Fecher T, Mayrhofer R, Krasbutter I, SansF Tscherning CC (2011) First GOCE gravity field models derived by three different approaches. J Geod 85(11):819–843. doi: 10.1007/s00190-011-0467-x CrossRefGoogle Scholar
  39. Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2012) The development and evaluation of the Earth Gravitational Model 2008 (EGM2008). J Geophys Res 117 (B04406). doi: 10.1029/2011JB008916
  40. Rummel R (2010) The interdisciplinary role of space geodesy—revisited. J Geodyn 49(3–4):112–115CrossRefGoogle Scholar
  41. Rummel R, Yi W, Stummer C (2011) GOCE gravitational gradiometry. J Geod 85(11):777–790CrossRefGoogle Scholar
  42. Rummel R (2012) Height unification using GOCE. J Geod Sci 2:355–362. doi: 10.2478/v10156-011-0047-2 Google Scholar
  43. Sandwell DT, Müller RD, Smith WHF, Garcia E, Francis R (2014) New global marine gravity model from CryoSat-2 and Jason-1 reveals buried tectonic structure. Science 346(6205):65–67. doi: 10.1126/science.1258213 CrossRefGoogle Scholar
  44. Save H, Bettadpur S, Tapley BD (2016) High resolution CSR GRACE RL05 mascons. J Geophys Res Solid Earth 121:7547–7569. doi: 10.1002/2016JB013007 CrossRefGoogle Scholar
  45. Schmidt M, Fengler M, Mayer-Guerr T, Eicker A, Kusche J, Sanchez L, Han SC (2007) Regional gravity field modeling in terms of spherical base functions. J Geod 81(1):17–38. doi: 10.1007/s00190-006-0101-5 CrossRefGoogle Scholar
  46. Sebera J, Pitonak M, Hamackova E, Novák P (2015) Comparative study of the spherical downward continuation. Surv Geophys 2:253–267. doi: 10.1007/s10712-014-9312-0 CrossRefGoogle Scholar
  47. Simons FJ, Dahlen FA (2006) Spherical Slepian functions and the polar gap in geodesy. Geophys J Int 166(3):1039–1061. doi: 10.1111/j.1365-245X.2006.03065.x CrossRefGoogle Scholar
  48. Slepian D, Pollak HO (1961) Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - I, Bell System Technical Journal 40Google Scholar
  49. Smith D (2007) The GRAV-D project: gravity for the Redefinition of the American Vertical Datum.
  50. Smith DA, Holmes SA, Li X, Guillaume Y, Wang YM, Brki B, Roman DR, Damiani TM (2013) Confirming regional 1 cm differential geoid accuracy from airborne gravimetry: the Geoid Slope Validation Survey of 2011. J Geod 87(1012):885–907. doi: 10.1007/s00190-013-0653-0 CrossRefGoogle Scholar
  51. Sneeuw N, van Gelderen M (1997) The polar gap; in: Sans, F.; Rummel, R. (eds.) Geodetic Boundary Value Problems in View of the One Centimeter Geoid, Lecture Notes in Earth Sciences, Vol. 65: 559-568, Springer, ISBN (Print) 978-3-540-62636-7, ISSN 0930-0317. doi: 10.1007/BFb0011717
  52. Sośnica K, Jäggi A, Meyer U, Thaller D, Beutler G, Arnold D, Dach R (2015) Time variable earths gravity field from SLR satellites. J Geod 89(10):945–960. doi: 10.1007/s00190-015-0825-1 CrossRefGoogle Scholar
  53. Studinger M, Bell R, Frearson N (2008) Comparison of AIRGrav and GT-1A airborne gravimeters for research applications. Geop 73(6):151–161. doi: 10.1190/1.2969664 Google Scholar
  54. Tapley BD, Bettadpur S, Watkins M, Reigber C (2004) The gravity recovery and climate experiment: mission overview and early results. Geophys Res Lett 31:L09607. doi: 10.1029/2004GL019920 CrossRefGoogle Scholar
  55. Tenzer R, Prutkin I, Klees R (2012) A comparison of different integral equation-based approaches for local gravity field modelling: case study for the Canadian Rocky Mountains. In: Kenyon S, Pacino MC, Marti U (eds) Geodesy for planet earth, International association of geodesy symposia, vol 136: 381388. doi: 10.1007/978-3-642-20338-1_46
  56. Tenzer R, Klees R (2008) The choice of the spherical radial basis functions in local gravity field modeling. Stud Geophys Geod 52:287–304. doi: 10.1007/s11200-008-0022-2 CrossRefGoogle Scholar
  57. Tscherning CC (2013) Geoid determination by 3D least-squares collocation. In: Sanso F, Sideris MG (eds) Geoid determination, lecture notes in earth system sciences 110. Springer, Berlin, pp 311–336. doi: 10.1007/978-3-540-74700-0_7 Google Scholar
  58. Wang YM, Saleh J, Li X, Roman DR (2012) The US Gravimetric Geoid of 2009 (USGG2009): model development and evaluation. J Geod 86(3):165–180. doi: 10.1007/s00190-011-0506-7 CrossRefGoogle Scholar
  59. Wang YM, Becker C, Mader G, Hanson T, Martin D, Holmes SA, Li X, Jiang T, Breidenbach S, Geoghegan C, Winester D (2015) Results of analysis of the geoid slope Validation Survey 2014 in Iowa, IUGG 2015Google Scholar
  60. Wittwer T (2009) Regional gravity field modelling with radial basis functions. Ph.D. thesis, Technical University of DelftGoogle Scholar
  61. Xu P (1998) Truncated SVD methods for discrete ill-posed problems. Geophys J Int 135:505–514. doi: 10.1046/j.1365-246X.1998.00652.x CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.DSTSilver SpringUSA

Personalised recommendations