Journal of Geodesy

, Volume 83, Issue 6, pp 495–508 | Cite as

Non-stationary covariance function modelling in 2D least-squares collocation

Review

Abstract

Standard least-squares collocation (LSC) assumes 2D stationarity and 3D isotropy, and relies on a covariance function to account for spatial dependence in the observed data. However, the assumption that the spatial dependence is constant throughout the region of interest may sometimes be violated. Assuming a stationary covariance structure can result in over-smoothing of, e.g., the gravity field in mountains and under-smoothing in great plains. We introduce the kernel convolution method from spatial statistics for non-stationary covariance structures, and demonstrate its advantage for dealing with non-stationarity in geodetic data. We then compared stationary and non- stationary covariance functions in 2D LSC to the empirical example of gravity anomaly interpolation near the Darling Fault, Western Australia, where the field is anisotropic and non-stationary. The results with non-stationary covariance functions are better than standard LSC in terms of formal errors and cross-validation against data not used in the interpolation, demonstrating that the use of non-stationary covariance functions can improve upon standard (stationary) LSC.

Keywords

Least squares collocation (LSC) Non-stationary covariance function modelling Elliptical kernel convolution Gravity field interpolation Darling fault, Australia 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Armstrong M (1998) Basic linear geostatistics. Springer, BerlinGoogle Scholar
  2. Atkinson PM, Lloyd CD (2007) Non-stationary variogram models for geostatistical sampling optimisation: an empirical investigation using elevation data, Comput & Geosci 33: 1285–1300. doi:10.1016/j.cageo.2007.05.011 Google Scholar
  3. Chilès JP, Delfiner P (1999) Geostatistics. Wiley, New YorkCrossRefGoogle Scholar
  4. Cressie N (1993) Statistics for spatial data. Wiley, New YorkGoogle Scholar
  5. Dermanis A (1984) Kriging and collocation—a comparison. Manuscr Geod 9: 159–167Google Scholar
  6. Deutsch CV, Journel AG (1998) GSLIB. Oxford University Press, OxfordGoogle Scholar
  7. Duquenne H, Everaerts M, Lambot P (2005) Merging a gravimetric model of the geoid with GPS/levelling data: an example in Belgium. In: Jekeli C, Bastos L, Fernandes J (eds) Gravity, geoid and space missions. Springer, Berlin, pp 131–136CrossRefGoogle Scholar
  8. Featherstone WE, Sproule DM (2006) Fitting AusGeoid98 to the Australian Height Datum using GPS-levelling and least squares collocation: application of a cross-validation technique. Surv Rev 38(301): 573–582Google Scholar
  9. Fieguth PW, Karl WC, Willsky AS, Wunsch C (1995) Multiresolution optimal interpolation and statistical analysis of TOPEX/Poseidon satellite altimetry. IEEE Trans Geosci Remote Sens 33(2): 280–292. doi:10.1109/36.377928 CrossRefGoogle Scholar
  10. Flury J (2006) Short-wavelength spectral properties of the gravity field from a range of regional data sets. J Geod 79(10-11): 624–640. doi:10.1007/s00190-005-0011-y CrossRefGoogle Scholar
  11. Forsberg R (1986) Spectral properties of the gravity field in the nordic countries. Boll Geodesia e Sc Aff XLV: 361–384Google Scholar
  12. Fuentes M (2001) A high frequency Kriging approach for non-stationary environmental processes. Environmetrics 12(5): 469–483. doi:10.1002/env.473 CrossRefGoogle Scholar
  13. Fuentes M, Smith R (2003) A new class of non-stationary spatial models. Technical report, Department of Statistics, North Carolina State UniversityGoogle Scholar
  14. Gaposchkin EM (1973) Standard Earth III-1973. Special report 353, Smithsonian Astrophysical Observatory CambridgeGoogle Scholar
  15. Goad CC, Tscherning CC, Chin MM (1984) Gravity empirical covariance values for the continental United States. J Geophys Res 89(B9): 7962–7968CrossRefGoogle Scholar
  16. Goos JM, Featherstone WE, Kirby JF, Holmes SA (2003) Experiments with two different approaches to gridding terrestrial gravity anomalies and their effect on regional geoid computation. Surv Rev 37(288): 92–112Google Scholar
  17. Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, OxfordGoogle Scholar
  18. Higdon D, Swall J, Kern J (1999) Non-stationary spatial modelling. In: Bernardo JM, Berger JO, Dawid AP, Smith AFM (eds) Bayesian statistics 6. Oxford University Press, Oxford, pp 761–768Google Scholar
  19. Kearsley W (1977) Non-stationary estimation in gravity prediction problem. Report 256, Department of Geodetic Science, The Ohio State University, ColumbusGoogle Scholar
  20. Keller W (1998) Collocation in reproducing kernel Hilbert spaces of a multiscale analysis. Phys Chem Earth 23(1): 25–29CrossRefGoogle Scholar
  21. Keller W (2000) A wavelet approach to non-stationary collocation. In: Schwarz KP (eds) Geodesy beyond 2000. Springer, Berlin, pp 208–214Google Scholar
  22. Keller W (2002) A wavelet solution to 1D non-stationary collocation with extension to the 2D case. In: Sideris MG (eds) Gravity, geoid and geodynamics 2000. Springer, Berlin, pp 79–84Google Scholar
  23. Kirby JF, Featherstone WE (2002) High-resolution grids of gravimetric terrain correction and complete Bouguer corrections over Australia. Explor Geophys 33: 161–165CrossRefGoogle Scholar
  24. Kirby JF (2003) On the combination of gravity anomalies and gravity disturbances for geoid determination in Western Australia. J Geod 77(7-8): 433–439. doi:10.1007/s00190-003-0334-5 CrossRefGoogle Scholar
  25. Knudsen P (2005) Patching local empirical covariance functions—a problem in altimeter data processing. In: Sansò F (eds) A window on the future of geodesy. Springer, Berlin, pp 483–487CrossRefGoogle Scholar
  26. Kotsakis C, Sideris MG (1999) The long road from deterministic collocation to multiresolution approximation. Report 1999.5. Department of Geodesy and Geoinformatics, Stuttgart UniversityGoogle Scholar
  27. Kotsakis C (2000) The multiresolution character of collocation. J Geod 74(3–4): 275–290. doi:10.1007/s001900050286 CrossRefGoogle Scholar
  28. Krarup T (1969) A contribution to the mathematical foundation of physical geodesy, Rep 44. Danish Geodetic Institute, CopenhagenGoogle Scholar
  29. Krige DG (1951) A statistical approach to some basic mine valuation problems on the Witwatersrand. J Chem Metall Min Soc S Afr 52: 119–139Google Scholar
  30. Lambeck K (1987) The Perth Basin: a possible framework for its formation and evolution. Explor Geophys 18(2): 124–128. doi:10.1071/EG987124 CrossRefGoogle Scholar
  31. 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 , Olson TR (1998) The development of the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) geopotential model EGM96. Technical report NASA/TP-1998- 206861, National Aeronautics and Space Administration, Greenbelt, pp 575Google Scholar
  32. Mahalanobis PC (1936) On the generalized distance in statistics. Proc Natl Inst Sci India 12: 49–55Google Scholar
  33. Matheron G (1962) Traité de Géostatistique Appliquée, Tome I, Mémoires du Bureau de Recherches Géologiques et Minières, No. 14. Editions Technip, ParisGoogle Scholar
  34. Moreaux G, Tscherning CC, Sansò F (1999) Approximation of harmonic covariance functions on the sphere by non harmonic locally supported functions. J Geod 73: 555–567CrossRefGoogle Scholar
  35. Moritz H (1980) Advanced physical geodesy. Abacus, Tunbridge WellsGoogle Scholar
  36. Nychka DW, Wikle C, Royle JA (2002) Multiresolution models for non-stationary spatial covariance functions. Stat Modell 2(4): 315–331. doi:10.1191/1471082x02st037oa CrossRefGoogle Scholar
  37. Paciorek CJ (2003) Non-stationary Gaussian processes for regression and spatial modelling. Ph.D. thesis, Carnegie Mellon University, PittsburghGoogle Scholar
  38. Paciorek CJ, Schervish MJ (2006) Spatial modelling using a new class of non-stationary covariance functions. Environmetrics 17(5): 483–506. doi:10.1002/env.785 CrossRefGoogle Scholar
  39. Rapp RH (1964) The prediction of point and mean gravity anomalies through the use of the digital computer. Report 43, Department of Geodetic Sciences, The Ohio State University, ColumbusGoogle Scholar
  40. Reguzzoni M, Sansò F, Venuti G (2005) The theory of general Kriging, with application to the determination of a local geoid. Geophys J Int 162(4): 303–314. doi:10.1111/j.1365-246X.2005.02662.x CrossRefGoogle Scholar
  41. Robert CP, Casella G (2004) Monte carlo statistical methods, 2nd edn. Springer, BerlinGoogle Scholar
  42. Sampson PD, Guttorp P (1992) Nonparametric estimation of non-stationary spatial covariance structure. J Am. Stat Assoc 87(417): 108–119. doi:10.2307/2290458 CrossRefGoogle Scholar
  43. Sampson P, Damian D, Guttorp P (2001) Advances in modeling and inference for environmental processes with non-stationary spatial covariance. In: Monestiez P, Allard D, Froidvaux R (eds) GeoENV 2000: Geostatistics for environmental applications. Kluwer, Dordrecht, pp 17–32Google Scholar
  44. Sansò F, Venuti G, Tscherning CC (2000) A theorem of insensivity of the collocation solution to variations of the metric in the interpolation space. In: Schwarz KP (eds) International association of geodesy symposia, vol 121. Springer, Heidelberg, pp 233–240Google Scholar
  45. Schaffrin B, Felus Y (2005) On total least-squares adjustment with constraints. In: Sansò F (eds) International association of geodesy symposia, vol 128. Springer, Heidelberg, pp 417–421Google Scholar
  46. Schwarz KP, Lachapelle G (1980) Local characteristics of the gravity anomaly covariance function. Bull Géod 54(1): 21–36CrossRefGoogle Scholar
  47. Swall JL (1999) Non-stationary spatial modelling using a process convolution approach. Ph.D. thesis, Institute of Statistics and Decision Sciences, Duke University, DurhamGoogle Scholar
  48. Tscherning CC, Rapp RH (1974) Closed covariance expressions for gravity anomalies, geoid undulations, and deflections of the vertical implied by anomaly degree variance models. Report 208, Department of Geodetic Sciences, The Ohio State University, ColumbusGoogle Scholar
  49. Tscherning CC (1975) Application of collocation for the planning of gravity surveys. J Geod 49(2): 183–198Google Scholar
  50. Tscherning CC, Sansò F, Arabelos D (1987) Merging regional geoids—preliminary considerations and experiences. Boll di Geodesia e Sci Aff XLVI(3): 191–206Google Scholar
  51. Tscherning CC (1991) Density-gravity covariance functions produced by overlapping rectangular blocks of constant density. Geophys J Int 105(3): 771–776. doi:10.1111/j.1365-246X.1991.tb00811.x CrossRefGoogle Scholar
  52. Tscherning CC (1994) Geoid determination by least-squares collocation using GRAVSOFT. Lectures notes for the international school for the determination and use of the Geoid. DIIAR, Politecnico di Milano, MilanoGoogle Scholar
  53. Tscherning CC (1999) Construction of anisotropic covariance functions using Riesz-representers. J Geod 73(6): 332–336. doi:10.1007/s001900050250 CrossRefGoogle Scholar
  54. Vyskǒcil V (1970) On the covariance and structure functions of anomalous gravity field. Stud Geophys et Geod 14(2): 174–177. doi:10.1007/BF02585616 CrossRefGoogle Scholar
  55. Xu PL (1991) Least squares collocation with incorrect prior information. Z Verm 116: 266–273Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Western Australian Centre for Geodesy, The Institute for Geoscience ResearchCurtin University of TechnologyPerthAustralia

Personalised recommendations