Abstract
In collocation applications, the prior covariance matrices or weight matrices between the signals and the observations should be consistent to their uncertainties; otherwise, the solution of collocation will be distorted. To balance the covariance matrices of the signals and the observations, a new adaptive collocation estimator is thus derived in which the corresponding adaptive factor is constructed by the ratio of the variance components of the signals and the observations. A maximum likelihood estimator of the variance components is thus derived based on the collocation functional model and stochastic model. A simplified Helmert type estimator of the variance components for the collocation is also introduced and compared to the derived maximum likelihood type estimator. Reasonable and consistent covariance matrices of the signals and the observations are arrived through the adjustment of the adaptive factor. The new adaptive collocation with related adaptive factor constructed by the derived variance components is applied in a transformation between the geodetic height derived by GPS and orthometric height. It is shown that the adaptive collocation is not only simple in calculation but also effective in balancing the contribution of observations and the signals in the collocation model.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Amiri-Simkooei AR (2007) Least-squares variance component estimation: theory and GPS applications. PhD Thesis, Delft University of Technology, Publication on Geodesy, 64, Netherlands Geodetic Commission, Delft
Bian S, Menz J (2000) Collocation: a synonym of kriging in geodesy. Math Geol 5: 43–
Caspary WF (1987) Concepts of network and deformation analysis. School of Surveying, the University of New South Wales, Kensington, pp 97–111
Featherstone WE, Sproule DM (2006) Fitting AUSGeio98 to the Australian height datum using GPS-leveling and least squares collocation: Application of a cross-validation technique. Survey Rev 38(301): 573–582
Felus YA, Saalfeld A, Schaffrin B (2005) Delauney triangulation structured kringing for surface interpolation. Survey Land Inf Sci 65(1): 27–36
Grafarend E (1980) An introduction to the variance–covariance-component estimation of Helmert type. Z Vermessungswes 4: 161–180
Journel AG, Huijbregt J (1989) Mining geo-statistics. Academic Press,
Karniefi AD (1990) Application of kriging technique to areal precipitation mapping in Arizona. GeoJournal 22(4): 391–398
Koch KR (1977) Least squares adjustment and collocation. Bull Géod 51: 127–135
Koch KR (1986) Maximum likelihood estimate of variance components. Bull Géod 60:329–328. Ideas by Pope AJ
Koch KR (1987) Bayesian inference for variance components. Manuscr Geod 12: 309–313
Koch KR (2000) Einführung indie Bayes-Statistik, Springer, Berlin, Heidelberg, New York, Barcelona, Hongkong, London, Mailand, Paris, Singapur, Tokio, pp 146
Koch KR, Kusche J (2002) Regularization of geopotential determination from satellite data by variance components. J Geod 76: 259–268
Krarup T (1978) Some remarks about collocation. In: Moritz H, Suenkel H (eds) Approximate methods in geodesy. H. Wichmann Verlag, Karlsruhe, pp 193–209
Mikhail EM, Ackermann F (1976) Observation and least squares, Harper and Row, New York, pp 418–421
Moritz H (1980) Statistical foundations of collocation. Boll Geod Sci Affini 2: 131–150
Oliver MA, Webster R (1990) Kriging: method of interpolation for geographical information system. Int J Geogr Inf Sys 4(3): 313–332
Ou Z (1989) Estimation of variance and covariance components. Bull Géod 63: 139–
Rao CR (1971) Estimation of variance and covariance components—MINQUE theory. J Multivar Anal 1: 257–275
Robeson SM (1997) Spherical methods for spatial interpolation: review and evaluation. Cartogr Geog Inf sys 24(1): 3–20
Schaffrin B (1981) Best invariant covariance component estimators and its application to the generalized multivariate adjustment of heterogeneous deformation observations. Bull Géod 55: 73–85
Schaffrin B (1983) Varianz–kovarianz-komponentenschätzung bei der Ausgleichung heterogener Wiederholungsmessungen. C282, Deutsche Geodatische Kommission, Munchen
Schaffrin B (1986) On robust collocation. In: Sanso F (ed) Proceedings of the first marussi symposium on mathematical geodesy Milano, 3–6 June 1985, pp 343–361
Searle SR, Casella G, McCulloch CE (1992) Variance components. Wiley, New York
Shen YZ, Liu DJ (2002) An unbiased estimate of the variance of unit weight after regularization. Geomat Inf Sci. Wuhan Univ 27:604–606 (in Chinese)
Sjöberg LE (1984) Non negative variance component estimation in the Gauss–Helmet adjustment model. Manuscr Geod 9: 247–280
Teunissen PJG, Amiri-Simkooei AR (2008) Least-squares variance component estimation. J Geod 82: 65–82
Tscherning CC (1978) Collocation and least squares methods as a tool for handling gravity field dependent data obtained through space research techniques. In: Hieber S, Guyenne TD (eds) on space oceanography, navigation and geodynamics European workshop. European Space Agency, Paris, pp 141–149
Xu PL, Shen Y, Fukuda Y, Liu YM (2006) Variance components estimation in linear inverse ill-posed models. J Geod 80: 69–81
Yang Y (1991) Robust Bayesian estimation. Bull Geod 65(3): 145–150
Yang Y (1992) Robustfying collocation. Manuscr Geod 17(1): 21–28
Yang Y, Gao W (2005) Comparison of adaptive factor in Kalman filters on navigation results. J Navigat 58: 471–478
Yang Y, Gao W (2006) An optimal adaptive Kalman filter. J Geod 80: 177–183
Yang Y, He H, Xu T (2001) Adaptively robust filtering for kinematic geodetic positioning. J Geod 75: 109–116
Yang Y, Liu N (2002) A new resolution of collocation by two minimization steps. Acta Geod Cartogr Sin 31(3):192–195 (in Chinese)
Yang Y, Xu T (2003) An adaptive Kalman filter based on sage window weights and variance components. J Navigat 56: 231–240
You RJ, Hwang HW (2006) Coordinate transformation between two geodetic datums of Taiwan by least-squares collocation. J Survey Eng 132(2): 64–70
Yu Z (1992) A generalization theory of estimation of variance-covariance components. Manuscr Geod 17: 295–301
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yang, Y., Zeng, A. & Zhang, J. Adaptive collocation with application in height system transformation. J Geod 83, 403–410 (2009). https://doi.org/10.1007/s00190-008-0226-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00190-008-0226-9