Abstract
We propose a methodology for the combination of a gravimetric (quasi-) geoid with GNSS-levelling data in the presence of noise with correlations and/or spatially varying noise variances. It comprises two steps: first, a gravimetric (quasi-) geoid is computed using the available gravity data, which, in a second step, is improved using ellipsoidal heights at benchmarks provided by GNSS once they have become available. The methodology is an alternative to the integrated processing of all available data using least-squares techniques or least-squares collocation. Unlike the corrector-surface approach, the pursued approach guarantees that the corrections applied to the gravimetric (quasi-) geoid are consistent with the gravity anomaly data set. The methodology is applied to a data set comprising 109 gravimetric quasi-geoid heights, ellipsoidal heights and normal heights at benchmarks in Switzerland. Each data set is complemented by a full noise covariance matrix. We show that when neglecting noise correlations and/or spatially varying noise variances, errors up to 10% of the differences between geometric and gravimetric quasi-geoid heights are introduced. This suggests that if high-quality ellipsoidal heights at benchmarks are available and are used to compute an improved (quasi-) geoid, noise covariance matrices referring to the same datum should be used in the data processing whenever they are available. We compare the methodology with the corrector-surface approach using various corrector surface models. We show that the commonly used corrector surfaces fail to model the more complicated spatial patterns of differences between geometric and gravimetric quasi-geoid heights present in the data set. More flexible parametric models such as radial basis function approximations or minimum-curvature harmonic splines perform better. We also compare the proposed method with generalized least-squares collocation, which comprises a deterministic trend model, a random signal component and a random correlated noise component. Trend model parameters and signal covariance function parameters are estimated iteratively from the data using non-linear least-squares techniques. We show that the performance of generalized least-squares collocation is better than the performance of corrector surfaces, but the differences with respect to the proposed method are still significant.
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References
Akaike H (1973) Information theory and an extension of the maximum likelihood principle. In: Petrov BN, Csáki F (eds) Second international symposium on information theory. Akadémiai Kiadó, Budapest, pp 267–281
Baarda W (1981) S-transformations and criterion matrices; Publications on Geodesy 18 (vol 5 no 1), second revised edition. Delft, The Netherlands
Brent RP (1973) Algorithms for minimization without derivatives. Prentice-Hall, Englewood Cliffs
Craven P, Wahba G (1979) Smoothing noisy data with spline functions. Numer Math 31: 377–403
de Bruijne AJT, Haagmans RHN, de Min EJ (1997) A preliminary North Sea Geoid model GEONZ97. Rep MDGAP-9735, Directoraat-Generaal Rijkswaterstaat, Meetkundige Dienst, Delft, The Netherlands
de Min E (1996) De geoide voor Nederland. Nederlandse Commissie voor Geodesie, Groene Serie, vol 34. Delft, The Netherlands
Denker H, Torge W (1998) The European gravimetric quasigeoid EGG97—An IAG supported continental enterprise. In: Forsberg R, Feissel M, Dietrich R (eds) Geodesy on the Move—Gravity, Geoid, Geodynamics and Antarctica. IAG Symp Vol 119: 249–254. Springer, Berlin
Featherstone WE (1998) Do we need a gravimetric geoid or a model of the base of the Australian height datum to transform GNSS heights? Aust Surv 43: 273–280
Featherstone WE (2000) Refinement of a gravimetric geoid using GNSS and levelling data. J Surv Eng 126: 27–56
Fotopoulos G (2005) Calibration of geoid error models via a combined adjustment of ellipsoidal, orthometric and gravimetric geoid height data. J Geod 79: 111–123
Goovaerts P (1997) Gostatistics for natural resources evaluation. Oxford University Press, New York
Grebenitcharsky RS, Rangelova EV, Sideris MG (2005) Transformation between gravimetric and GNSS/levelling-derived geoids using additional gravity information. J Geodyn 39: 527–544
Heck B (1997) Formulation and linearization of boundary value problems: from observables to a mathematical model. In: Sanso F, Rummel R (eds) Geodetic boundary value problems in view of the one centimeter geoid. Lecture notes in earth sciences. Springer, Berlin, pp 121–160
Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman, San Francisco
Isaaks EH, Srivastava RM (1990) An introduction to applied geostatistics. Oxford University Press, USA
Kellogg OD (1929) Foundations of potential theory. Springer, Berlin
Klees R, Wittwer T (2007) A data-adaptive design of a spherical basis function network for gravity field modelling. In: Tregoning P, Rizos C (eds) Dynamic planet—monitoring and understanding a dynamic planet with geodetic and oceanographic tools, International Association of Geodesy Symposia, vol 130. Springer, Berlin, pp 323–328
Klees R, Tenzer R, Prutkin I, Wittwer T (2008) A data-driven approach to local gravity field modelling using spherical radial basis functions. J Geod 82: 457–471
Matern B (1986) Spatial variation. Lecture notes in statistics, 2nd edn, vol 36. Springer, New York
Meissl P (1971) Preparations for the numerical evaluation of second-order Molodensky-type formulas. Report 163, Dept Geod Sci & Surv, Ohio State University, Columbus
Meissl P (1976) Hilbert spaces and their application to geodetic least-squares problems. Boll Geod Sci Affini 35: 49–80
Molodensky MS, Eremeev VF, Yurkina MI (1962) Methods for study of the external gravitational field and figure of the earth. Israeli Programme for the Translation of Scientific Publications, Jerusalem
Moritz H (1989) Advanced physical geodesy, 2nd edn. Wichmann, Karlsruhe
Nahavandchi N, Soltanpour A (2006) Improved determination of heights using a conversion surface by combining gravimetric quasi-geoid /geoid and GNSS-levelling height differences. Stud Geophys Geod 50: 165–180
Pardo-Iguzquiza E, Dowd PA (2001) VARIOG2D: a computer program for estimating the semi-variogram and its uncertainty. Comput Geosci 27: 549–561
Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2008) An Earth gravitational model to degree 2160: EGM2008. Paper presented at the EGU General Assembly 2008, Vienna, Austria
Prutkin I, Klees R (2008) On the non-uniqueness of local quasi-geoids computed from terrestrial gravity anomalies. J Geod 82: 147–156
Smith WHF, Wessel P (1990) Gridding with continuous curvature splines in tension. Geophysics 55: 293–305
Tscherning CC (1978) Introduction to functional analysis with a view to its application in approximation theory. In: Moritz H, Sünkel H (eds) Approximation methods in geodesy. Wichmann H, Karlsruhe, pp 157–192
van Loon J (2008) Functional and stochastic modelling of satellite gravity data, 231 p. PhD thesis, Delft University of Technology, Delft, The Netherlands
Webster R, Olivier MA (2007) Geostatistics for environmental scientists, 2nd edn. Wiley, New York
Wessel P, Smith WHF (1991) Free software helps map an display data. EOS Trans AGU 72: 441
Wong L, Gore R (1969) Accuracy of geoid heights from modified Stokes kernels. Geophys J R Astron Soc 18: 81–91
Acknowledgments
Urs Marti from the Federal Office of Topography Swisstopo provided the test data sets. Software written by Jasper van Loon has been used to compute some corrector surface models. This support is gratefully acknowledged. We also acknowledge the detailed comments of three anonymous reviewers and of the editor, which helped us improving the quality of the manuscript.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Klees, R., Prutkin, I. The combination of GNSS-levelling data and gravimetric (quasi-) geoid heights in the presence of noise. J Geod 84, 731–749 (2010). https://doi.org/10.1007/s00190-010-0406-2
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DOI: https://doi.org/10.1007/s00190-010-0406-2