Accuracy of unmodified Stokes’ integration in the R-C-R procedure for geoid computation
-
Zahra Ismail
and
Olivier Jamet
Abstract
Geoid determinations by the Remove-Compute-Restore (R-C-R) technique involves the application of Stokes’ integral on reduced gravity anomalies. Numerical Stokes’ integration produces an error depending on the choice of the integration radius, grid resolution and Stokes’ kernel function.
In this work, we aim to evaluate the accuracy of Stokes’ integral through a study on synthetic gravitational signals derived from EGM2008 on three different landscape areas with respect to the size of the integration domain and the resolution of the anomaly grid. The influence of the integration radius was studied earlier by several authors. Using real data, they found that the choice of relatively small radii (less than 1°) enables to reach an optimal accuracy. We observe a general behaviour coherent with these earlier studies. On the other hand, we notice that increasing the integration radius up to 2° or 2.5° might bring significantly better results. We note that, unlike the smallest radius corresponding to a local minimum of the error curve, the optimal radius in the range 0° to 6° depends on the terrain characteristics. We also find that the high frequencies, from degree 600, improve continuously with the integration radius in both semi-mountainous and mountain areas.
Finally, we note that the relative error of the computed geoid heights depends weakly on the anomaly spherical harmonic degree in the range from degree 200 to 2000. It remains greater than 10 % for any integration radii up to 6°. This result tends to prove that a one centimetre accuracy cannot be reached in semi-mountainous and mountainous regions with the unmodified Stokes’ kernel.
Acknowledgements
This work was supported by a doctoral scholarship from the University of Tichrine, Latakia, Syria.
References
[1] T. Krarup. A Contribution to the Mathematical Foundation of Physical Geodesy. In Kai Borre, editor, Mathematical Foundation of Geodesy, pages 29–90. Springer Berlin Heidelberg, 2006.Search in Google Scholar
[2] H. Yildiz, R. Forsberg, J. Ågren, C. C. Tscherning, and L. E. Sjöberg. Comparison of Remove-Compute-Restore and Least Squares Modification of Stokes’ Formula Techniques to Quasi-geoid Determination over the Auvergne Testarea. Journal of Geodetic Science, 2:53–64, 2011.10.2478/v10156-011-0024-9Search in Google Scholar
[3] I. Panet, Y. Kuroishi, and M. Holschneider. Wavelet Modelling of the Gravity Field by Domain Decomposition Methods: An Example Over Japan. Geophysical Journal International, 184:203–219, 2011.Search in Google Scholar
[4] H. Duquenne. A New Solution for the Quasigeoid in FranceQGF98. In Proceedings of the 2nd Continental Workshop on the Geoid in Europe, Budapest, volume 98 of Finnish Geodetic Institut reports, pages 251–255, March 1998.Search in Google Scholar
[5] A. KiliCoglu, C. A. Diren, H. Yildiz, M. Bolme, B. AAktug, M. Simav, and O. Lenk. Regional Gravimetric Quasi-Geoid Model and Transformation Surface to National Height System for Turkey (THG-09). Studia Geophysica et Geodaetica, 55(4):557–578, 2011.10.1007/s11200-010-9023-zSearch in Google Scholar
[6] N. Srinivas, V. M. Tiwari, J. S. Tarial, S. Prajapti, A. E.Meshram, B. Singh, and B. Nagarajan. Gravimetric Geoid of a Part of South India and its Comparison with Global Geopotential Models and GPS-Levelling Data. Journal of Earth System Science, 121(4):1025–1032, 2012.10.1007/s12040-012-0205-7Search in Google Scholar
[7] P. Valty and H. Duquenne. Quasi-Geoid of New Caledonia: Computation, Results and Analysis. In Stelios P. Mertikas, editor, Gravity, Geoid and Earth Observation, volume 135 of International Association of Geodesy Symposia, pages 427–435. Springer Berlin Heidelberg, 2010.10.1007/978-3-642-10634-7_57Search in Google Scholar
[8] R. Forsberg. Modelling the Fine Structure of the Geoid: Methods, Data Requirements and Some Results. Surveys in Geophysics, 14:403–418, 1993.Search in Google Scholar
[9] A. H. W Kearsley. Data Requirements for Determining Precise Relative Geoid Heights from Gravimetry. Journal of Geophysical Research, 91(B9):9193–9201, Aug 1986.10.1029/JB091iB09p09193Search in Google Scholar
[10] A. H. W Kearsley. The Determination of Precise Geoid Height Differences Using Ring Integration. Bollettino Di Geodesia E Scienze Affini, (2):151–174, 1986.Search in Google Scholar
[11] A. H. W Kearsley. Tests on the Recovery of Precise Geoid Height Differences from Gravimetry. Journal of Geophysical Research, 93(B6):6559–6570, June 1988.10.1029/JB093iB06p06559Search in Google Scholar
[12] M. B. Higgins, R. Forsberg, and A. H. W. Kearsley. The Effects of Varying Cap Sizes on Geoid Computations: Experiences with FFT and Ring Integration. In Rene Forsberg, Martine Feissel, and Reinhard Dietrich, editors, Geodesyon the Move, volume 119 of International Association of Geodesy Symposia, pages 201–206. Springer Berlin Heidelberg, 1998.10.1007/978-3-642-72245-5_28Search in Google Scholar
[13] A. Ellmann. The Geoid for the Baltic Countries Determined by the Least Squares Modification of Stokes Formula. PhD thesis, Royal Institute of Technology (KTH), Stockholm, 2004. Doctoral Dissertation in Geodesy No.1061.Search in Google Scholar
[14] C. Hwang, C. G. Wang, and Y. S. Hsiao. Terrain Correction Computation Using Gaussian Quadrature. Computers and Geosciences, 29:1259–1268, 2003.10.1016/j.cageo.2003.08.003Search in Google Scholar
[15] P. Valty, H. Duquenne, and I. Panet. Auvergne Dataset: Testing-Several Geoid Computation Methods. In Geodesy for Planet Earth, volume 136 of International Association of Geodesy Symposia, pages 465–472. Springer Berlin Heidelberg,2012.10.1007/978-3-642-20338-1_56Search in Google Scholar
[16] N. K. Pavlis, S. A. Holmes, S. C. Kenyon, and John J. K. Factor. The Development and Evaluation of the Earth Gravitational Model 2008 (EGM2008). Journal of Geophysical Research: Solid Earth, 117(B4), Apr. 2012.10.1029/2011JB008916Search in Google Scholar
[17] W. E. Featherstone. Tests of Two Forms of Stokes’s Integral Using a Synthetic Gravity Field Based on Spherical Harmonics. In Erik W. Grafarend, Friedrich W. Krumm, and Volker S. Schwarze, editors, Geodesy-The Challenge of the 3rd Millennium, pages 163–171. Springer Berlin Heidelberg, 2003.10.1007/978-3-662-05296-9_17Search in Google Scholar
[18] O. Esan. Spectral Analysis of Gravity Field Data and Errorsin View of Sub-Decimeter Geoid Determination in Canada. PhD thesis, The University of Calgary, Calgary, Alberta, Canada, 2000.Search in Google Scholar
[19] W. E. Featherstone, J. F. Kirby, C. Hirt, M. S. Filmer, S. J. Claessens, N. J. Brown, G. Hu, and G. M. Johnston. The AUSGeoid09 model of the Australian Height Datum. Journal of Geodesy, 85(3):133–150, 2011.10.1007/s00190-010-0422-2Search in Google Scholar
[20] J. Krynski and A. Lyszkowicz. Centimetre Quasigeoid Modellingin Poland using Heterogeneous Data. In Gravity Fieldof the Earth, volume 18 of IAG Proceedings of the 1st InternationalSymposium of the International Gravity Field Service(IGFS), pages 37–42. Istanbul, Turkey, 2006.Search in Google Scholar
[21] G.G Stokes. On the Variation of Gravity at the Surface of the Earth. Trans. Cambridge Philosophical Society VIII, 1849.Search in Google Scholar
[22] W. A. Heiskanen and H. Moritz. Physical geodesy, volume 86. Springer-Verlag, 1967.10.1007/BF02525647Search in Google Scholar
[23] L. E. Sjöberg. A General Model for Modifying Stokes’s Formula and its Least-Squares Solution. Journal of Geodesy, 77(7–8):459–464, 2003.10.1007/s00190-003-0346-1Search in Google Scholar
[24] M. Eshagh. A Strategy Towards an EGM08-based Fennoscandian Geoid Model. Journal of Applied Geophysics, 87: 53–59, 2012.10.1016/j.jappgeo.2012.08.008Search in Google Scholar
[25] Digital Relief of the Surface of the Earth. Data Announcement88-MGG-02, NOAA, National Geophysical Data Center, Boulder, Colorado, 1988.Search in Google Scholar
[26] R. Forsberg & C. C Tscherning. Geodetic Gravity Field Modelling Programs, 2003.Search in Google Scholar
[27] J. Flury. Short-Wavelength Spectral Properties of the Gravity Field from a Range of Regional Data Sets. Journal of Geodesy, 79(10–11):624–640, 2006.10.1007/s00190-005-0011-ySearch in Google Scholar
[28] W. H. F. Smith R. Scharroo J. Luis Wessel, P. and F. Wobbe. Generic Mapping Tools: Improved Version Released. EOSTrans. AGU, 94:409–410, 2013.Search in Google Scholar
[29] R. H. Rapp. The Earth’s Gravity Field to Degree and Order180 Using Seasat Altimeter Data, Terrestrial Ggravity Data and other Data. Technical Report 32, Dep. of Geod. Sci. and Surv., Ohio State University, Colombus, 1981.10.21236/ADA113098Search in Google Scholar
[30] F. G. Lemoine, S. C. Kenyon, J. K. Factor, R. G. Trimmer, N. K. Pavlis, D. S. Chinn, C. M. Cox, S. M. Klosko, S. B.Luthcke, M. H. Torrence, Y. M. Wang, R. G. Williamson, E. C. Pavlis, R. H. Rapp, and T. R. Olson. The Development of the Joint NASA GSFC and NIMA Geopotential ModelEGM96. Technical Report TP-1998–206861, NASA Goddard Space Flight Center, Greenbelt, Maryland, 20771 USA, July 1998.Search in Google Scholar
[31] H. Duquenne. A Data Set to Test Geoid Computation Methods. Harita Dergisi, (18):61–65, 2007. Proc. 1st international symposium of the International Gravity Field Service’ Gravity field of the Earth’, Istanbul, Turkey, Aug. 2006.Search in Google Scholar
[32] H.T Moritz. Advanced Physical Geodesy. Sammlung Wichmann: NeueFolge, Buchreihe. Wichmann, 1980.Search in Google Scholar
© 2015 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Research Articles
- Comparison between multi-constellation ambiguity-fixed PPP and RTK for maritime precise navigation
- Performance of the High Sensitivity Open Source Multi-GNSS Assisted GNSS Reference Server.
- An Improved Between-Satellite Single-Difference Precise Point Positioning Model for Combined GPS/Galileo Observations
- Accuracy of unmodified Stokes’ integration in the R-C-R procedure for geoid computation
- Outlier Detection by means of Monte Carlo Estimation including resistant Scale Estimation
Articles in the same Issue
- Frontmatter
- Research Articles
- Comparison between multi-constellation ambiguity-fixed PPP and RTK for maritime precise navigation
- Performance of the High Sensitivity Open Source Multi-GNSS Assisted GNSS Reference Server.
- An Improved Between-Satellite Single-Difference Precise Point Positioning Model for Combined GPS/Galileo Observations
- Accuracy of unmodified Stokes’ integration in the R-C-R procedure for geoid computation
- Outlier Detection by means of Monte Carlo Estimation including resistant Scale Estimation
