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A study on the evaluation of the geoid-quasigeoid separation term over Pakistan with a solution of first and second order height terms

Abstract

An attempt has been made to evaluate the geoid-quasigeoid separation term over Pakistan by using solutions of terms involving first and second order terrain heights. The first term, involving the Bouguer anomaly, has a significant value and requires being incorporated in any case for determination of the geoid from the quasigoidal solution. The results of the study show that the second term of separation, which involves the vertical gravity anomaly gradient, is significant only in areas with very high terrain elevations and reaches a maximum value of 2–3 cm. The integration radius of 18 km for the evaluation of the vertical gravity anomaly gradient was found to be adequate for the near zone contribution in the case of the vertical gravity anomaly gradient. The Earth Gravity Model EGM96 height anomaly gradient terms were evaluated to assess the magnitude of the model dependent part of the separation term. The density of the topographic masses was estimated with the linear operator of vertical gravity anomaly gradient using the complete Bouguer anomaly data with an initial arbitrary density of 2.67 g/cm3 to study the effect of variable Bouguer density on the geoid-quasigeoid separation. The density estimates seem to be reasonable except in the area of very high relief in the northern parts. The effect of variable density is significant in the value of the Bouguer anomaly-dependent geoid-quasigeoid separation and needs to be incorporated for its true applicability in the geoid-quasigeoid separation determination. The geoid height (N) was estimated from the geoid-quasigeoid separation term plus global part of height anomaly and terrain-dependant correction terms. The results were compared with the separation term computed from EGM96-derived gravity anomalies and terrain heights to estimate its magnitude and the possible amount of commission and omission effects.

References

  • Andersen, O. B., L. Anne, Vest, and P. Knudsen, The KMS04 Multi- Mission mean Sea Surface, Proceedings of the Workshop GOCINA: Improving modeling of ocean transport and climate prediction in the North Atlantic region using GOCE gravimetry, April 13–15, 2005, Novotel, Luxembourg, 2005.

    Google Scholar 

  • Bian, S., Some cubature formulas for singular integrals in geodesy, J. Geod., 71, 443–453, 1997.

    Article  Google Scholar 

  • Bian, S. and X. Dong, On the singular integration in physical geodesy, Manuscr. Geod., 16, 283–287, 1991.

    Google Scholar 

  • Bursa, M., Report of Special Commission SC3, Fundamental constants, International Association of Geodesy, Paris, 1995.

    Google Scholar 

  • Fösrste, C., F. Flechtner, R. Schmidt, U. Meyer, R. Stubenvoll, F. Barthelmes, R. König, K. H. Neumayer, M. Rothacher, Ch. Reigber, R. Biancale, S. Bruinsma, J.-M. Lemoine, and J. C. Raimondo, A new high resolution global gravity field model derived from combination of GRACE and CHAMP mission and altimetry/gravimetry surface gravity data, Poster presented at EGU General Assem. 2005, Vienna, Austria, 24–29, April, 2005, 2005.

    Google Scholar 

  • Fösrste, C., F. Flechtner, R. Schmidt, R. König, U. Meyer, R. Stubenvoll, M. Rothacher, F. Barthelmes, H. Neumayer, R. Biancale, S. Bruinsma, J.-M. Lemoine, and S. Loyer, A mean global gravity field model from the combination of satellite mission and altimetry/gravimetry surface data—EIGEN-Gl04C, Geophys. Res. Abst., 8, 03462, 2006.

    Google Scholar 

  • GETECH, GETECH report on South East Asia Gravity project (SEAGP), GETECH Group plc., Kitson House, Elmete Hall Elmete Lane, Roundhay University of Leeds, LS8 2LJ, U.K., 1995.

    Google Scholar 

  • GRAVSOFT, A system for geodetic gravity field modelling, C. C. Tscherning, Department of Geophysics, Juliane Maries Vej 30, DK-2100 Copenhagen N. R. Forsberg and P. Knudsen, Kort og Matrikelstyrelsen, Rentemestervej-8, DK-2400 Copenhagen NV, 2005.

    Google Scholar 

  • Heiskanen, W. A. and H. Moritz, Physical Geodesy, Freeman, San Francisco, 1967.

    Google Scholar 

  • Helmut, L., A generalized form of Nettletons’s density determination, Geophys. Prospect., 15, 247–258, 1965.

    Google Scholar 

  • Huang, J., P. Vanicek, S. Pagiatakis, and W. Brink, Effect of topographical mass density variation on gravity and geoid in the Canadian Rocky Mountains, J. Geodyn., 74, 805–815, 2001.

    Article  Google Scholar 

  • Hunegnaw, A., The effect of lateral density variation on local geoid determination, Proc. IAG 2001 Sci. Assem., Budapest, Hungary, 2001.

    Google Scholar 

  • Kiamehr, R., The impact of lateral density variation model in the determination of precise gravimetric geoid in mountainous areas: a case study of Iran, Geophys. J. Int., 167, 521–527, 2006.

    Article  Google Scholar 

  • Kuhn, M., GeoidBestimmung unter verwendung verschiedener dichtehypothesen. Deutsche Geodatische Kommission, in Dissertationen, Heft Nr. 520, edited by C. Reihe, Munchen, Gaermany, 2000a.

    Google Scholar 

  • Kuhn, M., Density modelling for geoid determination. GGG2000, July 31- August 4, 2000, Alberta, Canada, 2000b.

    Google Scholar 

  • Kuhtreiber, N., Precise geoid determination using a density variation model, Phys. Chem. Earth, 23(1), 59–63, 1998.

    Article  Google Scholar 

  • Lemoine, F. G., D. E. Smith, R. Smith, L. Kunz, N. K. Pavlis, S. M. Klosko, D. S. Chinn, M. H. Torrence, R. G. Williamson, C. M. Cox, K. E. Rachlin, Y. M. Wang, E. C. Pavlis, S. C. Kenyon, R. Salman, R. Trimmer, R. H. Rapp, and R. S. Nerem, The development of thr NASA, GSFC and NIMA joint geopotential model, in Gravzly, Geozd, and Marzne Geod., edited by Segawa, Fugimoto and Okubo, IAG Synzposza 117, Springer-Verlag, Berlin, 461–470, 1997.

    Chapter  Google Scholar 

  • Lisitzin, E., Sea level changes, Elsevier, Amsterdam, 1974.

    Google Scholar 

  • Martinec, Z., Effect of lateral density variations of topographical masses in view of improving geoid model accuracy over Canada, Contract report for Geodetic Survey of Canada, Ottawa, Canada, 1993.

    Google Scholar 

  • Martinec, Z., P. Vanicek, A. Mainville, and M. Veronneau, The effect of lake water on geoidal height, Manuscr. Geod., 20, 193–203, 1995.

    Google Scholar 

  • Molodensky, M. S., V. F. Eremeev, and M. I. Yurkina, Methods for the study of the external gravitational field and figure of the Earth, Israeli Program for Scientific Translations, Jerusalem, 1962.

    Google Scholar 

  • Nahavandchi, H., Two different methods of geoidal height determinations using a spherical harmonics representation of the geopotential, topographic corrections and height anomaly-geoidal height difference, J. Geod., 76, 345–352, 2002.

    Article  Google Scholar 

  • Nahavandchi, H. and L. E. Sjöberg, Terrain correction to power H3 in gravimetric geoid determination, J. Geod., 72, 124–135, 1998.

    Article  Google Scholar 

  • Nafe, L. E. and C. L. Drake, Physical properties of marine sediments, in The sea Interscience, edited by Hill, 794–815, 1963.

    Google Scholar 

  • Nettleton, L. L., Elementary gravity and magnetic for geologists and seismologists, SEG Monogr. Ser. l, 121, 1971.

  • Noor, E., J. Chen, L. Yulin, and J. Zhang, Report on data processing/ adjustment regarding “A” & “AB” order GPS networks of Pakistan June 15-24, 1997, Survey of Pakistan Rawalpindi, 1997.

    Google Scholar 

  • Omang, O. C. D. and R. Forsberg, How to handle topography in practical geoid determination: three examples, J. Geod., 74, 458–466, 2000.

    Article  Google Scholar 

  • Pagiatakis, S. D. and C. Armenakis, Gravimetric geoid modelling with GIS, Int. Geoid Serv. Bull., 8, 105–112, 1999.

    Google Scholar 

  • Rapp, R. H., Methods for the computation of geoid undulations from potential coe.cients, Bull. Geod., 101, 283–297, 1971.

    Article  Google Scholar 

  • Rapp, R. H., A FORTRAN Program for the computation of gravimetric quantities from high degree spherical harmonic expansions, Rep. 334, Dept. of Geodetic Science and Surveying, The Ohio State University, Columbus, Ohio, 1982.

    Google Scholar 

  • Rapp, R. H., Global geoid determination, in Geoid and Its Geophysical Interpretations, edited by Vanicek and Christou, p. 57–76, CRC Press, Boca Raton. FL, 1994a.

    Google Scholar 

  • Rapp, R. H., The use of potential coefficient models in computing geoid undulations. Intenational School for the determination and use of the geoid, Int. Geoid Serv., DIIAR-Politecnico di Milano, 71–99, 1994b.

    Google Scholar 

  • Rapp, R. H., Use of potential coefficient models for geoid undulation determinations using a spherical harmonic representation of the height anomaly/geoid undulation difference, J. Geod., 71, 282–289, 1997.

    Article  Google Scholar 

  • Reigber, Ch., P. Schwintzer, R. Stubenvoll, R. Schmidt, F. Flechtner, U. Meyer, R. König, H. Neumayer, Ch. Förste, F. Barthelmes, S. Y. Zhu, G. Balmino, R. Biancale, J.-M. Lemoine, H. Meixner, and J. C. Raimondo, A High Resolution Global Gravity Field Model Combining CHAMP and GRACE Satellite Mission and Surface Data: EIGEN-CG01C, 2004.

    Google Scholar 

  • Sadiq, M. and Z. Ahmad, A comparative study of the geoid-quasigeoid separation term C at two different locations with different topographic distributions, Newton’s Bulletin, 3, 1–10, International Geoid Service www.iges.polimi.it, 2006.

    Google Scholar 

  • Sadiq, M. and Z. Ahmad, On the selection of optimal global geopotential model for geoid modeling: a case study in Pakistan, Internal report#11, Dept. of Earth Sciences QAU, Islamabad, 2007.

    Google Scholar 

  • Sadiq, M., Z. Ahmad, and M. Ayub, Vertical gravity anomaly gradient effect on the geoid-quasigeoid separation and an optimal integration radius in planer approximation, Studia Geophys. Geod., 2008 (submitted).

    Google Scholar 

  • Sjöberg, L. E., On the error of spherical harmonic development of gravity at the surface of the Earth, Rep. 257, Department of Geodetic Science, The Ohio State University, Columbus, 1977.

    Google Scholar 

  • Sjöberg, L. E., On the total terrain effects in geoid and quasigeoid determinations using Helmert second condensation method, Rep. 36, Division of Geodesy, Royal Institute of Technology, Stockholm, 1994.

    Google Scholar 

  • Sjöberg, L. E., On the quasigeoid to geoid separation, Manuscr. Geod., 20, 182–192, 1995.

    Google Scholar 

  • Thorarinsson, F. and S. G. Magnusson, Bouguer density determination by fractal analysis, Geophys., 55, 932–935, 1990.

    Article  Google Scholar 

  • Torge, W., Geodesy, 3rd ed., 2001.

    Book  Google Scholar 

  • Tziavos, I. N. and W. E. Featherstone, First results of using digital density data in gravimetric geoids computation in Australia, IAG Symposia, GGG2000, Springer Verlag, Berlin Heidelberg, 123, 335–340, 2000.

    Google Scholar 

  • USGS, EROS, SRTM30 Digital Elevation Model DATA Center, SIOUX Fall, SD 57198-0001, http://srtm.usgs.gov/data/obtainingdata.html/data/obtainingdata.html.

  • Vanicek, P., M. Naja, Z. Martinec, L. Harrie, and L. E. Sjöberg, Higher order reference field in the generalized Stokes-Helmert scheme for geoid computation, J. Geod., 70, 176–182, 1995.

    Article  Google Scholar 

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Correspondence to Muhammad Sadiq.

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Sadiq, M., Ahmad, Z. & Akhter, G. A study on the evaluation of the geoid-quasigeoid separation term over Pakistan with a solution of first and second order height terms. Earth Planet Sp 61, 815–823 (2009). https://doi.org/10.1186/BF03353192

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Key words

  • Geoid
  • quasigeoid
  • C1 & C2 correction terms
  • gravity anomaly
  • height anomaly
  • vertical gravity anomaly gradient