Reformulation of the Vening-Meinesz Moritz Inverse Problem of Isostasy for Isostatic Gravity Disturbances

Affiliation(s)

National School of Surveying, University of Otago, Dunedin, New Zealand.

Division of Geodesy and Geoinformatics, Royal Institute of Technology (KTH), Stockholm, Sweden.

National School of Surveying, University of Otago, Dunedin, New Zealand.

Division of Geodesy and Geoinformatics, Royal Institute of Technology (KTH), Stockholm, Sweden.

ABSTRACT

The isostatic gravity anomalies have been traditionally used to solve the inverse problems of isostasy. Since gravity measurements are nowadays carried out together with GPS positioning, the utilization of gravity disturbances in various regional gravimetric applications becomes possible. In global studies, the gravity disturbances can be computed using global geopotential models which are currently available to a relatively high accuracy and resolution. In this study we facilitate the definition of the isostatic gravity disturbances in the Vening-Meinesz Moritz inverse problem of isostasy for finding the Moho depths. We further utilize uniform mathematical formalism in the gravimetric forward modelling based on methods for a spherical harmonic analysis and synthesis of gravity field. We then apply both mathematical procedures to determine globally the Moho depths using the isostatic gravity disturbances. The results of gravimetric inversion are finally compared with the global crustal seismic model CRUST2.0; the RMS fit of the gravimetric Moho model with CRUST2.0 is 5.3 km. This is considerably better than the RMS fit of 7.0 km obtained after using the isostatic gravity anomalies.

The isostatic gravity anomalies have been traditionally used to solve the inverse problems of isostasy. Since gravity measurements are nowadays carried out together with GPS positioning, the utilization of gravity disturbances in various regional gravimetric applications becomes possible. In global studies, the gravity disturbances can be computed using global geopotential models which are currently available to a relatively high accuracy and resolution. In this study we facilitate the definition of the isostatic gravity disturbances in the Vening-Meinesz Moritz inverse problem of isostasy for finding the Moho depths. We further utilize uniform mathematical formalism in the gravimetric forward modelling based on methods for a spherical harmonic analysis and synthesis of gravity field. We then apply both mathematical procedures to determine globally the Moho depths using the isostatic gravity disturbances. The results of gravimetric inversion are finally compared with the global crustal seismic model CRUST2.0; the RMS fit of the gravimetric Moho model with CRUST2.0 is 5.3 km. This is considerably better than the RMS fit of 7.0 km obtained after using the isostatic gravity anomalies.

Cite this paper

R. Tenzer and M. Bagherbandi, "Reformulation of the Vening-Meinesz Moritz Inverse Problem of Isostasy for Isostatic Gravity Disturbances,"*International Journal of Geosciences*, Vol. 3 No. 5, 2012, pp. 918-929. doi: 10.4236/ijg.2012.325094.

R. Tenzer and M. Bagherbandi, "Reformulation of the Vening-Meinesz Moritz Inverse Problem of Isostasy for Isostatic Gravity Disturbances,"

References

[1] W. A. Heiskanen and H. Moritz, “Physical Geodesy,” Freeman W. H., New York, 1967.

[2] J. H. Pratt, “On the Attraction of the Himalaya Mountains and of the Elevated Regions beyond upon the Plumb-Line in India,” Transactions of the Royal Society of London, Vol. 145, 1855, pp. 53-100.

[3] J. F. Hayford, “The Figure of the Earth and Isostasy from Measurements in the United States,” Government Printing Office, Washington DC, 1909.

[4] J. F. Hayford and W. Bowie, “The Effect of Topography and Isostatic Compensation upon the Intensity of Gravity,” BiblioBazaar, Charleston, 1912.

[5] G. B. Airy, “On the Computations of the Effect of the Attraction of the Mountain Masses as Disturbing the Apparent Astronomical Latitude of Stations in Geodetic Surveys,” Transactions of the Royal Society of London, Vol. 145, 1855, pp. 101-104.

[6] W. A. Heiskanen and F. A. Vening Meinesz, “The Earth and Its Gravity Field,” McGraw-Hill Book Company Inc., New York, 1958.

[7] F. A. Vening Meinesz, “Une Nouvelle Méthode Pour la Réduction Isostatique Régionale de l’Intensité de la Pesanteur,” Bulletin Geodesique, Vol. 29, No. 1, 1931, pp. 33-51. doi:10.1007/BF03030038

[8] H. Moritz, “The Figure of the Earth,” Wichmann H., Karlsruhe, 1990.

[9] L. E. Sj?berg, “Solving Vening Meinesz-Moritz Inverse Problem in Isostasy,” Geophysical Journal International, Vol. 179, No. 3, 2009, pp. 1527-1536. doi:10.1111/j.1365-246X.2009.04397.x

[10] L. E. Sj?berg and M. Bagherbandi, “A Method of Estimating the Moho Density Contrast With A Tentative Application by EGM08 and CRUST2.0,” Acta Geophysica, Vol. 59, No. 3, 2011, pp. 502-525. doi:10.2478/s11600-011-0004-6

[11] M. Bagherbandi and L. E. Sj?berg, “Comparison of Crustal Thickness from Two Isostatic Models versus CRUST2.0,” Studia Geophysica et Geodartica, Vol. 55, No. 4, 2011, pp. 641-666. doi:10.1007/s11200-010-9030-0

[12] C. Bassin, G. Laske and T. G. Masters, “The Current Limits of Resolution for Surface Wave Tomography in North America,” EOS Transactions, American Geophysical Union, Vol. 81, 2000.

[13] M. K. Kaban, P. Schwintzer and S. A. Tikhotsky, “A Global Isostatic Gravity Model of the Earth,” Geophysical Journal International, Vol. 136, No. 3, 1999, pp. 519- 536. doi:10.1046/j.1365-246x.1999.00731.x

[14] M. K. Kaban, P. Schwintzer and Ch. Reigber, “A New Isostatic Model of the Lithosphere and Gravity Field,” Journal of Geodesy, Vol. 78, No. 6, 2004, pp. 368-385. doi:10.1007/s00190-004-0401-6

[15] R. Tenzer, Hamayun and P. Vajda, “Global Maps of the CRUST2.0 Crustal Components Stripped Gravity Disturbances,” Journal of Geophysical Research, Vol. 114, No. B5, 2009, pp. 281-297.

[16] R. Tenzer, V. Gladkikh, P. Vajda and P. Novák, “Spatial and Spectral Analysis of Refined Gravity Data for Modelling the Crust-Mantle Interface and Mantle-Lithosphere Structure,” Surveys in Geophysics, Vol. 33, No. 5, 2012, pp. 817-839. doi:10.1007/s10712-012-9173-3

[17] C. Braitenberg, S. Wienecke and Y. Wang, “Basement Structures from Satellite-Derived Gravity Field: South China Sea ridge,” Journal of Geophysical Research, Vol. 111, 2006, Article ID: B05407. doi:10.1029/2005JB003938

[18] S. Wienecke, C. Braitenberg and H.-J. G?tze, “A New Analytical Solution Estimating the Flexural Rigidity in the Central Andes,” Geophysical Journal International, Vol. 169, No. 3, 2007, pp. 789-794. doi:10.1111/j.1365-246X.2007.03396.x

[19] P. Vajda, P. Vaní?ek, P. Novák, R. Tenzer and A. Ellmann, “Secondary Indirect Effects in Gravity Anomaly Data Inversion or Interpretation,” Journal of Geophysical Research, Vol. 112, 2007, Article ID: B06411.

[20] R. Tenzer, P. Vajda and P. Hamayun, “A Mathematical Model of the Bathymetry-Generated External Gravitational Field,” Contributions to Geophysics and Geodesy, Vol. 40, No. 1, 2010, pp. 31-44. doi:10.2478/v10126-010-0002-8

[21] R. Tenzer, A. Abdalla, P. Vajda and P. Hamayun, “The Spherical Harmonic Representation of the Gravitational Field Quantities Generated by the Ice Density Contrast,” Contributions to Geophysics and Geodesy, Vol. 40, No. 3, 2010, pp. 207-223. doi:10.2478/v10126-010-0009-1

[22] R. Tenzer, P. Novák, P. Vajda, V. Gladkikh and P. Hamayun, “Spectral Harmonic Analysis and Synthesis of Earth’s Crust Gravity Field,” Computational Geosciences, Vol. 16, No. 1, 2012, pp. 193-207. doi:10.1007/s10596-011-9264-0

[23] R. Tenzer, Hamayun and P. Vajda, “A Global Correlation of the Step-Wise Consolidated Crust-Stripped Gravity Field Quantities with the Topography, Bathymetry, and the CRUST2.0 Moho Boundary,” Contributions to Geophysics and Geodesy, Vol. 39, No. 2, 2009, pp. 133-147. doi:10.2478/v10126-009-0006-4

[24] N. K. Pavlis, S. A. Holmes, S. C. Kenyon and J. K. Factor, “The Development and Evaluation of the Earth Gravitational Model 2008 (EGM2008),” Journal of Geophysical Research, Vol. 117, 2012, Article ID: B04406.

[25] H. Moritz, “Advanced Physical Geodesy,” Abacus Press, Tunbridge Wells, 1980.

[26] N. K. Pavlis, J. K. Factor and S. A. Holmes, “Terrain-Related Gravimetric Quantities Computed for the Next EGM,” In: A. Kili?oglu and R. Forsberg, Eds., Gravity Field of the Earth. Proceedings of the 1st International Symposium of the International Gravity Field Service (IGFS), Harita Dergisi, No. 18, General Command of Mapping, Ankara, Turkey, 2007.

[27] W. J. Hinze, “Bouguer Reduction Density, Why 2.67?” Geophysics, Vol. 68, No. 5, 2003, pp. 1559-1560. doi:10.1190/1.1620629

[28] R. Tenzer, P. Novák and V. Gladkikh, “The Bathymetric Stripping Corrections to Gravity Field Quantities for a Depth-Dependant Model of the Seawater Density,” Marine Geodesy, Vol. 35, No. 2, 2012, pp. 198-220. doi:10.1080/01490419.2012.670592

[29] V. Gladkikh and R. Tenzer, “A Mathematical Model of the Global Ocean Saltwater Density Distribution,” Pure and Applied Geophysics, Vol. 169, No. 1-2, 2011, pp. 249-257. doi:10.1007/s00024-011-0275-5

[30] D. R. Johnson, H. E. Garcia and T. P. Boyer, “World Ocean Database 2009 Tutorial,” In: S. Levitus, Ed., NODC Internal Report 21, NOAA Printing Office, Silver Spring, 2009, 18 p.

[31] V. V. Gouretski and K. P. Koltermann, “Berichte des Bundesamtes für Seeschifffahrt und Hydrographie,” Nr. 35, 2004.

[32] R. Tenzer, P. Novák and V. Gladkikh, “On the Accuracy of the Bathymetry-Generated Gravitational Field Quantities for a Depth-Dependent Seawater Density Distribution,” Studia Geophysica et Geodaetica, Vol. 55, No. 4, 2011, pp. 609-626. doi:10.1007/s11200-010-0074-y

[33] S. Ekholm, “A Full Coverage, High-Resolution, Topographic Model of Greenland, Computed from a Variety of Digital Elevation Data,” Journal of Geophysical Research, Vol. B10, No. 21, 1996, pp. 961-972.

[34] J. D. Cutnell and W. J. Kenneth, “Physics,” 3rd Edition, Wiley, New York, 1995.

[35] R. Tenzer, Hamayun and P. Vajda, “Global Map of the Gravity Anomaly Corrected for Complete Effects of the Topography, and of Density Contrasts of Global Ocean, Ice, and Sediments,” Contributions to Geophysics and Geodesy, Vol. 38, No. 4, 2008, pp. 357-370.

[36] R. Tenzer, Hamayun, P. Novák, V. Gladkikh and P. Vajda, “Global Crust-Mantle Density Contrast Estimated from EGM2008, DTM2008, CRUST2.0, and ICE-5G,” Pure and Applied Geophysics, Vol. 169, No. 9, 2012, pp. 1663-1678. doi:10.1007/s00024-011-0410-3

[37] D. H. Eckhardt, “The Gains of Small Circular, Square and Rectangular Filters for Surface waves on a Sphere,” Journal of Geodesy Vol. 57, No. 1-4, 1983, pp. 394-409.doi:10.1007/BF02520942

[38] O. ?adek and Z. Martinec, “Spherical harmonic expansion of the earth’s crustal thickness up to degree and order 30,” Studia Geophisica et Geodeatica, Vol. 35, No. 3, 1991, pp. 151-165. doi:10.1007/BF01614063

[39] M. Grad, T. Tiira and ESC Working Group, “The Moho Depth Map of the European Plate,” Geophysical Journal Internatioanl, Vol. 176, No. 1, 2009, pp. 279-292. doi:10.1111/j.1365-246X.2008.03919.x

[40] M. Bagherbandi and L. E. Sj?berg, “Non-Isostatic Effects on Crustal Thickness: A Study Using CRUST2.0 in Fennoscandia,” Physics of the Earth and Planetary Interiors, Vol. 200-201, 2012, pp. 37-44. doi:10.1016/j.pepi.2012.04.001

[1] W. A. Heiskanen and H. Moritz, “Physical Geodesy,” Freeman W. H., New York, 1967.

[2] J. H. Pratt, “On the Attraction of the Himalaya Mountains and of the Elevated Regions beyond upon the Plumb-Line in India,” Transactions of the Royal Society of London, Vol. 145, 1855, pp. 53-100.

[3] J. F. Hayford, “The Figure of the Earth and Isostasy from Measurements in the United States,” Government Printing Office, Washington DC, 1909.

[4] J. F. Hayford and W. Bowie, “The Effect of Topography and Isostatic Compensation upon the Intensity of Gravity,” BiblioBazaar, Charleston, 1912.

[5] G. B. Airy, “On the Computations of the Effect of the Attraction of the Mountain Masses as Disturbing the Apparent Astronomical Latitude of Stations in Geodetic Surveys,” Transactions of the Royal Society of London, Vol. 145, 1855, pp. 101-104.

[6] W. A. Heiskanen and F. A. Vening Meinesz, “The Earth and Its Gravity Field,” McGraw-Hill Book Company Inc., New York, 1958.

[7] F. A. Vening Meinesz, “Une Nouvelle Méthode Pour la Réduction Isostatique Régionale de l’Intensité de la Pesanteur,” Bulletin Geodesique, Vol. 29, No. 1, 1931, pp. 33-51. doi:10.1007/BF03030038

[8] H. Moritz, “The Figure of the Earth,” Wichmann H., Karlsruhe, 1990.

[9] L. E. Sj?berg, “Solving Vening Meinesz-Moritz Inverse Problem in Isostasy,” Geophysical Journal International, Vol. 179, No. 3, 2009, pp. 1527-1536. doi:10.1111/j.1365-246X.2009.04397.x

[10] L. E. Sj?berg and M. Bagherbandi, “A Method of Estimating the Moho Density Contrast With A Tentative Application by EGM08 and CRUST2.0,” Acta Geophysica, Vol. 59, No. 3, 2011, pp. 502-525. doi:10.2478/s11600-011-0004-6

[11] M. Bagherbandi and L. E. Sj?berg, “Comparison of Crustal Thickness from Two Isostatic Models versus CRUST2.0,” Studia Geophysica et Geodartica, Vol. 55, No. 4, 2011, pp. 641-666. doi:10.1007/s11200-010-9030-0

[12] C. Bassin, G. Laske and T. G. Masters, “The Current Limits of Resolution for Surface Wave Tomography in North America,” EOS Transactions, American Geophysical Union, Vol. 81, 2000.

[13] M. K. Kaban, P. Schwintzer and S. A. Tikhotsky, “A Global Isostatic Gravity Model of the Earth,” Geophysical Journal International, Vol. 136, No. 3, 1999, pp. 519- 536. doi:10.1046/j.1365-246x.1999.00731.x

[14] M. K. Kaban, P. Schwintzer and Ch. Reigber, “A New Isostatic Model of the Lithosphere and Gravity Field,” Journal of Geodesy, Vol. 78, No. 6, 2004, pp. 368-385. doi:10.1007/s00190-004-0401-6

[15] R. Tenzer, Hamayun and P. Vajda, “Global Maps of the CRUST2.0 Crustal Components Stripped Gravity Disturbances,” Journal of Geophysical Research, Vol. 114, No. B5, 2009, pp. 281-297.

[16] R. Tenzer, V. Gladkikh, P. Vajda and P. Novák, “Spatial and Spectral Analysis of Refined Gravity Data for Modelling the Crust-Mantle Interface and Mantle-Lithosphere Structure,” Surveys in Geophysics, Vol. 33, No. 5, 2012, pp. 817-839. doi:10.1007/s10712-012-9173-3

[17] C. Braitenberg, S. Wienecke and Y. Wang, “Basement Structures from Satellite-Derived Gravity Field: South China Sea ridge,” Journal of Geophysical Research, Vol. 111, 2006, Article ID: B05407. doi:10.1029/2005JB003938

[18] S. Wienecke, C. Braitenberg and H.-J. G?tze, “A New Analytical Solution Estimating the Flexural Rigidity in the Central Andes,” Geophysical Journal International, Vol. 169, No. 3, 2007, pp. 789-794. doi:10.1111/j.1365-246X.2007.03396.x

[19] P. Vajda, P. Vaní?ek, P. Novák, R. Tenzer and A. Ellmann, “Secondary Indirect Effects in Gravity Anomaly Data Inversion or Interpretation,” Journal of Geophysical Research, Vol. 112, 2007, Article ID: B06411.

[20] R. Tenzer, P. Vajda and P. Hamayun, “A Mathematical Model of the Bathymetry-Generated External Gravitational Field,” Contributions to Geophysics and Geodesy, Vol. 40, No. 1, 2010, pp. 31-44. doi:10.2478/v10126-010-0002-8

[21] R. Tenzer, A. Abdalla, P. Vajda and P. Hamayun, “The Spherical Harmonic Representation of the Gravitational Field Quantities Generated by the Ice Density Contrast,” Contributions to Geophysics and Geodesy, Vol. 40, No. 3, 2010, pp. 207-223. doi:10.2478/v10126-010-0009-1

[22] R. Tenzer, P. Novák, P. Vajda, V. Gladkikh and P. Hamayun, “Spectral Harmonic Analysis and Synthesis of Earth’s Crust Gravity Field,” Computational Geosciences, Vol. 16, No. 1, 2012, pp. 193-207. doi:10.1007/s10596-011-9264-0

[23] R. Tenzer, Hamayun and P. Vajda, “A Global Correlation of the Step-Wise Consolidated Crust-Stripped Gravity Field Quantities with the Topography, Bathymetry, and the CRUST2.0 Moho Boundary,” Contributions to Geophysics and Geodesy, Vol. 39, No. 2, 2009, pp. 133-147. doi:10.2478/v10126-009-0006-4

[24] N. K. Pavlis, S. A. Holmes, S. C. Kenyon and J. K. Factor, “The Development and Evaluation of the Earth Gravitational Model 2008 (EGM2008),” Journal of Geophysical Research, Vol. 117, 2012, Article ID: B04406.

[25] H. Moritz, “Advanced Physical Geodesy,” Abacus Press, Tunbridge Wells, 1980.

[26] N. K. Pavlis, J. K. Factor and S. A. Holmes, “Terrain-Related Gravimetric Quantities Computed for the Next EGM,” In: A. Kili?oglu and R. Forsberg, Eds., Gravity Field of the Earth. Proceedings of the 1st International Symposium of the International Gravity Field Service (IGFS), Harita Dergisi, No. 18, General Command of Mapping, Ankara, Turkey, 2007.

[27] W. J. Hinze, “Bouguer Reduction Density, Why 2.67?” Geophysics, Vol. 68, No. 5, 2003, pp. 1559-1560. doi:10.1190/1.1620629

[28] R. Tenzer, P. Novák and V. Gladkikh, “The Bathymetric Stripping Corrections to Gravity Field Quantities for a Depth-Dependant Model of the Seawater Density,” Marine Geodesy, Vol. 35, No. 2, 2012, pp. 198-220. doi:10.1080/01490419.2012.670592

[29] V. Gladkikh and R. Tenzer, “A Mathematical Model of the Global Ocean Saltwater Density Distribution,” Pure and Applied Geophysics, Vol. 169, No. 1-2, 2011, pp. 249-257. doi:10.1007/s00024-011-0275-5

[30] D. R. Johnson, H. E. Garcia and T. P. Boyer, “World Ocean Database 2009 Tutorial,” In: S. Levitus, Ed., NODC Internal Report 21, NOAA Printing Office, Silver Spring, 2009, 18 p.

[31] V. V. Gouretski and K. P. Koltermann, “Berichte des Bundesamtes für Seeschifffahrt und Hydrographie,” Nr. 35, 2004.

[32] R. Tenzer, P. Novák and V. Gladkikh, “On the Accuracy of the Bathymetry-Generated Gravitational Field Quantities for a Depth-Dependent Seawater Density Distribution,” Studia Geophysica et Geodaetica, Vol. 55, No. 4, 2011, pp. 609-626. doi:10.1007/s11200-010-0074-y

[33] S. Ekholm, “A Full Coverage, High-Resolution, Topographic Model of Greenland, Computed from a Variety of Digital Elevation Data,” Journal of Geophysical Research, Vol. B10, No. 21, 1996, pp. 961-972.

[34] J. D. Cutnell and W. J. Kenneth, “Physics,” 3rd Edition, Wiley, New York, 1995.

[35] R. Tenzer, Hamayun and P. Vajda, “Global Map of the Gravity Anomaly Corrected for Complete Effects of the Topography, and of Density Contrasts of Global Ocean, Ice, and Sediments,” Contributions to Geophysics and Geodesy, Vol. 38, No. 4, 2008, pp. 357-370.

[36] R. Tenzer, Hamayun, P. Novák, V. Gladkikh and P. Vajda, “Global Crust-Mantle Density Contrast Estimated from EGM2008, DTM2008, CRUST2.0, and ICE-5G,” Pure and Applied Geophysics, Vol. 169, No. 9, 2012, pp. 1663-1678. doi:10.1007/s00024-011-0410-3

[37] D. H. Eckhardt, “The Gains of Small Circular, Square and Rectangular Filters for Surface waves on a Sphere,” Journal of Geodesy Vol. 57, No. 1-4, 1983, pp. 394-409.doi:10.1007/BF02520942

[38] O. ?adek and Z. Martinec, “Spherical harmonic expansion of the earth’s crustal thickness up to degree and order 30,” Studia Geophisica et Geodeatica, Vol. 35, No. 3, 1991, pp. 151-165. doi:10.1007/BF01614063

[39] M. Grad, T. Tiira and ESC Working Group, “The Moho Depth Map of the European Plate,” Geophysical Journal Internatioanl, Vol. 176, No. 1, 2009, pp. 279-292. doi:10.1111/j.1365-246X.2008.03919.x

[40] M. Bagherbandi and L. E. Sj?berg, “Non-Isostatic Effects on Crustal Thickness: A Study Using CRUST2.0 in Fennoscandia,” Physics of the Earth and Planetary Interiors, Vol. 200-201, 2012, pp. 37-44. doi:10.1016/j.pepi.2012.04.001