Abstract
The availability of high-resolution global digital elevation data sets has raised a growing interest in the feasibility of obtaining their spherical harmonic representation at matching resolution, and from there in the modelling of induced gravity perturbations. We have therefore estimated spherical Bouguer and Airy isostatic anomalies whose spherical harmonic models are derived from the Earth’s topography harmonic expansion. These spherical anomalies differ from the classical planar ones and may be used in the context of new applications. We succeeded in meeting a number of challenges to build spherical harmonic models with no theoretical limitation on the resolution. A specific algorithm was developed to enable the computation of associated Legendre functions to any degree and order. It was successfully tested up to degree 32,400. All analyses and syntheses were performed, in 64 bits arithmetic and with semi-empirical control of the significant terms to prevent from calculus underflows and overflows, according to IEEE limitations, also in preserving the speed of a specific regular grid processing scheme. Finally, the continuation from the reference ellipsoid’s surface to the Earth’s surface was performed by high-order Taylor expansion with all grids of required partial derivatives being computed in parallel. The main application was the production of a 1′ × 1′ equiangular global Bouguer anomaly grid which was computed by spherical harmonic analysis of the Earth’s topography–bathymetry ETOPO1 data set up to degree and order 10,800, taking into account the precise boundaries and densities of major lakes and inner seas, with their own altitude, polar caps with bedrock information, and land areas below sea level. The harmonic coefficients for each entity were derived by analyzing the corresponding ETOPO1 part, and free surface data when required, at one arc minute resolution. The following approximations were made: the land, ocean and ice cap gravity spherical harmonic coefficients were computed up to the third degree of the altitude, and the harmonics of the other, smaller parts up to the second degree. Their sum constitutes what we call ETOPG1, the Earth’s TOPography derived Gravity model at 1′ resolution (half-wavelength). The EGM2008 gravity field model and ETOPG1 were then used to rigorously compute 1′ × 1′ point values of surface gravity anomalies and disturbances, respectively, worldwide, at the real Earth’s surface, i.e. at the lower limit of the atmosphere. The disturbance grid is the most interesting product of this study and can be used in various contexts. The surface gravity anomaly grid is an accurate product associated with EGM2008 and ETOPO1, but its gravity information contents are those of EGM2008. Our method was validated by comparison with a direct numerical integration approach applied to a test area in Morocco–South of Spain (Kuhn, private communication 2011) and the agreement was satisfactory. Finally isostatic corrections according to the Airy model, but in spherical geometry, with harmonic coefficients derived from the sets of the ETOPO1 different parts, were computed with a uniform depth of compensation of 30 km. The new world Bouguer and isostatic gravity maps and grids here produced will be made available through the Commission for the Geological Map of the World. Since gravity values are those of the EGM2008 model, geophysical interpretation from these products should not be done for spatial scales below 5 arc minutes (half-wavelength).
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References
Amante C, Eakins BW (2009) ETOPO1 1 arc-minute global relief model: procedures, data sources and analysis. NOAA Technical Memorandum NESDIS NGDC-24, Boulder (Co)
Balmino G (1994) Gravitational potential harmonics from the shape of an homogeneous body. Cel Mech Dyn Astr 60(3): 331–364
Balmino G (2003) Ellipsoidal corrections to spherical harmonics of surface phenomena gravitational effets. Festschrift zum 70. Geburtstag von H. Moritz, Publication of Graz Techn. University, pp 21–30
Bosch W (1983) Effiziente Algorithmen zur Berechnung von Raster-Punkwerten von Kugelfunktionsentwicklungen. Memorandum, D.G.F.I., Munich
Commission for the Geological Map of the World-CGMW (2010) Resolutions of the CGMW General Assembly, UNESCO, Paris
Gerstl M (1980) On the recursive computation of the integrals of the associated Legendre functions. Manuscr Geod 5: 181–199
Heiskanen WA, Moritz H (1967) Physical Geodesy. Freeman and co, San Francisco
Hofmann-Wellenhof B, Moritz H (2005) Physical Geodesy. Springer, Berlin
Holmes SA, Featherstone WE (2002) A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions. J Geod 76: 279–299
Jekeli C (1981) Alternative methods to smooth the Earth’s gravity field, Rep N0.310, Department of Geodetic Science and Surveying, Ohio State University, Columbus
Jekeli C, Lee KJ, Kwon JH (2007) On the computation and approximation of ultra-high-degree spherical harmonic series. J Geod 81: 603–615
Kuhn M, Featherstone WE (2003) On the optimal spatial resolution of crustal mass distributions for forward gravity field modelling. In: Gravity and geoid 2002, Proceedings, pp 195–200
Kuhn M, Featherstone WE, Kirby JF (2009) Complete spherical Bouguer gravity anomalies over Australia. Aust J Earth Sci 56: 213–223
NGA (1999) Gravity station data format and anomaly computations. Technical Report, Geospatial Sciences Division, St Louis (Mo)
Pavlis NK (1988) Modeling and estimation of a low degree geopotential model from terrestrial gravity data. Rep. N0. 386, Dpt. of Geodetic Sc. and Surveying, Ohio State Univ., Columbus
Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2008) An Earth Gravitational Model to Degree 2160: EGM2008. EGU General Assembly, Vienna, Austria, 13–18 April 2008
Ramillien G (2002) Gravity/magnetic potential of uneven shell topography. J Geod 76(3): 139–149
Rapp RH, Pavlis NK (1990) The development and analysis of geopotential coefficient Models to spherical harmonic degree 360. J Geophys Res 95(B13): 21885–21911
Rummel R, Rapp RH, Sünkel H, Tscherning CC (1988) Comparisons of global topographic-isostatic models to the Earth’s observed gravity Field. Rep. N0. 388, Dpt. of Geodetic Sc. and Surveying, Ohio State Univ., Columbus
Torge W (2001) Geodesy, 3rd edn. de Gruyter, The Netherlands
Tsoulis D (2001) Terrain correction computations for a densely sampled DTM in the Bavarian Alps. J geod 75(5/6): 291–307
Wenzel G (1998) Ultra-high degree geopotential models GPM98A, B and C to degree 1800. Joint meeting of the Intern. Gravity Commission and Intern. Geoid Commission, 7–12 September, Trieste
Wieczorek MA (2007) Gravity and topography of the terrestrial planets. In: Treatise on geophysics, vol 10. Elsevier, Amsterdam, pp 165–206
Wieczorek MA, Phillips RJ (1998) Potential anomalies on a sphere: applications to the thickness of the lunar crust. J Geophys Res 103(E1): 1715–1724
Wigner EP (1959) Group theory and its application to quantum mechanics of atomic spectra. Academic Press, New-York
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Balmino, G., Vales, N., Bonvalot, S. et al. Spherical harmonic modelling to ultra-high degree of Bouguer and isostatic anomalies. J Geod 86, 499–520 (2012). https://doi.org/10.1007/s00190-011-0533-4
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DOI: https://doi.org/10.1007/s00190-011-0533-4