Abstract
In the numerical evaluation of geodetic convolution integrals, whether by quadrature or discrete/fast Fourier transform (D/FFT) techniques, the integration kernel is sometimes computed at the centre of the discretised grid cells. For singular kernels—a common case in physical geodesy—this approximation produces significant errors near the computation point, where the kernel changes rapidly across the cell. Rigorously, mean kernels across each whole cell are required. We present one numerical and one analytical method capable of providing estimates of mean kernels for convolution integrals. The numerical method is based on Gauss-Legendre quadrature (GLQ) as efficient integration technique. The analytical approach is based on kernel weighting factors, computed in planar approximation close to the computation point, and used to convert non-planar kernels from point to mean representation. A numerical study exemplifies the benefits of using mean kernels in Stokes’s integral. The method is validated using closed-loop tests based on the EGM2008 global gravity model, revealing that using mean kernels instead of point kernels reduces numerical integration errors by a factor of ~5 (at a grid-resolution of 10 arc min). Analytical mean kernel solutions are then derived for 14 other commonly used geodetic convolution integrals: Hotine, Eötvös, Green-Molodensky, tidal displacement, ocean tide loading, deflection-geoid, Vening-Meinesz, inverse Vening-Meinesz, inverse Stokes, inverse Hotine, terrain correction, primary indirect effect, Molodensky’s G1 term and the Poisson integral. We recommend that mean kernels be used to accurately evaluate geodetic convolution integrals, and the two methods presented here are effective and easy to implement.
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References
Abramowitz IA, Stegun MA (1972) Handbook of mathematical functions. Dover Publications, New York
Alberts B, Klees R (2004) A comparison of methods for the inversion of airborne gravity data. J Geod 78(1): 55–65. doi:10.1007/s00190-003-0366-x
Asgharzadeh MF, von Frese RB, Kim HR, Leftwich TE, Kim JW (2007) Spherical prism gravity effects by Gauss-Legendre quadrature integration. Geophys J Int 169(1): 1–11. doi:10.1111/j.1365-246X.2007.03214.x
Bláha T, Hirsch M, Keller W, Scheinert M (1996) Application of a spherical FFT approach in airborne gravimetry. J Geod 70(11): 663–672. doi:10.1007/BF00867145
Bos MS, Baker TF (2005) An estimate of the errors in gravity ocean tide loading computations. J Geod 79(1-3): 50–63. doi:10.1007/s00190-005-0442-5
Boyarsky EA, Afanasyeva LV, Koneshov VN, Rozhkov YE (2010) On the calculation of the vertical deflection and the geoid undulation from gravity anomalies. Phys Solid Earth 46(6): 538–543
Claessens SJ, Hirt C, Amos MJ, Featherstone WE, Kirby JF (2011) The NZGEOID09 New Zealand quasigeoid model. Surv Rev 43(319): 2–15. doi:10.1179/003962610X12747001420780
Conte SD, de Boor C (1972) Elementary numerical analysis—an algorithmic approach. McGraw-Hill, Kogakusha
de Min E (1994) On the numerical evaluation of Stokes’ integral. Int Geoid Serv Bull 3: 41–46
de Min E (1996) De Geoide voor Nederland. Nederlandse Commissie voor Geodesie Publikatie 34 (Dutch geodetic commission publication no 34), Delft
El Habiby MM (2007) Wavelet representation of geodetic operators, UCGE report no 20250, University of Calgary
Engels H (1980) Numerical quadrature and cubature. Academic Press, London
Featherstone WE, Olliver JG (1997) A method to validate gravimetric geoid computation software based on Stokes’s integral. J Geod 71(9): 571–576. doi:10.1007/s001900050125
Featherstone WE (2002) Tests of two forms of Stokes’s integral using a synthetic gravity field based on spherical harmonics. In: Grafarend EW, Krumm FW, Schwarze VS (eds) Geodesy—the challenge for the third millennium. Springer, Berlin, pp 163–171
Featherstone WE, Kirby JF, Hirt C, Filmer MS, Claessens SJ, Brown N, Hu G, Johnston GM (2011) The AUSGeoid2009 model of the Australian Height Datum. J Geod 85(3): 133–150. doi:10.1007/s00190-010-0422-2
Forsberg R (1984) A study of terrain reductions, density anomalies and geophysical inversion methods in gravity field modelling. Report 355, Department of Geodetic Science and Surveying, Ohio State University, Columbus
Forsberg R (1985) Gravity field terrain effect computations by FFT. Bull Géodésique 59(4): 342–360. doi:10.1007/BF02521068
Golub GH, Welsch JH (1969) Calculation of Gauss quadrature rules. Math Comp 23(106): 221–230, s1–s10
Haagmans R, de Min E, van Gelderen M (1993) Fast evaluation of convolution integrals on the sphere using 1D FFT, and a comparison with existing methods for Stokes’ integral. manuscripta geodaetica 18(5): 227–241
Hammer S (1939) Terrain corrections for gravimeter stations. Geophysics 4(3): 184–194. doi:10.1190/1.1440495
Hamming RW (1986) Numerical methods for scientists and engineers, 2nd edn. Dover Publications, New York (reprint)
Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman, San Francisco
Hipkin RG (1988) Bouguer anomalies and the geoid: a reassessment of Stokes’s method. Geophys J Int 92: 53–66. doi:10.1111/j.1365-246X.1988.tb01120.x
Holmes, SA, Pavlis NK (2008) Spherical harmonic synthesis software harmonic_synth. http://earth-info.nga.mil/GandG/wgs84/gravitymod/new_egm/new_egm.html
Hotine M (1969) Mathematical geodesy, ESSA Monograph No 2. US Department of Commerce, Washington, DC
Huang J, Vaníček P, Novák P (2000) An alternative algorithm to FFT for the numerical evaluation of Stokes’s integral. Studia Geophysica et Geodaetica 44(3): 374–380. doi:10.1023/A:1022160504156
Hwang C (1998) Inverse Vening Meinesz formula and deflection-geoid formula: applications to the predictions of gravity and geoid over the South China Sea. J Geod 72(5): 304–312. doi:10.1007/s001900050169
Kearsley AHW (1986) Data requirements for determining precise relative geoid heights from gravimetry. J Geophys Res 91(B9): 9193–9201
Klees R (1996) Numerical calculation of weakly singular surface integrals. J Geod 70(11): 781–797
Lehmann R (1997) Fast space-domain evaluation of geodetic surface integrals. J Geod 71(9): 533–540
Lether FG, Wenston PR (1995) Minimax approximation to the zeros of P_n(x) and Gauss-Legendre quadrature. J Comp Appl Math 59(2): 245–252. doi:10.1016/0377-0427(94)00030-5
Makhloof AA, Ilk KH (2008) Far-zone effects for different topographic-compensation models based on a spherical harmonic expansion of the topography. J Geod 82(10): 613–635
Molodensky MS, Yeremeyev VF, Yurkina MI (1962) Methods for study of the external gravitational field and figure of the Earth. Translated from Russian, Isreali Programme for Scientific Translations, Jerusalem
Moritz H (1980) Advanced physical geodesy. Wichmann Verlag, Karlsruhe
Newton’s Bulletin (2009) Newton’s Bulletin Issue n° 4, April 2009. Publication of the International Association of Geodesy and International Gravity Field Service. ISSN: 1810-8555
Novák P, Vaníček P, Véronneau M, Holmes SA, Featherstone WE (2001) On the accuracy of modified Stokes’s integration in high-frequency gravimetric geoid determination. J Geod 74(9): 644–654. doi:10.1007/s001900000126
Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2008) An Earth gravitational model to degree 2160: EGM2008. Presented at the 2008 General Assembly of the European Geoscience Union, Vienna, Austria, 13–18 April 2008
Press WH, Teukolsky SA, Vetterling WT, Flannary BP (2003) Numerical recipes in Fortran 77—the art of scientific computing, 2nd edn. Fortran numerical recipes, vol 1. Cambridge University Press, Cambridge
Press WH, Teukolsky SA, Vetterling WT, Flannary BP (2002) Numerical recipes in C—the art of scientific computing, 2nd edn. Cambridge University Press, Cambridge
Sampietro D, Sona G, Venuti G (2007) Residual terrain correction on the sphere by an FFT Algorithm. In: Proceedings of the 1st International Symposium on international gravity field service, Aug 28–Sep 1, Istanbul, Turkey, Harita Dergesi Special Issue, pp 306–311
Schwarz K-P, Sideris MG, Forsberg R (1990) The use of FFT techniques in physical geodesy. Geophys J Int 100(3): 485–514. doi:10.1111/j.1365-246X.1990.tb00701.x
Sideris MG, Li YC (1993) Gravity field convolution without windowing and edge effects. Bull Géodésique 67(2): 107–118. doi:10.1007/BF01371374
Sideris MG, She BB (1995) A new high-resolution geoid for Canada and part of the US by the 1D-FFT method. Bull Géodésique 69(2): 92–108. doi:10.1007/BF00819555
Sormann H (2009) Numerische Methoden in der Physik Institut für Theoretische Physik-Computational Physics, TU Graz, Austria. http://itp.tugraz.at/LV/sormann/NumPhysik/Skriptum/
Stark PA (1970) Introduction to numerical methods. MacMillian, New York
Stoer J, Bulirsch R (1980) Introduction to numerical analysis. Springer, New York
Strang van Hees G (1990) Stokes formula using fast Fourier techniques. manuscripta geodaetica 15(4): 235–239
Stroud AH (1971) Approximate calculation of multiple integrals. Prentice-Hall, New York
Torge W (2001) Geodesy, 3rd edn. De Gruyter, Berlin
Tscherning CC (2003) Proposal for the precise definition of mean values of gravity field quantities. Newton’s Bulletin, no. 1. International Geoid Service, pp 11–13
Tziavos IN (1996) Comparisons of spectral techniques for geoid computations over large areas. J Geod 70(6): 357–373. doi:10.1007/BF00868188
Val’ko M, Mojzeš M, Janák J, Papčo J (2008) Comparison of two different solutions to Molodensky’s G 1 term. Studia Geophysica et Geodaetica 52(1): 71–86. doi:10.1007/s11200-008-0006-2
van Gelderen (1991) The geodetic boundary value problem in two dimensions and its iterative solution, publication no 35, Netherlands Geodetic Commission, Amsterdam
van Gysen H (1994) Thin-plate spline quadrature of geodetic integrals. Bull Geodesique 68(3): 173–179
Vaníček P, Kleusberg A (1987) The Canadian geoid—Stokesian approach. manuscripta geodaetica 12(2): 86–98
Vaníček P, Krakiwsky EJ (1986) Geodesy: the concepts, 2nd edn. Elsevier, Amsterdam
Vening-Meinesz FA (1928) A formula expressing the deflection of the plumb-lines in the gravity anomalies and some formulae for the gravity field and the gravity potential outside the geoid. Proc. Koninkl. Akad. Wetenschaft 31:315–331 (Amsterdam)
von Winckel G (2004) Legendre-Gauss Quadrature Weights and Nodes. Matlab function lgwt. http://www.mathworks.com/matlabcentral/fileexchange/4540
Wessel P, Smith WHF (1998) New, improved version of the Generic Mapping Tools released. EOS Trans. AGU 79:579
Wild-Pfeiffer F (2008) A comparison of different mass elements for use in gravity gradiometry. J Geod 82(10): 637–653. doi:10.1007/s00190-008-0219-8
Zhang C (1993) Recovery of gravity information from satellite altimetry data and associated forward geopotential models. UCGE report no 20058, University of Calgary
Zhang C (1995) A general formula and its inverse formula for gravimetric transformations by use of convolution and deconvolution techniques. J Geod 70(1–2): 51–64. doi:10.1007/BF00863418
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Hirt, C., Featherstone, W.E. & Claessens, S.J. On the accurate numerical evaluation of geodetic convolution integrals. J Geod 85, 519–538 (2011). https://doi.org/10.1007/s00190-011-0451-5
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DOI: https://doi.org/10.1007/s00190-011-0451-5