Efficiency investigation of tesseroid based methods for computing gravimetric terrain correction

Author

Assistant Professor, Civil Engineering Faculty, Shahrood University of Technology, Iran

Abstract

The gravitational effect of topographical masses is one of the important component of the gravity field, which plays a key role in geophysical and geodetic studies. For geophysical interpretations, it is necessary to eliminate the effect of topography as a disturbing factor from the observed gravity data. In geodetic applications, the solution of geodetic boundary problem such as Stokes requires mass free space above the geoid. In present study efficiency of different tesseroid based methods are compared with well-known rectangular prism to evaluate the gravimetric terrain corrections up to distance of 1.5 arc-degree known as the Hayford-Bowie zone. For this purpose, the mathematical formula: the vertical derivative of Newton integral and the digital elevation model (DEM) are used as data. In computing the topographic effect, we are involved with the two factors: 1- the integral element (point, line, plane, rectangular prism, tesseroid, etc.) and 2- geometry of topography (planar, spherical and ellipsoidal), which causes some difficulties to understand the subject. Finite element method is a general and standard method for estimating the terrain correction. In this method, the gravitational topographic effect is evaluated as the total gravitational effect of the smaller elements. 
Tesseroid is the geometrical body bounded by two concentric spheres. This element uses the spherical geometry of topography which introduces relative error of about 1% (Novak and Grafarend, 2005). By choosing this element, the Newton integral and its radial derivatives do not have an analytic solution, and numerical integration must be applied. The rectangular prism element, has been used frequently to compute terrain correction in various studies. It uses planar geometry and has an analytical solution for Newton's integral and its derivatives. Recently many studies investigated tesseroid based method to compute the potential and attraction of topographic masses, see, [Fukushima, 2017; Grombein et al., 2013; Heck and Seitz, 2007; Uieda et al., 2016]. Fukushima's method utilizes the 3D numerical double-exponential integration method, HS's method uses the Tylor series up to term 2 and the PM method is the zero term approximation of HS method. The simulation studies demonstrated the higher accuracy of tesseroid based methods compared to the method of prism in the literature. However, their performance is not tested for gravimetric terrain correction. The main goal of this study is the investigation of efficiency, in terms of speed and accuracy, of four tesseroid methods: Fukushima, Martinec-Vanicek (MV), Heck-Seitz (HS), point mass (PM) compared with prism in Hayford-Bowie zone.
To investigate the computation accuracy, we used bounded spherical shell with constant thinness and density for which the analytical exact solution exists. The thinness of the shell have been chosen 1000 meter and the computation point is located on the origin of bounded spherical shell on the equator in the spherical coordinate (0,0,1000). The computation of terrain correction are discretized in different zones: innermost, inner and outer correspond respectively to ,  and  and with different sizes. The contribution of innermost zone is over 75% of total effect. Numerical results indicate the success of the prism for topographic effect in all three zones, especially for masses in neighborhoods of computation points, than those methods based on tesseroid. To overcome the effect of Earth's curvature, the elevation of computation point is corrected using a simple formula. Also, our calculations show that, in innermost zone, the topography should be discretized in 30 meter elements to achieve 10 Gal level of accuracy.

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References

گلی، م. و نجفی علمداری، م.، 1393، تقریب بیضویِ اثرات توپوگرافی در مدل‌سازی میدان گرانی زمین، مجله فیزیک زمین و فضا، 40(2)، 113-124.
Asgharzadeh, M. F., Von Frese, R. R. B., Kim, H. R., Leftwich T. E. and Kim, J. W., 2007, Spherical prism gravity effects by Gauss-Legendre quadrature integration. Geophysical Journal International, 169, 1-11.
Claessens, S. and Hirt, C., 2013, Ellipsoidal topographic potential: New solutions for spectral forward gravity modeling of topography with respect to a reference ellipsoid, Journal of Geophysical Research: Solid Earth. 118, 5991-6002.
Forsberg, R., 1984, A study of terrain reductions, density anomalies and geophysical inversion methods in gravity field modelling, Report 355, Department of Geodetic Science. The Ohio State University, Columbus.
Forsberg, R., 1985, Gravity field terrain effect computations by FFT. Journal of Geodesy, 59: 342-360.
Sansò, F. and Rummel, R., 1997, Geodetic boundary value problems in view of the one centimeter geoid. Lecture Notes in Earth Sciences, Berlin Springer Verlag, 65.
Fukushima, T., 2017, Accurate computation of gravitational field of a tesseroid, revised.
Grombein, T., Seitz, K. and Heck, B., 2013, Optimized formulas for the gravitational field of a tesseroid. Journal of Geodesy, 877, 645-660.
Heck, B. and Seitz, K., 2007, A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling. Journal of Geodesy, 81, 121-136.
 Hinze, W. J., 2003, Bouguer reduction density, why 2.67? Geophysics, 68, 1559-1560.
Huang, J., 2002, Computational methods for the discrete downward continuation of the Earth gravity and effects of lateral topographical mass density variation of gravity and geoid, Ph.D thesis, UNB.
Hwang, C., Wang C.-G. and Hsiao, Y.-S., 2003, Terrain correction computation using Gaussian quadrature. Computers & Geosciences 29, 1259-1268.
Krynski, J., Mank, M. and Grzyb, M., 2005, Evaluation of digital terrain models in Poland in view of a cm geoid modelling. Geodesy and Cartography, 54, 155-175.
LaFehr, T. R., 1991, Standardization in gravity reduction. Geophysics, 56, 1170-1178.
Mader, K., 1951, Das Newtonsche Raumpotential prismatischer Kper und seine Ableitungen bis zur dritten Ordnung. sterr Z Vermess Sonderheft.
Martinec, Z., 1998, Boundary-value problems for gravimetric determination of a precise geoid. Heidelberg, Springer.
Martinec, Z. and Vanicek, P., 1994, Direct topographical effect of Helmert's condensation for a spherical geoid. Man. Geod. 19, 257-268.
Martinec, Z., Vanicek, P., Mainville, A. and Veronneau, M., 1995, Evaluation of topographical effects in precise geoid computation from densely sampled heights. Journal of Geodesy, 20, 193-203.
Nagy, D., 1966, The gravitational attraction of a right rectangular prism. Geophysics, 31, 362-371.
Nagy, D., Papp G. and Benedek, J., 2000, The gravitational potential and its derivatives for the prism. Journal of Geodesy 74, 552-560.
Novak, P. and Grafarend, E. W., 2005, The ellipsoidal representation of the topographical potential and its vertical gradient. Journal of Geodesy, 78, 691-706.
Novák, P., Vaníček, P., Martinec, Z. and Véronneau, M., 2001, Effects of the spherical terrain on gravity and the geoid. Journal of Geodesy, 75, 491-504.
Nowell, D. A. G., 1999, Gravity terrain corrections †an overview. Journal of Applied Geophysics, 42, 117-134.
Smith, D. A., 2000, The gravitational attraction of any polygonally shaped vertical prism with inclined top and bottom faces. Journal of Geodesy, 74, 414-420.
Smith, D. A., Robertson, D. S. and Milbert, D. G., 2001, Gravitational attraction of local crustal masses in spherical coordinates. Journal of Geodesy, 74, 783-795.
Sun, W., 2002, A formula for gravimetric terrain corrections using powers of topographic height. Journal of Geodesy, 76, 399-406.
Takahasi, H. and Mori, M., 1974, Double exponential formulas for numerical integration. Publications of the Research Institute for Mathematical Sciences, 93, 721-741.
Tenzer, R., Novac, P., Janak, J., Huang, J., Najafi-Almadari, M., Vajda, P. and Santos, M., 2003, A review of the UNB Stokes-Helmert approach for precise geoid determination In Honoring The Academic Life Of Petr Vanicek.
Tsoulis, D., 2003, Terrain modeling in forward gravimetric problems, a case study on local terrain effects. Journal of Applied Geophysics, 54, 145-160.
Tsoulis, D., Novák, P. and Kadlec, M., 2009, Evaluation of precise terrain effects using high-resolution digital elevation models. J. Geophys. Res., 114, B02404.
Tsoulis, D. V., 1998, A combination method for computing terrain corrections. Physics and Chemistry of The Earth 23, 53-58.
Uieda, L., Barbosa, V. and Braitenberg, C., 2016, Tesseroids: Forward-modeling gravitational fields in spherical coordinates. GEOPHYSICS, 815, F41-F48.
Vaníček, P., Kingdon, R., Kuhn, M., Ellmann, A., Featherstone, W. E., Santos, M. C., Martinec, Z., Hirt, C. and Avalos-Naranjo, D., 2013, Testing Stokes-Helmert geoid model computation on a synthetic gravity field: experiences and shortcomings, Studia Geophysica et Geodaetica, 573, 369-400.
Vanícek, P., Tenzer, R., Sjoberg, L., Martinec, Z. and Featherstone, W., 2004, New views of the spherical Bouguer gravity anomaly. Geophysical Journal International, 13, 460-472.
Wild-Pfeiffer, F., 2008, A comparison of different mass elements for use in gravity gradiometry. Journal of Geodesy, 82, 637-653.
Yamamoto, A., 2002, Spherical terrain corrections for gravity anomaly using a digital elevation model gridded with nodes at every 50 m. Journal of the Faculty of Science, Hokkaido University. Series 7, Geophysics, 11, 845-880.
Zahorec, P., 2015, Inner zone terrain correction calculation using interpolated heights. Contributions to Geophysics and Geodesy. 45, 219.
Journal of the Earth and Space Physics
Volume 44, Issue 3
November 2018
Pages 595-606

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