Solving the gradiometric boundary value problem by the finite element method
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Sep 21, 2024
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Abstract
We present an innovative numerical approach for gravity field modelling where the second derivatives of disturbing potential measured at the level of satellite orbits can be directly taken into account. To that goal, we create the computational domain bounded by the chosen region on the Earth's surface, corresponding boundary at the level of chosen satellites and additional four side boundaries. The boundary value problem consists of the Laplace equation for the unknown disturbing potential accompanied by the first derivatives of the disturbing potential given on the approximation of the Earth's surface, the second derivatives of the disturbing potential, e.g. from the GOCE measurements, given on the upper boundary away from the Earth, and the disturbing potential given on four side boundaries. To solve such a problem, we have derived the numerical scheme based on the finite element and the finite difference methods. We test its order of convergence by one theoretical experiment, and then we present gravity field modelling in Europe using EGM2008 data.
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Macák, M., Mikula, K., Minarechová, Z., & Čunderlík, R.
(2024).
Solving the gradiometric boundary value problem by the finite element method.
Proceedings Of The Conference Algoritmy, , 26 - 35.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/2141/1024
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References
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[16] M. Macák, Z. Minarechová, L. Tomek, R. Čunderlı́k and K. Mikula, Solving the fixed gravimetric boundary value problem by the finite element method using mapped infinite elements, Computational Geosciences, (2023).
[17] P. Meissl, The use of finite elements in physical geodesy, Report 313, Geodetic Science and Surveying, The Ohio State University, (1981).
[18] Z. Minarechová, M. Macák, R. Čunderlı́k and K. Mikula, On the finite element method for solving the oblique derivative boundary value problems and its application in local gravity field modelling, Journal of Geodesy, Vol. 95, 70, (2021).
[19] P. Novák, M. Šprlák and M. Pitoňák, On determination of the geoid from measured gradients of the Earth’s gravity field potential, Earth-Science Reviews, Vol. 221, 103773, (2021).
[20] N. K. Pavlis, S. A. Holmes, S. C. Kenyon and J. K. Factor, The development and evalua- tion of the Earth Gravitational Model 2008 (EGM2008), Journal of Geophysical Research, 117, B04406, (2012).
[21] J. N. Reddy, An Introduction to the Finite Element Method, 3rd Edition, McGraw-Hill Education, New York, ISBN: 9780072466850, (2006).
[22] R. Rummel and O. L. Colombo, Gravity Field Determination from Satellite Gradiometry, Bulletin Geodesique 57, 233–246, (1985).
[23] B. Shaofeng and C. Dingbo, The finite element method for the geodetic boundary value problem, Manuscr Geod 16:353–359, (1991).
[24] M. Šprlák, J. Sebera, M. Vaľko and P. Novák, Spherical integral formulas for upward/downward continuation of gravitational gradients onto gravitational gradients, Journal of Geodesy, Vol. 88, 2, 179-197, (2014).
[25] C. C. Tscherning, R. Forsberg and M. Vermeer, Methods for Regional Gravity Field Modelling from SST and SGG Data, Report 90: 2, Finnish Geodetic Institute, Helsinki, (1990).
[26] Z. Yin and N. Sneeuw, Modeling the gravitational field by using CFD techniques, J Geod 95, 68 (2021).
[2] A. Bjerhammar and L. Svensson, On the geodetic boundary value problem for a fixed boundary surface, A satellite approach, Bull Geod., 57 (1-4), 382-393, (1983).
[3] S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, Springer- Verlag, New York, (2002).
[4] M. Brovelli, F. Migliaccio and F. Sansó, A BVP Approach to the Reduction of Spaceborne Gradiometry: Theory and Simulations, LAG 110:169–180, (1991).
[5] R. Čunderlı́k, M. Kollár and K. Mikula, 1D along-track pre-processing of the GOCE gravity gradients and nonlinear filtering of the radial components Vzz in spatial domain, Contributions to Geophysics and Geodesy, 53(4), 333-351, (2023).
[6] M. Eshagh, Alternative expression for gravity gradients in local north-oriented frame and
tensor spherical harmonics, Acta Geophysica, Vol. 58, 215-243, (2010).
[7] Z. Fašková, R.Čunderlı́k, J. Janák, K. Mikula and M. Šprlák, Gravimetric quasigeoid in Slovakia by the finite element method, Kybernetika, Vol. 43, No. 6: 789-796, (2007).
[8] Z. Fašková, R.Čunderlı́k and K. Mikula, Finite element method for solving geodetic boundary value problems, J Geod 84(2): 135-144, (2010).
[9] R. Floberghagen, M. Fehringer, D. Lamarre, D. Muzi, B. Frommknecht, Ch. Steiger, J. Pineiro, and A. da Costa, Mission design, operation and exploitation of the gravity Field and steady-state ocean circulation explorer mission, J. Geod., vol. 85, 749-758, (2011).
[10] P. Holota, Boundary Value Problems and Invariants of the Gravitational Tensor in Satellite Gradiometry, Lecture Notes in Earth Sciences, Vol. 25, 447–457, (1986).
[11] P. Holota, Coerciveness of the linear gravimetric boundary-value problem and a geometrical interpretation, J Geod 71: 640-651, (1997).
[12] K. R. Koch and A. J. Pope, Uniqueness and existence for the geodetic boundary value problem using the known surface of the earth, Bull. Geod., 46, 467-476, (1972).
[13] G. M. Lieberman, Oblique Derivative Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, ISBN: 978-981-4452-32-8, (2013).
[14] Z. C. Luo, The Theory and Methodology for the Determination of the Earth’s Gravity Field from Satellite Gravity Gradiometry Data, Dissertation (in Chinese), Wuhan Technical University of Surveying and Mapping, (1996).
[15] M. Macák, Z. Minarechová, R. Čunderlı́k and K. Mikula, The finite element method as a tool to solve the oblique derivative boundary value problem in geodesy, Tatra Mountains Mathematical Publications. Vol. 75, no. 1, 63-80, (2020).
[16] M. Macák, Z. Minarechová, L. Tomek, R. Čunderlı́k and K. Mikula, Solving the fixed gravimetric boundary value problem by the finite element method using mapped infinite elements, Computational Geosciences, (2023).
[17] P. Meissl, The use of finite elements in physical geodesy, Report 313, Geodetic Science and Surveying, The Ohio State University, (1981).
[18] Z. Minarechová, M. Macák, R. Čunderlı́k and K. Mikula, On the finite element method for solving the oblique derivative boundary value problems and its application in local gravity field modelling, Journal of Geodesy, Vol. 95, 70, (2021).
[19] P. Novák, M. Šprlák and M. Pitoňák, On determination of the geoid from measured gradients of the Earth’s gravity field potential, Earth-Science Reviews, Vol. 221, 103773, (2021).
[20] N. K. Pavlis, S. A. Holmes, S. C. Kenyon and J. K. Factor, The development and evalua- tion of the Earth Gravitational Model 2008 (EGM2008), Journal of Geophysical Research, 117, B04406, (2012).
[21] J. N. Reddy, An Introduction to the Finite Element Method, 3rd Edition, McGraw-Hill Education, New York, ISBN: 9780072466850, (2006).
[22] R. Rummel and O. L. Colombo, Gravity Field Determination from Satellite Gradiometry, Bulletin Geodesique 57, 233–246, (1985).
[23] B. Shaofeng and C. Dingbo, The finite element method for the geodetic boundary value problem, Manuscr Geod 16:353–359, (1991).
[24] M. Šprlák, J. Sebera, M. Vaľko and P. Novák, Spherical integral formulas for upward/downward continuation of gravitational gradients onto gravitational gradients, Journal of Geodesy, Vol. 88, 2, 179-197, (2014).
[25] C. C. Tscherning, R. Forsberg and M. Vermeer, Methods for Regional Gravity Field Modelling from SST and SGG Data, Report 90: 2, Finnish Geodetic Institute, Helsinki, (1990).
[26] Z. Yin and N. Sneeuw, Modeling the gravitational field by using CFD techniques, J Geod 95, 68 (2021).