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Gravity effects of polyhedral bodies with linearly varying density

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Abstract

We extend a recent approach for computing the gravity effects of polyhedral bodies with uniform density by the case of bodies with linearly varying density and by consistently taking into account the relevant singularities. We show in particular that the potential and the gravity vector can be given an expression in which singularities are ruled out, thus avoiding the introduction of small positive numbers advocated by some authors in order to circumvent undefined operations. We also prove that the entries of the second derivative exhibit a singularity if and only if the observation point is aligned with an edge of a face of the polyhedron. The formulas presented in the paper have been numerically checked with alternative ones derived on the basis of different approaches, already established in the literature, and intensively tested by computing the gravity effects induced by real asteroids with arbitrarily assigned density variations.

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References

  • Banerjee, B., DasGupta, S.P.: Gravitational attraction of a rectangular parallelepiped. Geophysics 42, 1053–1055 (1977)

    Article  ADS  Google Scholar 

  • Barnett, C.T.: Theoretical modeling of the magnetic and gravitational fields of an arbitrarily shaped three-dimensional body. Geophysics 41, 1353–1364 (1976)

    Article  ADS  Google Scholar 

  • Bowen, R.M., Wang, C.C.: Introduction to Vectors and Tensors, Vol 2: Vector and Tensor Analysis. http://hdl.handle.net/1969.1/3609 (2006)

  • Chai, Y., Hinze, W.J.: Gravity inversion of an interface above which the density contrast varies exponentially with depth. Geophysics 53, 837845 (1988)

    Article  Google Scholar 

  • D’Urso, M.G., Russo, P.: A new algorithm for point-in polygon test. Surv. Rev. 284, 410–422 (2002)

    Article  Google Scholar 

  • D’Urso, M.G.: New expressions of the gravitational potential and its derivates for the prism. In: Sneeuw, N., Novák, P., Crespi, M., Sansò, F. (eds.) VII Hotine-Marussi International Symposium on Mathematical Geodesy. Springer, Berlin, Heidelberg (2012)

    Google Scholar 

  • D’Urso, M.G.: On the evaluation of the gravity effects of polyhedral bodies and a consistent treatment of related singularities. J. Geod. 87, 239–252 (2013a)

    Article  ADS  Google Scholar 

  • D’Urso M.G.: Some remarks on the computation of the gravitational potential of masses with linearly varying density. In: VIII Hotine-Marussi Symposium, Rome, June 17–21 (2013b)

  • D’Urso, M.G., Marmo, F.: Vertical stress distribution in isotropic half-spaces due to surface vertical loadings acting over polygonal domains. Zeit. Ang. Math. Mech. (2013c). doi:10.1002/zamm.201300034

  • D’Urso, M.G., Marmo, F.: On a generalized Love’s problem. Comput. Geosci. 61, 144–151 (2013d)

    Article  ADS  Google Scholar 

  • D’Urso, M.G.: Analytical computation of gravity effects for polyhedral bodies. J. Geod. 88, 13–29 (2014)

    Article  ADS  Google Scholar 

  • Gallardo-Delgado, L.A., Perez-Flores, M.A., Gomez-Trevino, E.: A versatile algorithm for joint inversion of gravity andmagnetic data. Geophysics 68, 949–959 (2003)

    Article  ADS  Google Scholar 

  • Garcìa-Abdeslem, J.: Gravitational attraction of a rectangular prism with depth dependent density. Geophysics 57, 470–473 (1992)

    Article  ADS  Google Scholar 

  • Garcìa-Abdeslem, J.: Gravitational attraction of a rectangular prism with density varying with depth following a cubic polynomial. Geophysics 70, J39–J42 (2005)

    Article  ADS  Google Scholar 

  • Hamayun, P., Prutkin, I., Tenzer, R.: The optimum expression for the gravitational potential of polyhedral bodies having a linearly varying density distribution. J. Geod. 83, 1163–1170 (2009)

    Article  ADS  Google Scholar 

  • Hansen, R.O.: An analytical expression for the gravity field of a polyhedral body with linearly varying density. Geophysics 64, 75–77 (1999)

    Article  ADS  Google Scholar 

  • Heiskanen, W.A., Moritz, H.: Physical Geodesy. Freeman, San Francisco (1967)

    Google Scholar 

  • Hoffmann, C.M.: Geometric and solid modeling. http://www.cs.purdue.edu/homes/cmh/distribution/books/geo.html (2002)

  • Holstein, H., Ketteridge, B.: Gravimetric analysis of uniform polyhedra. Geophysics 61(2), 357–364 (1996)

    Article  ADS  Google Scholar 

  • Holstein, H.: Gravimagnetic anomaly formulas for polyhedra of spatially linear media. Geophysics 68, 157–167 (2003)

  • Jiancheng, H., Wenbin, S.: Comparative study on two methods for calculating the gravitational potential of a prism. Geo-spat. Inf. Sci. 13, 60–64 (2010)

    Article  Google Scholar 

  • Kellogg, O.D.: Foundations of Potential Theory. Springer, Berlin, Heidelberg, New York (1929)

    Book  Google Scholar 

  • Koch, K.R.: Die Topographische Schwere- und Lotabweichungs Reduktion für Aufpunkte in Geneigtem Gelände. Allg. Verm. Nachr. 1, 438–441 (1965)

    Google Scholar 

  • Kwok, Y.-K.: Gravity gradient tensor due to a polyhedron with polygonal facets. Geophys. Prospect. 39, 435–443 (1991)

    Article  ADS  Google Scholar 

  • MacMillan, W.D.: Theoretical Mechanics, vol. 2: the Theory of the Potential. Mc-Graw-Hill, New York (1930)

    Google Scholar 

  • Mader, K.: Das Newtonsche Raumpotential Prismatischer körper und Seine Ableitungen bis zur Dritten Ordnung. Österr Zeits Vermes, Sonderheft 11 (1951)

  • Matlab version 7.10.0: The MathWorks Inc., Natick, Massachusetts (2012)

  • Nagy, D.: The gravitational attraction of a right rectangular prism. Geophysics 31, 362–371 (1966)

    Article  ADS  Google Scholar 

  • Nagy, D., Papp, G., Benedek, J.: The gravitational potential and its derivatives for the prism. J. Geod. 74, 553–560 (2000)

    Article  ADS  Google Scholar 

  • Okabe, M.: Analytical: expressions for gravity anomalies due to homogeneous polyhedral bodies and translation into magnetic anomalies. Geophysics 44, 730–741 (1979)

    Article  ADS  Google Scholar 

  • Paul, M.K.: The gravity effect of a homogeneous polyhedron for three-dimensional interpretation. Pure Appl. Geophys. 112, 553–561 (1974)

    Article  ADS  Google Scholar 

  • Petrovic̀, S.: Determination of the potential of homogeneous polyhedral bodies using line integrals. J. Geod. 71, 44–52 (1996)

    Article  ADS  MATH  Google Scholar 

  • Pohanka, V.: Optimum expression for computation of the gravity field of a homogeneous polyhedral body. Geophys. Prospect. 36, 733–751 (1988)

    Article  ADS  Google Scholar 

  • Pohanka, V.: Optimum expression for computation of the gravity field of a polyhedral body with linearly increasing density. Geophys. Prospect. 46, 391–404 (1998)

    Article  ADS  Google Scholar 

  • Sessa, S., D’Urso, M.G.: Employment of Bayesian networks for risk assessment of excavation processes in dense urban areas. In: Proceedings of 11th International Conference on ICOSSAR 2013, pp. 30163–30169 (2013)

  • Smith, D.A.: The gravitational attraction of any polygonally shaped vertical prism with inclined top and bottom faces. J. Geod. 74, 414–420 (2000)

    Article  ADS  MATH  Google Scholar 

  • Strakhov, V.N., Lapina, M.I., Yefimov, A.B.: A solution to forward problems in gravity and magnetism with new analytical expression for the field elements of standard approximating body. Izv Earth Sci. 22, 471–482 (1986)

    Google Scholar 

  • Tang, K.T.: Mathematical Methods for Engineers and Scientists. Springer, Berlin, Heidelberg, New York (2006)

    Google Scholar 

  • Tsoulis, D.: A Note on the Gravitational Field of the Right Rectangular Prism. Boll. Geod. Sci. Aff. LIX-1:21–35 (2000)

  • Tsoulis, D., Petrovic̀, S.: On the singularities of the gravity field of a homogeneous polyhedral body. Geophysics 66, 535–539 (2001)

    Article  ADS  Google Scholar 

  • Tsoulis, D., Wziontek, H., Petrovic̀, S.: A bilinear approximation of the surface relief in terrain correction computations. J. Geod. 77, 338–344 (2003)

    Article  ADS  MATH  Google Scholar 

  • Tsoulis, D.: Analytical computation of the full gravity tensor of a homogeneous arbitrarily shaped polyhedral source using line integrals. Geophysics 77, F1–F11 (2012)

    Article  ADS  Google Scholar 

  • Waldvogel, J.: The Newtonian potential of homogeneous polyhedra. J. Appl. Math. Phys. 30, 388–398 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  • Werner, R.A.: The gravitational potential of a homogeneous polyhedron. Celest. Mech. Dyn. Astron. 59, 253–278 (1994)

    Article  ADS  MATH  Google Scholar 

  • Werner, R.A., Scheeres, D.J.: Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769 Castalia. Celest. Mech. Dyn. Astron. 65, 313–344 (1997)

    Article  ADS  MATH  Google Scholar 

  • Zhou, X.: 3D vector gravity potential and line integrals for the gravity anomaly of a rectangular prism with 3D variable density contrast. Geophysics 74, I43–I53 (2009)

    Article  Google Scholar 

  • Zhou, X.: Analytic solution of the gravity anomaly of irregular 2D masses with density contrast varying as a 2D polynomial function. Geophysics 75, I11–I19 (2010)

    Article  Google Scholar 

Download references

Acknowledgments

The author wishes to express its deep gratitude to the Associate Editor, prof. Erricos C. Pavlis, and to the anonymous reviewers for careful suggestions and useful comments which resulted in an improved version of the original manuscript.

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  1. DICeM—Università di Cassino e del Lazio Meridionale, via G. Di Biasio 43, 03043, Cassino, FR, Italy

    M. G. D’Urso

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  1. M. G. D’Urso
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Correspondence to M. G. D’Urso.

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D’Urso, M.G. Gravity effects of polyhedral bodies with linearly varying density. Celest Mech Dyn Astr 120, 349–372 (2014). https://doi.org/10.1007/s10569-014-9578-z

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  • DOI: https://doi.org/10.1007/s10569-014-9578-z

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