# A spectral-domain approach for gravity forward modelling of 2D bodies

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## Abstract

We provide the spectral-domain solutions for the gravity forward modelling of a 2D body with constant, polynomial and exponential density distributions, including the gravitational attraction and its arbitrary-order derivatives. The gravity effects may be directly obtained from the derivatives of a constructed scalar quantity and the surface integral of the density function over the 2D body. Then, the surface integrals of the spectral expansion coefficients of the scalar and the density function are converted to the line integrals along the boundary of the mass body using the Gauss divergence theorem and thus can be evaluated by numerical integration. The surface integrals for the exponential density model are expressed as the infinite expansions of line integrals and converge fast. Approximating the mass body to a 2D polygon, the surface integrals can be evaluated by simple analytic formulas for the constant density model and by linear recursive relations for the polynomial density model. The numerical implementation shows that the spectral-domain algorithm of this paper can produce high accurate forward results, e.g. 13–16 digits achievable precision for the gravity vector. Although the spectral-domain method is only suitable for forward computation of the external gravity effects of the 2D body, it is numerically stable for arbitrary observation point outside the smallest enclosed circle. The closed-form solutions for the 2D body with constant or polynomial density distribution are high precision to evaluate the gravity anomaly at the point near or inside the body, but may be lower precision when the computation point moves away from the body. The spectral-domain algorithm can handle the polynomial and exponential density model in both horizontal and vertical directions.

## Keywords

Gravity effect 2D mass representation Spectral-domain method Polynomial density Exponential density Polygon## Notes

### Acknowledgements

We are grateful to the editors and two anonymous reviewers for their valuable comments and suggestions that improved the manuscript. This work is supported by National Natural Science Foundation of China under Grant Nos. 41631072, 41774021 and 41771487.

## References

- Artemjev M, Kaban M, Kucherinenko V, Demyanov G, Taranov V (1994) Subcrustal density inhomogeneities of Northern Eurasia as derived from the gravity data and isostatic models of the lithosphere. Tectonophysics 240:249–280Google Scholar
- Balmino G (1994) Gravitational potential harmonics from the shape of an homogeneous body. Celest Mech Dyn Astron 60:331–364Google Scholar
- Bott M (1960) The use of rapid digital computing methods for direct gravity interpretation of sedimentary basins. Geophys J Int 3:63–67Google Scholar
- Chai Y, Hinze WJ (1988) Gravity inversion of an interface above which the density contrast varies exponentially with depth. Geophysics 53:837–845Google Scholar
- Chakravarthi V, Sundararajan N (2004) Ridge-regression algorithm for gravity inversion of fault structures with variable density. Geophysics 69:1394–1404Google Scholar
- Chao BF, Rubincam DP (1989) The gravitational field of phobos. Geophys Res Lett 16:859–862Google Scholar
- Chappell A, Kusznir N (2008) An algorithm to calculate the gravity anomaly of sedimentary basins with exponential density-depth relationships. Geophys Prospect 56:249–258Google Scholar
- Chen C, Chen Y, Bian S (2019a) Evaluation of the spherical harmonic coefficients for the external potential of a polyhedral body with linearly varying density. Celest Mech Dyn Astron 131:1–28Google Scholar
- Chen C, Ouyang Y, Bian S (2019b) Spherical harmonic expansions for the gravitational field of a polyhedral body with polynomial density contrast. Surv Geophys 40:197–246Google Scholar
- Cordell L (1973) Gravity analysis using an exponential density-depth function; San Jacinto Graben, California. Geophysics 38:684–690Google Scholar
- Cunningham LE (1970) On the computation of the spherical harmonic terms needed during the numerical integration of the orbital motion of an artificial satellite. Celest Mech 2:207–216Google Scholar
- D’Urso MG (2012) New expressions of the gravitational potential and its derivatives for the prism. In: VII Hotine-Marussi symposium on mathematical geodesy. Springer, Berlin, pp 251–256Google Scholar
- D’Urso MG (2013) On the evaluation of the gravity effects of polyhedral bodies and a consistent treatment of related singularities. J Geod 87:239–252Google Scholar
- D’Urso MG (2014a) Analytical computation of gravity effects for polyhedral bodies. J Geod 88:13–29Google Scholar
- D’Urso MG (2014b) Gravity effects of polyhedral bodies with linearly varying density. Celest Mech Dyn Astron 120:349–372Google Scholar
- D’Urso MG (2015) The gravity anomaly of a 2D polygonal body having density contrast given by polynomial functions. Surv Geophys 36:391–425Google Scholar
- D’Urso MG, Trotta S (2015) Comparative assessment of linear and bilinear prism-based strategies for terrain correction computations. J Geod 89:199–215Google Scholar
- D’Urso MG, Trotta S (2017) Gravity anomaly of polyhedral bodies having a polynomial density contrast. Surv Geophys 38:781–832Google Scholar
- Fukushima T (2016) Numerical integration of gravitational field for general three-dimensional objects and its application to gravitational study of grand design spiral arm structure. Mon Not R Astron Soc 463:1500–1517Google Scholar
- Fukushima T (2017) Precise and fast computation of the gravitational field of a general finite body and its application to the gravitational study of asteroid eros. Astron J 154:145Google Scholar
- Fukushima T (2018a) Accurate computation of gravitational field of a tesseroid. J Geod 92:1371–1386Google Scholar
- Fukushima T (2018b) Recursive computation of gravitational field of a right rectangular parallelepiped with density varying vertically by following an arbitrary degree polynomial. Geophys J Int 215:864–879Google Scholar
- García-Abdeslem J (1992) Gravitational attraction of a rectangular prism with depth-dependent density. Geophysics 57:470–473Google Scholar
- García-Abdeslem J (2005) The gravitational attraction of a right rectangular prism with density varying with depth following a cubic polynomial. Geophysics 70:J39–J42Google Scholar
- Gradshteyn IS, Ryzhik IM (2007) Table of integrals, series, and products, 7th edn. Academic Press, LondonGoogle Scholar
- Grombein T, Seitz K, Heck B (2013) Optimized formulas for the gravitational field of a tesseroid. J Geod 87:645–660Google Scholar
- Hamayun Prutkin I, Tenzer R (2009) The optimum expression for the gravitational potential of polyhedral bodies having a linearly varying density distribution. J Geod 83:1163–1170Google Scholar
- Hansen R (1999) An analytical expression for the gravity field of a polyhedral body with linearly varying density. Geophysics 64:75–77Google Scholar
- Hansen R, Wang X (1988) Simplified frequency-domain expressions for potential fields of arbitrary three-dimensional bodies. Geophysics 53:365–374Google Scholar
- Heck B, Seitz K (2007) A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling. J Geod 81:121–136Google Scholar
- Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman, San FranciscoGoogle Scholar
- Hofmann-Wellenhof B, Moritz H (2006) Physical geodesy, 2nd edn. Springer, BerlinGoogle Scholar
- Holstein H (2003) Gravimagnetic anomaly formulas for polyhedra of spatially linear media. Geophysics 68(1):157–167Google Scholar
- Holstein H, Ketteridge B (1996) Gravimetric analysis of uniform polyhedra. Geophysics 61(2):357–364Google Scholar
- Holstein H, Sherratt E, Anastasiades C (2007) Gravimagnetic anomaly formulae for triangulated homogeneous polyhedra. In: 69th EAGE conference and exhibition incorporating SPE EUROPEC 2007Google Scholar
- Hubbert MK (1948) A line-integral method of computing the gravimetric effects of two-dimensional masses. Geophysics 13:215–225Google Scholar
- Jamet O, Thomas E (2004) A linear algorithm for computing the spherical harmonic coefficients of the gravitational potential from a constant density polyhedron. In: Proceedings of the 2nd international GOCE user workshop, GOCE. The Geoid and Oceanography, ESA-ESRIN, Frascati, Italy, Citeseer, pp 8–10Google Scholar
- Jiang L, Zhang J, Feng Z (2017) A versatile solution for the gravity anomaly of 3D prism-meshed bodies with depth-dependent density contrast. Geophysics 82:G77–G86Google Scholar
- Jiang L, Liu J, Zhang J, Feng Z (2018) Analytic expressions for the gravity gradient tensor of 3D prisms with depth-dependent density. Surv Geophys 39:337–363Google Scholar
- Kellogg OD (1929) Foundations of potential theory. Springer, BerlinGoogle Scholar
- Laurie DP (1997) Calculation of Gauss–Kronrod quadrature rules. Math Comput 66:1133–1145Google Scholar
- Litinsky VA (1989) Concept of effective density: key to gravity depth determinations for sedimentary basins. Geophysics 54:1474–1482Google Scholar
- Martín-Atíenza B, Garcia-Abdeslem J (1999) 2-D gravity modeling with analytically defined geometry and quadratic polynomial density functions. Geophysics 64:1730–1734Google Scholar
- Martinec Z, Pěč K, Burša M (1989) The phobos gravitational field modeled on the basis of its topography. Earth Moon Planets 45(3):219–235Google Scholar
- Murthy IR, Rao DB (1979) Gravity anomalies of two-dimensional bodies of irregular cross-section with density contrast varying with depth. Geophysics 44:1525–1530Google Scholar
- Nagy D, Papp G, Benedek J (2000) The gravitational potential and its derivatives for the prism. J Geod 74:552–560Google Scholar
- Netteton LL (1940) Geophysical prospecting for oil. McGraw-Hill Book Company Inc., New YorkGoogle Scholar
- Pedersen LB (1978) Wavenumber domain expressions for potential fields from arbitrary 2-, 21/2-, and 3-dimensional bodies. Geophysics 43:626–630Google Scholar
- Petrovskaya M, Vershkov A (2010) Construction of spherical harmonic series for the potential derivatives of arbitrary orders in the geocentric earth-fixed reference frame. J Geod 84:165–178Google Scholar
- Pohánka V (1998) Optimum expression for computation of the gravity field of a polyhedral body with linearly increasing density. Geophys Prospect 46:391–404Google Scholar
- Rao DB (1985) Analysis of gravity anomalies over an inclined fault with quadratic density function. Pure Appl Geophys 123:250–260Google Scholar
- Rao DB (1986a) Gravity anomalies of a trapezoidal model with quadratic density function. Proc Indian Acad Sci Earth Planet Sci 95:275–284Google Scholar
- Rao DB (1986b) Modelling of sedimentary basins from gravity anomalies with variable density contrast. Geophys J R Astron Soc 84:207–212Google Scholar
- Rao DB (1990) Analysis of gravity anomalies of sedimentary basins by an asymmetrical trapezoidal model with quadratic density function. Geophysics 55:226–231Google Scholar
- Rao DB, Prakash M, Babu NR (1993) Gravity interpretation using Fourier transforms and simple geometrical models with exponential density contrast. Geophysics 58:1074–1083Google Scholar
- Rao CV, Chakravarthi V, Raju M (1994) Forward modeling: gravity anomalies of two-dimensional bodies of arbitrary shape with hyperbolic and parabolic density functions. Comput Geosci 20:873–880Google Scholar
- Rao CV, Raju M, Chakravarthi V (1995) Gravity modelling of an interface above which the density contrast decreases hyperbolically with depth. J Appl Geophys 34:63–67Google Scholar
- Ren Z, Chen C, Pan K, Kalscheuer T, Maurer H, Tang J (2017a) Gravity anomalies of arbitrary 3D polyhedral bodies with horizontal and vertical mass contrasts. Surv Geophys 38:479–502Google Scholar
- Ren Z, Zhong Y, Chen C, Tang J, Pan K (2017b) Gravity anomalies of arbitrary 3D polyhedral bodies with horizontal and vertical mass contrasts up to cubic order. Geophysics 83:G1–G13Google Scholar
- Ren Z, Zhong Y, Chen C, Tang J, Kalscheuer T, Maurer H, Li Y (2018) Gravity gradient tensor of arbitrary 3D polyhedral bodies with up to third-order polynomial horizontal and vertical mass contrasts. Surv Geophys 39:901–935Google Scholar
- Riley KF, Hobson MP, Bence SJ (2006) Mathematical methods for physics and engineering: a comprehensive guide. Cambridge University Press, CambridgeGoogle Scholar
- Ruotoistenmäki T (1992) The gravity anomaly of two-dimensional sources with continuous density distribution and bounded by continuous surfaces. Geophysics 57:623–628Google Scholar
- Silva JB, Costa DC, Barbosa VC (2006) Gravity inversion of basement relief and estimation of density contrast variation with depth. Geophysics 71:J51–J58Google Scholar
- Talwani M, Worzel JL, Landisman M (1959) Rapid gravity computations for two-dimensional bodies with application to the Mendocino submarine fracture zone. J Geophys Res 64:49–59Google Scholar
- Tsoulis D (2000) A note on the gravitational field of the right rectangular prism. Boll Geod Sci Affin 59:21–35Google Scholar
- Tsoulis D, Petrović S (2001) On the singularities of the gravity field of a homogeneous polyhedral body. Geophysics 66:535–539Google Scholar
- Tsoulis D, Jamet O, Verdun J, Gonindard N (2009) Recursive algorithms for the computation of the potential harmonic coefficients of a constant density polyhedron. J Geod 83:925–942Google Scholar
- Uieda L, Barbosa VC, Braitenberg C (2016) Tesseroids: forward-modeling gravitational fields in spherical coordinate. Geophysics 81:F41–F48Google Scholar
- van Gelderen M (1992) The geodetic boundary value problem in two dimensions and its iterative solution. PhD thesis, Faculty of Civil Engineering and Geosciences, Technische Universiteit Delft, DelftGoogle Scholar
- Vermeille H (2011) An analytical method to transform geocentric into geodetic coordinates. J Geod 85:105–117Google Scholar
- Werner RA (1997) Spherical harmonic coefficients for the potential of a constant-density polyhedron. Comput Geosci 23:1071–1077Google Scholar
- Werner RA, Scheeres DJ (1997) Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769 castalia. Celest Mech Dyn Astron 65:313–344Google Scholar
- Wu L (2018a) Comparison of 3-D Fourier forward algorithms for gravity modelling of prismatic bodies with polynomial density distribution. Geophys J Int 215:1865–1886Google Scholar
- Wu L (2018b) Efficient modeling of gravity fields caused by sources with arbitrary geometry and arbitrary density distribution. Surv Geophys 39:401–434Google Scholar
- Wu L (2019) Fourier-domain modeling of gravity effects caused by polyhedral bodies. J Geod 93:635–653Google Scholar
- Wu L, Chen L (2016) Fourier forward modeling of vector and tensor gravity fields due to prismatic bodies with variable density contrast variable density contrast. Geophysics 81:G13–G26Google Scholar
- Zhang Y, Chen C (2018) Forward calculation of gravity and its gradient using polyhedral representation of density interfaces: an application of spherical or ellipsoidal topographic gravity effect. J Geod 92:205–218Google Scholar
- Zhang J, Jiang L (2017) Analytical expressions for the gravitational vector field of a 3-D rectangular prism with density varying as an arbitrary-order polynomial function. Geophys J Int 210:1176–1190Google Scholar
- Zhang J, Zhong B, Zhou X, Dai Y (2001) Gravity anomalies of 2-D bodies with variable density contrast. Geophysics 66:809–813Google Scholar
- Zhou X (2008) 2D vector gravity potential and line integrals for the gravity anomaly caused by a 2D mass of depth-dependent density contrast. Geophysics 73:I43–I50Google Scholar
- Zhou X (2009a) 3D vector gravity potential and line integrals for the gravity anomaly of a rectangular prism with 3D variable density contrast. Geophysics 74:I43–I53Google Scholar
- Zhou X (2009b) General line integrals for gravity anomalies of irregular 2D masses with horizontally and vertically dependent density contrast. Geophysics 74:I1–I7Google Scholar
- Zhou X (2010) Analytic solution of the gravity anomaly of irregular 2D masses with density contrast varying as a 2D polynomial function. Geophysics 75:I11–I19Google Scholar