Surveys in Geophysics

, Volume 39, Issue 3, pp 401–434 | Cite as

Efficient Modeling of Gravity Fields Caused by Sources with Arbitrary Geometry and Arbitrary Density Distribution

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Abstract

We present a brief review of gravity forward algorithms in Cartesian coordinate system, including both space-domain and Fourier-domain approaches, after which we introduce a truly general and efficient algorithm, namely the convolution-type Gauss fast Fourier transform (Conv-Gauss-FFT) algorithm, for 2D and 3D modeling of gravity potential and its derivatives due to sources with arbitrary geometry and arbitrary density distribution which are defined either by discrete or by continuous functions. The Conv-Gauss-FFT algorithm is based on the combined use of a hybrid rectangle-Gaussian grid and the fast Fourier transform (FFT) algorithm. Since the gravity forward problem in Cartesian coordinate system can be expressed as continuous convolution-type integrals, we first approximate the continuous convolution by a weighted sum of a series of shifted discrete convolutions, and then each shifted discrete convolution, which is essentially a Toeplitz system, is calculated efficiently and accurately by combining circulant embedding with the FFT algorithm. Synthetic and real model tests show that the Conv-Gauss-FFT algorithm can obtain high-precision forward results very efficiently for almost any practical model, and it works especially well for complex 3D models when gravity fields on large 3D regular grids are needed.

Keywords

Convolution-type integrals Gravity forward modeling Conv-Gauss-FFT algorithm Arbitrary geometry Arbitrary density distribution 

Notes

Acknowledgements

The authors are very grateful to two anonymous reviewers for their critique, helpful comments, and valuable suggestions which improved the manuscript significantly. This study was funded by the National Natural Science Foundation of China under Grant No. 41504089.

References

  1. Asgharzadeh MF, von Frese RRB, Kim HR, Leftwich TE, Kim JW (2007) Spherical prism gravity effects by Gauss–Legendre quadrature integration. Geophys J Int 169(1):1–11.  https://doi.org/10.1111/j.1365-246X.2007.03214.x CrossRefGoogle Scholar
  2. Barnett CT (1976) Theoretical modeling of magnetic and gravitational-fields of an arbitrarily shaped 3-dimensional body. Geophysics 41(6):1353–1364.  https://doi.org/10.1190/1.1440685 CrossRefGoogle Scholar
  3. Bhattacharyya B (1966) Continuous spectrum of the total-magnetic-field anomaly due to a rectangular prismatic body. Geophysics 31(1):97–121CrossRefGoogle Scholar
  4. Bhattacharyya B, Navolio M (1975) Digital convolution for computing gravity and magnetic anomalies due to arbitrary bodies. Geophysics 40(6):981–992CrossRefGoogle Scholar
  5. Blakely RJ (1996) Potential theory in gravity and magnetic applications. Cambridge University Press, CambridgeGoogle Scholar
  6. Cai YG, Wang CY (2005) Fast finite-element calculation of gravity anomaly in complex geological regions. Geophys J Int 162(3):696–708.  https://doi.org/10.1111/j.1365-246X.2005.02711.x CrossRefGoogle Scholar
  7. Cai HZ, Zhdanov M (2015) Application of Cauchy-type integrals in developing effective methods for depth-to-basement inversion of gravity and gravity gradiometry data. Geophysics 80(2):G81–G94.  https://doi.org/10.1190/GEO2014-0332.1 CrossRefGoogle Scholar
  8. Casenave F, Métivier L, Pajot-Métivier G, Panet I (2016) Fast computation of general forward gravitation problems. J Geod 90(7):655–675.  https://doi.org/10.1007/s00190-016-0900-2 CrossRefGoogle Scholar
  9. Chai Y, Hinze WJ (1988) Gravity inversion of an interface above which the density contrast varies exponentially with depth. Geophysics 53(6):837–845CrossRefGoogle Scholar
  10. Chai Y, Jia J (1990) Parker’s formulas in different forms and their applications to oil gravity survey. Oil Geophys Prospect 25(3):321–332Google Scholar
  11. Chakravarthi V, Raghuram HM, Singh SB (2002) 3-D forward gravity modeling of basement interfaces above which the density contrast varies continuously with depth. Comput Geosci 28(1):53–57.  https://doi.org/10.1016/S0098-3004(01)00080-2 CrossRefGoogle Scholar
  12. Chappell A, Kusznir N (2008) An algorithm to calculate the gravity anomaly of sedimentary basins with exponential density-depth relationships. Geophys Prospect 56(2):249–258.  https://doi.org/10.1111/j.1365-2478.2007.00674.x CrossRefGoogle Scholar
  13. Chenot D, Debeglia N (1990) 3-dimensional gravity or magnetic constrained depth inversion with lateral and vertical variation of contrast. Geophysics 55(3):327–335.  https://doi.org/10.1190/1.1442840 CrossRefGoogle Scholar
  14. Conway JT (2015) Analytical solution from vector potentials for the gravitational field of a general polyhedron. Celest Mech Dyn Astron 121(1):17–38.  https://doi.org/10.1007/s10569-014-9588-x CrossRefGoogle Scholar
  15. Cooley JW, Tukey JW (1965) An algorithm for the machine calculation of complex Fourier series. Math Comput 19(90):297–301CrossRefGoogle Scholar
  16. Cordell L (1973) Gravity analysis using an exponential density-depth function—San-Jacinto-Graben, California. Geophysics 38(4):684–690.  https://doi.org/10.1190/1.1440367 CrossRefGoogle Scholar
  17. Deng XL, Grombein T, Shen WB, Heck B, Seitz K (2016) Corrections to “a comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling” (Heck and Seitz, 2007) and “optimized formulas for the gravitational field of a tesseroid” (Grombein et al., 2013). J Geod 90(6):585–587.  https://doi.org/10.1007/s00190-016-0907-8 CrossRefGoogle Scholar
  18. Driscoll JR, Healy DM (1994) Computing Fourier transforms and convolutions on the 2-sphere. Adv Appl Math 15(2):202–250CrossRefGoogle Scholar
  19. D’Urso MG (2013) On the evaluation of the gravity effects of polyhedral bodies and a consistent treatment of related singularities. J Geod 87(3):239–252.  https://doi.org/10.1007/s00190-012-0592-1 CrossRefGoogle Scholar
  20. D’Urso MG (2014a) Analytical computation of gravity effects for polyhedral bodies. J Geod 88(1):13–29.  https://doi.org/10.1007/s00190-013-0664-x CrossRefGoogle Scholar
  21. D’Urso MG (2014b) Gravity effects of polyhedral bodies with linearly varying density. Celest Mech Dyn Astron 120(4):349–372.  https://doi.org/10.1007/s10569-014-9578-z CrossRefGoogle Scholar
  22. D’Urso MG (2015) The gravity anomaly of a 2D polygonal body having density contrast given by polynomial functions. Surv Geophys 36(3):391–425.  https://doi.org/10.1007/s10712-015-9317-3 CrossRefGoogle Scholar
  23. D’Urso MG (2016) A remark on the computation of the gravitational potential of masses with linearly varying density. In: VIII Hotine-Marussi symposium on mathematical geodesy, vol 142, pp 205–212.  https://doi.org/10.1007/1345_2015_138
  24. D’Urso MG, Trotta S (2015) Comparative assessment of linear and bilinear prism-based strategies for terrain correction computations. J Geod 89(3):199–215.  https://doi.org/10.1007/s00190-014-0770-4 CrossRefGoogle Scholar
  25. D’Urso MG, Trotta S (2017) Gravity anomaly of polyhedral bodies having a polynomial density contrast. Surv Geophys 38(4):781–832.  https://doi.org/10.1007/s10712-017-9411-9 CrossRefGoogle Scholar
  26. Farquharson C, Mosher C (2009) Three-dimensional modelling of gravity data using finite differences. J Appl Geophys 68(3):417–422. . http://www.sciencedirect.com/science/article/pii/S0926985109000500
  27. Forsberg R (1985) Gravity field terrain effect computations by FFT. Bull Geod 59(4):342–360CrossRefGoogle Scholar
  28. 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(4):145.  https://doi.org/10.3847/1538-3881/aa88b8 CrossRefGoogle Scholar
  29. Gallardo LA, Perez-Flores MA, Gomez-Trevino E (2005) Refinement of three-dimensional multilayer models of basins and crustal environments by inversion of gravity and magnetic data. Tectonophysics 397(1–2):37–54.  https://doi.org/10.1016/j.tecto.2004.10.010 CrossRefGoogle Scholar
  30. Gallardo-Delgado LA, Perez-Flores MA, Gomez-Trevino E (2003) A versatile algorithm for joint 3D inversion of gravity and magnetic data. Geophysics 68(3):949–959.  https://doi.org/10.1190/1.1581067 CrossRefGoogle Scholar
  31. Garcia-Abdeslem J (1992) Gravitational attraction of a rectangular prism with depth-dependent density. Geophysics 57(3):470–473CrossRefGoogle Scholar
  32. Garcia-Abdeslem J (2003) 2D modeling and inversion of gravity data using density contrast varying with depth and source-basement geometry described by the Fourier series. Geophysics 68(6):1909–1916.  https://doi.org/10.1190/1.1635044 CrossRefGoogle Scholar
  33. Garcia-Abdeslem J (2005) The gravitational attraction of a right rectangular prism with density varying with depth following a cubic polynomial. Geophysics 70(6):J39–J42.  https://doi.org/10.1190/1.2122413 CrossRefGoogle Scholar
  34. Garcia-Abdeslem J, Romo JM, Gomez-Trevino E, Ramirez-Hernandez J, Esparza-Hernandez FJ, Flores-Luna CF (2005) A constrained 2D gravity model of the Sebastián Vizcaíno Basin, Baja California Sur, Mexico. Geophys Prospect 53(6):755–765CrossRefGoogle Scholar
  35. Granser H (1987) 3-dimensional interpretation of gravity-data from sedimentary basins using an exponential density depth function. Geophys Prospect 35(9):1030–1041.  https://doi.org/10.1111/j.1365-2478.1987.tb00858.x CrossRefGoogle Scholar
  36. Grombein T, Seitz K, Heck B (2013) Optimized formulas for the gravitational field of a tesseroid. J Geod 87(7):645–660.  https://doi.org/10.1007/s00190-013-0636-1 CrossRefGoogle Scholar
  37. Gruber C, Novák P, Sebera J (2011) FFT-based high-performance spherical harmonic transformation. Studia Geophys Geod 55(3):489–500CrossRefGoogle Scholar
  38. Guptasarma D, Singh B (1999) New scheme for computing the magnetic field resulting from a uniformly magnetized arbitrary polyhedron. Geophysics 64(1):70–74.  https://doi.org/10.1190/1.1444531 CrossRefGoogle Scholar
  39. Guspi F (1992) 3-dimensional Fourier gravity inversion with arbitrary density contrast. Geophysics 57(1):131–135CrossRefGoogle Scholar
  40. 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(12):1163–1170.  https://doi.org/10.1007/s00190-009-0334-1 CrossRefGoogle Scholar
  41. Hansen RO (1999) An analytical expression for the gravity field of a polyhedral body with linearly varying density. Geophysics 64(1):75–77.  https://doi.org/10.1190/1.1444532 CrossRefGoogle Scholar
  42. Hansen R, Wang X (1988) Simplified frequency-domain expressions for potential fields of arbitrary three-dimensional bodies. Geophysics 53(3):365–374CrossRefGoogle Scholar
  43. 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(2):121–136.  https://doi.org/10.1007/s00190-006-0094-0 CrossRefGoogle Scholar
  44. Hirt C, Featherstone WE, Claessens SJ (2011) On the accurate numerical evaluation of geodetic convolution integrals. J Geod 85(8):519–538.  https://doi.org/10.1007/s00190-011-0451-5 CrossRefGoogle Scholar
  45. Holstein H (2003) Gravimagnetic anomaly formulas for polyhedra of spatially linear media. Geophysics 68(1):157–167.  https://doi.org/10.1190/1.1543203 CrossRefGoogle Scholar
  46. Holstein H, Ketteridge B (1996) Gravimetric analysis of uniform polyhedra. Geophysics 61(2):357–364.  https://doi.org/10.1190/1.1443964 CrossRefGoogle Scholar
  47. Jia Z, Wu SG (2011) Potential fields and their partial derivatives produced by a 2D homogeneous polygonal source: a summary with some revisions. Geophysics 76(4):L29–L34.  https://doi.org/10.1190/1.3587221 CrossRefGoogle Scholar
  48. 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(4):G77–G86.  https://doi.org/10.1190/geo2016-0394.1 CrossRefGoogle Scholar
  49. Lee TC, Biehler S (1991) Inversion modeling of gravity with prismatic mass bodies. Geophysics 56(9):1365–1376.  https://doi.org/10.1190/1.1443156 CrossRefGoogle Scholar
  50. Li X, Chouteau M (1998) Three-dimensional gravity modeling in all space. Surv Geophys 19(4):339–368.  https://doi.org/10.1023/A:1006554408567 CrossRefGoogle Scholar
  51. Li YC, Sideris MG (1994) Improved gravimetric terrain corrections. Geophys J Int 119(3):740–752.  https://doi.org/10.1111/j.1365-246X.1994.tb04013.x CrossRefGoogle Scholar
  52. Martin-Atienza B, Garcia-Abdeslem J (1999) 2-D gravity modeling with analytically defined geometry and quadratic polynomial density functions. Geophysics 64(6):1730–1734CrossRefGoogle Scholar
  53. Mohlenkamp MJ (1999) A fast transform for spherical harmonics. J Fourier Anal Appl 5(2–3):159–184CrossRefGoogle Scholar
  54. Murthy IR, Rao DB (1979) Gravity anomalies of two-dimensional bodies of irregular cross-section with density contrast varying with depth. Geophysics 44(9):1525–1530CrossRefGoogle Scholar
  55. Nagy D (1966) The gravitational attraction of a right rectangular prism. Geophysics 31(2):362–371CrossRefGoogle Scholar
  56. Nagy D, Papp G, Benedek J (2000) The gravitational potential and its derivatives for the prism. J Geod 74(7–8):552–560CrossRefGoogle Scholar
  57. Okabe M (1979) Analytical expressions for gravity anomalies due to homogeneous polyhedral bodies and translations into magnetic anomalies. Geophysics 44(4):730–741CrossRefGoogle Scholar
  58. Parker R (1973) The rapid calculation of potential anomalies. Geophys J R Astron Soc 31(4):447–455CrossRefGoogle Scholar
  59. Parker RL (1995) Improved Fourier terrain correction. 1. Geophysics 60(4):1007–1017.  https://doi.org/10.1190/1.1443829 CrossRefGoogle Scholar
  60. Pedersen LB (1978) Wavenumber domain expressions for potential fields from arbitrary 2-, 21/2-, and 3-dimensional bodies. Geophysics 43(3):626–630CrossRefGoogle Scholar
  61. Pohanka V (1998) Optimum expression for computation of the gravity field of a polyhedral body with linearly increasing density. Geophys Prospect 46(4):391–404.  https://doi.org/10.1046/j.1365-2478.1998.960335.x CrossRefGoogle Scholar
  62. Rao DB (1986) Modelling of sedimentary basins from gravity anomalies with variable density contrast. Geophys J R Astron Soc 84(1):207–212CrossRefGoogle Scholar
  63. Rao CV, Pramanik AG, Kymar GVRK, Raju MLML (1994) Gravity interpretation of sedimentary basins with hyperbolic density contrast. Geophys Prospect 42(7):825–839.  https://doi.org/10.1111/j.1365-2478.1994.tb00243.x CrossRefGoogle Scholar
  64. 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(2):479–502.  https://doi.org/10.1007/s10712-016-9395-x CrossRefGoogle Scholar
  65. Ren Z, Tang J, Kalscheuer T, Maurer H (2017b) Fast 3D large-scale gravity and magnetic modeling using unstructured grids and an adaptive multilevel fast multipole method. J Geophys Res 122(1):79–109CrossRefGoogle Scholar
  66. Ren Z, Zhong Y, Chen C, Tang J, Pan K (2017c) Gravity anomalies of arbitrary 3D polyhedral bodies with horizontal and vertical mass contrasts up to cubic order. Geophysics.  https://doi.org/10.1190/geo2017-0219.1 Google Scholar
  67. Rokhlin V, Tygert M (2005) Fast algorithms for spherical harmonic expansions. Society for Industrial and Applied MathematicsGoogle Scholar
  68. Sanso F, Sideris MG (2013) Geoid determination: theory and methods. Lecture notes in earth system sciences. Springer, BerlinGoogle Scholar
  69. Sideris MG, Li YC (1993) Gravity-field convolutions without windowing and edge effects. Bull Geod 67(2):107–118.  https://doi.org/10.1007/BF01371374 CrossRefGoogle Scholar
  70. Smith DA (2000) The gravitational attraction of any polygonally shaped vertical prism with inclined top and bottom faces. J Geod 74(5):414–420CrossRefGoogle Scholar
  71. 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(1):49–59CrossRefGoogle Scholar
  72. Tontini FC, Cocchi L, Carmisciano C (2009) Rapid 3-D forward model of potential fields with application to the Palinuro Seamount magnetic anomaly (southern Tyrrhenian Sea, Italy). J Geophys Res Solid Earth 114(B02):103.  https://doi.org/10.1029/2008JB005907 Google Scholar
  73. Tsoulis DV (1998) A combination method for computing terrain corrections. Phys Chem Earth 23(1):53–58CrossRefGoogle Scholar
  74. Tsoulis D (2001) Terrain correction computations for a densely sampled DTM in the Bavarian Alps. J Geod 75(5–6):291–307.  https://doi.org/10.1007/s001900100176 CrossRefGoogle Scholar
  75. Tsoulis D (2012) Analytical computation of the full gravity tensor of a homogeneous arbitrarily shaped polyhedral source using line integrals. Geophysics 77(2):F1–F11.  https://doi.org/10.1190/GEO2010-0334.1 CrossRefGoogle Scholar
  76. Tsoulis D, Petrovic S (2001) On the singularities of the gravity field of a homogeneous polyhedral body. Geophysics 66(2):535–539.  https://doi.org/10.1190/1.1444944 CrossRefGoogle Scholar
  77. Tsoulis D, Wziontek H, Petrovic S (2003) A bilinear approximation of the surface relief in terrain correction computations. J Geod 77(5–6):338–344.  https://doi.org/10.1007/s00190-003-0332-7 CrossRefGoogle Scholar
  78. 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(10):925–942.  https://doi.org/10.1007/s00190-009-0310-9 CrossRefGoogle Scholar
  79. Tygert M (2006) Fast algorithms for spherical harmonic expansions. II. J Comput Phys 227(8):4260–4279CrossRefGoogle Scholar
  80. Uieda L, Barbosa VCF, Braitenberg C (2016) Tesseroids: forward-modeling gravitational fields in spherical coordinates. Geophysics 81(5):F41–F48CrossRefGoogle Scholar
  81. Vogel CR (2002) Computational methods for inverse problems, vol 23. SIAM, PhiladelphiaCrossRefGoogle Scholar
  82. Werner RA (2017) The solid angle hidden in polyhedron gravitation formulations. J Geod 91(3):307–328.  https://doi.org/10.1007/s00190-016-0964-z CrossRefGoogle Scholar
  83. Wu XZ (1981) Computation of spectrum of potential field due to 3-dimensional bodies (homogeneous models). Chin J Geophys Chin Ed 24(3):336–348Google Scholar
  84. Wu XZ (1983) The computation of spectrum of potential-field due to 3-D arbitrary bodies with physical parameters varying with depth. Chin J Geophys Chin Ed 26(2):177–187Google Scholar
  85. Wu L (2016) Efficient modelling of gravity effects due to topographic masses using the Gauss-FFT method. Geophys J Int 205(1):160–178CrossRefGoogle Scholar
  86. Wu L, Chen L (2016) Fourier forward modeling of vector and tensor gravity fields due to prismatic bodies with variable density contrast. Geophysics 81(1):G13–G26CrossRefGoogle Scholar
  87. Wu L, Lin Q (2017) Improved Parker’s method for topographic models using Chebyshev series and low rank approximation. Geophys J Int 209(2):1296–1325CrossRefGoogle Scholar
  88. Wu L, Tian G (2014) High-precision Fourier forward modeling of potential fields. Geophysics 79(5):G59–G68.  https://doi.org/10.1190/GEO2014-0039.1 CrossRefGoogle Scholar
  89. 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(2):1176–1190.  https://doi.org/10.1093/gji/ggx230 CrossRefGoogle Scholar
  90. Zhang Y, Wong YS (2015) BTTB-based numerical schemes for three-dimensional gravity field inversion. Geophys J Int 203(1):243–256.  https://doi.org/10.1093/gji/ggv301 CrossRefGoogle Scholar
  91. Zhang JZ, Zhong BS, Zhou XX, Dai Y (2001) Gravity anomalies of 2-D bodies with variable density contrast. Geophysics 66(3):809–813.  https://doi.org/10.1190/1.1444970 CrossRefGoogle Scholar
  92. Zhang EH, Shi L, Li YH, Wang QS, Han CW (2015) 3D interface inversion of gravity data in the frequency domain using a parabolic density-depth function and the application in Sichuan-Yunnan region. Chin J Geophys Chin Ed 58(2):556–565Google Scholar
  93. Zhdanov MS, Liu XJ (2013) 3-D Cauchy-type integrals for terrain correction of gravity and gravity gradiometry data. Geophys J Int 194(1):249–268.  https://doi.org/10.1093/gji/ggt120 CrossRefGoogle Scholar
  94. Zhou XB (2008) 2D vector gravity potential and line integrals for the gravity anomaly caused by a 2D mass of depth-dependent density contrast. Geophysics 73(6):I43–I50.  https://doi.org/10.1190/1.2976116 CrossRefGoogle Scholar
  95. Zhou XB (2009a) 3D vector gravity potential and line integrals for the gravity anomaly of a rectangular prism with 3D variable density contrast. Geophysics 74(6):I43–I53.  https://doi.org/10.1190/1.3239518 CrossRefGoogle Scholar
  96. Zhou XB (2009b) General line integrals for gravity anomalies of irregular 2D masses with horizontally and vertically dependent density contrast. Geophysics 74(2):I1–I7.  https://doi.org/10.1190/1.3073761 CrossRefGoogle Scholar
  97. Zhou X (2010) Analytic solution of the gravity anomaly of irregular 2D masses with density contrast varying as a 2D polynomial function. Geophysics 75(2):I11–I19.  https://doi.org/10.1190/1.3294699 CrossRefGoogle Scholar
  98. Zuber MT, Smith DE, Cheng AF, Garvin JB, Aharonson O, Cole TD, Dunn PJ, Guo Y, Lemoine FG, Neumann GA, Rowlands DD, Torrence MH (2000) The shape of 433 Eros from the NEAR-Shoemaker laser rangefinder. Science 289(5487):2097. http://science.sciencemag.org/content/289/5487/2097.abstract

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Authors and Affiliations

  1. 1.Center for Optics and Optoelectronics Research (COOR), Collaborative Innovation Center for Information Technology in Biological and Medical Physics, College of ScienceZhejiang University of TechnologyHangzhouChina

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