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- 1. Computation of the gravity gradient tensor due to topographic masses using tesseroids Leonardo Uieda 1 Naomi Ussami 2 Carla F Braitenberg 3 1. Observatorio Nacional, Rio de Janeiro, Brazil 2. Universidade de São Paulo, São Paulo, Brazil 3. University of Trieste, Trieste, Italy. August 9, 2010
- 2. Outline The Gravity Gradient Tensor (GGT) What is a tesseroid Why use tesseroids Numerical issues Modeling topography with tesseroids Topographic effect in the Paraná Basin region Further applications Concluding remarks
- 3. Gravity Gradient Tensor
- 4. Gravity Gradient Tensor Hessian matrix of gravitational potential
- 5. Gravity Gradient Tensor Hessian matrix of gravitational potential ∂2V ∂2V ∂2V ∂x 2 ∂x∂y ∂x∂z gxx gxy gxz 2 ∂ V ∂2V ∂ 2V GGT = gyx gyy gyz = gzx gzy gzz ∂y ∂x ∂y 2 ∂y ∂z 2 ∂2V 2V ∂ V ∂ ∂z∂x ∂z∂y ∂z 2
- 6. Gravity Gradient Tensor Hessian matrix of gravitational potential ∂2V ∂2V ∂2V ∂x 2 ∂x∂y ∂x∂z gxx gxy gxz 2 ∂ V ∂2V ∂ 2V GGT = gyx gyy gyz = gzx gzy gzz ∂y ∂x ∂y 2 ∂y ∂z 2 ∂2V 2V ∂ V ∂ ∂z∂x ∂z∂y ∂z 2 Volume integrals
- 7. Gravity Gradient Tensor Hessian matrix of gravitational potential ∂2V ∂2V ∂2V ∂x 2 ∂x∂y ∂x∂z gxx gxy gxz 2 ∂ V ∂2V ∂ 2V GGT = gyx gyy gyz = gzx gzy gzz ∂y ∂x ∂y 2 ∂y ∂z 2 ∂2V 2V ∂ V ∂ ∂z∂x ∂z∂y ∂z 2 Volume integrals ˆ gij (x, y , z) = Kernel(x, y , z, x , y , z ) dΩ Ω
- 8. Gravity Gradient Tensor Can discretize volume Ω using:
- 9. Gravity Gradient Tensor Can discretize volume Ω using: Rectangular prisms
- 10. Gravity Gradient Tensor Can discretize volume Ω using: Rectangular prisms Tesseroids (spherical prisms)
- 11. What is a tesseroid?
- 12. What is a tesseroid? Z Tesseroid r φ X λ Y
- 13. What is a tesseroid?Delimited by: Z 2 meridians r λ1 φ X λ Y
- 14. What is a tesseroid?Delimited by: Z 2 meridians r λ2 φ X λ Y
- 15. What is a tesseroid?Delimited by: Z 2 meridians 2 parallels φ1 r φ X λ Y
- 16. What is a tesseroid?Delimited by: Z 2 meridians 2 parallels φ2 r φ X λ Y
- 17. What is a tesseroid?Delimited by: Z 2 meridians 2 parallels 2 concentric spheres r1 r φ X λ Y
- 18. What is a tesseroid?Delimited by: Z 2 meridians 2 parallels 2 concentric spheres r2 r φ X λ Y
- 19. Why use tesseroids?
- 20. Why use tesseroids? Earth Core Matle Crust
- 21. Why use tesseroids? Earth Core Matle Crust
- 22. Why use tesseroids? Want to model the geologic body Observation Point Geologic body
- 23. Why use tesseroids? Flat Earth Observation Point
- 24. Why use tesseroids? Flat Earth + Rectangular Prisms Observation Point
- 25. Why use tesseroids? Flat Earth Good for small + Rectangular Prisms regions (Rule of thumb: < Observation 2500 km) Point
- 26. Why use tesseroids? Flat Earth Good for small + Rectangular Prisms regions (Rule of thumb: < Observation 2500 km) Point and close observation point
- 27. Why use tesseroids? Flat Earth Good for small + Rectangular Prisms regions (Rule of thumb: < Observation 2500 km) Point and close observation point Not very accurate for larger regions
- 28. Why use tesseroids? Spherical Earth Observation Point
- 29. Why use tesseroids? Spherical Earth + Rectangular Prisms Observation Point
- 30. Why use tesseroids? Spherical Earth + Rectangular Prisms Observation Point
- 31. Why use tesseroids? Spherical Earth Usually accurate enough (if mass of + Rectangular Prisms prisms = mass of tesseroids) Observation Point
- 32. Why use tesseroids? Spherical Earth Usually accurate enough (if mass of + Rectangular Prisms prisms = mass of tesseroids) Observation Point Involves many coordinate changes
- 33. Why use tesseroids? Spherical Earth Usually accurate enough (if mass of + Rectangular Prisms prisms = mass of tesseroids) Observation Point Involves many coordinate changes Computationally slow
- 34. Why use tesseroids? Spherical Earth Observation Point
- 35. Why use tesseroids? Spherical Earth + Tesseroids Observation Point
- 36. Why use tesseroids? Spherical Earth + Tesseroids Observation Point
- 37. Why use tesseroids? As accurate as Spherical Earth Spherical Earth + + Tesseroids rectangular prisms Observation Point
- 38. Why use tesseroids? As accurate as Spherical Earth Spherical Earth + + Tesseroids rectangular prisms Observation But faster Point
- 39. Why use tesseroids? As accurate as Spherical Earth Spherical Earth + + Tesseroids rectangular prisms Observation But faster Point As shown in Wild-Pfeiffer (2008)
- 40. Why use tesseroids? As accurate as Spherical Earth Spherical Earth + + Tesseroids rectangular prisms Observation But faster Point As shown in Wild-Pfeiffer (2008) Some numerical problems
- 41. Numerical issues
- 42. Numerical issues Gravity Gradient Tensor (GGT) volume integrals solved:
- 43. Numerical issues Gravity Gradient Tensor (GGT) volume integrals solved: Analytically in the radial direction
- 44. Numerical issues Gravity Gradient Tensor (GGT) volume integrals solved: Analytically in the radial direction Numerically over the surface of the sphere
- 45. Numerical issues Gravity Gradient Tensor (GGT) volume integrals solved: Analytically in the radial direction Numerically over the surface of the sphere Using the Gauss-Legendre Quadrature (GLQ)
- 46. Numerical issues At 250 km height with Gauss-Legendre Quadrature (GLQ) order 2
- 47. Numerical issues At 50 km height with Gauss-Legendre Quadrature (GLQ) order 2
- 48. Numerical issues At 50 km height with Gauss-Legendre Quadrature (GLQ) order 10
- 49. Numerical issues General rule:
- 50. Numerical issues General rule: Distance to computation point > Distance between nodes
- 51. Numerical issues General rule: Distance to computation point > Distance between nodes Increase number of nodes
- 52. Numerical issues General rule: Distance to computation point > Distance between nodes Increase number of nodes Divide the tesseroid in smaller parts
- 53. Modeling topography with tesseroids
- 54. Modeling topography with tesseroids Computer program: Tesseroids
- 55. Modeling topography with tesseroids Computer program: Tesseroids Python programming language
- 56. Modeling topography with tesseroids Computer program: Tesseroids Python programming language Open Source (GNU GPL License)
- 57. Modeling topography with tesseroids Computer program: Tesseroids Python programming language Open Source (GNU GPL License) Project hosted on Google Code
- 58. Modeling topography with tesseroids Computer program: Tesseroids Python programming language Open Source (GNU GPL License) Project hosted on Google Code http://code.google.com/p/tesseroids
- 59. Modeling topography with tesseroids Computer program: Tesseroids Python programming language Open Source (GNU GPL License) Project hosted on Google Code http://code.google.com/p/tesseroids Under development:
- 60. Modeling topography with tesseroids Computer program: Tesseroids Python programming language Open Source (GNU GPL License) Project hosted on Google Code http://code.google.com/p/tesseroids Under development: Optimizations using C coded modules
- 61. Modeling topography with tesseroids To model topography:
- 62. Modeling topography with tesseroids To model topography: Digital Elevation Model (DEM) ⇒ Tesseroid model
- 63. Modeling topography with tesseroids To model topography: Digital Elevation Model (DEM) ⇒ Tesseroid model 1 Grid Point = 1 Tesseroid
- 64. Modeling topography with tesseroids To model topography: Digital Elevation Model (DEM) ⇒ Tesseroid model 1 Grid Point = 1 Tesseroid Top centered on grid point
- 65. Modeling topography with tesseroids To model topography: Digital Elevation Model (DEM) ⇒ Tesseroid model 1 Grid Point = 1 Tesseroid Top centered on grid point Bottom at reference surface
- 66. Topographic effect in the Paraná Basin region
- 67. Topographic effect in the Paraná Basin region Digital Elevation Model (DEM) Grid:
- 68. Topographic effect in the Paraná Basin region Digital Elevation Model (DEM) Grid: ETOPO1
- 69. Topographic effect in the Paraná Basin region Digital Elevation Model (DEM) Grid: ETOPO1 10’ x 10’ Grid
- 70. Topographic effect in the Paraná Basin region Digital Elevation Model (DEM) Grid: ETOPO1 10’ x 10’ Grid ~ 23,000 Tesseroids
- 71. Topographic effect in the Paraná Basin region Digital Elevation Model (DEM) Grid: ETOPO1 10’ x 10’ Grid ~ 23,000 Tesseroids Density = 2.67 g × cm−3
- 72. Topographic effect in the Paraná Basin region Digital Elevation Model (DEM) Grid: ETOPO1 10’ x 10’ Grid ~ 23,000 Tesseroids Density = 2.67 g × cm−3 Computation height = 250 km
- 73. Topographic effect in the Paraná Basin region
- 74. Topographic effect in the Paraná Basin region Height of 250 km
- 75. Topographic effect in the Paraná Basin region Topographic effect in the region has the same order of magnitude as a 2◦ × 2◦ × 10 km tesseroid (100 Eötvös)
- 76. Topographic effect in the Paraná Basin region Topographic effect in the region has the same order of magnitude as a 2◦ × 2◦ × 10 km tesseroid (100 Eötvös) Need to take topography into account when modeling (even at 250 km altitudes)
- 77. Further applications
- 78. Further applications Satellite gravity data = global coverage
- 79. Further applications Satellite gravity data = global coverage + Tesseroid modeling:
- 80. Further applications Satellite gravity data = global coverage + Tesseroid modeling: Regional/global inversion for density (Mantle)
- 81. Further applications Satellite gravity data = global coverage + Tesseroid modeling: Regional/global inversion for density (Mantle) Regional/global inversion for relief of an interface (Moho)
- 82. Further applications Satellite gravity data = global coverage + Tesseroid modeling: Regional/global inversion for density (Mantle) Regional/global inversion for relief of an interface (Moho) Joint inversion with seismic tomography
- 83. Concluding remarks
- 84. Concluding remarks Developed a computational tool for large-scale gravity modeling with tesseroids
- 85. Concluding remarks Developed a computational tool for large-scale gravity modeling with tesseroids Better use tesseroids than rectangular prisms for large regions
- 86. Concluding remarks Developed a computational tool for large-scale gravity modeling with tesseroids Better use tesseroids than rectangular prisms for large regions Take topographic effect into consideration when modeling density anomalies within the Earth
- 87. Concluding remarks Developed a computational tool for large-scale gravity modeling with tesseroids Better use tesseroids than rectangular prisms for large regions Take topographic effect into consideration when modeling density anomalies within the Earth Possible application: tesseroids in regional/global gravity inversion
- 88. Thank you
- 89. References WILD-PFEIFFER, F. A comparison of different mass elements for use in gravity gradiometry. Journal of Geodesy, v. 82 (10), p. 637 - 653, 2008.

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