Computation of the gravity gradient tensor due to topographic masses using tesseroids

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The GOCE satellite mission has the objective of measuring the Earth's gravitational field with an unprecedented accuracy through the measurement of the gravity gradient tensor (GGT). One of the several applications of this new gravity data set is to study the geodynamics of the lithospheric plates, where the flat Earth approximation may not be ideal and the Earth's curvature should be taken into account. In such a case, the Earth could be modeled using tesseroids, also called spherical prisms, instead of the conventional rectangular prisms. The GGT due to a tesseroid is calculated using numerical integration methods, such as the Gauss-Legendre Quadrature (GLQ), as already proposed by Asgharzadeh et al. (2007) and Wild-Pfeiffer (2008). We present a computer program for the direct computation of the GGT caused by a tesseroid using the GLQ. The accuracy of this implementation was evaluated by comparing its results with the result of analytical formulas for the special case of a spherical cap with computation point located at one of the poles. The GGT due to the topographic masses of the Parana basin (SE Brazil) was estimated at 260 km altitude in an attempt to quantify this effect on the GOCE gravity data. The digital elevation model ETOPO1 (Amante and Eakins, 2009) between 40º W and 65º W and 10º S and 35º S, which includes the Paraná Basin, was used to generate a tesseroid model of the topography with grid spacing of 10' x 10' and a constant density of 2670 kg/m3. The largest amplitude observed was on the second vertical derivative component (-0.05 to 1.20 Eötvos) in regions of rough topography, such as that along the eastern Brazilian continental margins. These results indicate that the GGT due to topographic masses may have amplitudes of the same order of magnitude as the GGT due to density anomalies within the crust and mantle.

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Computation of the gravity gradient tensor due to topographic masses using tesseroids

  1. 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. 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. 3. Gravity Gradient Tensor
  4. 4. Gravity Gradient Tensor Hessian matrix of gravitational potential
  5. 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. 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. 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. 8. Gravity Gradient Tensor Can discretize volume Ω using:
  9. 9. Gravity Gradient Tensor Can discretize volume Ω using: Rectangular prisms
  10. 10. Gravity Gradient Tensor Can discretize volume Ω using: Rectangular prisms Tesseroids (spherical prisms)
  11. 11. What is a tesseroid?
  12. 12. What is a tesseroid? Z Tesseroid r φ X λ Y
  13. 13. What is a tesseroid?Delimited by: Z 2 meridians r λ1 φ X λ Y
  14. 14. What is a tesseroid?Delimited by: Z 2 meridians r λ2 φ X λ Y
  15. 15. What is a tesseroid?Delimited by: Z 2 meridians 2 parallels φ1 r φ X λ Y
  16. 16. What is a tesseroid?Delimited by: Z 2 meridians 2 parallels φ2 r φ X λ Y
  17. 17. What is a tesseroid?Delimited by: Z 2 meridians 2 parallels 2 concentric spheres r1 r φ X λ Y
  18. 18. What is a tesseroid?Delimited by: Z 2 meridians 2 parallels 2 concentric spheres r2 r φ X λ Y
  19. 19. Why use tesseroids?
  20. 20. Why use tesseroids? Earth Core Matle Crust
  21. 21. Why use tesseroids? Earth Core Matle Crust
  22. 22. Why use tesseroids? Want to model the geologic body Observation Point Geologic body
  23. 23. Why use tesseroids? Flat Earth Observation Point
  24. 24. Why use tesseroids? Flat Earth + Rectangular Prisms Observation Point
  25. 25. Why use tesseroids? Flat Earth Good for small + Rectangular Prisms regions (Rule of thumb: < Observation 2500 km) Point
  26. 26. Why use tesseroids? Flat Earth Good for small + Rectangular Prisms regions (Rule of thumb: < Observation 2500 km) Point and close observation point
  27. 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. 28. Why use tesseroids? Spherical Earth Observation Point
  29. 29. Why use tesseroids? Spherical Earth + Rectangular Prisms Observation Point
  30. 30. Why use tesseroids? Spherical Earth + Rectangular Prisms Observation Point
  31. 31. Why use tesseroids? Spherical Earth Usually accurate enough (if mass of + Rectangular Prisms prisms = mass of tesseroids) Observation Point
  32. 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. 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. 34. Why use tesseroids? Spherical Earth Observation Point
  35. 35. Why use tesseroids? Spherical Earth + Tesseroids Observation Point
  36. 36. Why use tesseroids? Spherical Earth + Tesseroids Observation Point
  37. 37. Why use tesseroids? As accurate as Spherical Earth Spherical Earth + + Tesseroids rectangular prisms Observation Point
  38. 38. Why use tesseroids? As accurate as Spherical Earth Spherical Earth + + Tesseroids rectangular prisms Observation But faster Point
  39. 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. 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. 41. Numerical issues
  42. 42. Numerical issues Gravity Gradient Tensor (GGT) volume integrals solved:
  43. 43. Numerical issues Gravity Gradient Tensor (GGT) volume integrals solved: Analytically in the radial direction
  44. 44. Numerical issues Gravity Gradient Tensor (GGT) volume integrals solved: Analytically in the radial direction Numerically over the surface of the sphere
  45. 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. 46. Numerical issues At 250 km height with Gauss-Legendre Quadrature (GLQ) order 2
  47. 47. Numerical issues At 50 km height with Gauss-Legendre Quadrature (GLQ) order 2
  48. 48. Numerical issues At 50 km height with Gauss-Legendre Quadrature (GLQ) order 10
  49. 49. Numerical issues General rule:
  50. 50. Numerical issues General rule: Distance to computation point > Distance between nodes
  51. 51. Numerical issues General rule: Distance to computation point > Distance between nodes Increase number of nodes
  52. 52. Numerical issues General rule: Distance to computation point > Distance between nodes Increase number of nodes Divide the tesseroid in smaller parts
  53. 53. Modeling topography with tesseroids
  54. 54. Modeling topography with tesseroids Computer program: Tesseroids
  55. 55. Modeling topography with tesseroids Computer program: Tesseroids Python programming language
  56. 56. Modeling topography with tesseroids Computer program: Tesseroids Python programming language Open Source (GNU GPL License)
  57. 57. Modeling topography with tesseroids Computer program: Tesseroids Python programming language Open Source (GNU GPL License) Project hosted on Google Code
  58. 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. 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. 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. 61. Modeling topography with tesseroids To model topography:
  62. 62. Modeling topography with tesseroids To model topography: Digital Elevation Model (DEM) ⇒ Tesseroid model
  63. 63. Modeling topography with tesseroids To model topography: Digital Elevation Model (DEM) ⇒ Tesseroid model 1 Grid Point = 1 Tesseroid
  64. 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. 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. 66. Topographic effect in the Paraná Basin region
  67. 67. Topographic effect in the Paraná Basin region Digital Elevation Model (DEM) Grid:
  68. 68. Topographic effect in the Paraná Basin region Digital Elevation Model (DEM) Grid: ETOPO1
  69. 69. Topographic effect in the Paraná Basin region Digital Elevation Model (DEM) Grid: ETOPO1 10’ x 10’ Grid
  70. 70. Topographic effect in the Paraná Basin region Digital Elevation Model (DEM) Grid: ETOPO1 10’ x 10’ Grid ~ 23,000 Tesseroids
  71. 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. 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. 73. Topographic effect in the Paraná Basin region
  74. 74. Topographic effect in the Paraná Basin region Height of 250 km
  75. 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. 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. 77. Further applications
  78. 78. Further applications Satellite gravity data = global coverage
  79. 79. Further applications Satellite gravity data = global coverage + Tesseroid modeling:
  80. 80. Further applications Satellite gravity data = global coverage + Tesseroid modeling: Regional/global inversion for density (Mantle)
  81. 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. 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. 83. Concluding remarks
  84. 84. Concluding remarks Developed a computational tool for large-scale gravity modeling with tesseroids
  85. 85. Concluding remarks Developed a computational tool for large-scale gravity modeling with tesseroids Better use tesseroids than rectangular prisms for large regions
  86. 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. 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. 88. Thank you
  89. 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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