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

84d34651c3931a54310a57484a109821?s=47 Leonardo Uieda
October 15, 2010

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

84d34651c3931a54310a57484a109821?s=128

Leonardo Uieda

October 15, 2010
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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 GGT =

      gxx gxy gxz gyx gyy gyz gzx gzy gzz   =           ∂2V ∂x2 ∂2V ∂x∂y ∂2V ∂x∂z ∂2V ∂y∂x ∂2V ∂y2 ∂2V ∂y∂z ∂2V ∂z∂x ∂2V ∂z∂y ∂2V ∂z2          
  6. Gravity Gradient Tensor Hessian matrix of gravitational potential GGT =

      gxx gxy gxz gyx gyy gyz gzx gzy gzz   =           ∂2V ∂x2 ∂2V ∂x∂y ∂2V ∂x∂z ∂2V ∂y∂x ∂2V ∂y2 ∂2V ∂y∂z ∂2V ∂z∂x ∂2V ∂z∂y ∂2V ∂z2           Volume integrals
  7. Gravity Gradient Tensor Hessian matrix of gravitational potential GGT =

      gxx gxy gxz gyx gyy gyz gzx gzy gzz   =           ∂2V ∂x2 ∂2V ∂x∂y ∂2V ∂x∂z ∂2V ∂y∂x ∂2V ∂y2 ∂2V ∂y∂z ∂2V ∂z∂x ∂2V ∂z∂y ∂2V ∂z2           Volume integrals gij(x, y, z) = ˆ Ω Kernel(x, y, z, x , y , z ) dΩ
  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 X Y r φ λ

    Tesseroid
  13. What is a tesseroid? Delimited by: 2 meridians Z X

    Y r φ λ 1 λ
  14. What is a tesseroid? Delimited by: 2 meridians Z X

    Y r φ λ 2 λ
  15. What is a tesseroid? Delimited by: 2 meridians 2 parallels

    Z X Y r φ λ 1 φ
  16. What is a tesseroid? Delimited by: 2 meridians 2 parallels

    Z X Y r φ λ φ 2
  17. What is a tesseroid? Delimited by: 2 meridians 2 parallels

    2 concentric spheres Z X Y r φ λ 1 r
  18. What is a tesseroid? Delimited by: 2 meridians 2 parallels

    2 concentric spheres Z X Y r φ λ 2 r
  19. Why use tesseroids?

  20. Why use tesseroids? Earth Matle Core Crust

  21. Why use tesseroids? Earth Matle Core Crust

  22. Why use tesseroids? Want to model the geologic body Observation

    Point Geologic body
  23. Why use tesseroids? Observation Point Flat Earth

  24. Why use tesseroids? Observation Point Flat Earth + Rectangular Prisms

  25. Why use tesseroids? Good for small regions (Rule of thumb:

    < 2500 km) Observation Point Flat Earth + Rectangular Prisms
  26. Why use tesseroids? Good for small regions (Rule of thumb:

    < 2500 km) and close observation point Observation Point Flat Earth + Rectangular Prisms
  27. Why use tesseroids? Good for small regions (Rule of thumb:

    < 2500 km) and close observation point Not very accurate for larger regions Observation Point Flat Earth + Rectangular Prisms
  28. Why use tesseroids? Observation Point Spherical Earth

  29. Why use tesseroids? Spherical Earth + Rectangular Prisms Observation Point

  30. Why use tesseroids? Observation Point Spherical Earth + Rectangular Prisms

  31. Why use tesseroids? Usually accurate enough (if mass of prisms

    = mass of tesseroids) Observation Point Spherical Earth + Rectangular Prisms
  32. Why use tesseroids? Usually accurate enough (if mass of prisms

    = mass of tesseroids) Involves many coordinate changes Observation Point Spherical Earth + Rectangular Prisms
  33. Why use tesseroids? Usually accurate enough (if mass of prisms

    = mass of tesseroids) Involves many coordinate changes Computationally slow Observation Point Spherical Earth + Rectangular Prisms
  34. Why use tesseroids? Observation Point Spherical Earth

  35. Why use tesseroids? Observation Point Spherical Earth + Tesseroids

  36. Why use tesseroids? Observation Point Spherical Earth + Tesseroids

  37. Why use tesseroids? As accurate as Spherical Earth + rectangular

    prisms Observation Point Spherical Earth + Tesseroids
  38. Why use tesseroids? As accurate as Spherical Earth + rectangular

    prisms But faster Observation Point Spherical Earth + Tesseroids
  39. Why use tesseroids? As accurate as Spherical Earth + rectangular

    prisms But faster As shown in Wild-Pfeiffer (2008) Observation Point Spherical Earth + Tesseroids
  40. Why use tesseroids? As accurate as Spherical Earth + rectangular

    prisms But faster As shown in Wild-Pfeiffer (2008) Some numerical problems Observation Point Spherical Earth + Tesseroids
  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.