Computation of the gravity gradient tensor due to topographic masses using tesseroids
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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 2 V x 2 g xx g xy g xz GGT = g yx g yy g yz = 2 V g zx g zy g zz y x 2 V z x 2 V x y 2 V y 2 2 V z y 2 V x z 2 V y z 2 V z 2
6 Gravity Gradient Tensor Hessian matrix of gravitational potential 2 V x 2 g xx g xy g xz GGT = g yx g yy g yz = 2 V g zx g zy g zz y x 2 V z x Volume integrals 2 V x y 2 V y 2 2 V z y 2 V x z 2 V y z 2 V z 2
7 Gravity Gradient Tensor Hessian matrix of gravitational potential 2 V x 2 g xx g xy g xz GGT = g yx g yy g yz = 2 V g zx g zy g zz y x 2 V z x Volume integrals 2 V x y 2 V y 2 2 V z y ˆ g ij (x, y, z) = Kernel(x, y, z, x, y, z ) dω Ω 2 V x z 2 V y z 2 V z 2
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 X λ φ Y
13 What is a tesseroid? Delimited by: Z 2 meridians r X λ 1 λ φ Y
14 What is a tesseroid? Delimited by: Z 2 meridians r X λ 2 λ φ Y
15 What is a tesseroid? Delimited by: Z 2 meridians 2 parallels φ 1 r X λ φ Y
16 What is a tesseroid? Delimited by: Z 2 meridians 2 parallels φ 2 r X λ φ Y
17 What is a tesseroid? Delimited by: Z 2 meridians 2 parallels 2 concentric spheres r 1 r X λ φ Y
18 What is a tesseroid? Delimited by: Z 2 meridians 2 parallels 2 concentric spheres r 2 r X λ φ 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? Good for small regions (Rule of thumb: < 2500 km) Flat Earth + Rectangular Prisms Observation Point
26 Why use tesseroids? Good for small regions (Rule of thumb: < 2500 km) Flat Earth + Rectangular Prisms Observation Point and close observation point
27 Why use tesseroids? Good for small regions (Rule of thumb: < 2500 km) Flat Earth + Rectangular Prisms Observation 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? Usually accurate enough (if mass of prisms = mass of tesseroids) Spherical Earth + Rectangular Prisms Observation Point
32 Why use tesseroids? Usually accurate enough (if mass of prisms = mass of tesseroids) Involves many coordinate changes Spherical Earth + Rectangular Prisms Observation Point
33 Why use tesseroids? Usually accurate enough (if mass of prisms = mass of tesseroids) Involves many coordinate changes Spherical Earth + Rectangular Prisms Observation Point 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 + rectangular prisms Spherical Earth + Tesseroids Observation Point
38 Why use tesseroids? As accurate as Spherical Earth + rectangular prisms But faster Spherical Earth + Tesseroids Observation Point
39 Why use tesseroids? As accurate as Spherical Earth + rectangular prisms But faster Spherical Earth + Tesseroids Observation Point As shown in Wild-Pfeiffer (2008)
40 Why use tesseroids? As accurate as Spherical Earth + rectangular prisms But faster Spherical Earth + Tesseroids Observation 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
59 Modeling topography with tesseroids Computer program: Tesseroids Python programming language Open Source (GNU GPL License) Project hosted on Google Code Under development:
60 Modeling topography with tesseroids Computer program: Tesseroids Python programming language Open Source (GNU GPL License) Project hosted on Google Code 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 5 S 10 S 15 S 20 S 25 S 30 S 35 S 40 S 75 W 70 W 65 W 60 W 55 W 50 W 45 W 40 W 35 W Height [m]
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 km tesseroid (10 0 Eötvös)
76 Topographic effect in the Paraná Basin region Topographic effect in the region has the same order of magnitude as a km tesseroid (10 0 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 , 2008.
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