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.

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.