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, Brazil2. Universidade de São Paulo, São Paulo, Brazil
3. University of Trieste, Trieste, Italy.
August 9, 2010
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
Gravity Gradient Tensor
Gravity Gradient Tensor
I Hessian matrix of gravitational potential
GGT =
gxx gxy gxzgyx gyy gyzgzx 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
I Volume integrals
gij(x , y , z) =
ˆΩ
Kernel(x , y , z, x ′, y ′, z ′) dΩ
Gravity Gradient Tensor
I Hessian matrix of gravitational potential
GGT =
gxx gxy gxzgyx gyy gyzgzx 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
I Volume integrals
gij(x , y , z) =
ˆΩ
Kernel(x , y , z, x ′, y ′, z ′) dΩ
Gravity Gradient Tensor
I Hessian matrix of gravitational potential
GGT =
gxx gxy gxzgyx gyy gyzgzx 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
I Volume integrals
gij(x , y , z) =
ˆΩ
Kernel(x , y , z, x ′, y ′, z ′) dΩ
Gravity Gradient Tensor
I Hessian matrix of gravitational potential
GGT =
gxx gxy gxzgyx gyy gyzgzx 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
I Volume integrals
gij(x , y , z) =
ˆΩ
Kernel(x , y , z, x ′, y ′, z ′) dΩ
Gravity Gradient Tensor
I Can discretize volume Ω using:
I Rectangular prisms
I Tesseroids (spherical prisms)
Gravity Gradient Tensor
I Can discretize volume Ω using:
I Rectangular prisms
I Tesseroids (spherical prisms)
Gravity Gradient Tensor
I Can discretize volume Ω using:
I Rectangular prisms
I Tesseroids (spherical prisms)
What is a tesseroid?
What is a tesseroid?
Z
XY
r
φ
λ
Tesseroid
What is a tesseroid?
Delimited by:
I 2 meridians
I 2 parallels
I 2 concentricspheres
Z
XY
r
φ
λ
1λ
What is a tesseroid?
Delimited by:
I 2 meridians
I 2 parallels
I 2 concentricspheres
Z
XY
r
φ
λ
2λ
What is a tesseroid?
Delimited by:
I 2 meridians
I 2 parallels
I 2 concentricspheres
Z
XY
r
φ
λ
1φ
What is a tesseroid?
Delimited by:
I 2 meridians
I 2 parallels
I 2 concentricspheres
Z
XY
r
φ
λ
φ2
What is a tesseroid?
Delimited by:
I 2 meridians
I 2 parallels
I 2 concentricspheres
Z
XY
r
φ
λ
1r
What is a tesseroid?
Delimited by:
I 2 meridians
I 2 parallels
I 2 concentricspheres
Z
XY
r
φ
λ
2r
Why use tesseroids?
Why use tesseroids?
Earth
Matle
Core
Crust
Why use tesseroids?
Earth
Matle
Core
Crust
Why use tesseroids?
Want to model the geologic body
ObservationPoint
Geologic body
Why use tesseroids?
I Good for smallregions(Rule of thumb: <2500 km)
I and closeobservation point
I Not very accuratefor larger regions
ObservationPoint
Flat Earth
Why use tesseroids?
I Good for smallregions(Rule of thumb: <2500 km)
I and closeobservation point
I Not very accuratefor larger regions
ObservationPoint
Flat Earth+ Rectangular Prisms
Why use tesseroids?
I Good for smallregions(Rule of thumb: <2500 km)
I and closeobservation point
I Not very accuratefor larger regions
ObservationPoint
Flat Earth+ Rectangular Prisms
Why use tesseroids?
I Good for smallregions(Rule of thumb: <2500 km)
I and closeobservation point
I Not very accuratefor larger regions
ObservationPoint
Flat Earth+ Rectangular Prisms
Why use tesseroids?
I Good for smallregions(Rule of thumb: <2500 km)
I and closeobservation point
I Not very accuratefor larger regions
ObservationPoint
Flat Earth+ Rectangular Prisms
Why use tesseroids?
I Usually accurateenough (if mass ofprisms = mass oftesseroids)
I Involves manycoordinatechanges
I Computationallyslow
ObservationPoint
Spherical Earth
Why use tesseroids?
I Usually accurateenough (if mass ofprisms = mass oftesseroids)
I Involves manycoordinatechanges
I Computationallyslow
Spherical Earth+ Rectangular Prisms
ObservationPoint
Why use tesseroids?
I Usually accurateenough (if mass ofprisms = mass oftesseroids)
I Involves manycoordinatechanges
I Computationallyslow
ObservationPoint
Spherical Earth+ Rectangular Prisms
Why use tesseroids?
I Usually accurateenough (if mass ofprisms = mass oftesseroids)
I Involves manycoordinatechanges
I Computationallyslow
ObservationPoint
Spherical Earth+ Rectangular Prisms
Why use tesseroids?
I Usually accurateenough (if mass ofprisms = mass oftesseroids)
I Involves manycoordinatechanges
I Computationallyslow
ObservationPoint
Spherical Earth+ Rectangular Prisms
Why use tesseroids?
I Usually accurateenough (if mass ofprisms = mass oftesseroids)
I Involves manycoordinatechanges
I Computationallyslow
ObservationPoint
Spherical Earth+ Rectangular Prisms
Why use tesseroids?
I As accurate asSpherical Earth +rectangular prisms
I But faster
I As shown inWild-Pfeiffer(2008)
I Some numericalproblems
ObservationPoint
Spherical Earth
Why use tesseroids?
I As accurate asSpherical Earth +rectangular prisms
I But faster
I As shown inWild-Pfeiffer(2008)
I Some numericalproblems
ObservationPoint
Spherical Earth + Tesseroids
Why use tesseroids?
I As accurate asSpherical Earth +rectangular prisms
I But faster
I As shown inWild-Pfeiffer(2008)
I Some numericalproblems
ObservationPoint
Spherical Earth + Tesseroids
Why use tesseroids?
I As accurate asSpherical Earth +rectangular prisms
I But faster
I As shown inWild-Pfeiffer(2008)
I Some numericalproblems
ObservationPoint
Spherical Earth + Tesseroids
Why use tesseroids?
I As accurate asSpherical Earth +rectangular prisms
I But faster
I As shown inWild-Pfeiffer(2008)
I Some numericalproblems
ObservationPoint
Spherical Earth + Tesseroids
Why use tesseroids?
I As accurate asSpherical Earth +rectangular prisms
I But faster
I As shown inWild-Pfeiffer(2008)
I Some numericalproblems
ObservationPoint
Spherical Earth + Tesseroids
Why use tesseroids?
I As accurate asSpherical Earth +rectangular prisms
I But faster
I As shown inWild-Pfeiffer(2008)
I Some numericalproblems
ObservationPoint
Spherical Earth + Tesseroids
Numerical issues
Numerical issues
I Gravity Gradient Tensor (GGT) volumeintegrals solved:
I Analytically in the radial direction
I Numerically over the surface of thesphere
I Using the Gauss-LegendreQuadrature (GLQ)
Numerical issues
I Gravity Gradient Tensor (GGT) volumeintegrals solved:
I Analytically in the radial direction
I Numerically over the surface of thesphere
I Using the Gauss-LegendreQuadrature (GLQ)
Numerical issues
I Gravity Gradient Tensor (GGT) volumeintegrals solved:
I Analytically in the radial direction
I Numerically over the surface of thesphere
I Using the Gauss-LegendreQuadrature (GLQ)
Numerical issues
I Gravity Gradient Tensor (GGT) volumeintegrals solved:
I Analytically in the radial direction
I Numerically over the surface of thesphere
I Using the Gauss-LegendreQuadrature (GLQ)
Numerical issues
At 250 km height with Gauss-Legendre Quadrature(GLQ) order 2
Numerical issues
At 50 km height with Gauss-Legendre Quadrature(GLQ) order 2
Numerical issues
At 50 km height with Gauss-Legendre Quadrature(GLQ) order 10
Numerical issues
I General rule:
I Distance to computation point > Distancebetween nodes
I Increase number of nodes
I Divide the tesseroid in smaller parts
Numerical issues
I General rule:
I Distance to computation point > Distancebetween nodes
I Increase number of nodes
I Divide the tesseroid in smaller parts
Numerical issues
I General rule:
I Distance to computation point > Distancebetween nodes
I Increase number of nodes
I Divide the tesseroid in smaller parts
Numerical issues
I General rule:
I Distance to computation point > Distancebetween nodes
I Increase number of nodes
I Divide the tesseroid in smaller parts
Modeling topographywith tesseroids
Modeling topography with tesseroids
Computer program: Tesseroids
I Python programming language
I Open Source (GNU GPL License)
I Project hosted on Google Code
I http://code.google.com/p/tesseroids
I Under development:
I Optimizations using C coded modules
Modeling topography with tesseroids
Computer program: Tesseroids
I Python programming language
I Open Source (GNU GPL License)
I Project hosted on Google Code
I http://code.google.com/p/tesseroids
I Under development:
I Optimizations using C coded modules
Modeling topography with tesseroids
Computer program: Tesseroids
I Python programming language
I Open Source (GNU GPL License)
I Project hosted on Google Code
I http://code.google.com/p/tesseroids
I Under development:
I Optimizations using C coded modules
Modeling topography with tesseroids
Computer program: Tesseroids
I Python programming language
I Open Source (GNU GPL License)
I Project hosted on Google Code
I http://code.google.com/p/tesseroids
I Under development:
I Optimizations using C coded modules
Modeling topography with tesseroids
Computer program: Tesseroids
I Python programming language
I Open Source (GNU GPL License)
I Project hosted on Google CodeI http://code.google.com/p/tesseroids
I Under development:
I Optimizations using C coded modules
Modeling topography with tesseroids
Computer program: Tesseroids
I Python programming language
I Open Source (GNU GPL License)
I Project hosted on Google CodeI http://code.google.com/p/tesseroids
I Under development:
I Optimizations using C coded modules
Modeling topography with tesseroids
Computer program: Tesseroids
I Python programming language
I Open Source (GNU GPL License)
I Project hosted on Google CodeI http://code.google.com/p/tesseroids
I Under development:I Optimizations using C coded modules
Modeling topography with tesseroids
To model topography:
I Digital Elevation Model (DEM)⇒ Tesseroidmodel
I 1 Grid Point = 1 Tesseroid
I Top centered on grid point
I Bottom at reference surface
Modeling topography with tesseroids
To model topography:
I Digital Elevation Model (DEM)⇒ Tesseroidmodel
I 1 Grid Point = 1 Tesseroid
I Top centered on grid point
I Bottom at reference surface
Modeling topography with tesseroids
To model topography:
I Digital Elevation Model (DEM)⇒ Tesseroidmodel
I 1 Grid Point = 1 Tesseroid
I Top centered on grid point
I Bottom at reference surface
Modeling topography with tesseroids
To model topography:
I Digital Elevation Model (DEM)⇒ Tesseroidmodel
I 1 Grid Point = 1 Tesseroid
I Top centered on grid point
I Bottom at reference surface
Modeling topography with tesseroids
To model topography:
I Digital Elevation Model (DEM)⇒ Tesseroidmodel
I 1 Grid Point = 1 Tesseroid
I Top centered on grid point
I Bottom at reference surface
Topographic effect in theParaná Basin region
Topographic effect in the Paraná Basin region
Digital Elevation Model (DEM) Grid:
I ETOPO1
I 10’ x 10’ Grid
I ~ 23,000 Tesseroids
I Density = 2.67 g × cm−3
I Computation height = 250 km
Topographic effect in the Paraná Basin region
Digital Elevation Model (DEM) Grid:
I ETOPO1
I 10’ x 10’ Grid
I ~ 23,000 Tesseroids
I Density = 2.67 g × cm−3
I Computation height = 250 km
Topographic effect in the Paraná Basin region
Digital Elevation Model (DEM) Grid:
I ETOPO1
I 10’ x 10’ Grid
I ~ 23,000 Tesseroids
I Density = 2.67 g × cm−3
I Computation height = 250 km
Topographic effect in the Paraná Basin region
Digital Elevation Model (DEM) Grid:
I ETOPO1
I 10’ x 10’ Grid
I ~ 23,000 Tesseroids
I Density = 2.67 g × cm−3
I Computation height = 250 km
Topographic effect in the Paraná Basin region
Digital Elevation Model (DEM) Grid:
I ETOPO1
I 10’ x 10’ Grid
I ~ 23,000 Tesseroids
I Density = 2.67 g × cm−3
I Computation height = 250 km
Topographic effect in the Paraná Basin region
Digital Elevation Model (DEM) Grid:
I ETOPO1
I 10’ x 10’ Grid
I ~ 23,000 Tesseroids
I Density = 2.67 g × cm−3
I Computation height = 250 km
Topographic effect in the Paraná Basin region
Topographic effect in the Paraná Basin region
Height of 250 km
Topographic effect in the Paraná Basin region
I Topographic effect in the region has the
same order of magnitude as a2 × 2 × 10 km tesseroid (100 Eötvös)
I Need to take topography into account when
modeling (even at 250 km altitudes)
Topographic effect in the Paraná Basin region
I Topographic effect in the region has the
same order of magnitude as a2 × 2 × 10 km tesseroid (100 Eötvös)
I Need to take topography into account when
modeling (even at 250 km altitudes)
Further applications
Further applications
I Satellite gravity data = global coverage
I + Tesseroid modeling:
I Regional/global inversion for density(Mantle)
I Regional/global inversion for relief of aninterface (Moho)
I Joint inversion with seismic tomography
Further applications
I Satellite gravity data = global coverage
I + Tesseroid modeling:
I Regional/global inversion for density(Mantle)
I Regional/global inversion for relief of aninterface (Moho)
I Joint inversion with seismic tomography
Further applications
I Satellite gravity data = global coverage
I + Tesseroid modeling:
I Regional/global inversion for density(Mantle)
I Regional/global inversion for relief of aninterface (Moho)
I Joint inversion with seismic tomography
Further applications
I Satellite gravity data = global coverage
I + Tesseroid modeling:
I Regional/global inversion for density(Mantle)
I Regional/global inversion for relief of aninterface (Moho)
I Joint inversion with seismic tomography
Further applications
I Satellite gravity data = global coverage
I + Tesseroid modeling:
I Regional/global inversion for density(Mantle)
I Regional/global inversion for relief of aninterface (Moho)
I Joint inversion with seismic tomography
Concluding remarks
Concluding remarks
I Developed a computational tool forlarge-scale gravity modeling with tesseroids
I Better use tesseroids than rectangularprisms for large regions
I Take topographic effect into considerationwhen modeling density anomalies within theEarth
I Possible application: tesseroids inregional/global gravity inversion
Concluding remarks
I Developed a computational tool forlarge-scale gravity modeling with tesseroids
I Better use tesseroids than rectangularprisms for large regions
I Take topographic effect into considerationwhen modeling density anomalies within theEarth
I Possible application: tesseroids inregional/global gravity inversion
Concluding remarks
I Developed a computational tool forlarge-scale gravity modeling with tesseroids
I Better use tesseroids than rectangularprisms for large regions
I Take topographic effect into considerationwhen modeling density anomalies within theEarth
I Possible application: tesseroids inregional/global gravity inversion
Concluding remarks
I Developed a computational tool forlarge-scale gravity modeling with tesseroids
I Better use tesseroids than rectangularprisms for large regions
I Take topographic effect into considerationwhen modeling density anomalies within theEarth
I Possible application: tesseroids inregional/global gravity inversion
Thank you
References
I WILD-PFEIFFER, F. A comparison of different masselements for use in gravity gradiometry. Journal ofGeodesy, v. 82 (10), p. 637 - 653, 2008.