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.