iMAGINE Seminar:
Satellite gravity gradients
05.06.2014
Time Topic
09:00-09:20 Welcome Haagmans – Bouman - Ebbing
09:20-10:10 Gravity and gravity gradients – from Gz to a tensor Ebbing
10:10-10.30 Coffee break
10:30-11:20 Satellite gravity data and global gravity models– an Bouman
overview
11:15-12.10 Geological mapping and 3D modelling using satellite Ebbing
gradients
12:10-13:00 Lunch
13:00-13.45 Global crustal models and implications for heat-flow Ebbing
modelling
13.45-14.10 ESA gravity gradients data sets – How to access? Bouman - Haagmans
14:10-14.30 Coffee break
14:30-15:00 Feedback and discussion
2 Satellite seminar, Tromsø 05.06.2014
STSE projects
http://goce4interior.dgfi.badw.de/
(DGFI - Munich, TNO - Utrecht, NGU -
Trondheim, CAU - Kiel)
‣Formula will likely be continued (3D Earth project (TBC))
ESA GOCE & Swarm | Roger Haagmans | Tromso, Norway | 03/06/2014 | EOP | Slide 4
ESA UNCLASSIFIED – For Official Use
GOCE+ Height Unification
GOCE | Christoph Steiger, Massimo Romanazzo, Björn Frommknecht, Roger Haagmans | San Francisco | 13/12/2013 | EOP | Slide 5
ESA UNCLASSIFIED – For Official Use
GOCE+ time variations
GOCE | Christoph Steiger, Massimo Romanazzo, Björn Frommknecht, Roger Haagmans | San Francisco | 13/12/2013 | EOP | Slide 6
ESA UNCLASSIFIED – For Official Use
GOCE+ projects --> new products for
the community
GOCE | Christoph Steiger, Massimo Romanazzo, Björn Frommknecht, Roger Haagmans | San Francisco | 13/12/2013 | EOP | Slide 7
ESA UNCLASSIFIED – For Official Use
GOCE+ air density response to Earthquake
GOCE | Christoph Steiger, Massimo Romanazzo, Björn Frommknecht, Roger Haagmans | San Francisco | 13/12/2013 | EOP | Slide 8
ESA UNCLASSIFIED – For Official Use
GOCE+ Geophysics
Team 1: DGFI (DE), NGU (NO), TNO (NL)
NE Atlantic margin
Saudi Arabia
Team 2: UWB (CZ), AAS (AT), AUT (GR), DIAS (IR), GIS (DE), TUD (NL)
Mid-Atlantic ridge
Africa
Full gravity gradient products (GRACE/GOCE) 4th QRT 2014 for users
GOCE | Christoph Steiger, Massimo Romanazzo, Björn Frommknecht, Roger Haagmans | San Francisco | 13/12/2013 | EOP | Slide 9
ESA UNCLASSIFIED – For Official Use
GOCE -Gravity field and steady-
state Ocean Circulation Explorer
Mission period
17 March 2009 – 11 November 2013
Gradiometer; 3 pairs of 3-axis, servo-
controlled, capacitive accelerometers
(each pair separated by a distance of
about 0.5 m).
Launcher: Rockot (converted SS-
19), from Plesetsk, Russia.
Gravity potential, gravity or gravity gradient
Topography [m] in the Gravity gradients
Indonesian Archipelago
Signal attenuates at
satellite height
Q
rQ
GOCE
ψ0
R
CHAMP
GRACE Satellite orbit
Observed Gravity data Gravity potential (shapes the Geoid)
(Haagmans, Prijatna, Omang, 2003)
ESA GOCE & Swarm | Roger Haagmans | Tromso, Norway | 03/06/2014 | EOP | Slide 11
ESA UNCLASSIFIED – For Official Use
Regional modelling and data reduction ?
What is the influence of ignoring
the rest of the globe on gravity
(scalar) and gravity gradients
(“local 3D” view)?
What is this influence near the
surface and at satellite altitude?
Can it be useful to use gravity
near the surface and gravity
gradients at satellite level as
complementary information for
regional modelling?
Application of satellite gradients
How can we model satellite data?
Is a topographic reduction necessary?
Is there any additional benefit using (satellite) gradients? All
signal is theoretically in the potential?
At which height should I use the data?
Downward continued to surface? At satellite height?
Do we need to consider a spherical Earth, if using satellites?
Do satellite gradients have a sensitivity beyond global models or
terrestrial data?
13 Satellite seminar, Tromsø 05.06.2014
Time Topic
09:00-09:20 Welcome Haagmans – Bouman - Ebbing
09:20-10:10 Gravity and gravity gradients – from Gz to a Ebbing
tensor
10:10-10.30 Coffee break
10:30-11:20 Satellite gravity data and global gravity models– an Bouman
overview
11:15-12.10 Geological mapping and 3D modelling using satellite Ebbing
gradients
12:10-13:00 Lunch
13:00-13.45 Global crustal models and implications for heat-flow Ebbing
modelling
13.45-14.10 ESA gravity gradients data sets – How to access? Bouman - Haagmans
14:10-14.30 Coffee break
14:30-15:00 Feedback and discussion
14 Satellite seminar, Tromsø 05.06.2014
FTG: WIDE BANDWIDTH & IMPROVED SNR MEASUREMENT
Gradiometry measurements allow high resolution from dynamic
GGI : HIGH RESOLUTION DATA FROM A DYNAMIC ENVIRONMENT
environment
ARKeX Theoretical World: Proof mass mounted on a spring
Conventional Gravity Perfect measure of the Earth’s gravity
Integrated
Gravity Interpretation
Real World: Cannot detect difference between plane’s motion and gravity signal
Imaging
Seismic
Conventional Gravity GPS cannot correct high frequency signal, filtered profile
BlueQube
Gravity Better
Gradiometry Solution Real World:
Two proof masses, measures difference in acceleration
Gravity Gradiometry “between” two masses
Cancels out effect of motion of aircraft/boat
Satellite seminar, Tromsø
Gravity: Force between two masses (Newton)
m1 ⋅ m2
r1 − r2 m1
F (r1 ) = −γ 2 ⋅ F
r1 − r2 r1 − r2
r1
r
Magnitude Direction m2
r2
γ = 6.67 ⋅10 −11 Nm
2
kg 2
F (r1 ) = m 1 ⋅ g (r1 )
g x
m2 r1 − r2
g (r1 ) = −γ 2 ⋅ = g y
r1 − r2 r1 − r2
gz
Gravity is a vector quantity with components in the x, y, z directions (gx, gy, gz)
Gravity → density → geology (Geophysics)
r − r0 g (r )
g (r ) = −γ ∫∫∫ ρ (r0 ) 3 dV0
V0
r r − r0
V0 r − r0
r0
ρ (r0 )
g (r ) ⇒ ρ (r0 )
gravity density
m
Unit : 1 mGal = 10 −5
s2 gz
Gravity tells us information about geology.
Scalar potential, vector acceleration, tensor gradient
scalar field vector field tensor field
potential acceleration gradient
∂U
= gx
Gxx Gxy Gxz
U(x,y,z) ∂x
∂U
G
yx G yy G yz
= gy
∂y Gzx Gzy Gzz
∂U
= gz
∂z
Gxx
spatial spatial
derivative derivative Gxy
gx
X X X Gxz Gyx
Y Y Y Gyy
gy
Z Z Z Gyz
U
Gzx
gz=g Gzy
Gzz
People also use U or T or G to denote the gravity and gradient components.
Coordinate systems might be (X, Y, Z) or (N, E, D) as shown in subscripts.
Gravity is derivative of potential
Gravitational field isconservative (work depends on end points, not on path taken).
Vector gravity field g ( x, y, z ) may be derived from scalar potential U ( x, y , z ).
∇U(x,y,z) = g(x,y,z)
∂U
= gx Derivative of potential
∂x x y z
∂U
∂y
= gy (g , g , g )
x y z
∂U
= gz
∂z
X
Z
Gravity is derivative of potential
Gravitational field isconservative (work depends on end points, not on path taken).
Vector gravity field g ( x, y, z ) may be derived from scalar potential U ( x, y , z ).
∇U(x,y,z) = g(x,y,z)
∂U
= gx Derivative of potential
∂x x y z
∂U
∂y
= gy (g , g , g )
x y z
∂U
= gz
∂z
X
Z
and gravity gradient is derivative of gravity.
Gradient: change in gravity components along three axial directions.
Cube Response
∂ ∂ 2
Gxx = gx = U
∂x ∂x∂x
∂ ∂2
Gxy = gx = U
∂y ∂x∂y … and so on
Gxx Gxy Gxz
Derivative
x y z
x Gxx Gxy Gxz
Component
Gyx Gyy Gyz
Gyx Gyy Gyz
y
Gzx Gzy Gzz X
z
Y
Gzx Gzy Gzz
Z
Symmetry
The order of differentiation doesn’t Cube Response
matter:
∂ 2U ∂ 2U
Gxy = = = G yx
∂y∂x ∂x∂y Gxx Gxy Gxz
Gxz = Gzx
G yz = Gzy
The gravity gradient tensor is Gyx Gyy Gyz
SYMMETRIC.
X
Y
Gzx Gzy Gzz
Z
Laplace’s Equation Zero Trace
Laplace’s Equation applies in Cube Response
source-free regions:
∂ 2
U ∂ 2
U ∂ 2
U
∇U=
2
+ + 2 =0
∂x 2
∂y 2
∂z Gxx Gxy Gxz
Gxx + G yy + Gzz = 0
The trace of the gravity gradient tensor is Gyy Gyz
zero
Only 5 independent components of the
gravity gradient tensor. X
Y
Gzz
Z
Gravity gradients
Example 1: Gzy and Gzx
Cube Response
( g z ( y2 ) − g z ( y1 )) ∂g z
Gzy = y2 − y1 =
∂y
( g z ( x2 ) − g z ( x1 )) ∂g z
Gzx = x2 − x1 =
∂x
Gzy
Gzx
X
∂g z
gz Y
∂y
Z
Example 2: Gyy and Gxx
( g y ( y2 )− g y ( y1 ))
Gyy = y2 − y1
Gyy
Y
Gxx
Z
Example 3: Gzz
( g z ( z2 )− g z ( z1 ))
Gzz = z2 − z1
G2 is closer to cube and measures stronger gravity than G1
G1 G2 Gzz =
( G2 − G1 )
∆z
∆z
G2
G1
Z
Units of Gravity and Gravity Gradiometry
Survey Type Measurement Fundamental Unit Unit
Gravity Survey Acceleration Meters/Second/Second mGal
Gravity Gradiometry Change in Meters/Second/Second/Meter Eötvös
Survey Acceleration
Gravity
gz
1 mgal = 10-5 m/s2
Gravity Gradiometry
Gzz, Gzx, Gxx, Gzy, Gyy, Gxy
1E (Eötvös) = 0.1 mGal / km = 0.1 microGal / m
Some considerations: Coordinate systems
(Des Fitzgerald, 2011)
• GOCE
- NWU (north, west, up)
29 Satellite seminar, Tromsø 05.06.2014
Gradient data and rotational invariants
• Gradients are dependent on the orientation of the coordinate
system, which may differ from the orientation of random geological
features
• Conventionally, the vertical and horizontal gradient are used for
interpretation:
• Invariants have the advantage to be independent from the
coordinate system and help to delineate the outline of density
contrasts.
• Pedersen and Rasmussen (1990) demonstrated the use of
rotational invariants of the gravity tensor:
• These rotational invariants are independent from the orientation of
flight lines and facilitate to detect sources randomly orientated in
any coordinate system
30 Satellite seminar, Tromsø 05.06.2014
Gradient data and rotational invariants
• Gradients are dependent on the orientation of the coordinate
system, which may differ from the orientation of random geological
features
• Conventionally, the vertical and horizontal gradient are used for
interpretation:
• Invariants have the advantage to be independent from the
coordinate system and help to delineate the outline of density
contrasts.
• Pedersen and Rasmussen (1990) demonstrated the use of
rotational invariants of the gravity tensor:
• These rotational invariants are independent from the orientation of
the flight lines and facilitate to detect sources randomly orientated
in any coordinate system
31 Satellite seminar, Tromsø 05.06.2014
Gradient data and rotational invariants
• Gradients are dependent on the orientation of the coordinate
system, which may differ from the orientation of random geological
features
• Conventionally, the vertical and horizontal gradient are used for
interpretation:
• Invariants have the advantage to be independent from the
coordinate system and help to delineate the outline of density
contrasts.
• Pedersen and Rasmussen (1990) demonstrated the use of
rotational invariants of the gravity tensor:
• These rotational invariants are independent from the orientation of
the flight lines and facilitate to detect sources randomly orientated
in any coordinate system
Martin Panzner, NTNU
32 Satellite seminar, Tromsø 05.06.2014
Gradient data and rotational invariants
• Gradients are dependent on the orientation of the coordinate
system, which may differ from the orientation of random geological
features
• Conventionally, the vertical and horizontal gradient are used for
interpretation:
• Invariants have the advantage to be independent from the
coordinate system and help to delineate the outline of density
contrasts.
• Pedersen and Rasmussen (1990) demonstrated
I1 the use ofI2
rotational invariants of the gravity tensor:
• These rotational invariants are independent from the orientation of
the flight lines and facilitate to detect sources randomly orientated
in any coordinate system
33 Satellite seminar, Tromsø 05.06.2014
Martin Panzner, NTNU
Gradient data and rotational invariants
• Gradients are dependent on the orientation of the coordinate system, which may
differ from the orientation of random geological features
• Conventionally, the vertical and horizontal gradient are used for interpretation:
Horizontal gradientG
• Invariants have the advantage to be independent from the coordinate system and
help to delineate the outline of density contrasts.
• Pedersen and Rasmussen (1990) demonstrated the use of rotational invariants of
the gravity tensor:
• These rotational invariants are independent from the orientation of the flight lines
and facilitate to detect sources randomly orientated in any coordinate system
Martin Panzner, NTNU
34 Satellite seminar, Tromsø 05.06.2014
a) Gzz b) Horizontal gradient IGxzGyz
c) I1 d) I2
35 Satellite seminar, Tromsø Martin Panzner, NTNU 05.06.2014
Gradient data and rotational invariants
• Pedersen and Rasmussen (1990) showed that the invariant ratio
lies between zero and unity for any potential field. When the causative body as
seen from the observation point looks more and more 3D like, then I increases
and eventually approaches unity.
• For a strict 2D case, I is equal to zero for all measurement points
36 Satellite seminar, Tromsø 05.06.2014
Example: Vredefort
Beiki 2011
Example Vredefort
Beiki 2011
Gzz Dimensionality I
Beiki 2011
http://www.gradiometry.com/
Advantage of using gradients
Gradients are less sensitive to regional fields/gradients
Higher sensitivity to near-surface structures
Gradients are useful to calculate invariants from Cartesian models
and to compare to global gravity gradients
Rotational invariant
No tensor rotation necessary
=> What about satellite gravity gradients?
41 Satellite seminar, Tromsø 05.06.2014
Flying
height
AGG surveys:
35-300m above
terrain
height
Flying
height
AGG surveys:
35-300m above
terrain
GOCE satellite:
224-270 km
above
height
Flat Earth
vs.
Spherical
calculations
44
Flat Earth
vs.
Spherical
calculations
45
Flat Earth
vs.
Spherical
calculations
46
Flat Earth vs. Spherical calculations
Uieda 2011
47
Flat Earth vs. Spherical calculations
Marussi Tensor
The Marussi tensor expresses the
gradients in relation to the radial distance
48 r, the latitude φ and longitude λ.
Application of satellite gradients
How can we model satellite data?
Is a topographic reduction necessary?
Is there any additional benefit using (satellite) gradients? All
signal is theoretically in the potential?
At which height should I use the data?
Downward continued to surface? At satellite height?
Do we need to consider a spherical Earth, if using satellites?
Do satellite gradients have a sensitivity beyond global models or
terrestrial data?
49 Satellite seminar, Tromsø 05.06.2014
Topographic and Bouguer correction
Complete Bouguer correction is defined as
1) Gravity effect of Bouguer slab
2) spherical correction
3) Terrain correction
Topographic and Bouguer correction
Complete Bouguer correction is defined as
1) Gravity effect of Bouguer slab
2) spherical correction
3) Terrain correction
h describes height above reference level or water depth and is different for each station.
Topographic and Bouguer correction
Complete Bouguer correction is defined as
1) Gravity effect of Bouguer slab
2) spherical correction
3) Terrain correction
Topographic and Bouguer correction
Complete Bouguer correction is defined as
1) Gravity effect of Bouguer slab
2) spherical correction
3) Terrain correction
Topographic and Bouguer correction
Bouguer calculation is always possible if only station height is known,
can be quickly calculated.
Terrain correction requires high-resolution topography for surface data
(25 or 50 m for local sources which have highest effect)
Topographic and Bouguer correction
BUT: NOT WORKING WELL FOR GRADIENTS
(Slab of constant thickness has not effect on gradients, terrain important)
=>TOPOGRAPHIC MASS REDUCTION (Computational demanding)
Topographic and Bouguer correction
For satellites: point of obervation above topography (e.g. 250 km)
=> no high-resolution topography needed
Topographic and Bouguer correction
TOPOGRAPHIC MASS REDUCTION (for gravity and gradients) possible
Gravity gradients due to
topography at satellite
height
Uieda 2011
GOCE data in 225 km height
59 Satellite seminar, Tromsø 05.06.2014
Topographic effect in 225 km height
60 Satellite seminar, Tromsø 05.06.2014
Topographic reduced GOCE data
61 Satellite seminar, Tromsø 05.06.2014
Application of satellite gradients
How can we model satellite data?
Is a topographic reduction necessary?
Is there any additional benefit using (satellite) gradients? All
signal is theoretically in the potential?
At which height should I use the data?
Downward continued to surface? At satellite height?
Do we need to consider a spherical Earth, if using satellites?
Do satellite gradients have a sensitivity beyond global models or
terrestrial data?
62 Satellite seminar, Tromsø 05.06.2014
Time Topic
09:00-09:20 Welcome Haagmans – Bouman - Ebbing
09:20-10:10 Gravity and gravity gradients – from Gz to a tensor Ebbing
10:10-10.30 Coffee break
10:30-11:20 Satellite gravity data and global gravity models– Bouman
an overview
11:15-12.10 Geological mapping and 3D modelling using satellite Ebbing
gradients
12:10-13:00 Lunch
13:00-13.45 Global crustal models and implications for heat-flow Ebbing
modelling
13.45-14.10 ESA gravity gradients data sets – How to access? Bouman - Haagmans
14:10-14.30 Coffee break
14:30-15:00 Feedback and discussion
63 Satellite seminar, Tromsø 05.06.2014
