M. Weigelt, W. Keller
GRACE GRAVITY FIELD SOLUTIONS
USING THE DIFFERENTIAL
GRAVIMETRY APPROACH
Geophysical implications of the gravity field
Geodesy
Continental Hydrology
Oceanography
Solid
Earth
Geoid
Glaciology
Earth
core
Tides
Atmosphere
2
Geophysical implications of the gravity field
ESA
3
Observation system
GRACE = Gravity Recovery
and Climate Experiment
•
Initial orbit height: ~ 485 km
•
Inclination: ~ 89°
•
Key technologies:
– GPS receiver
– Accelerometer
– K-Band Ranging System
•
Observed quantity:
– range
– range rate
CSR, UTexas
4
Gravity field modelling
with
gravitational constant times mass of the Earth
radius of the Earth
spherical coordinates of the calculation point
Legendre function
degree, order
(unknown) spherical harmonic coefficients
5
Spectral representations
Degree RMS
Spherical
harmonic
spectrum
Spherical
harmonic
spectrum of ITG-GRACE2010s
-6
0
-7
20
degree n
-8
40
-9
60
-10
-11
80
-80
-60
-40
-20
0
20
Snm - order m - Cnm
40
60
80
-12
6
Outline
• GRACE geometry
• Solution strategies
– Variational equations
– Differential gravimetry approach
• What about the Next-Generation-GRACE?
7
Geometry of the GRACE system
Geometry of the GRACE system
Differentiation
Rummel et al. 1978
Integration
9
Solution Strategies
Solution strategies
Variational equations
In-situ observations
Classical
Energy Integral
(Reigber 1989, Tapley 2004)
(Han 2003, Ramillien et al. 2010)
Celestial mechanics approach
Differential gravimetry
(Beutler et al. 2010, Jäggi 2007)
(Liu 2010)
Short-arc method
LoS Gradiometry
(Keller and Sharifi 2005)
(Mayer-Gürr 2006)
…
…
Numerical integration
Analytical integration
11
Equation of motion
Basic equation: Newton’s equation of motion
where
gi are all gravitational and non-gravitational disturbing forces
In the general case: ordinary second order non-linear differential equation
Double integration yields:
12
Linearization
For the solution, linearization using a Taylor expansion is necessary:
Types of partial derivatives:
• initial position
Homogeneous
solution
• initial velocity
• residual gravity field
coefficients
• additional parameter
…
Inhomogeneous
solution
13
Homogeneous solution
Homogeneous solution needs
the partial derivatives:
Derivation by integration of the variational equation
Double integration !
14
Inhomogeneous solution (one variant)
Solution of the inhomogeneous part by the method of the variation of the
constant (Beutler 2006):
with zi (t) being the columns of the matrix of the variational equation of the
homogeneous solution at each epoch.
Estimation of
®i by solving the equation system at each epoch:
15
Application to GRACE
In case of GRACE, the observables are range and range rate:
Chain rule needs to be applied:
16
Limitations
• additional parameters
– compensate errors in the initial conditions
– counteract accumulation of errors
• outlier detection difficult
• limited application to local areas
• high computational effort
• difficult estimation of corrections to the initial conditions in case
of GRACE (twice the number of unknowns, relative observation)
17
In-situ observations:
Differential gravimetry approach
Instantaneous relative reference frame
Position
Velocity
alongtrack
alongtrack
261,5
1
0.5
[m/s]
[km]
[km]
[m/s]
11
261,0
0.5
230,1
0
230,0
50
100
100
time [min]
time
260,5
0
-1
259,5
00
crosstrack
crosstrack
0.5
0.5
-0.5
260,0
-0.5
229,9
-1
00
radial
radial
[m/s]
[km]
1
230,2
00
-0.5
-0.5
50
100
50
100
time[min]
[min]
time
-1
-1
00
50
100
50
100
time [min]
time
[min]
19
In-situ observation
Range observables:
Multiplication with unit vectors:
GRACE
20
Relative motion between two epochs
absolute motion neglected!
GPS
Epoch 1
K-Band
Epoch 2
21
Limitation
Combination of highly
precise K-Band
observations with
comparably low accurate
GPS relative velocity
22
Residual quantities
• Orbit fitting using the homogeneous solution of the variational
equation with a known a priori gravity field
estimated orbit
true orbit
GPS-observations
• Avoiding the estimation of empirical parameters by using short arcs
23
Residual quantities
ratio ≈ 1:2
ratio ≈ 25:1
24
Approximated solution
25
Next generation GRACE
Next generation GRACE
• New type of intersatellite
distance measurement
based on laser
interferometry
• Noise reduction by a
factor 10 expected
M. Dehne, Quest
27
Solution for next generation GRACE
28
Variational equations for velocity term
• Reduction to residual quantity insufficient
• Modeling the velocity term by variational equations:
• Application of the method of the variations of the constants
29
Results
• Only minor improvements
by incorporating the
estimation of corrections
to the spherical harmonic
coefficients due to the
velocity term
• Limiting factor is the orbit
fit to the GPS positions
 additional estimation of
corrections to the initial
conditions necessary
30
Summary
• The primary observables of the GRACE system (range & range
rate) are connected to gravity field quantities through variational
equations (numerical integration) or through in-situ observations
(analytical integration).
• Variational equations pose a high computational effort.
• In-situ observations demand the combination of K-band and
GPS information.
• Next generation GRACE instruments pose a challenge to
existing solution strategies.
31