Master Thesis
Jekeli’s Approach for Regional GRACE Gravity
Field Refinement
Haochen Li
Institute of Geodesy, University Stuttgart
Duration of the Thesis: 6 months
Completion: October 2014
Supervisor: Prof. Dr.-Ing. Wolfgang Keller
Dr.-Ing. Markus Antoni
Introduction
The Earth’s gravity field reflects the inner mass distribution of the Earth, the recovery of it, no
matter globally or regionally, is always a fascinating topic in geophysics area.
In the passing decades, several gravity field missions have been executed to reveal the details
of Earth’s gravity field. Among them is the successful terrestrial Gravity Recovery and
Climate Experiment (GRACE) mission which is based on the principle of low-low satellite to
satellite tracking (ll-SST). It provides data to derive the gravity field not only globally but also
regionally.
In dealing with ll-SST data to recover the gravity field, mainly two kinds of approaches can
be applied, namely the potential difference approach and the variational equation approach.
Jekeli’s approach is a possible realization of potential difference approach. Since the GRACE
data has a higher data density in the polar area, the feasibility of Jekeli’s approach in regional
gravity field recovery will be tested in this thesis, while the recovery of regional gravity field
by variational equation approach will be set as a reference. Finally, a new discovery in GRACE
data will be discussed, namely sensitivity problem of GRACE data.
Methodology
The two methods applied in this thesis are Jekeli’s approach and variational equation approach.
In the application of Jekeli’s approach, a proper regional gravity field model should also be
applied so that it can combine with the Jekeli’s approach to recover the regional gravity field.
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1. Regional gravity field model
The potential of the mass disturbance δM in region S0 is expressed as:
T ( ηc , X ) = √
GδM
(( Rσ)2
+ ∥X∥2 − 2σηc X)
(0.1)
For a group of mass distribution δMi , the potential is express as equation 0.2:
n
∑ T (ηci ), X) = ∑ αi ψ(ηci , X).
(0.2)
αi = GδMi
(0.3)
i =1
i =1
Let
αi are the parameters which will be estimated in this thesis.
2. Jekeli’s approch
The equation of Jekel’s approach is as followed:
V1 − V2 ≡ V21 ≈ | Ẋ 2 |ρ̇21 .
(0.4)
In order to obtain a more precise expression, the effect of earth rotation must be taken into
consideration. Combining with equation ??, the potential difference between two satellites can
be expressed as:
V1 − V2 ≡ V21 ≈ | Ẋ 2 |ρ̇21 − ωe ( X11 Ẋ21 − X21 Ẋ11 − X12 Ẋ22 + X22 Ẋ12 ) − E0 .
(0.5)
3. Variational equation approach
Variational equation approach is based on the Partial derivative of observables which can
be derived from variational equation. With the help of introduced reference potential and
variational equation, the residuals of range rate between two satellites can be expressed as:


α
1
[
]
 . 
∂ρ̇
∂ρ̇
δρ̇21 = ∂α
(0.6)
, · · · , ∂αn ·  ..  .
1
αn
with
∂ρ̇ ∂X 1
∂ρ̇ ∂ Ẋ 1
∂ρ̇ ∂X 2
∂ρ̇ ∂ Ẋ 2
∂ρ̇
=
+
+
+
∂α
∂X 1 ∂α
∂X 2 ∂α
∂ Ẋ 1 ∂α
∂ Ẋ 2 ∂α
∂ρ̇
(0.7)
where ∂X can be easily derived according to the geometry of satellite while the ∂X
∂α term (containing the derivative of position and velocity w.r.t. α) can be obtained through variational
2
equation.
Results
1. Jekeli’s approach
The selected single ground track is as followed:
Groundtrack in Greenland area
Arc of sat1
Arc of sat2
Point1
90
Point2
85
latitude [degree]
Point3
Point4
80
75
70
65
60
−60
−50
−40
−30
−20
longitude [degree]
Figure 0.1: Single ground track
The estimation results in noise free scenario are:
Table 0.1: Noise free scenario − JKA
Original
α1
α2
α3
α4
= 10000m3 s−2
= 20000m3 s−2
= 15000m3 s−2
= 30000m3 s−2
Estimated
α̂1
α̂2
α̂3
α̂4
= 1.1301 × 104 m3 s−2
= 2.0571 × 104 m3 s−2
= 1.4494 × 104 m3 s−2
= 3.1931 × 103 m3 s−2
The figure of observations is as followed:
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Observations (potential difference)
0.01
corrected observation
orignal observation
0.005
0
−0.005
δV[m2/s2]
−0.01
−0.015
−0.02
−0.025
−0.03
−0.035
−0.04
0
50
100
150
200
250
t[s]
Figure 0.2: Observations−JKA
The blue curve in Fig.0.2 is the approximately potential difference residuals while the red curve
is the corrected observations. The black points stand for the time when the first satellite passes
the local mass point. To check the result, the curve of corrected observations are compared with
the curve of weight sum matrix, the comparison result is as followed:
Comparison
0.03
corrected observation
Weight sum of design matrix
Weight sum of design matrix with shift
0.02
0
2
2
δV[m /s ]
0.01
−0.01
−0.02
−0.03
−0.04
0
50
100
150
200
250
t[s]
Figure 0.3: Comparison−JKA
In Fig.0.3, the curve of corrected observation (red) and the curve the weight sum of design
matrix (blue) are almost coincide with each other, which means the estimation works well
and indicates that there is no system error, but the results in Table 0.1 are still not as good
as expected. The relative error is still at the level of 10%. There should be still something
unmolded, which would be a new topic for further research, and this would not be discussed in
this thesis. Despite this, it can be reflected through the estimation error. The figure of estimation
error is as followed:
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Estimation error
−4
7
x 10
6
5
Vobs − Vestimated [m2/s2]
4
3
2
1
0
−1
−2
−3
0
50
100
150
200
250
t[s]
Figure 0.4: Estimation error−JKA
Comparing Fig.0.3 and Fig.0.4, the estimation error is 1% of the observations. It shows clearly
that there is more work need to be done.
2. Variational equation approach
The results of variational equation approach for single arc in noise free situation is as followed:
This result is not bad. Since the feasibility of variational equation approach in regional scenario
Table 0.2: Noise free scenario - VA
Original
α1
α2
α3
α4
Estimated
= 10000m3 s−2
= 20000m3 s−2
= 15000m3 s−2
= 30000m3 s−2
α̂1
α̂2
α̂3
α̂4
= 10000.06m3 s−2
= 20000.08m3 s−2
= 14999.99m3 s−2
= 30000.04m3 s−2
has been proved. It can be a reference for the results of Jekeli’s approach.
The figure of observations is as followed:
Observations (potential difference)
−6
2
x 10
observation
1
ρdot[m/s]
0
−1
−2
−3
−4
0
50
100
150
200
250
t[s]
Figure 0.5: Observations−VA
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The figure of comparison between observations and weight sum of design matrix is :
Comparison
−6
x 10
2
observation
Weight sum of design matrix
1
ρdot[m/s]
0
−1
−2
−3
−4
0
50
100
150
200
250
t[s]
Figure 0.6: Comparison−VA
In Fig.0.6, the two curves coincide perfectly with each other. How perfect it is can be reflected
on the estimation error.
The estimation error is shown in the following figure:
Estimation error
−12
5
x 10
4
ρobs − ρestimated [m2/s2]
3
2
1
0
−1
−2
−3
−4
0
50
100
150
200
t[s]
Figure 0.7: Estimation error−VA
The estimation error is at the level of 10−12 . It is random error.
3. Sensitivity problem
The results are as followed:
6
250
Figure 0.8: Sensitivity quantify of JKA in longitude (noise free)
Figure 0.9: Sensitivity quantify of JKA in latitude (noise free)
Comparing Fig.0.8 and Fig.0.9, it is clear that, in Jekel’s approach the observations is more
sensitive in north-south direction than in east-west direction, as the relative error in northsouth direction is one magnitude smaller than it in east-west direction.
Figure 0.10: Sensitivity quantify of JKA in longitude (with noise)
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Figure 0.11: Sensitivity quantify of JKA in latitude (with noise)
Comparing Fig.0.10 and Fig.0.11, the result of noise scenario is the same with the former noise
free comparison. How about the results in variational equation approach?
Figure 0.12: Sensitivity quantify of VA in longitude (noise free)
Figure 0.13: Sensitivity quantify of VA in latitude (noise free)
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Comparing Fig.0.12 and Fig.0.13, it turns out that the sensitivity problem also exists in
variational equation approach, but the magnitude is far less than in Jekeli’s approach.
Figure 0.14: Sensitivity quantify of VA in longitude (with free)
Figure 0.15: Sensitivity quantify of VA in latitude (with free)
Comparing Fig.0.8 to Fig.0.15, the sensitivity problem exists in the GRACE data, and particularly obvious in the Jekeli’s approach. No matter in Jekeli’s approach and variational equation
approach, the sensitivity problem is more significant in longitude change direction than in latitude change direction.
Conclusion
Through this whole thesis, the following conclusions can be drawn:
1. Even though Jekeli’s approach are applied short arc wise in the Greenland area where the
data density is higher, compared with variational equation approach, Jekeli’s approach is
still less precise in local case, which indicates that Jekelis approach cannot compete with
variational equation approach in regional scenario;
2. There is sensitivity problem exists in the GRACE data, which means the data is more
sensitive on the south-north distributed mass than the west-east distributed one. This is
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reasonable, since the GRACE satellites has a near polar orbit. The range rate between two
satellite reflects more south-north change than the west-east one.
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