Estimation of Spherical Harmonic Geopotential Coefficients for Drag-free Satellite-to-Satellite Tracking Missions
Seong Hyeon Hong, John W. Conklin
University of Florida
Abstract
Estimation Problem Formulation
This research is based on the estimation of the spherical harmonic geopotential coefficients
for satellite-to-satellite tracking missions. Main goals for this research are,
1. To develop methods for determining effects of acceleration noise and orbit selection on
geopotential estimation errors for Low-Low Satellite-to-Satellite Tracking mission.
2. Compare the statistical covariance of geopotential estimates to actual estimation error, so
that the statistical error can be used in mission design, which is far less computationally
intensive compared to a full non-linear estimation process.
3. Utilize acceleration noise measurements made using the UF precision torsion pendulum to
analyze the effects of actual instrument noise on geopotential estimation.
For the estimation process the measurement data which consists of the range rate between
the two satellites and the positions of each satellite is generated. The measurement noise and
the residual acceleration noise are designed and embedded within this simulated
measurement and during the estimation process. Noise models are not restricted to be just
white, but colored noise models were used in the simulation based on the expected
performance of current technology. The two sets of different coefficients that were used, one
to create the measurement data and another one as a priori information, were acquired from
the Center of Space Research at the University of Texas at Austin. Similar to the GRACE
mission, a single set of coefficients were estimated with the period of 30 days. The estimation
process was broken down into small batches, one batch containing 3 hours of data to save the
computation time and to decrease the memory space associated with the simulation.
Therefore the Kalman batch estimation process derived from the least squares estimation was
applied for the simulation. The resulting geoid height error and the covariance are compared
to the actual result from the GRACE mission.
Introduction
Recursive Least Squares
Estimation Equations
System Equations
K 𝑘 = P𝑘 H𝑘 T (P𝑧 −1 + H𝑘 P𝑘 H𝑘 T )−1
θ𝑘+1 = Iθ𝑘 + Φ𝑘 C𝑘 ν𝑘
θ𝑘+1 = θ𝑘 + K 𝑘 (Z𝑘 − Z0 − H𝑘 θ𝑘 − θ0 )
Z𝑘 − Z0 = A𝑘 Φ𝑘 (θ𝑘 − θ0 ) + ϵ𝑘
Orbit Configuration
Semi Major Axis
Eccentricity
Inclination
Argument of Perigee
Right Ascension of the Ascending Node
Separation Distance
Magnitude
6830 km
0.001
88.95
133
155.8
220 km
The advantage of having a pair of satellites in measuring the gravity field is the high accuracy
acquired by the precise range rate measurements between the two satellites. Instead of
measuring the perturbation of the satellites just by high-end GPS receiver, laser ranging
system is applied which measures the range rate between the two satellites. Laser ranging
system has comparably less noise (factor of 100) than the K-Band ranging system [6], which
was applied in GRACE mission.
Therefore, there are two types of measurement data, range rate and position. The data is
acquired in every 10 seconds, therefore for each hour 360 sets are measured and each set
consists of 7 individual data. Below is the vector of one set of measurement data at time t.
θ: state vector (estimating parameters)
Z: measurement vector
Φ: State transition matrix
ν: state noise vector
ϵ: measurement noise vector
I: identity matrix
A: state to measurement transformation matrix
C: state noise to state transformation matrix
k: index of epoch and 0 represents the initial epoch
r1 : position 1
r2 : position 2
𝜌(𝑡)
𝑥1 (𝑡)
𝑦1 (𝑡)
z = 𝑧1 (𝑡)
𝑥2 (𝑡)
𝑦2 (𝑡)
𝑧2 (𝑡)
www.PosterPresentations.com
Earth’s gravitational potential is expressed with nth degree mth order Legendre polynomial
nth degree polynomial coefficients lead to (n2 – 22 + 2*n – 1) number of parameters
Size of the degree is limited by the computational capabilities
1 set of measurement data has 7 elements, 1 day of measurement contains 8640  7 data
points
Data Size
(days)
Degree of
Coefficient
Case A
10
20
Case B
10
30
Case C
30
20
Noise Design
The result shows that more data leads to smaller covariance and higher number of degree leads to
larger covariance. However, degree of the coefficients do not have as much effect on the
covariance compare to the data size.
Measurement Noise (𝛜)
•
•
•
•
Noise caused by the measuring instruments
Considered to be white (Gaussian distribution)
Considered to be WSS (wide sense stationary)
Consists of range rate and position noise
- Range rate noise is from the laser interferometer: ϵ𝜌 ~𝑁 0, 1𝑒 −11 𝑘𝑚/𝑠
~𝑁(0, (1𝑒 −5
𝑘𝑚)2 )
Estimation Without Acceleration Noise
2
[3]
The plot shows that the
statistical and actual error
curves tend to have same
magnitude as the
estimation process
proceeds.
This proves that the
measurement noise is
modeled correctly.
[3]
State Noise (𝛎)
• Noise caused by the uncertainty of the state model
- this uncertainty is caused by non-gravitational acceleration noise that is not included in
the model
• Considered to be colored
• Considered to be WSS only in period of 3 hours
Estimation With Acceleration Noise
𝟏+
𝟎.𝟎𝟎𝟓
𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚
× 𝟏𝟎−𝟏𝟎 [3]
The acceleration noise was
generated by the finite
impulse response(FIR)
method [4] and the
covariance of this noise
vector was calculated by the
interpolation of FIR [5].
With the same observation
as the above, this plot
proves that the
acceleration noise is
modeled correctly.
Therefor both
measurement and
acceleration noise are
modeled accurately.
Simulation Procedure
True Gravity Model
Measurement Error
Acceleration Error
Generate Positions
and Range Rate data
Generate Positions and
Range Rate data
Compute Information
Matrix of
Measurement and
Acceleration Noise
Future Work
- Determine accurate measurement and acceleration noise model and its effect on the
estimation process
- Evaluate the effect of orbit selection on estimating gravitational coefficients
- Utilize the acceleration measurement of the actual UF precision torsion pendulum
Estimate Orbit and
Gravity Model
Update A Priori Gravity
Model
Compare with True Gravity
Model
RESEARCH POSTER PRESENTATION DESIGN © 2012
-
H = AΦ
D = ΦC
P: covariance of state-estimate vector
𝑄: covariance of state noise vector
P𝑧 : covariance of measurement noise vector
K: Kalman Gain
A Priori Gravity Model
𝝆 : range rate
Relation between Estimation Uncertainty and Data size and Degree of Coefficients
P𝑘+1 = I − K 𝑘 H𝑘 P𝑘 + D𝑘 𝑄D𝑘 T
- Position noise is from the GPS sensors: ϵ𝑟1,2
Two current and successful Earth geodesy space missions, Gravity Recovery And Climate
Experiment (GRACE) and Gravity and Ocean Circulation Explorer (GOCE) demonstrated that
the geoid heights (or spherical harmonic geopotential coefficients) can be estimated with high
accuracy. The GRACE mission uses a pair of polar orbiting satellites and measures the range
rate between them to estimate the geoid height. On the other hand, the GOCE mission uses a
single drag free satellite and a gravity gradiometer to avoid all other perturbations but the
one caused by the gravity field. This research analyzes a space mission that is formed by
combining the ideas of the two missions stated above, which leads to a pair of drag free
satellites.
Preliminary Results
Reference
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