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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.

Abstract

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.

Introduction

Estimation Problem Formulation

Seong Hyeon Hong, John W. Conklin

University of Florida

Estimation of Spherical Harmonic Geopotential Coefficients for Drag-free Satellite-to-Satellite Tracking Missions

Preliminary Results

Future Work

Orbit Configuration MagnitudeSemi Major Axis 6830 km

Eccentricity 0.001Inclination 88.95

Argument of Perigee 133Right Ascension of the Ascending Node 155.8

Separation Distance 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 satel-lites. 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 mea-surement data at time t.

r1 : position 1

r2 : position 2

: range rate

Simulation Procedure

Generate Positions and Range Rate data

Measurement Error

Acceleration Error

Generate Positions and Range Rate data

Compute Information Matrix of Measurement and Acceleration Noise

Estimate Orbit and Gravity Model

Update A Priori Gravity Model

Compare with True Gravity Model

A Priori Gravity Model

True Gravity Model

Noise Design

Data Size (days)

Degree of Coefficient

Case A 10 20

Case B 10 30

Case C 30 20

Reference

System EquationsRecursive Least Squares

Estimation Equations

= = : covariance of state-estimate vector: covariance of state noise vector: covariance of measurement noise vector: Kalman Gain

: state vector (estimating parameters) : measurement vector: State transition matrix: state noise vector: measurement noise vector: identity matrix: state to measurement transformation matrix: state noise to state transformation matrix: index of epoch and 0 represents the initial epoch

z=¿

Measurement Noise ()

• 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: [3]- Position noise is from the GPS sensors: [3]

State Noise ()

• 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

[3]

- 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

Relation between Estimation Uncertainty and Data size and Degree of Coefficients

Estimation Without Acceleration Noise

- 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

1. Tapley B. D., Schutz B. E., Born George H.; Statistical Orbit Determination; Elsevier Academic Press, Amsterdam (2004)2. Montenbruck O., Gill E.; Satellite Orbits Models, Methods and Applications; Springer-Verlag Berlin Heidelberg, New York, 3 rd ed. (2005)3. Kim J.; Simulation Study of A Low-Low Satellite-to-Satellite Tracking Mission; Center for Space Research, Univ. of Texas at Austin, CSR-00-02

(2000)4. Kasdin N. J.; Discrete Simulation of Colored Noise and Stochastic Processes and 1/f Power Law Noise Generation; Proceedings of the IEEE, Vol. 83,

No. 5, 802-827 (1995)5. Stoyanov M., Gunzburger M., Burkardt J.; Pink Noise, 1/f Noise, and Their Effect on Solutions of Differential Equations; International Journal for

Uncertainty Quantification, Vol. 1 (3), 257-278 (2011)6. Heinzel G., Sheard B., Brause N., Danzmann K., Dehne M., Gerberding O., Mahrdt C., Muller V., Schutze D., Stede G., Klipstein W., Folkner W.,

Spero R., Nicklaus K., Gath P., Shaddock D.; Laser Ranging Interferometer for GRACE follow-on; Albert Einstein Institute, Jet Propulsion Laboratory, The Austrailian National University (2012)

Estimation With Acceleration Noise

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].

The result shows that more data leads to smaller covariance and higher number of de-gree leads to larger covariance. However, degree of the coefficients do not have as much effect on the covariance compare to the data size.

The plot shows that the statistical and actual er-ror curves tend to have same magnitude as the estimation process pro-ceeds. This proves that the measurement noise is modeled correctly.

With the same observa-tion as the above, this plot proves that the ac-celeration noise is modeled correctly. Therefor both mea-surement and accelera-tion noise are modeled accurately.