Kinematic velocity determination for the low-Earth-orbit satellites using the extended Kalman filter: a case study, the GRACE twin satellites

 Global Positioning System (GPS) receivers in gravimetric satellites continuously measure valuable information about 3D satellite position. However, the velocity of satellites, which has important applications in the satellite geodesy such as gravity field recovery, cannot be directly measured. These data are used in the energy integral method or other methods based on the Earth gravity field motion equation to determine the velocity or acceleration of satellite. In this study, the velocity vector is computed using the numerical differentiation and the Kalman filtering for the Gravity Recovery And Climate Experiment (GRACE) twin satellites. The Numerical results show that the Kalman filtering yields more accurate results than numerical differentiation when they are compared with the intersatellite range-rate measurements.
In the wake of the New Gravity Satellite era due to the launch of ChAllenging Minisatellite Payload (CHAMP), GRACE and GOCE, processing methods of enormously large orbit data has become the focus of the geodetic interest. The input data are different from earlier times as they contain some millions of continuous position data per satellite per year. The huge number of data arises from continuous observation from these satellites to the GPS system. This can be done due to the much higher altitude of the GPS satellites (20,000 km) compared to that of the gravity satellites (between 250 and 500 km). The latter is often referred to as Low Earth Orbiter, i.e. LEO. The GPS-LEO constellation as described above in technical terms is called High-Low Satellite to Satellite Tracking (High-Low SST). Thus some million position-data of the LEOs are the basis of the global gravity field determination techniques. The concept behind the Solutions is that satellites are in free-fall in the gravity field of the Earth. After modeling and removing all further force sources (e.g. gravitation of the Sun and the Moon and other planets, direct and indirect tides, surface forces (atmospheric drag, solar radiation pressure)) the remaining orbit is a trajectory in space, which is governed purely by the gravity field of the Earth. Therefore, the task is only to determine the force behind the motion. Conservation laws can be applied for satellites successfully. The Newton’s equation of law states the conservation of forces in a closed system. Applying it for a satellite requires information of the acceleration along the orbit. In this article the velocity vector is derived as a part of the unknown vector in Kalman filter algorithm.
Kalman filter is a well-known mathematical tool, which gives the answer to the most frequent engineering question: how can we get the best estimate of the system state from the noisy measurements? The Kalman filter is a data processing algorithm that estimates the state of a system from noisy measurements using least-squares. It gives the optimal system state estimate together with a measure of how precise is the state estimate compared to true state. The Kalman filter performs optimal solution for a linear process with uncorrelated, white, zero mean Gaussian process and measurement disturbances. The Kalman filter is a “one-step back recursive filter”, meaning that there is no need to store past measurements for the purpose of computing the present time.
We assume that the discrete random kinematic process to be estimated can be modeled with two main Kalman filter equations:
Process equation
Measurement equation
Where x is the system state vector, A is the state transition matrix and W is the process noise. From this discrete time linear formation of the Kalman filter, the discrete time nonlinear formation of the Extended Kalman Filter is based. For the state space model for the Extended Kalman Filter (EKF), the above linear equations are replaced by one nonlinear function:
 
 
In this study, the velocity vector is computed using numerical differentiation and the Kalman filtering for the GRACE twin satellites. The numerical analysis shows that the Extended Kalman filtering yields the optimal solution. The comparison is performed based on the intersatellite range-rate measurements.