**Authors**

**Abstract**

The Earth’s gravity field dedicated missions, CHAMP, GRACE and GOCE have opened up a new era for the study of the Earth’s gravity field. The missions measure different functionals on the Earth’s gravitational potential which are represented in terms of the Stokes Coefficients or the so-called geopotential models. The mapping process of the observations into the coefficients is called recovery of the Earth’s gravity field.

Different approaches have been introduced by the research institutions and universities for recovery of the field. In principle, these methods can be categorized into two integral- and acceleration-based groups. The integral-based approach utilizes the integral form of the satellite equation of motion, i.e., Jacobi’s integral whereas the acceleration method is set up based on the Newton’s gravitational equation. The main difference between the methods is the scalar and vectorial form of the observation equations. Moreover, the Jacobi’s integral is only need the position and velocity observations where the acceleration approach is written based on the position, velocity and acceleration measurements.

This papers aims to numerically compare the methods for the recovery of the Earth’s gravity field in the GRACE mission. The vectorial form the high-low satellite-to-satellite observations in the GRACE mission is used for the recovery purposes where the observation equations in the integral approach are derived from the scalar form the observations. Our numerical investigation shows that the acceleration approach can lead to a higher accuracy if the full form of the observation is used. However, the recovery can be done with comparable accuracy to the integral approach only using the radial component of the acceleration differences in the high-low mode.

The aliasing problem is the next point that has been addressed in this paper. The recovered geopotential models are usually affected by the high-degree frequency of the gravitational field. The acceleration approach is much more sensitive to the presence of the omission error due the imperfect modeling of the high-frequency constituents. Using the numerically derived second-order time derivative of the GPS-based position observations is might be the main reason of the higher sensitivity.

Different approaches have been introduced by the research institutions and universities for recovery of the field. In principle, these methods can be categorized into two integral- and acceleration-based groups. The integral-based approach utilizes the integral form of the satellite equation of motion, i.e., Jacobi’s integral whereas the acceleration method is set up based on the Newton’s gravitational equation. The main difference between the methods is the scalar and vectorial form of the observation equations. Moreover, the Jacobi’s integral is only need the position and velocity observations where the acceleration approach is written based on the position, velocity and acceleration measurements.

This papers aims to numerically compare the methods for the recovery of the Earth’s gravity field in the GRACE mission. The vectorial form the high-low satellite-to-satellite observations in the GRACE mission is used for the recovery purposes where the observation equations in the integral approach are derived from the scalar form the observations. Our numerical investigation shows that the acceleration approach can lead to a higher accuracy if the full form of the observation is used. However, the recovery can be done with comparable accuracy to the integral approach only using the radial component of the acceleration differences in the high-low mode.

The aliasing problem is the next point that has been addressed in this paper. The recovered geopotential models are usually affected by the high-degree frequency of the gravitational field. The acceleration approach is much more sensitive to the presence of the omission error due the imperfect modeling of the high-frequency constituents. Using the numerically derived second-order time derivative of the GPS-based position observations is might be the main reason of the higher sensitivity.

**Keywords**