# SLR, GRACE and Swarm Gravity Field Determination and Combination
## 2 Materials and Methods
### Grace  
The GRACE satellites were launched in March 2002 into polar orbits at an initial orbit altitude of 500 km [1]. During 15 years, they provided information about temporal variations of the Earth gravity field at unprecedented spatial and temporal resolution. The mission ended in October 2017 due to battery failure after decay to an orbit altitude of 330 km. The satellites were equipped with GPS receivers for orbit determination [57]. The key instrument for gravity field recovery was a K-band inter-satellite link which provided range measurements with micrometer accuracy [69]. Due to long-periodic systematic errors that cancel out by differentiation [70], most analysis centers use range-rates for gravity field recovery that were derived from the original range observations by numerical differentiation. Surface forces acting on the GRACE satellites were measured with onboard accelerometers in order to separate them from the gravitational forces [49]. We estimated monthly gravity fields from K-band range-rates and kinematic orbits that were used instead of the original GPS phase observations for efficiency reasons (the equivalence of both approaches was demonstrated by [PERSON] et al. [71]). Therefore, in the first step, kinematic satellite positions were determined using the Bernese GNSS Software 5.2 (AIUB, Bern, Switzerland) [72]. The kinematic orbits were introduced as pseudo-observations together with their epoch-wise covariances [56]. Normal equations (NEQs) were computed for both observation types on a daily basis, combined and summed up to monthly batches. All but the gravity field parameters were pre-eliminated from the combined daily NEQs. Note that the K-band observations only contain line-of-sight, i.e., predominantly along-track information and a K-band only orbit solution therefore would be rank-deficit [70]. The accelerometer provides biased accelerations in the instrument frame that is closely aligned to the radial, along-track, and cross-track direction of the co-rotating orbital tripod. To account for the biases, daily constant accelerations are estimated in radial and cross-track directions. In along-track, four polynomial parameters are estimated per day to also absorb temperature induced variations in the accelerometer measurements that are in conflict with the ultra-sensitive K-band observations [20]. However, the key element of the Celestial Mechanics Approach (CMA) that was developed at AIUB for orbit and gravity field determination [48] is the estimation of rather frequent pseudo-stochastic accelerations (or pulses) to compensate deficiencies in the a priori force model [51]. Accelerations are estimated at 15 min intervals in all three directions of the orbital tripod. They are constrained to zero in order to minimize absorption of gravity signal. The parameterization applied for GRACE processing is summarized in Table 4.  
For the a priori force model, the static part of AIUB-GRACE03S is used, but compared to the SLR-processing with increased spherical harmonic resolution to take into account the high sensitivity of the K-band observable. In addition, the background models for tidal effects and AOD were chosen consistently with the SLR-processing (details on the force model can be found in Table 2). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **LAGEOS-1/2** & **SLR LEOs** \\ \hline Station coordinates & 30 days & 30 days \\ Earth orientation parameters & piecewise-linear, daily & piecewise-linear, daily \\ Geocenter coordinates & 30 days & 30 days \\ Gravity field & degree/order 10 & degree/order 10 \\ Range biases & selected sites & all sites \\ Initial state & 10 days & 1 day \\ Const. acceleration along-track & 10 days & none (air drag modelled) \\ Air drag scaling factor & none & 1 day \\ Periodic along-track & 10 days & 1 day \\ Periodic cross-track & none & 1 day \\ Pseudo-stochastic pulses & none & once per revolution along-track \\ \hline \hline \end{tabular}
\end{table}