# SLR, GRACE and Swarm Gravity Field Determination and Combination
## 2 Materials and Methods
### Combination of Swarm and SLR on the Normal Equation Level  
To fill the gap between GRACE and GRACE-FO, we propose a combination of gravity fields derived by multi-SLR and Swarm analysis as detailed above. A very simple combination of SLR and GRACE, in fact the replacement of C\({}_{20}\) estimates in the monthly GRACE gravity fields by values derived from SLR, is already common practice [76]. In contrast to this, we perform a combination of Swarm and SLR on the NEQ level, i.e., we solve the equation  
\[(w_{SLR}\mathbf{N}_{SLR}+w_{Swarm}\mathbf{N}_{Swarm})\mathbf{dx}=w_{SLR}\mathbf{b}_{SLR}+w_{ Suarm}\mathbf{b}_{Swarm} \tag{2}\]  
for the vector of unknown gravity field coefficients \(\mathbf{dx}\), where \(\mathbf{N}_{SLR}\) and \(\mathbf{N}_{Swarm}\) are the normal equation matrices of SLR and Swarm, \(\mathbf{b}_{SLR}\) and \(\mathbf{b}_{Swarm}\) are the corresponding right-hand side vectors of the individual normal equation systems, and \(w_{SLR}\) and \(w_{Swarm}\) are weighting factors. The solution of the combined normal equation system is superior to a combination on a solution level because correlations between gravity field coefficients and other force model, orbit, instrument and reference frame parameters are taken into account [34]. Since all satellite data have been processed consistently \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **GRACE** & **Swarm** \\ \hline Gravity field & degree/order 60 or 90 & degree/order 70 \\ Initial state & 1 day & 1 day \\ Const. acceleration radial & 1 day & 1 day \\ Const. acceleration along-track & none & 1 day \\ Polynomial order 3 along-track & 1 day & none \\ Const. acceleration cross-track & 1 day & 1 day \\ Accelerometer scaling factors & 1 day, in all 3 directions & none \\ Pseudo-stochastic accelerations & 15 minutes, constrained: \(1\times 10^{-7}\frac{\text{m}}{\text{s}^{2}}\) & 15 minutes, constrained: \(7\times 10^{-6}\frac{\text{m}}{\text{s}^{2}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Orbit parameterization of Earth observation satellites and time intervals for which the individual parameters are estimated. and all but the gravity field parameters were pre-eliminated from the individual NEQs, the combination is straightforward. The key question of the combination is the ratio of the weights \(w_{Swarm}\) and \(w_{SLR}\) assigned to the different observation techniques [38]. We first derive monthly weights by variance component estimation (VCE) [77]. As shown in Figure 1, the ratio \(w_{Swarm}:w_{SLR}\) of the weights derived by VCE ranges from 20 to 95. To assess the plausibility of these weights, we derive alternative weights based on the accuracy estimates of the SLR ranges, which typically vary around 2 cm (see Table 1), and the GPS L1 phase observations of Swarm, for which [PERSON] et al. [78] provide accuracy estimates close to 3 mm in case of strong ionosphere activity and close to 2 mm in case of low ionosphere activity (as long as no advanced observation screening in the region of the geomagnetic equator is performed). Based on these accuracy assumptions, a ratio of weights in the range from 44 to 100 can be expected, which is quite close to what is determined by VCE. The slightly positive trend in the relative weights visible in Figure 1 is explained by the decaying orbit altitude of the Swarm satellites and the corresponding increase in sensitivity to the gravity field. A seasonal variation probably is related to the seasonal tracking characteristics of the SLR stations and the fact that much more stations are located in the northern than in the southern hemisphere. Test combinations were performed based on constant weighting ratios \(w_{Swarm}:w_{SLR}=100\):1, 10:1, 2.5:1 or 1:1.