# SLR, GRACE and Swarm Gravity Field Determination and Combination
## 2 Materials and Methods
### Slr  
An important prerequisite for the determination of the long wavelength part of the gravity field is the exact modeling of the surface forces acting on the satellites. To simplify the modeling of the surface forces, most of the geodetic SLR satellites have a low area-to-mass ratio [9]. The high-flying SLR satellites LAGEOS 1 and 2 are mainly sensitive to the Earth's flattening C\({}_{20}\) and its variations with time [60]. Higher spectral resolution of the gravity field can be achieved by combined processing of the fleet of geodetic SLR satellites at lower orbits (SLR-LEOs) and at different inclinations of the orbital plane (see [10; 11; 12; 19]). We exploit the two LAGEOS satellites and the following SLR-LEOs for the determination of large scale temporal gravity field variations: Starlette, AJISAI, Stella, Larets, LARES, and the old Earth observation satellite Beacon-C that is carrying a laser retro-reflector array and is included due to its low orbit inclination. The orbit characteristics of all satellites used are compiled in Table 1. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Satellite** & **Launch Date** & **Orbit Altitude** & **Inclination** & **Observation Type** & **A Priori Error** \\ \hline Beacon-C & 1965 & 940–1300 km & 41.23\({}^{\circ}\) & SLR & 50 mm \\ Starlette & 1975 & 800–1100 km & 49.84\({}^{\circ}\) & SLR & 20 mm \\ LAGEOS-1 & 1976 & 5860 km & 109.90\({}^{\circ}\) & SLR & 8 mm \\ AJISAI & 1986 & 1500 km & 50.04\({}^{\circ}\) & SLR & 25 mm \\ LAGEOS-2 & 1992 & 5620 km & 52.67\({}^{\circ}\) & SLR & 8 mm \\ Stella & 1993 & 810 km & 98.57\({}^{\circ}\) & SLR & 20 mm \\ Larets & 2003 & 690 km & 97.77\({}^{\circ}\) & SLR & 30 mm \\ LARES & 2012 & 1440 km & 69.56\({}^{\circ}\) & SLR & 15 mm \\ GRACE A and B & 2002 & 500 km in 03/2002 & 89\({}^{\circ}\) & GPS & 2 mm \\  & & & & K-band & 0.3 μm/s\({}^{2}\) \\ Swarm A and C & 2013 & 460 km in 04/2014 & 87.4\({}^{\circ}\) & GPS & 2–3 mm \\ Swarm B & 2013 & 530 km in 04/2014 & 88.0\({}^{\circ}\) & GPS & 2–3 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Orbit characteristics of satellites used and a priori observation errors, as determined by variance analysis of residuals (Beacon-C is down-weighted due to large surface forces acting on the non-spherical satellite). The same a priori model of gravitational forces is consistently used for the SLR, GRACE and Swarm processing. To avoid affecting the derived temporal gravity field variations by a priori information, we use a static a priori gravity field. Furthermore, the background force model consists of solid Earth tides, ocean tides, ocean pole tides, and de-aliasing of ocean and atmosphere mass variations (AOD) [61]. Specific for the SLR satellites are models of the surface forces for air drag, solar radiation pressure and albedo that take into account the properties of the individual SLR satellites. The force model constituents and the resolution of their expansion in spherical harmonics (if applicable) are listed in Table 2.  
Even if SLR is sensitive to gravity field variations at higher degrees, at monthly resolution, only spherical harmonics coefficients (SHC) of degrees 2 to 5 and of degree 6 and order 1 can be determined from SLR alone, i.e., without regularization. At higher degrees SLR-only gravity field solutions suffer from the strong correlations between individual SHC [19]. A prerequisite for the separation of individual SHC, even at the low degree of 5, is that SLR LEOs orbiting at different inclinations are used [10]. In this context Beacon-C and LARES are helpful due to their rather exotic orbit inclinations (Table 1). However, the correct localization of the mass loss signal is not possible at this low resolution [33]. Therefore, we perform a combined solution with Swarm (see Section 2.4). In combination with Swarm, the SHC are decorrelated by the Swarm observations. To exploit the sensitivity of SLR beyond degree 5 in the combination and to avoid omission or commissioning errors [67], the SHC are set up to degree and order 10.