# SLR, GRACE and Swarm Gravity Field Determination and Combination
## 2 Materials and Methods  
The dominating force acting on a satellite is due to Earth gravity. Satellite orbits at low altitudes are sensitive to mass distribution and redistribution and therefore to mass transport in the Earth system. The long-term mean gravity field of the Earth has been determined by the dedicated gravimetric satellite mission GFZ-1 [39], by CHAMP [40] that also observed the magnetosphere, and by GRACE [41; 42] and GOCE [43; 44], again dedicated to gravimetry. Temporal variations in gravity have mainly been derived from GRACE (see [PERSON] et al. [45] for an overview), but also from CHAMP [46] and other Earth observing LEOs such as the three satellites of the Swarm mission [47] or from the fleet of geodetic SLR satellites (e.g., [10; 11; 12]). Prerequisite for gravity field determination is the precise observation of the satellite orbits. Today this may either happen by laser ranging or by GPS- or Doris-tracking of the satellites (the latter being irrelevant here due to the choice of missions). Critical for orbit modeling and signal separation is the knowledge of all forces acting on the satellites, the so-called background force model, and the set of parameters estimated to represent the orbit, improve the a priori force model and absorb model or instrument errors, the so-called orbit parameterization. In detail, the force model consists of a priori models of the gravitational forces by the Earth and third bodies, ocean, atmosphere and solid Earth tides, satellite specific models of the surface forces such as air drag, solar radiation pressure and Earth albedo, and an empirical or pseudo-stochastic part [48]. In case of dedicated gravity missions, the surface forces are normally measured by accelerometers onboard the satellites [49] (due to technical reasons only the sum of all forces plus an unknown but constant bias can be observed). At AIUB, the gravity field determination from satellite observations is treated as a generalized orbit determination problem [48]. The satellites' orbits are co-estimated with the instrument specific parameters like accelerometer scale factors and with the parameters updating the a priori background forces, i.e., the weight coefficients \(C_{lm}\) and \(S_{lm}\) of the expansion of the Earth's gravitational potential \(V\) in spherical harmonics [50]  
\[V(r,\phi,\lambda)=\frac{GM}{R}\sum_{l=0}^{\infty}\left(\frac{R}{r}\right)^{l+ 1}\sum_{m=0}^{l}P_{lm}(\sin\phi)\ (C_{lm}\cos m\lambda+S_{lm}\sin m\lambda)\, \tag{1}\]  
where \(r,\phi,\lambda\) are the spherical coordinates in an Earth-fixed reference frame, \(GM\) is the product of the gravitational constant and the Earth's mass, \(R\) is the semi-major axis of the Earth and \(P_{lm}\) are the fully normalized Legendre functions of degree \(l\) and order \(m\). In case of SLR, air drag scale factors, station and geocenter coordinates, and Earth orientation parameters are also co-estimated [19]. Deficiencies of the force model are mitigated by pseudo-stochastic accelerations or pulses [51]. In order to reduce the absorption of gravity field signal by these parameters, their frequency has to be tailored to the orbit altitude and observation sampling of the specific satellite and their magnitude has to be limited by constraints depending on the satellites' environment at orbit altitude. By extensive cross-validation with other analysis centers in the frame of the European Gravity Service for Improved Emergency Management (EGSIEM) [52] project, it could be shown that mass trends and the amplitude of seasonal variations in the AIUB gravity fields are not affected by the pseudo-stochastic parameterization. Since a separate determination of sub-groups of the parameter space leads to a regularization favoring the a priori force model applied [53] orbit, stochastic and force model parameters have to be determined together in one adjustment process.