# SLR, GRACE and Swarm Gravity Field Determination and Combination
## 2 Materials and Methods
### Spectral Resolution, Signal Leakage, and Filter Loss  
The Earth gravity field is commonly represented by a spherical harmonic expansion (Equation (1)), truncated at a certain maximum degree \(l_{max}<\infty\). The maximum degree (and consequently order) of this expansion determines the spatial scale of the represented signal. Monthly GRACE gravity fields are available to degree and order 90 (corresponding to a spatial scale of 460 km at the equator), Swarm monthly gravity fields are expanded to degree/order 70 but contain significant time-variable signal only up to about degree 13 (approx. 3100 km at the equator) [47]. SLR-derived gravity fields are even limited to degree 6 (approx. 6700 km at the equator) due to correlations between individual SHC at higher degrees that cannot be determined in unconstrained solutions given the inhomogenous observation coverage [10]. The truncation of the spherical harmonic expansion at a certain maximum degree causes signal leakage (e.g., [23; 24; 25; 26]). To demonstrate this effect, we simulate a mass layer with uniform mass distribution within the island of Greenland, while the mass over the rest of the globe is set to zero. The simulated mass distribution is expanded in a series of spherical harmonics and reconstructed from the SHC truncating the series at various values of \(l_{max}\). In Figure 3a, the reconstructed mass distributions are shown and the percentage of the integrated mass still contained inside the shorelines of Greenland is listed. Figure 2: Contribution per spherical harmonics coefficient of Swarm (**left**) and SLR (**right**) satellites and mean contribution per degree (**middle column**) in case of relative weighting 100:1 (**top**), 10:1 (**second row**), 2.5:1 (**third row**) or equal weight (**bottom row**). The signal content is also attenuated by filters to suppress the noise inherent to the gravity field models. A very simple filter that is commonly applied to GRACE monthly gravity fields is a Gaussian filter with 300 km filter radius [27]. In the spatial domain, the Gaussian filter represents a weighted average, the weighting function being bell-shaped with 300 km the half-width radius. In the spectral domain, the filter corresponds to degree-dependent scaling factors that quickly approach zero for higher degrees (from 1 at degree 0, the scaling factor has already dropped to 0.01 at degree 60). The experiment on the signal leakage by truncation of the spherical harmonic expansion is repeated, additionally applying a 300 km Gaussian filter for smoothing (Figure 3b). Due to the filter attenuation at medium to high degrees, the signal loss is even more dramatic than in the unfiltered case. More advanced filter types like the non-isotropic filter proposed by [PERSON] et al. [28] and the decorrelation filter DDK [29] were designed to minimize the filter loss, but the principle problem of signal attenuation persists. Both experiments show that gravity fields truncated at different maximum degrees cannot be compared. At the very limited resolution of the SLR-derived gravity fields, only a fraction of the original mass is localized inside the borders of Greenland. The true mass localization and change can be recovered approximately applying iterative forward-modeling approaches [25, 26], which require additional assumptions like mass loss being concentrated in fast flowing sections of ice streams, or at the coast. Nevertheless, leakage (in and out) and filter loss remain major limitations to the quantification of mass transport from satellite gravimetry. In the following, we truncate all gravity fields to the same low degree for the sake of comparison. Furthermore, we refrain from the use of filters. To derive mass from SHC, first the dimensionless SHC are scaled by the factors provided by [PERSON] et al. [30] to transform them to EWH. Then, global 1\({}^{\circ}\)-grids of EWH are computed from the scaled SHC by spherical harmonic synthesis (Equation (1)). The series expansion is truncated at the maximum degree of choice (either 6 or 10).