# Hybrid Electrostatic-Atomic Accelerometer for Future Space Gravity Missions
## 3 Results and Discussions
### Numerical Gravity Performance Simulations
#### 3.1.3 Bias and Scale Factor Impact  
The impact of accelerometer bias and particularly scale factor uncertainty on the gravity field retrieval was investigated. First, it was concluded that accelerometer measurements suffering from a constant observation offset would only affect the retrieval of the term \(C_{00}\) and, consequently, some further (primarily zonal) coefficient groups. However, since \(C_{00}\) is fixed and linked to the total Earth's mass, the gravity solution is scaled accordingly, thus fully removing any effects resulting from the bias. The following is dedicated to evaluating the impact of improved knowledge or stability of the hybrid accelerometer's scale factor with respect to the one of a stand-alone EA. We consider that the retrieval error \(\Delta\) has  
Figure 13: Relative improvement of the formal errors for _Case 3_ (**a**) and _Case 4_ (**b**) towards the EA scenario for a **GRACE-type mission** under consideration of the static gravity field signal and a hybrid accelerometer with a \(1/f^{3}\) EA noise slope, as well as an EA corner frequency of 1 mHz. Global geoid height errors are shown for EA-only (**c**), _Case 3_ (**a**), and _Case 4_ (**b**).  
two dominant contributions, one coming from the measured signal impacted by the scale factor uncertainty and the other coming from the instrument noise, and the objective is to determine which one dominates in terms of the ASD of acceleration noise. For this analysis, we define the relative scale factor knowledge as \(2\times 10^{-3}\) for the EA and \(10^{-5}\) for the hybrid measurements, assumed to be invariant over the observation period. We consider first a mission flying in drag-free mode as specified in the NGGM requirements [1], where the thrusters compensate the drag signal at a certain level, leaving the so-called residual drag to be measured by the accelerometer. In this case, the contribution of the scale factor error to \(\Delta\) is fully masked by the instrument noise in both cases of a stand-alone EA and a hybrid instrument.  
We consider now a mission where no drag compensation is applied at all, with the level of the drag acceleration signal given in Figure 14, extrapolated from [71], here slightly more conservative, especially with regard to the cross-track and radial signals' spectra.  
It can be seen in Figure 15a, where a stand-alone EA is considered, that the total retrieval error is dominated clearly by the scale factor error from 0.2 mHz to 10 mHz, leading to an increased retrieval error (cf. Figure 15b).  
When a hybrid instrument is now considered (cf. Figure 16, where a _Case 2_ 1D-hybridization is analyzed), the impact of the scale factor error is no longer significant relative to the instrument noise. If we consider particularly a 1D-hybridization along-track and a _Case 2_ CAI noise level, the impact of the scale factor error is no longer significant relative to the instrument noise. We can see in Figure 16a that the residual signal in the cross  
Figure 14: Drag signal model at a 361 km altitude, extrapolated from [71].  
Figure 15: Retrieval error with scale factor and instrument noise contributors, considering a stand-alone EA (**a**) and the corresponding LOS projection (**b**).  
track and radial direction features partially higher amplitudes than the instrument noise level, but this effect is fully mitigated once the line-of-sight (LOS) projection is computed. This performance already poses a significant improvement compared to a stand-alone EA.