# Hybrid Electrostatic-Atomic Accelerometer for Future Space Gravity Missions
## 2 Materials and Methods
### Hybrid Instrument Simulator
#### 2.3.4 Hybridization Algorithm  
The objective of the hybridization algorithm is to combine both electrostatic and atomic measurements so that the EA can benefit from the long-term stability of the CAI measurement.  
The principle of the algorithm is described in Figure 10. The two inputs are the acceleration measurement of the EA (\(a_{EA}\)) at 1 kHz and the signal of the CAI (\(\overline{P}_{i}\)) at 1/\(T_{c}\), that is to say, 0.25 Hz for a cycle time \(T_{c}=\) 4 s.  
The fringe index of the CAI is determined thanks to the EA measurement. For this purpose, the EA measurement is convoluted by the response of the atom interferometer (cf. Equation (4)) and corrected from the EA bias giving a debiased EA measurement (\(a_{EA,i}^{hyb}\)).  
The estimation of the EA bias is performed iteratively by comparing the value of acceleration provided by the CAI (\(a_{CAI,i}\)) and the value of acceleration provided by the EA for the current cycle after convolution with the atomic response function and corrected by the bias estimated at the previous iteration (\(a_{EA,i}^{hyb}\)).  
For the interferometric cycle \(i\), the estimation \(b_{i}\) of the EA bias is:  
\[b_{i+1}=b_{i}+G_{b}\times\Big{(}a_{EA,i}^{hyb}-a_{CAI,i}\Big{)} \tag{8}\]  
where \(G_{b}\) is the hybridization gain to be adjusted to give more or less weight to the current CAI measurement to determine the EA bias. Basically, this gain should be optimized taking into account the noise spectra of the CAI and the EA and more particularly the frequency where the two ASDs of acceleration noise intersect. This gain determines the convergence time of the hybridization algorithm to evaluate the EA bias. Note that during the convergence process, the acceleration is still continuously provided by the hybrid instrument.  
In the CAI, the atoms' acceleration is measured once per cycle. The interaction phase is surrounded by two dead times (see Figure 2), dedicated to the trapping and detection of the atoms. Because of these measurement dead times, the atom interferometer is subjected to errors. To avoid this effect, the EA measurements are used to fill the dead times. The hybrid acceleration measurement, after this processing filter, is for cycle \(i\):  
\[a_{Hyb,i}=a_{CAI,i}+\Big{[}a_{EA}*(f_{a,T_{c}}-f_{a,T})/T^{2}\Big{]}(t_{i}) \tag{9}\]  
where \(f_{a,T}\) is the acceleration response function of the CAI for an interrogation time \(T\) and \(f_{a,T_{c}}\) for an interrogation time \(T_{c}\).  
The two possible outputs of interest are the continuous electrostatic acceleration measurement corrected with its bias estimation at each cycle, that is \(a_{EA}-b_{i}\), and the  
Figure 10: Global structure of the hybridization algorithm.  
interferometric measurement after determination of the fringe index and removal of dead times, \(a_{\mathcal{C},\Lambda,i}^{hub}\). The former is the one that we mainly considered in this paper and that we used for the numerical gravity performance simulations. The latter could be useful in the case where the CAI offers better noise performance compared to the EA and when the mission allows a lower measurement sampling rate.