# Combination Analysis of Future Polar-Type Gravity Mission and GRACE Follow-On
## 2 Numerical Simulation Scheme
### Numerical Simulation Procedures  
The simulation procedures involving the forward and the backward simulation steps [18] are shown in Figure 1. In the forward simulation step, we apply a set of true background models to generate the simulated observations. The true background models are listed in Table 2, where EIGEN6C4 [56] is used as the true static gravity field model and EOT11a (including main wave and interpolated secondary wave) [57] is treated as the true ocean tide model. For non-tidal variation signals, the AOHIS component of ESM is adopted to represent the integrated signal of atmosphere (A), oceans (O), terrestrial hydrological water (H), continental ice-sheets (I), and the solid earth (S) [24]. The maximum degree and order for all models is chosen as 100 [18]. Given the initial orbit elements defined in Table 1, the satellite orbits for both GFO and FPG can be generated by numerical integration [50] (pp. 117-146). The combined eighth-order Runge-Kutta and eighth-order \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Orbit** & **Altitude** & **Inclination** & **Eccentricity** & **Inter-Satellite** \\
**Configuration** & **(km)** & **(deg.)** & **Eccentricity** & **Distance (km)** \\ \hline _GFO_ & 490 & 89 & 0.0015 & 220 \\ \hline _FPG-87_ & & 87 & & \\ _FPG-89_ & & 89 & & \\ _FPG-91_ & & 91 & 0.0015 & 220 \\ _FPG-93_ & & 93 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Initial orbit configurations of GFO and FPG for simulation study Gauss-Jackson integrator with an integration step of 5 s is applied to generate the error-free data, which include satellite orbit positions and inter-satellite range-rates. To produce more realistic observations, above error-free data should be added with sensor noises. In this work, the sensor noises are treated as Gaussian white noise with zero expectation [31; 32], while more sophisticated noise models can be found in [58]. The standard deviations (SD) of major onboard sensors' noises, listed in Table 3, are based on their performances in real GRACE data. As shown, 1-cm SD is selected for orbit positions [31; 32; 33]. With regard to the range-rate data, we add noise with SD of 0.2 \(\upmu\)m/s to KBR measurements in GFO and FPG [18; 45], while with SD of 10 nm/s to LRI measurements in GFO [18]. Although the non-gravitational forces are not simulated during orbit integration, the accelerometer's noise with SD of 0.3 nm/s\({}^{2}\) is directly simulated during the integration of orbit [52] (pp. 50-51). We simulate full-year observations in 2006 for all orbit configurations, which are subsequently used in the backward simulation step for gravity field model recovery. In the backward simulation step, we recover the monthly gravity field model up to 100 d/o [18] by the dynamic approach using above simulated observations. For parameterizations, we choose the orbit arc length as 6 h for the initial position and velocity estimation [44; 45; 46], and estimate the accelerometer parameters (bias and linear drift) every 1.5 h (orbital period) [45]. Observations' weight matrices in Equation (9) are constructed based on corresponding SD values listed in Table 3 with the same variance of unit weight. The reference background models, used for reference orbit and partial derivatives integration in Equations (5) and (7), are listed in Table 2. As shown, FES2004 [59] is applied as reference ocean tide model, and DEAL plus AOerr components of ESM [27] are used to represent the AO components of AOHIS in true background models [24]. Since the emphasis of the article is on temporal gravity model recovery, the true and reference static models are identical [21; 33; 35]. The differences between true and reference background models are designed to introduce the temporal aliasing errors to the recovered gravity model [18; 21; 33; 37]. As shown in Figure 1, the temporal signal we want to recover in each monthly model is the average HIS of this month, which is the integrated signal of terrestrial hydrological water (H), continental ice-sheets (I), and the solid earth (S) [18; 24].