# Combination Analysis of Future Polar-Type Gravity Mission and GRACE Follow-On
## 2 Numerical Simulation Scheme
### Evaluation Metrics for Recovered Gravity Field Model  
In a simulated world, the quality of the recovered gravity field model can be easily evaluated by comparing the true and the recovered fields. In this article, the evaluation metrics for recovered gravity field models are derived based on monthly SHC differences \(\{\Delta C_{lm},\Delta S_{lm}\}\), which are calculated by subtracting the recovered model from the true static model (EIGEN6C4 100 d/o) plus the true monthly HIS signal (100 d/o) as shown in Equation (12) according to [18],  
\[\Delta C_{lm}=\overline{\Sigma}_{lm}^{EIGEN6C4+HIS}-\overline{\Sigma}_{lm}^{ REC},\Delta S_{lm}=\overline{\Sigma}_{lm}^{EIGEN6C4+HIS}-\overline{\Sigma}_{lm}^{ REC} \tag{12}\]  
where the superscripts _EIGEN6C4_, _HIS_, and _REC_ represent the SHC from EIGEN6C4, monthly HIS and the recovered models in simulations. Given SHC differences, the degree geoid height error (DGBE) and cumulative geoid height error (CGHE) of the recovered model can be calculated by  
\[\begin{split} DGHE(n)&=R_{E}\cdot\sqrt{\sum\limits_{m=0 }^{l}{(\Delta C_{lm}^{2}+\Delta S_{lm}^{2})}},(l=n)\\ CGHE(n)&=R_{E}\cdot\sqrt{\sum\limits_{l=2}^{n}{ \sum\limits_{m=0}^{l}{(\Delta C_{lm}^{2}+\Delta S_{lm}^{2})}}}\end{split} \tag{13}\]  
where \(DGHE(n)\) represents the DGHE of degree \(n\) and \(CGHE(n)\) represents the CGHE up to degree \(n\). One can also construct spatial-domain metrics by the spatial differences plot (SDP) [60] (pp. 24-25). For the \(k\)-th grid point on the Earth surface, its geoid height difference can be calculated by Equation (14) [60] (p. 22), with (1) and (12),  
\[\Delta N_{k}(\varphi_{k},\lambda_{k})=R_{E}\cdot\sum\limits_{l=2}^{L_{MAX}} \sum\limits_{m=0}^{l}{(\Delta C_{lm}\cos m\lambda_{k}+\Delta S_{lm}\sin m \lambda_{k})\cdot\overline{P}_{lm}(\sin\varphi_{k})} \tag{14}\]  
where \(\Delta N_{k}\) stands for the geoid height difference, and other symbols are the same as those defined in Equation (1); \(L_{MAX}\) is chosen as 100. Based on Equation (14), the SDP can be created by plotting all grids' \(\Delta N_{k}\) within a specific spatial range \(\mathbf{\Omega}\) on the world map given corresponding geographic locations. \(\mathbf{\Omega}\) can be global or regional, e.g., oceans, Amazon basin. The grid size (interval) is chosen as 1-by-1 degree in the followings. Additionally, the latitude-weighted root mean square (wRMS) [60] (pp. 24-25) of above SDP can be calculated by Equation (15),  
\[wRMS(\mathbf{\Omega})=\sqrt{\frac{\sum\limits_{\forall k\in\mathbf{\Omega}}{ (\Delta N_{k}^{2}\cdot\cos\varphi_{k})}}{\sum\limits_{\forall k\in\mathbf{ \Omega}}{\cos\varphi_{k}}}} \tag{15}\]  
where all grids within \(\mathbf{\Omega}\) are summed based on their area weights reflected by latitudes.  
Since metrics in (13) and (15) represent absolute errors (positive values) of the recovered model, comparisons between any two recovered models, in terms of DGHE, CGHE, or wRMS, can be carried out by calculating error reduction rate (ERR) of the model \(i\) over the model \(j\) as shown in Equation (16),  
\[ERR_{j}^{i}(q)=\frac{q_{j}-q_{i}}{q_{j}}\times 100\% \tag{16}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Obs. Type** & **Orbit Pos.** & **GFO/FPG-KBR** & **GFO-LRI** & **Accelerometer** \\ \hline SD Value & 1 cm & 0.2 \(\mu\)m/s & 10 nm/s & 0.3 nm/s\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: The standard deviations (SD) of white Gaussian sensor noises used for the generation of simulated observationswhere \(q\) can be any value of DGHE, CGHE and wRMS.