# Impact of Groundtrack Pattern of a Single Pair Mission on the Gravity Recovery Quality
## 3 Results
### Global Gravity Recovery Solutions  
The global Root Mean Square (RMS) error of the mission scenarios of Table 1 are illustrated in Figures 1-6 for different recovery cases \(L_{max}\) = 32, 64, CNR and MCNR. The choices of \(L_{max}\) = 32, 64 are arbitrary, but selected as two values below CNR (\(L_{max}\) = 32) and between CNR and MCNR (\(L_{max}\) = 64) for 6-day solutions (see the next sections). The error is calculated in terms of geoid height for 3-day to 9-day gravity solutions by each mission scenario. As the figures show, the scenario 1 with long full repeat cycle (32 days), but shortest sub-cycle span (1 day), performs the worst. For the MCNR recovery, however, the last scenario (scenario 12) with 8 days repeat cycle, generates extremely large errors on the 8 th day. That is clearly not expected, since 8 days is the full repeat cycle of this scenario; therefore, one expects the best performance from the sampling viewpoint for that time-span. The potential reason for such a big error may come from high spatial aliasing of the solution (8-day recovery corresponds to \(L_{max}\) = 128 for MCNR) and the geometry of the orbit (short full-repeat cycle scenario).  
For the case \(L_{max}\) = 32 (Figure 1), with the increase of the time-pan of the solutions, the recovery error decreases. That is caused by more sampling in space, and therefore, less spatial aliasing, even though the temporal aliasing gradually increases, but not significantly. The same reason also applies for the case \(L_{max}\) = 64 (Figures 2 and 3). Figure 2, however, shows extremely high errors for solutions shorter than 4 days. In fact, the error is caused by poor conditioning of the normal matrices (singularity problem). The figure is deliberately brought to show that beyond MCNR (here spherical harmonic, SH, degree 48 for 3-day solutions), the recovery is very poor which is, in particular, the case for some orbit configurations.  
For \(L_{max}\) = CNR and MCNR, however, it is generally different, i.e., for most of the scenarios, the longer the recovery time-interval, the bigger the error (see Figures 4-6). However, different scenarios perform differently. For some scenarios of the case \(L_{max}\) = CNR, the error increase is significant. The reason for the increase should be associated with the spatial aliasing (by higher spherical harmonic coefficient degree and order to be solved), which is more predominant for some mission scenarios, most likely caused by the geometry of their orbits, and hence, the time evolution of their groundtracks. A similar statement is also valid for the case MCNR (Figures 5 and 6), although more fluctuations are seen in the recovery error of few scenarios. In particular, Figure 5 illustrates very large errors for some orbit scenarios. We think that these extreme error values are caused by severe spatial aliasing through  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Scenario** & \(\beta\)/\(\alpha\) **(ev./Day)** & **Altitude (km)** & **Sub-Cycle (Days)** & **Orbit Type** \\ \hline
1 & 511/32 & 263.4 & 1 & drifting \\
2 & 509/32 & 280.8 & 11 & skipping \\
3 & 493/31 & 281.7 & 10 & skipping \\
4 & 206/13 & 297.6 & 6 & skipping \\
5 & 507/32 & 298.4 & 13 & skipping \\
6 & 95/6 & 301.3 & 1 & drifting \\
7 & 364/23 & 303.3 & 6 & skipping \\
8 & 205/13 & 319.4 & 4 & skipping \\
9 & 503/32 & 333.8 & 7 & skipping \\
10 & 485/31 & 354.9 & 14 & skipping \\
11 & 391/25 & 356.4 & 11 & skipping \\
12 & 125/8 & 360.7 & 3 & skipping \\ \hline \hline \end{tabular}
\end{table}
Table 1: Satellite mission scenarios of this paper study.  
Figure 4: Error RMS in terms of geoid height for the gravity solutions of the Table 1 mission scenarios for maximum spherical harmonic degree and order CNR.  
Figure 3: The zoom-in graph of the Figure 2.