# Next-Generation Gravity Missions: Sino-European Numerical Simulation Comparison Exercise
## 4 Results and Discussions  
The simulation scenarios described in Chapter 2 have been processed by all contributing groups applying the methods described in Chapter 3. In this chapter, the results will be compared, assessed, and discussed. Figure 4 shows the differences of the gravity field solutions of Scenarios A and B to their corresponding references solutions, which are represented by AOHIS mean fields over the 9- and 7-day period, respectively. They are expressed in terms of the degree root mean square (RMS) error of equivalent water heights (EWH), which is computed from fully normalized coefficients of a spherical harmonic series expansion \(\left\{\overline{C}_{nm},\overline{S}_{nm}\right\}\) of degree \(n\) and order \(m\), by [43]  
\[\sigma_{n}(EWH)=\frac{a\rho_{e}}{3\rho_{w}}\,\frac{2n+1}{1+k_{n}}\sqrt{\sum_{m =0}^{n}\,\overline{C}_{nm}^{2}+\,\overline{S}_{nm}^{2}} \tag{4}\]  
where \(\rho_{w}\) and \(\rho_{e}\) represent the average densities of water and Earth, respectively, \(a\) the semi-major axis of the Earth, and \(k_{n}\) the load Love number of degree \(n\). \(\left\{\overline{C}_{nm},\overline{S}_{nm}\right\}\) can either represent the full signal, or the differences between estimated and true coefficients. The EWH expresses the height of a mass-equivalent column of water per unit area. It is used here for comparing the results of five processing centres, because it is one of the standard quantities to express mass change processes in the Earth system, and was also used in the reference document [4] for this purpose. Figure 4 shows all 9- and 7-daily solutions obtained by the TUM processor for a two-month period as thin lines, as well as the mean of the two-month period as solid lines. Even though Scenario B is based on an inclined satellite with a lower orbit altitude, the performance of these two scenarios is very similar. This can be explained by the fact that Scenario A comprises 9-day solutions, while in Scenario B the solutions are based only on 7 days each. This lower number of observations seems to counteract the lower orbit altitude of the second pair. Since the performance of these two scenarios is very similar, in the following, when we will inter-compare the solutions of the five processing centers, we will concentrate on Scenario A. Figure 5a shows the degree RMS of the solutions with respect to the AOHIS input signal of the five processing centers for the first 9-day period. Even though there are several differences, they perform reasonably similarly. Apart from a slightly different error level over the whole SH spectrum, evidently the Tongji solutions perform better in the higher degrees, followed by TUM and WHU. A possible reason could be that both processing centers are using full covariance information as a stochastic model (cf. Table 3). In order to analyze the variability of the solutions among themselves, Figure 5b shows again the degree RMS, but now with the TUM solution as the reference. Evidently, the internal variability of the solutions is almost at the same level as their deviation from the input AOHJS signal. In order to analyze these differences in more detail, Figure 6 shows the SH coefficients of the input AOHIS signal averaged over the first 9 days of the analysis period (Figure 6a), as well as the coefficient differences of the five individual solutions from this AOHIS signal (Figure 6b-f). In general, there is a weakness in the estimation of the sectorial and near-sectorial coefficients, which is intrinsic to  
Figure 4: Degree root mean square (RMS) of Scenarios A and B computed with the TUM simulator. The light blue and green lines show the individual 9- and 7-day solutions, respectively, and the blue and red solid lines are the average performance over the two-month period. Figure 5: Degree RMS of first 9-day period of Scenario A of the five numerical simulators: (**a**) degree RMS with respect to the input AOHIS signal; (**b**) degree RMS with respect to the TUM solution. the in-line inter-satellite ranging concept.