# Next-Generation Gravity Missions: Sino-European Numerical Simulation Comparison Exercise
## 2 Closed-Loop Simulations and Input Data  
The basis of the inter-comparison exercise are so-called numerical closed-loop simulations. Compared to real-data processing, the advantage of these closed-loop simulations, which are performed in a completely synthetic simulation world, is that the truth is perfectly known. Therefore, the effect of any data set, instrument noise assumption, or processing strategy is directly reflected by deviation of the final result from the \"true world\", and can be studied separately from all other error contributors to the total error budget. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Simulation** & \
\begin{tabular}{c} **Semi-major** \\ **Axis Minas** \\ **6378 km** \\ \end{tabular} & **Inclination (\(\mathbf{\hat{r}}\))** & **Cycle/Retrieval** & **Start Epoch** &
\begin{tabular}{c} **Drift Rate** \\ (\(\mathbf{\hat{r}}\)**/cycle)** \\ \end{tabular} \\ \hline A & -355/–390 & 70,89 & 9 & 1 Jan 2002 & 1.5 \\ B & -355/–340 & 70,89 & 7 & 1 Jan 2002 & 1.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main orbit parameters of numerical simulation scenarios. Figure 2 shows an overview of the numerical simulation set-up. The simulation starts with the definition of the \"true world\" (upper left), using force models for the static gravity field as well as tidal and non-tidal temporal variations. All gravity-related models are parameterized as coefficients of a spherical harmonic (SH) series expansion. In parallel, specific orbit constellations and the corresponding orbit parameters have to be defined, in our case Bender-type double pairs. The corresponding orbits are generated by numerical integration. Based on the a-priori force models, error-free observations are computed along the orbit tracks, with a sampling period of 5 s. They are then superimposed by instrument errors with pre-specified spectral noise characteristics, resulting in \"true observations\", which mimic as realistically as possible the real observations. On the other hand, a reference world is built up (upper right). This is necessary because the functional model of gravity field estimation from low-low inter-satellite ranging observations is highly non-linear, and this requires a Taylor point as approximate prior knowledge. In the same way as for the \"true observations\", so-called \"reference observations\" are computed, which are based on force models that are usually different from those used for the \"true world\" scenario. The main goal of the gravity field determination is then to estimate residuals with respect to these a-priori models. Adding the residuals to the a-priori models should result in the ideal case (no errors in the system) in the true world models. In the case that errors do exist, such as instrument errors, but also aliasing errors due to the specific space-time sampling properties of the chosen orbit constellation, the estimated gravity field will deviate from the true world, and thus provides a clear hint on the achievable performance of the specific mission design. For the inter-comparison exercise, two Bender double-pair scenarios have been defined (Table 1). They are based on extensive prior studies on optimum orbit design of double-pair configuration with the goal to optimize the space-time sampling. The two scenarios A and B mainly differ in the orbit altitude of the polar satellite (390 vs. 340 km) and their repeat cycle (9 vs. 7 days). The orbits were designed such that ground tracks of the polar and inclined satellite pair drift at the same rate within the sub-cycle period, and (b) the retrieval period is always equal to the length of the sub-cycle, so that the gravity field retrieval is always supported by the densest possible ground track pattern. As a side  
Figure 2: Scheme of a closed-loop simulation set-up. The red blocks are related to orbit design and input data generation, the blue blocks are tightly linked to the gravity parameter estimation process, while the green block refers to the comparison and validation of the resulting gravity models.