# Gravity Field Recovery Using High-Precision, High-Low Inter-Satellite Links
## 3 Simulation Environment  
All simulations were executed with a full numerical mission simulator [33; 34], which has already been successfully applied to recover satellite-only gravitational field models from GOCE data [35]. The simulation environment is based on numerical orbit integration, following a multistep method for the numerical integration according to [PERSON] & [PERSON] 1976 [36], which applies a modified divided difference form of the Adams predict-evaluate-correct-evaluate (PECE) formulas and local extrapolation. According to this method, the order and the step size are adjusted to control the local error per unit step in a generalized sense. The generation of \"true\" dynamic orbits and, subsequently, the \"true\" GNSS high-low SST and low-low laser ranging SST observations, is done by adding different force models according to the \"true\" world of Table 2. The impact of orbit errors on the gravity field processing is taken into account as well by propagating 1 cm white noise of the integrated orbit positions of each satellite. The resulting erroneous dynamic orbits serve as computational points for the reference values of the observations and enable the computation of the GNSS high-low SST observations in three directions. In order to ensure the accuracy of the inter-satellite link, error-free dynamic orbits are used for the reference values of the low-low SST observations from the laser interferometer system, which are expressed in terms of range-rates. Figure 4: Amplitude spectral density (ASD) of the relative acceleration measurement error in terms of accelerations for the low–low pair formation (dark blue and magenta) and for MOBILE (dark green). The generation of the noise time-series was done by scaling the spectrum of normally distributed random time-series with their individual spectral model (light blue, red, and light green curves). The adopted gravity field approach is based on a modification of the integral equation approach from [PERSON] 1969 [37] where the orbit is divided into continuous short arcs of 6 h length, and the position vectors at the arc node points are set up as unknown parameters, which are estimated together with the gravity field coefficients. This technique has already been successfully applied in real data applications to recover satellite-only gravitational field models for CHAMP and GRACE [38] (ITSG-Grace2016) [39]. The functional model follows the typical formulation used for low-low SST missions like GRACE, which comprises a high-low SST and a low-low SST component. Position differences between two satellites are used for the computation of the reference values for the high-low SST part of the observation system, whereas the reference values for the low-low SST part are derived by projecting position and velocity differences between two satellites onto the line-of-sight, leading to the computation of inter-satellite range-rates. Table 2 gives an overview of the force and noise models used in the processing for the \"true\" and \"reference\" world. The static gravity field model is represented by the GOCC003s model, which is a satellite-only gravity field model based on GRACE, GOCE, and LAGEOS [40]. In order to simulate geophysical signals, ESA's updated Earth system model [41] has been used, which contains the five main geophysical signal components atmosphere (A), ocean (O), hydrology (H), ice (I), and solid Earth (S) with a time resolution of six hours, linearly interpolated to the epochs. The Earth system model covers the time period 1995-2006, and contains plausible variability and trends in both low-degree coefficients and the global mean eustatic sea level. It depicts reasonable mass variability all over the globe at a wide range of frequencies including multi-year trends, year-to-year variability, and seasonal variability, even at very fine spatial scales, which is important for a realistic representation of spatial aliasing and leakage.