# Gravity Field Recovery Using High-Precision, High-Low Inter-Satellite Links
## 1 Introduction  
In times of a changing climate the need for innovative observation techniques for capturing geophysical processes in the Earth system becomes increasingly urgent. In this context, the observation of the temporal gravity field by satellites from space play an important role when investigating, e.g., rapid changes in the cryosphere, oceans, water cycle, and solid Earth processes on a global scale. For the determination of temporal gravity fields, in the last decade satellite missions such as GRACE [1] or Challenging Minisatellite Payload (CHAMP) [2; 3] orbited around the globe and helped to get a better understanding of the Earth's mass flux signals. CHAMP was based on high-low satellite-to-satellite tracking (SST) exploiting the Global Positioning System (GPS) [4] over a time span of 10 years. The accuracy of the CHAMP orbit information of 2-3 cm [5] derived from GPS allowed for resolving only the long wave range of the time varying gravity field, with spatial scales of \(\approx\)1000 km, e.g., Baur 2013 [6]. Analyzing the perturbed orbit of other Low Earth Orbiters (LEO), such as the Swarm satellites, allows for a similar performance [7]. The GRACE mission reached spatial scales of the temporal gravity field of \(\approx\)300 km and below due to a combination of K-band microwave low-low inter-satellite ranging between two identical satellites following each other in the same orbit at a distance of about 220 km with micrometer precision, and high-low GPS satellite-to-satellite tracking plus accelerometer observations. These missions improved our knowledge of water mass variations on the continents, in the oceans, and the atmosphere to a great extent. Additionally, the static gravityfield retrieved from the Gravity field and steady-state Ocean Circulation Explorer (GOCE) [8] mission has improved our knowledge of the long-term static mass distribution, and has provided the physical reference surface of the geoid with centimeter precision with a spatial resolution down to 70-80 km. The observation of the Earth's gravity field will be continued by the GRACE Follow-On mission [9], which was successfully launched in May 2018. The instruments have been slightly modified compared to those used in GRACE, and additionally the GRACE Follow-On mission includes an inter-satellite laser ranging interferometer as a demonstrator [10], resulting in an increased accuracy of the SST observations to a few nanometers. One of the main error contributions when observing the time variable gravity field results from geophysical signals with periods shorter than the temporal resolution of the satellite mission, as they alias into the solutions. Traditionally, the temporal resolution of the GRACE data products has been 30 days [1], leading to temporal aliasing effects due to non-tidal mass variations, especially atmospheric, oceanic, and hydrological signals having large amplitudes in the high-frequency range [11], as well as tidal signals with mainly semi-diurnal and diurnal periods. The temporal aliasing errors in GRACE appear in terms of striping effects, which are caused by the anisotropic error behavior resulting from along-track inter-satellite ranging. This type of error pattern also remains for the GRACE Follow-On mission because temporal aliasing errors clearly dominate the error budget of gravity field retrieval [12; 13; 14; 15], while instrument errors of the laser interferometer only play a minor role in the total error budget. In general, there are different approaches of how to deal with temporal aliasing errors: The gravity field solutions retrieved by GRACE are typically treated with de-striping and filtering techniques (e.g., References [16; 17; 18; 19]), which are applied a posteriori in order to reduce the striping effects. A further de-aliasing method is proposed by [PERSON] et al. 2015 [20], where observations from GRACE are being processed using spherical cap mascons resulting in greater resolutions for smaller spatial regions.