# Impact of Groundtrack Pattern of a Single Pair Mission on the Gravity Recovery Quality
## 1 Introduction  
Since the launch of the Gravity Recovery and Climate Experiment (GRACE) mission, several research works, e.g., [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13] have investigated the performance of next generation satellite gravity missions. Inline (GRACE-like), alternative (advanced) formations (e.g., Pendulum, Cartwheel, LISA), as well as double-pair mission scenarios (two pairs of inline formation) have been studied. Furthermore, the European Space Agency (ESA) projects \"Assessment of a Next Generation Mission for Monitoring the Variations of Earth's Gravity\" [14] and \"Assessment of Satellite Constellations for Monitoring the Variations in Earth's Gravity Field\" [15] investigated the science requirements, performance criteria and design of future satellite gravity missions. These two ESA studies briefly discussed two advanced formations and Bender configurations. However, because of the complexity of the technical implementation of alternative formations such as Pendulum, Cartwheel andLISA, their employment as a future gravity mission is currently not of interest to the community [11], although the future technical improvements may bring them back to the consideration. Some of the aforementioned studies also suggested a range of parameter choices for double-pair satellite missions for full repeat period and sub-Nyquist solutions. Most of the aforementioned works chose individual constellation scenarios by rough assessments of the sampling behavior of the missions. However, the studies by [5; 6; 8; 15] investigated some performance criteria for their optimal double-pair mission scenario search strategies. In gravity satellite missions, two sampling descriptions exist, which dictate the quality of the gravity recovery change in space and time domains. First, a Heisenberg-like uncertainty theorem in satellite geodesy addresses the trade-off between spatial and temporal resolution by  
\[D_{space}\times D_{time}=\frac{2\pi\alpha}{\beta}=2\pi T_{revolution}=const. \tag{1}\]  
i.e., the product of spatial resolution \(D_{space}\) and the time resolution \(D_{time}\) is constant and equal to \(2\pi T_{revolution}\) with \(T_{revolution}\) as the revolution time [8; 16; 17]. Second, the Colombo-Nyquist rule (CNR) requires  
\[\beta\geq 2L_{max}\ (\text{or}\ 2M_{max}) \tag{2}\]  
where \(\beta\) is the number of satellite revolutions within a repeat period in nodal days (\(\alpha\)). The formulation means that the number of satellite revolutions in a repeat period should be equal to or larger than twice the maximum spherical harmonic degree (\(L_{max}\)) or order (\(M_{max}\)) to be detected [18; 19]. Indeed, this rule limits the spatial resolution of the gravity recovery. However, the works by [20; 21] argue that the spatial resolution can be improved, not by twice the maximum degree or order \(L_{max}\) or \(M_{max}\), but by a smaller degree or order values. In particular, the work by [21] discusses that those values can even reach \(L_{max}\) or \(M_{max}\) themselves, depending on the parity of the satellite repeat orbit. However, the description by [21] is not fully applicable, and requires further investigations. In fact, this study states that a recoverable gravity solution in a full repeat period can be achieved by the new modified rule when \(\beta-\alpha\) is odd for the near-polar orbits. Later, [8] took the benefit of the proposed modified Colombo-Nyquist rule (MCNR) to achieve sub-Nyquist gravity recovery, although it should be mentioned that the term \"sub-Nyquist\" in that work is not defined according to sampling distribution and groundtrack pattern evolution, but by fixed values defined by mission revolutions per nodal days. Therefore, in that context, one can consider Nyquist as the upper limit in sampling theorem. The short-time intervals solutions, called sub-Nyquist solutions, in consequence, may benefit from higher temporal resolution, and thus, less temporal aliasing, compared to the longer span solutions, e.g., defined by CNR.