Comment on egusphere-2023-172

Abstract

<strong class="journal-contentHeaderColor">Abstract.</strong> Spacecraft constellations consisting of multiple satellites are more and more becoming of interest not only for commercial, but also for space science missions. The proposed and accepted scientific multi-satellite missions to operate within Earth's magnetospheric environment, like HelioSwarm, require extending established methods for the analysis of multi-spacecraft data to more than four spacecraft. The wave telescope is one of those methods. It is used to detect waves and characterize turbulence from multi-point magnetic field data, by providing spectra in reciprocal position-space. The wave telescope can be applied to an arbitrary number of spacecraft already. However, the exact limits of the detection for such cases are not known if the spacecraft, acting as sampling points, are irregularly spaced. We extend the wave telescope technique to an arbitrary number of spatial dimensions and show how the characteristic upper detection limit in <em>k</em>-space imposed by aliasing, the spatial Nyquist limit, behaves for irregular spaced sampling points. This is done by analyzing wave telescope <em>k</em>-space spectra obtained from synthetic plane wave data in 1D up to 3D. As known from discrete Fourier transform methods, the spatial Nyquist limit can be expressed as the greatest common divisor in 1D. We extend this to arbitrary numbers of spatial dimensions and spacecraft. We show that the spatial Nyquist limit can be found by determining the shortest possible basis of the spacecraft distance vectors. This may be done using linear combination in position-space and transforming the obtained shortest basis to <em>k</em>-space. Alternatively, the shortest basis can be determined mathematically by applying the Modified Lenstra-Lenstra-Lov&aacute;sz algorithm (MLLL) combined with a lattice enumeration algorithm. Thus, we give a generalized solution to the determination of the spatial Nyquist limit for arbitrary numbers of spacecraft and dimensions without any need of a priori knowledge of the measured data. Additionally, we give first insights on the application to real-world data incorporating spacecraft position errors and minimizing <em>k</em>-space aliasing. As the wave telescope is an estimator for a multi-dimensional Fourier transform, the results of this analysis can be applied to Fourier transform itself or other Fourier transform estimators making use of irregular sampling points. Therefore, our findings are also of interest to other fields of signal processing.