Preprints
https://doi.org/10.5194/egusphere-2023-172
https://doi.org/10.5194/egusphere-2023-172
24 Feb 2023
 | 24 Feb 2023

The m-Dimensional Spatial Nyquist Limit Using the Wave Telescope for Larger Numbers of Spacecraft

Leonard Schulz, Karl-Heinz Glassmeier, Ferdinand Plaschke, Simon Toepfer, and Uwe Motschmann

Abstract. 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 k-space imposed by aliasing, the spatial Nyquist limit, behaves for irregular spaced sampling points. This is done by analyzing wave telescope k-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 k-space. Alternatively, the shortest basis can be determined mathematically by applying the Modified Lenstra-Lenstra-Lovász 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 k-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.

Journal article(s) based on this preprint

07 Nov 2023
The m-dimensional spatial Nyquist limit using the wave telescope for larger numbers of spacecraft
Leonard Schulz, Karl-Heinz Glassmeier, Ferdinand Plaschke, Simon Toepfer, and Uwe Motschmann
Ann. Geophys., 41, 449–463, https://doi.org/10.5194/angeo-41-449-2023,https://doi.org/10.5194/angeo-41-449-2023, 2023
Short summary

Leonard Schulz et al.

Interactive discussion

Status: closed

Comment types: AC – author | RC – referee | CC – community | EC – editor | CEC – chief editor | : Report abuse
  • RC1: 'Comment on egusphere-2023-172', Anonymous Referee #1, 28 Mar 2023
    • AC1: 'Reply on RC1', Leonard Schulz, 31 Aug 2023
  • RC2: 'Comment on egusphere-2023-172', Anonymous Referee #2, 03 Aug 2023
    • AC2: 'Reply on RC2', Leonard Schulz, 31 Aug 2023
  • RC3: 'Comment on egusphere-2023-172', Stephan C. Buchert, 04 Aug 2023
    • AC3: 'Reply on RC3', Leonard Schulz, 31 Aug 2023

Interactive discussion

Status: closed

Comment types: AC – author | RC – referee | CC – community | EC – editor | CEC – chief editor | : Report abuse
  • RC1: 'Comment on egusphere-2023-172', Anonymous Referee #1, 28 Mar 2023
    • AC1: 'Reply on RC1', Leonard Schulz, 31 Aug 2023
  • RC2: 'Comment on egusphere-2023-172', Anonymous Referee #2, 03 Aug 2023
    • AC2: 'Reply on RC2', Leonard Schulz, 31 Aug 2023
  • RC3: 'Comment on egusphere-2023-172', Stephan C. Buchert, 04 Aug 2023
    • AC3: 'Reply on RC3', Leonard Schulz, 31 Aug 2023

Peer review completion

AR: Author's response | RR: Referee report | ED: Editor decision | EF: Editorial file upload
ED: Publish subject to revisions (further review by editor and referees) (01 Sep 2023) by Georgios Balasis
AR by Leonard Schulz on behalf of the Authors (04 Sep 2023)  Author's response   Author's tracked changes   Manuscript 
ED: Referee Nomination & Report Request started (05 Sep 2023) by Georgios Balasis
RR by Stephan C. Buchert (11 Sep 2023)
RR by Anonymous Referee #1 (20 Sep 2023)
ED: Publish as is (21 Sep 2023) by Georgios Balasis
AR by Leonard Schulz on behalf of the Authors (22 Sep 2023)

Journal article(s) based on this preprint

07 Nov 2023
The m-dimensional spatial Nyquist limit using the wave telescope for larger numbers of spacecraft
Leonard Schulz, Karl-Heinz Glassmeier, Ferdinand Plaschke, Simon Toepfer, and Uwe Motschmann
Ann. Geophys., 41, 449–463, https://doi.org/10.5194/angeo-41-449-2023,https://doi.org/10.5194/angeo-41-449-2023, 2023
Short summary

Leonard Schulz et al.

Leonard Schulz et al.

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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.

Short summary
The upper detection limit in reciprocal space, the spatial Nyquist limit, is derived for arbitrary spatial dimensions for the wave telescope analysis technique. This is important as future space plasma missions will incorporate larger numbers of spacecraft (>4). Our findings are a key element in planning the spatial distribution of future multi-point spacecraft missions. The wave telescope is a multi-dimensional Fourier transform estimator; hence, this can be applied to other fields of research.