Articles | Volume 41, issue 2
https://doi.org/10.5194/angeo-41-449-2023
© Author(s) 2023. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/angeo-41-449-2023
© Author(s) 2023. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Regular paper
| 
07 Nov 2023
Regular paper |
| 07 Nov 2023
| 07 Nov 2023
The m-dimensional spatial Nyquist limit using the wave telescope for larger numbers of spacecraft
Institute of Geophysics and Extraterrestrial Physics, Technische Universität Braunschweig, Braunschweig, Germany
Karl-Heinz Glassmeier
Institute of Geophysics and Extraterrestrial Physics, Technische Universität Braunschweig, Braunschweig, Germany
Max Planck Institute of Solar System Research, Göttingen, Germany
Ferdinand Plaschke
Institute of Geophysics and Extraterrestrial Physics, Technische Universität Braunschweig, Braunschweig, Germany
Simon Toepfer
Institute for Theoretical Physics, Technische Universität Braunschweig, Braunschweig, Germany
Uwe Motschmann
Institute for Theoretical Physics, Technische Universität Braunschweig, Braunschweig, Germany
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Gerlinde Timmermann, David Fischer, Hans-Ulrich Auster, Ingo Richter, Benjamin Grison, and Ferdinand Plaschke
Geosci. Instrum. Method. Data Syst., 14, 447–458, https://doi.org/10.5194/gi-14-447-2025, https://doi.org/10.5194/gi-14-447-2025, 2025
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We've compared the amplitude spectral densities of a fluxgate magnetometer (FGM) and an anisotropic magnetoresistive (AMR) magnetometer during ground testing with the amplitude spectral densities obtained in different regions of near-Earth space. The FGM can measure the fields in the different space regions and their fluctuations within a frequency range of 1 mHz to 2.5 Hz. The AMR magnetometer is only suitable for more turbulent regions such as the magnetosheath due to its higher noise levels.
Bruce T. Tsurutani, Gurbax S. Lakhina, Rajkumar Hajra, Richard B. Horne, Masatomi Iizawa, Yasuhito Narita, Ingo von Borstel, Karl-Heinz Glassmeier, Volker Bothmer, Klaus Reinsch, Philipp Schulz, and Sami Solanki
EGUsphere, https://doi.org/10.5194/egusphere-2025-5536, https://doi.org/10.5194/egusphere-2025-5536, 2025
This preprint is open for discussion and under review for Annales Geophysicae (ANGEO).
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During the 10–11 May 2024 geomagnetic storm, the red auroral rays appear at higher altitudes and connect to green rays lower down. The effect is linked to energetic electrons precipitating into the atmosphere during the storm. As the electrons continue downward, they hit oxygen below 200 km altitude and produce green light (5577 Å), named Stable Auroral Green (SAG) arcs. These observations mark the first reported sightings of such detailed, combined features.
Lars Klingenstein, Niklas Grimmich, Yuri Shprits, Adrian Pöppelwerth, and Ferdinand Plaschke
EGUsphere, https://doi.org/10.5194/egusphere-2025-4530, https://doi.org/10.5194/egusphere-2025-4530, 2025
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We applied machine learning to investigate how the solar wind and Earth's geomagnetic activity control the position of the magnetopause, the boundary layer of Earth's magnetic field. Our results demonstrate that geomagnetic activity strongly influences this boundary and should be incorporated in predictive models. Using data from multiple spacecraft, we developed a simple mathematical description of the magnetopause distance that improves understanding of solar wind–magnetosphere interactions.
Yasuhito Narita, Daniel Schmid, and Uwe Motschmann
Ann. Geophys., 43, 417–425, https://doi.org/10.5194/angeo-43-417-2025, https://doi.org/10.5194/angeo-43-417-2025, 2025
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It is often the case that only magnetic field data are available for in situ planetary studies using spacecraft. Either plasma data are not available or the data resolution is limited. Nevertheless, the theory of plasma instability tells us how to interpret the magnetic field data (wave frequency) in terms of flow speed and beam velocity, generating the instability. We invent an analysis tool for Mercury's upstream waves as an example.
Niklas Grimmich, Adrian Pöppelwerth, Martin Owain Archer, David Gary Sibeck, Ferdinand Plaschke, Wenli Mo, Vicki Toy-Edens, Drew Lawson Turner, Hyangpyo Kim, and Rumi Nakamura
Ann. Geophys., 43, 151–173, https://doi.org/10.5194/angeo-43-151-2025, https://doi.org/10.5194/angeo-43-151-2025, 2025
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The boundary of Earth's magnetic field, the magnetopause, deflects and reacts to the solar wind, the energetic particles emanating from the Sun. We find that certain types of solar wind favour the occurrence of deviations between the magnetopause locations observed by spacecraft and those predicted by models. In addition, the turbulent region in front of the magnetopause, the foreshock, has a large influence on the location of the magnetopause and thus on the accuracy of the model predictions.
Niklas Grimmich, Ferdinand Plaschke, Benjamin Grison, Fabio Prencipe, Christophe Philippe Escoubet, Martin Owain Archer, Ovidiu Dragos Constantinescu, Stein Haaland, Rumi Nakamura, David Gary Sibeck, Fabien Darrouzet, Mykhaylo Hayosh, and Romain Maggiolo
Ann. Geophys., 42, 371–394, https://doi.org/10.5194/angeo-42-371-2024, https://doi.org/10.5194/angeo-42-371-2024, 2024
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In our study, we looked at the boundary between the Earth's magnetic field and the interplanetary magnetic field emitted by the Sun, called the magnetopause. While other studies focus on the magnetopause motion near Earth's Equator, we have studied it in polar regions. The motion of the magnetopause is faster towards the Earth than towards the Sun. We also found that the occurrence of unusual magnetopause locations is due to similar solar influences in the equatorial and polar regions.
Adrian Pöppelwerth, Georg Glebe, Johannes Z. D. Mieth, Florian Koller, Tomas Karlsson, Zoltán Vörös, and Ferdinand Plaschke
Ann. Geophys., 42, 271–284, https://doi.org/10.5194/angeo-42-271-2024, https://doi.org/10.5194/angeo-42-271-2024, 2024
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In the magnetosheath, a near-Earth region of space, we observe increases in plasma velocity and density, so-called jets. As they propagate towards Earth, jets interact with the ambient plasma. We study this interaction with three spacecraft simultaneously to infer their sizes. While previous studies have investigated their size almost exclusively statistically, we demonstrate a new method of determining the sizes of individual jets.
Tomas Karlsson, Ferdinand Plaschke, Austin N. Glass, and Jim M. Raines
Ann. Geophys., 42, 117–130, https://doi.org/10.5194/angeo-42-117-2024, https://doi.org/10.5194/angeo-42-117-2024, 2024
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The solar wind interacts with the planets in the solar system and creates a supersonic shock in front of them. The upstream region of this shock contains many complicated phenomena. One such phenomenon is small-scale structures of strong magnetic fields (SLAMS). These SLAMS have been observed at Earth and are important in determining the properties of space around the planet. Until now, SLAMS have not been observed at Mercury, but we show for the first time that SLAMS also exist there.
Yasuhito Narita, Daniel Schmid, and Simon Toepfer
Ann. Geophys., 42, 79–89, https://doi.org/10.5194/angeo-42-79-2024, https://doi.org/10.5194/angeo-42-79-2024, 2024
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The magnetosheath is a transition layer surrounding the planetary magnetosphere. We develop an algorithm to compute the plasma flow velocity and magnetic field for a more general shape of magnetosheath using the concept of potential field and suitable coordinate transformation. Application to the empirical Earth magnetosheath region is shown in the paper. The developed algorithm is useful when interpreting the spacecraft data or simulation of the planetary magnetosheath region.
Simon Toepfer, Karl-Heinz Glassmeier, and Uwe Motschmann
Ann. Geophys., 41, 253–267, https://doi.org/10.5194/angeo-41-253-2023, https://doi.org/10.5194/angeo-41-253-2023, 2023
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The present study discusses the modeling and interpretation of magnetospheric structures via electromagnetic knots for the first time. The mathematical foundations of electromagnetic knots are presented, and the formalism is reformulated in terms of the classical wave telescope technique. The method is tested against synthetically generated magnetic field data describing a plasmoid as a two-dimensional magnetic ring structure.
Yasuhito Narita, Simon Toepfer, and Daniel Schmid
Ann. Geophys., 41, 87–91, https://doi.org/10.5194/angeo-41-87-2023, https://doi.org/10.5194/angeo-41-87-2023, 2023
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Magnetopause is a shielding boundary of planetary magnetic field. Many mathematical models have been proposed to describe or to reproduce the magnetopause location, but they are restricted to the real-number functions. In this work, we analytically develop a magnetopause model in the complex-number domain, which is advantageous in deforming the magnetopause shape in a conformal (angle-preserving) way, and is suited to compare different models or map one model onto another.
Simon Toepfer, Ida Oertel, Vanita Schiron, Yasuhito Narita, Karl-Heinz Glassmeier, Daniel Heyner, Patrick Kolhey, and Uwe Motschmann
Ann. Geophys., 40, 91–105, https://doi.org/10.5194/angeo-40-91-2022, https://doi.org/10.5194/angeo-40-91-2022, 2022
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Revealing the nature of Mercury’s internal magnetic field is one of the primary goals of the BepiColombo mission. Besides the parametrization of the magnetic field contributions, the application of a robust inversion method is of major importance. The present work provides an overview of the most commonly used inversion methods and shows that Capon’s method as well as the Tikhonov regularization enable a high-precision determination of Mercury’s internal magnetic field up to the fifth degree.
Martin Volwerk, Beatriz Sánchez-Cano, Daniel Heyner, Sae Aizawa, Nicolas André, Ali Varsani, Johannes Mieth, Stefano Orsini, Wolfgang Baumjohann, David Fischer, Yoshifumi Futaana, Richard Harrison, Harald Jeszenszky, Iwai Kazumasa, Gunter Laky, Herbert Lichtenegger, Anna Milillo, Yoshizumi Miyoshi, Rumi Nakamura, Ferdinand Plaschke, Ingo Richter, Sebastián Rojas Mata, Yoshifumi Saito, Daniel Schmid, Daikou Shiota, and Cyril Simon Wedlund
Ann. Geophys., 39, 811–831, https://doi.org/10.5194/angeo-39-811-2021, https://doi.org/10.5194/angeo-39-811-2021, 2021
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On 15 October 2020, BepiColombo used Venus as a gravity assist to change its orbit to reach Mercury in late 2021. During this passage of Venus, the spacecraft entered into Venus's magnetotail at a distance of 70 Venus radii from the planet. We have studied the magnetic field and plasma data and find that Venus's magnetotail is highly active. This is caused by strong activity in the solar wind, where just before the flyby a coronal mass ejection interacted with the magnetophere of Venus.
Daniel Schmid, Yasuhito Narita, Ferdinand Plaschke, Martin Volwerk, Rumi Nakamura, and Wolfgang Baumjohann
Ann. Geophys., 39, 563–570, https://doi.org/10.5194/angeo-39-563-2021, https://doi.org/10.5194/angeo-39-563-2021, 2021
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In this work we present the first analytical magnetosheath plasma flow model for the space environment around Mercury. The proposed model is relatively simple to implement and provides the possibility to trace the flow lines inside the Hermean magnetosheath. It can help to determine the the local plasma conditions of a spacecraft in the magnetosheath exclusively on the basis of the upstream solar wind parameters.
Martin Volwerk, David Mautner, Cyril Simon Wedlund, Charlotte Goetz, Ferdinand Plaschke, Tomas Karlsson, Daniel Schmid, Diana Rojas-Castillo, Owen W. Roberts, and Ali Varsani
Ann. Geophys., 39, 239–253, https://doi.org/10.5194/angeo-39-239-2021, https://doi.org/10.5194/angeo-39-239-2021, 2021
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The magnetic field in the solar wind is not constant but varies in direction and strength. One of these variations shows a strong local reduction of the magnetic field strength and is called a magnetic hole. These holes are usually an indication that there is, or has been, a temperature difference in the plasma of the solar wind, with the temperature along the magnetic field lower than perpendicular. The MMS spacecraft data have been used to study the characteristics of these holes near Earth.
Horia Comişel, Yasuhito Narita, and Uwe Motschmann
Ann. Geophys., 39, 165–170, https://doi.org/10.5194/angeo-39-165-2021, https://doi.org/10.5194/angeo-39-165-2021, 2021
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Identification of a large-amplitude Alfvén wave decaying into a pair of
ion-acoustic and daughter Alfvén waves is one of the major goals in the
observational studies of space plasma nonlinearity.
Growth-rate maps
may serve as a useful tool for predictions of the wavevector spectrum of density
or magnetic field fluctuations in various scenarios for the
wave–wave coupling processes developing at different stages in
space plasma turbulence.
Yasuhito Narita, Ferdinand Plaschke, Werner Magnes, David Fischer, and Daniel Schmid
Geosci. Instrum. Method. Data Syst., 10, 13–24, https://doi.org/10.5194/gi-10-13-2021, https://doi.org/10.5194/gi-10-13-2021, 2021
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The systematic error of calibrated fluxgate magnetometer data is studied for a spinning spacecraft. The major error comes from the offset uncertainty when the ambient magnetic field is low, while the error represents the combination of non-orthogonality, misalignment to spacecraft reference direction, and gain when the ambient field is high. The results are useful in developing future high-precision magnetometers and an error estimate in scientific studies using magnetometer data.
Simon Toepfer, Yasuhito Narita, Daniel Heyner, Patrick Kolhey, and Uwe Motschmann
Geosci. Instrum. Method. Data Syst., 9, 471–481, https://doi.org/10.5194/gi-9-471-2020, https://doi.org/10.5194/gi-9-471-2020, 2020
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The Capon method serves as a powerful and robust data analysis tool when working on various kinds of ill-posed inverse problems. Besides the analysis of waves, the method can be used in a generalized way to compare actual measurements with theoretical models, such as Mercury's magnetic field analysis. In view to the BepiColombo mission this work establishes a mathematical basis for the application of Capon's method to analyze Mercury's internal magnetic field in a robust and manageable way.
Ovidiu Dragoş Constantinescu, Hans-Ulrich Auster, Magda Delva, Olaf Hillenmaier, Werner Magnes, and Ferdinand Plaschke
Geosci. Instrum. Method. Data Syst., 9, 451–469, https://doi.org/10.5194/gi-9-451-2020, https://doi.org/10.5194/gi-9-451-2020, 2020
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We propose a gradiometer-based technique for cleaning multi-sensor magnetic field data acquired on board spacecraft. The technique takes advantage on the fact that the maximum-variance direction of many AC disturbances on board spacecraft does not change over time. We apply the proposed technique to the SOSMAG instrument on board GeoKompsat-2A. We analyse the performance and limitations of the technique and discuss in detail how various disturbances are removed.
Cited articles
Achar, B. N. N.: Reciprocal lattice in two dimensions, Am. J. Phys., 54, 663–665, https://doi.org/10.1119/1.14513, 1986. a
Ajtai, M.: The Shortest Vector Problem in L2 is NP-Hard for Randomized Reductions (Extended Abstract), in: Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, STOC '98, 10–19, Association for Computing Machinery, New York, NY, USA, https://doi.org/10.1145/276698.276705, 1998. a
Angelopoulos, V.: The THEMIS Mission, Space Sci. Rev., 141, 5–34, https://doi.org/10.1007/s11214-008-9336-1, 2008. a
Baumjohann, W. and Treumann, R.: Basic Space Plasma Physics – Revised Edition, Imperial College Press, https://doi.org/10.1142/P850, 2012. a
Bendat, J. S. and Piersol, A. G.: Random Data: Analysis and Measurement Procedures, pp. 189 ff., John Wiley & Sons Inc., ISBN: 0-471-06470-X, 1971. a
Borovsky, J. E. and Valdivia, J. A.: The Earth's Magnetosphere: A Systems Science Overview and Assessment, Surv. Geophys., 39, 817–859, https://doi.org/10.1007/s10712-018-9487-x, 2018. a
Brillouin, L.: Les électrons libres dans les métaux et le role des réflexions de Bragg, J. Phys. Radium, 1, 377–400, https://doi.org/10.1051/jphysrad:01930001011037700, 1930. a, b
Broeren, T. and Klein, K. G.: Data-driven Uncertainty Quantification of the Wave Telescope Technique: General Equations and Demonstration Using HelioSwarm, Astrophys. J. Suppl. S., 266, 12, https://doi.org/10.3847/1538-4365/acc6c7, 2023. a
Bronshtein, I., Semendyayev, K., Musiol, G., and Muehlig, H.: Handbook of Mathematics, p. 323 ff., Springer, 5th Edn., ISBN: 978-3-540-72121-5, 2007. a
Burch, J. L., Moore, T. E., Torbert, R. B., and Giles, B. L.: Magnetospheric Multiscale Overview and Science Objectives, Space Sci. Rev., 199, 5–21, https://doi.org/10.1007/s11214-015-0164-9, 2016. a
Capon, J.: High-resolution frequency-wavenumber spectrum analysis, P. IEEE, 57, 1408–1418, https://doi.org/10.1109/PROC.1969.7278, 1969. a
Capon, J., Greenfield, R., and Kolker, R.: Multidimensional maximum-likelihood processing of a large aperture seismic array, P. IEEE, 55, 192–211, https://doi.org/10.1109/PROC.1967.5439, 1967. a
Chanteur, G.: Spatial Interpolation for Four Spacecraft: Theory, in: Analysis Methods for Multi-Spacecraft Data, edited by: Paschmann, G. and Daly, P. W., Vol. 1, chap. 14, 349–370, International Space Science Institute, 1998. a
Constantinescu, O. D., Glassmeier, K.-H., Motschmann, U., Treumann, R. A., Fornaçon, K.-H., and Fränz, M.: Plasma wave source location using CLUSTER as a spherical wave telescope, J. Geophys. Res.-Space, 111, A09221, https://doi.org/10.1029/2005JA011550, 2006. a
Dunlop, M., Southwood, D., Glassmeier, K.-H., and Neubauer, F.: Analysis of multipoint magnetometer data, Adv. Space Res., 8, 273–277, https://doi.org/10.1016/0273-1177(88)90141-X, 1988. a
Escoubet, C. P., Fehringer, M., and Goldstein, M.: Introduction The Cluster mission, Ann. Geophys., 19, 1197–1200, https://doi.org/10.5194/angeo-19-1197-2001, 2001. a
Eyer, L. and Bartholdi, P.: Variable stars: Which Nyquist frequency?, Astron. Astrophys. Sup., 135, 1–3, https://doi.org/10.1051/aas:1999102, 1999. a
Fincke, U. and Pohst, M.: Improved methods for calculating vectors of short length in a lattice, including a complexity analysis, Math. Comput., 44, 463–471, 1985. a
Frisch, U. and Kolmogorov, A. N.: Turbulence: the legacy of AN Kolmogorov, Cambridge university press, ISBN: 0 521 45103 5, 1995. a
Glassmeier, K.-H., Motschmann, U., Dunlop, M., Balogh, A., Acuña, M. H., Carr, C., Musmann, G., Fornaçon, K.-H., Schweda, K., Vogt, J., Georgescu, E., and Buchert, S.: Cluster as a wave telescope – first results from the fluxgate magnetometer, Ann. Geophys., 19, 1439–1447, https://doi.org/10.5194/angeo-19-1439-2001, 2001. a, b, c, d, e
Haykin, S. S.: Adaptive Filter Theory, 396–399, Prentice Hall Information and System Science Series, Prentice-Hall Inc., New Jersey, 2nd Edn., ISBN: 0-13-013236-5, 1991. a
Hoffstein, J., Pipher, J., and Silverman, J. H.: An Introduction to Mathematical Cryptography, Vol. 1, Springer, https://doi.org/10.1007/978-0-387-77993-5, 2008. a, b
Kirchner, J. W.: Aliasing in
Klein, K. and Spence, H. and the HelioSwarm Science Team: HelioSwarm: Leveraging Multi-Point, Multi-Scale Spacecraft Observations to Characterize Turbulence, EGU General Assembly 2021, online, 19–30 April 2021, EGU21-6812, https://doi.org/10.5194/egusphere-egu21-6812, 2021. a
Lenstra, A. K., Lenstra, H. W., and Lovász, L.: Factoring polynomials with rational coefficients, Math. Ann., 261, 515–534, 1982. a
Motschmann, U., Woodward, T. I., Glassmeier, K. H., Southwood, D. J., and Pinçon, J. L.: Wavelength and direction filtering by magnetic measurements at satellite arrays: Generalized minimum variance analysis, J. Geophys. Res.-Space, 101, 4961–4965, https://doi.org/10.1029/95JA03471, 1996. a, b, c, d, e
Narita, Y.: Plasma Turbulence in the Solar System, Springer Berlin, https://doi.org/10.1007/978-3-642-25667-7, 2012. a
Narita, Y.: A Note on Capon's Minimum Variance Projection for Multi-Spacecraft Data Analysis, Front. Phys., 7, 8, https://doi.org/10.3389/fphy.2019.00008, 2019. a
Narita, Y. and Glassmeier, K.-H.: Spatial aliasing and distortion of energy distribution in the wave vector domain under multi-spacecraft measurements, Ann. Geophys., 27, 3031–3042, https://doi.org/10.5194/angeo-27-3031-2009, 2009. a, b, c, d
Narita, Y., Glassmeier, K.-H., and Treumann, R. A.: Wave-Number Spectra and Intermittency in the Terrestrial Foreshock Region, Phys. Rev. Lett., 97, 191101, https://doi.org/10.1103/PhysRevLett.97.191101, 2006. a
Narita, Y., Glassmeier, K.-H., and Motschmann, U.: High-resolution wave number spectrum using multi-point measurements in space – the Multi-point Signal Resonator (MSR) technique, Ann. Geophys., 29, 351–360, https://doi.org/10.5194/angeo-29-351-2011, 2011. a
Narita, Y., Plaschke, F., Nakamura, R., Baumjohann, W., Magnes, W., Fischer, D., Vörös, Z., Torbert, R. B., Russell, C. T., Strangeway, R. J., Leinweber, H. K., Bromund, K. R., Anderson, B. J., Le, G., Chutter, M., Slavin, J. A., Kepko, E. L., Burch, J. L., Motschmann, U., Richter, I., and Glassmeier, K.-H.: Wave telescope technique for MMS magnetometer, Geophys. Res. Lett., 43, 4774–4780, https://doi.org/10.1002/2016GL069035, 2016. a
Narita, Y., Glassmeier, K.-H., and Motschmann, U.: The Wave Telescope Technique, J. Geophys. Res.-Space, 127, e2021JA030165, https://doi.org/10.1029/2021JA030165, 2022. a, b, c, d
Neubauer, F. M. and Glassmeier, K.-H.: Use of an array of satellites as a wave telescope, J. Geophys. Res.-Space, 95, 19115–19122, https://doi.org/10.1029/JA095iA11p19115, 1990. a, b
Nyquist, H.: Certain Topics in Telegraph Transmission Theory, Transactions of the American Institute of Electrical Engineers, 47, 617–644, https://doi.org/10.1109/T-AIEE.1928.5055024, 1928. a
Odlyzko, A. M.: The rise and fall of knapsack cryptosystems, in: Cryptology and Computational Number Theory, Vol. 42 of Proceedings of Symposia in Applied Mathematics, 75–88, American Mathematical Society, 1989. a
Pinçon, J.-L. and Glassmeier, K.-H.: Multi-Spacecraft Methods of Wave Field Characterisation, in: Multi-Spacecraft Analysis Methods Revisited, edited by: Paschmann, G. and Daly, P. W., Vol. 8, 47–54, International Space Science Institute, ISBN: 987-92-9221-937-6, 2008. a
Pinçon, J. L. and Lefeuvre, F.: Local characterization of homogeneous turbulence in a space plasma from simultaneous Measurements of field components at several points in space, J. Geophys. Res.-Space, 96, 1789–1802, https://doi.org/10.1029/90JA02183, 1991. a
Pinçon, J. L. and Motschmann, U.: Multi-Spacecraft Filtering: General Framework, in: Analysis Methods for Multi-Spacecraft Data, edited by: Paschmann, G. and Daly, P. W., Vol. 1, chap. 3, 65–78, International Space Science Institute, 1998. a
Plaschke, F., Glassmeier, K.-H., Constantinescu, O. D., Mann, I. R., Milling, D. K., Motschmann, U., and Rae, I. J.: Statistical analysis of ground based magnetic field measurements with the field line resonance detector, Ann. Geophys., 26, 3477–3489, https://doi.org/10.5194/angeo-26-3477-2008, 2008. a, b
Pohst, M.: A modification of the LLL reduction algorithm, J. Symb. Comput., 4, 123–127, https://doi.org/10.1016/S0747-7171(87)80061-5, 1987. a
Retino, A.: The Plasma Observatory: exploring particle energization in space plasmas through multi-point, multi-scale in situ measurements, in: 43rd COSPAR Scientific Assembly, 28 January–4 February, Vol. 43, p. 1091, 2021. a
Schulz, L.: The m-Dimensional Spatial Nyquist Limit Using the Wave Telescope for Larger Numbers of Spacecraft Dataset, Zenodo [data set], https://doi.org/10.5281/zenodo.7604102, 2023. a
Shannon, C.: Communication in the Presence of Noise, P. IRE, 37, 10–21, https://doi.org/10.1109/JRPROC.1949.232969, 1949. a
Shmueli, U.: A general introduction to space groups, Vol. B: Reciprocal Space of International Tables for Crystallography, 2–9, Springer, 3rd Edn., ISBN: 978-1-4020-8205-4, 2008. a
Toepfer, S., Narita, Y., Heyner, D., Kolhey, P., and Motschmann, U.: Mathematical foundation of Capon's method for planetary magnetic field analysis, Geosci. Instrum. Method. Data Syst., 9, 471–481, https://doi.org/10.5194/gi-9-471-2020, 2020. a, b
VanderPlas, J. T.: Understanding the Lomb–Scargle Periodogram, Astrophys. J. Suppl. S., 236, 16, https://doi.org/10.3847/1538-4365/aab766, 2018. a, b, c
Zhang, L., He, J., Narita, Y., and Feng, X.: Reconstruction Test of Turbulence Power Spectra in 3D Wavenumber Space with at Most 9 Virtual Spacecraft Measurements, J. Geophys. Res.-Space, 126, e2019JA027413, https://doi.org/10.1029/2019JA027413, 2021. a
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 power spectrum estimator; hence, this can be applied to other fields of research.
The upper detection limit in reciprocal space, the spatial Nyquist limit, is derived for...
