The m-Dimensional Spatial Nyquist Limit Using the Wave Telescope for Larger Numbers of Spacecraft
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.
Leonard Schulz et al.
Status: open (until 28 Apr 2023)
- RC1: 'Comment on egusphere-2023-172', Anonymous Referee #1, 28 Mar 2023 reply
Leonard Schulz et al.
Leonard Schulz et al.
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Comments on the manuscript by Leonhard Schulz et al. entitled "The m-Dimensional Spatial Nyquist Limit Using the Wave Telescope for Larger Numbers of Spacecraft"
### General comments
The study is concerned with spatial aliasing phenomena in spacecraft array wave vector estimation techniques when more than the minimum number of probes needed to identify all wave vector components are available. To generalize one-dimensional approaches employed in studies of the temporal aliasing effect for unevenly sampled time series, the authors apply concepts from number theory to determine a suitable elementary cell in the lattice of reciprocal vectors. While results are shown for the wave telescope technique, the underlying geometrical arguments are of general nature so that the approach should be applicable also to other wave vector estimators.
The overall quality of the manuscript is convincing, the logic is sound, the organization is clear, and the figures are of general publication standards. In the core sections 3.2 an 3.3, the development of the mathematics is rigorous. In sections 1, 2, and 3.1, some motivational arguments are confusing and need further clarification.
### Specific comments
Since the focus of this study is on the phenomenon of (spatial) aliasing, the authors could clarify its conceptual basis in more detail: How does the ambiguity called aliasing expresses itself, and what are the algebraic conditions? In the context of spacecraft array wave analysis, without reference to a specific estimator, one of the earliest considerations was offered by Dunlop et al. (1988) who based their argument on wave vector ambiguity in the representation of the phase, see also the presentation by Chanteur (1998) using reciprocal vectors based on mesocentric position vectors. The current setup of the paper, starting the presentation with a detailed review of the wave telescope technique in section 2.1, suggests that the spatial aliasing analysis is restricted to a specific method while it is more general.
Lines 114-120: With the addition of a conceptual characterization of spatial aliasing as suggested above, it may be worthwhile rewriting the beginning of section 2.2. In its present form, the first paragraph on Fourier transformation appears incomplete and partially incorrect (see below under "Technical corrections") while its main (and only?) purpose seems to be the introduction of aliasing and the Nyquist limit (Nyquist frequency, or critical frequency) in the context of regular (evenly sampled) time series. Furthermore, as textbooks on (array) signal processing (e.g., Bendat and Piersol, 1971; Pillai, 1989) and also the cited literature on the wave telescope or k-filtering technique (Pincon and Lefeuvre, 1991; Pincon and Motschmann, 1998) show, power spectral density (PSD) estimation can be accomplished by a range of different methods, with Fourier techniques only a special (and, due to the availability of the Fast Fourier Transform FFT for evenly sampled time series, a numerically most efficient) approach. Even in the key 1D aliasing references mentioned in this paper, PSD estimation takes center stage using a Bayesian perspective (Bretthorst, 2001), through a spectral window function, convolved with the true spectrum to yield the spectrum obtained after irregular sampling (Eyer and Bartholdi, 1999), or generalized periodogram estimation (Mignard, 2005).
Lines 149-153, "However, as an important remark, ... Brillouin zone ": Since this is labeled as an important remark, it seems worthwhile explaining it in more detail, and what exactly is contrary to comments made in earlier papers.
Lines 163-174: Please consider to rewrite the paragraph or at least amend it with additional explanations and clarifications. Apparently, the text is meant to introduce the aliasing problem for irregular sampling using the Fourier transform of time series, but the presentation remains unclear, and may be partially incorrect (see below under "Technical corrections").
Line 249, "The regular sampling point case ...": A brief characterization of the underlying three-S/C configuration for 2D wave vector estimation should be added. Does the term refer to an equilateral triangle, or to any configuration of three S/C in two dimensions?
Line 253, "Different deformations of the outer region of the maxima result from numerical artefacts": Please clarify. Are the numerical artefacts related to the necessarily finite representation of numbers in computer memory (machine accuracy)?
Line 256, "2D situation and four irregularly positioned S/C": How would a configuration of four regularly positioned S/C look like? Or does the term "irregular in the 2D context simply mean "more than three S/C"?
Line 388, "for all regularly spaced subsets of sampling points": Please clarify. Here, in 2D with 4 S/C, does "all regularly spaced subsets" refer to all subsets comprising three S/C? How would the procedure look like with 5 S/C (and still in 2D)?
### Technical corrections
Lines 114-120, "at these frequencies the discrete FT equals the values of the continuous FT", Eq. (9): The discrete Fourier Transform (FT) can be defined in different ways, depending on the purpose and the context (e.g., Eriksson, 1998). In the present manuscript, Eq. (9) yields Fourier analysis (forward FT). To clarify the purpose and the normalization used here, consider adding also the formula for Fourier synthesis (backward FT). The statement "the discrete FT equals the values of the continuous FT" does not seem correct for general time series as the continuous FT is based on the continuous signal b(t) and thus uses information also in between the sampled times. If the statement is supposed to be kept, the definition of the corresponding continuous FT needs to be added, and the reasoning explained.
Lines 163-174: As in the previous comment on Eq. (9), the formula in Eq. (16) needs clarification and additional explanations. Apparently, it is based on a FT formula (Fourier analysis) for nonuniform sampling which is amended by the addition (insertion) of zeros, but then the corresponding formula for (nonuniform) Fourier synthesis should be included. The Fourier transform of the time series after "zero adding" may formally produce the same output, but it remains unclear in which sense the result gives the coefficients of a harmonic series expansion (and in which form exactly), and how it can be interpreted as a starting point for PSD estimation. It appears as if the paragraph is supposed to provide a shortcut to the spectral window function approach presented by Eyer and Bartholdi (1999), but there the argument is more complete, explaining the connection to the true power spectrum through a convolution. Note that the concept of "zero adding" (inserting zeros into an irregular time series) differs from the popular technique of "zero padding", often used in the time series context to amend an evenly (!) sampled time series with zeros to arrive at a sample length that takes advantage of the computational efficient FFT.
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