• 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"

Correction to the first author name: It’s Leonard instead of Leonhard.

### 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.

The authors thank the reviewer for the general, positive assessment of the manuscript. Specific comments below will be addressed one by one in the following.

### 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.

Regarding the first part of this comment concerning aliasing: We agree that the topic of aliasing, which is elementary for the study, has been introduced too briefly. Therefore, we want to add additional explanation and formulae to the beginning of section 2.2. Please also refer to the answer made in the specific comments on line 114-120 below for more details on proposed changes.

Regarding the second part of the comment: Indeed the spatial aliasing and the findings of this study concerning it are not restricted to the wave telescope. This is mentioned on several occasions, e.g. in the abstract and conclusions. However, we agree that this fundamental aspect should be emphasized on other occasions as well, especially in section 2. Therefore, we would like to add an introductory paragraph to the section prior to the detailed review of the wave telescope technique. As the current study is focused on the wave telescope technique, the assessment of other estimators goes beyond the scope of the paper. Thus, we will explain the study’s sole focus on the wave telescope but emphasize the general nature of spatial aliasing. Additionally, we want to cite the two suggested references in section 2.2 in the sentence (lines 121-123) “The concept of aliasing and frequency domain periodicity can be directly transferred from frequency space to k-space in general (Dunlop et al., 1988; Chanteur, 1998), in particular for the wave telescope (Neubauer and Glassmeier, 1990; Pinçon and Motschmann, 1998; Glassmeier et al., 2001; Narita and Glassmeier, 2009).” We think this describes exactly what the reviewer is aiming at as well.

• 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).

Thank you for this important comment. As said above, we agree that the explanation of aliasing and its implications for time and frequency space should be explained more. Nevertheless the authors think that discrete Fourier transform (DFT) serves as a good explanatory example when introducing aliasing. The DFT is, as the reviewer noted, due to the availability of FFT a numerically most efficient transform and it is the most widely used and very basic transform when wanting to acquire the frequency domain. Additionally, the introduction of the DFT not only serves as an introduction for aliasing and the Nyquist limit but also as a precursor for the later introduction of the technique of zero adding for irregularly sampled time series, explained in section 3.1. Also DFT is needed for the preparatory work on time series when wanting to apply the wave telescope, as the time series b(t,r) first needs to be transformed to a frequency series b(w,r) (see equation 1). Therefore, the authors would like to leave the definition of the DFT in, but want to incorporate it into a more expanded explanation of aliasing and the Nyquist limit. Additionally, the authors want to add the backwards transform and remove one false claim made as requested in the technical corrections.

Considering the reviewers remarks that Fourier transform is only a special way to estimate a PSD: We agree to this remark. Combined with the remark of another reviewer, we want to replace the statement that the WT is a Fourier transform estimator by “With a plane wave model, the wave telescope can be interpreted as a power spectrum estimator substituting power spectral density estimation via a spatial Fourier transform (Motschmann et al., 1996; Plaschke et al., 2008; Narita and Glassmeier, 2009).” (line 92-93). Throughout the paper, there are numerous occasions, where the claim was made that the wave telescope estimates a Fourier transform. We want to replace this by “power spectrum estimation” as this describes more precisely what the wave telescope is doing. Additionally, we want to remark in the study (in the beginning of section 2.2.) that “In order to yield a spectrum, one normally estimates the power spectral density (PSD), which can be accomplished by a range of different methods. Nevertheless, here we want to focus on Fourier transform as an illustrative example...”.

• 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.

We understand that this statement has to be explained in more detail. In our paper, we adopt the definition of a unit cell from crystallography, for example as presented in (Souvignier, International Tables for Crystallography (2016). Vol. A, Section 1.3.2, pp. 22–28.). They present 2 differentiations of unit cells, one the one hand the “primitive unit cell”, which is synonymous to the parallelepiped construction of the periodic cell in our paper and in previous wave telescope publications. On the other hand there is the “Voronoi cell” equal to the first Brillouin zone, which is defined by a Wigner-Seitz cell approach (constructing the perpendicular bisector at half the distance to each lattice point and find the smallest volume of the intersecting bisectors lines or planes, depending on the number of dimensions). This cell is the locus of points in space that are closer to the origin lattice point than to any of the other lattice points (Souvignier, 2016). Only these point in reciprocal space constitute the points at which no aliasing is taking place. Thus it is the only unit cell within which there are no aliased points contrary to the parallelepiped cell.

All this is in contrast to specific statements in papers on the wave telescope:

Neubauer and Glassmeier 1990: “The subvolume closest to k= 0 can be described by (equation 12a)” -> not correct, as the subvolume closest to k=0 would be the 1^st Brillouin zone and not the primitive unit cell calculated in their equation 12a.

Narita and Glassmeier 2009: “Reciprocal vectors are not always mutually orthogonal but have in general various angles. In such a case the Brillouin zone forms a diamond-shaped cell in the wave vector domain.” à This is not the 1^st Brillouin zone, but the parallelepiped cell.

Narita et al 2022: “Therefore, it is conventional and safer to search for wavevectors within the first Brillouin zone spanned by the reciprocal vectors […]” à The Brillouin zone is not spanned by the reciprocal vectors, this again is the parallelepiped cell.

We want to replace the current wording with a compacted version of the above, stressing the difference between the first Brillouin zone and the primitive unit cell. As the concept of these two unit cells is well accepted in solid state physics and crystallography and the choosing of the right unit cell is important to the problem of aliasing, we think it is beneficial to stress this terminological differentiation to avoid future confusion. In addition, we have noted that in our study, further distinction is needed of “spatial Nyquist limit” and “periodic cell”. So far, this has not been clear throughout the paper and we want to change especially Figure captions to make this clearer, always bearing in mind the explanations made in the above paragraph.

• 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").

Please refer to the answer to the respective comment 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?

The terminology and difference between regular and irregular sampling points is explained in section 3.1 on the 1D case. Nevertheless, we understand from the reviewer’s comment, that there still is some ambiguity regarding the use of “regular sampling points”. Thus we want to reword the sentence in question to better show that regular sampling points refers to the number of spacecraft rather than their configuration. We also want to add that the regular sampled 2D case is only represented by a three spacecraft configuration not forming a line in space (thus covering all 2 spatial dimensions). Additionally, we will explain that the specific spacecraft configuration used as an example for such a regular sampling points case, resulting in the spectrum of Figure 5, is indeed an equilateral triangle, here.

• 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)?

We apologize for this incorrect statement, this has been overlooked in final checking of the manuscript and the sentence will be removed.

• 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"?

As said above (comment on line 249), the terms regular and irregular indeed refer to the number of spacecraft. In the paper, we want to explain this in more detail in the above paragraph (see the response to comment on line 249), thus we argue that at this point in the text it should be clear to the reader what is meant by irregular in the context of a 2D situation.

• 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)?

Again, we would like to refer to the response to the comment on line 249. There, we want to explain in more detail what is meant by regular and irregular spaced sampling points. Thus, this should then be clearer to the reader. However, to be even more precise, we would like to add that a regular spaced subset of sampling points is constituted by a subset of m+1 sampling points that span the m spatial dimensions.

### 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.

Thank you for this comment. The authors agree that the adding of the back transform (synthesis) of the FT adds value and will integrate this. Also we agree that the statement “the discrete FT equals the values of the continuous FT” is indeed incorrect. The authors apologize for this error. Please refer to the above answer to the specific comment on lines 114-120 for additional changes the authors want to implement regarding this paragraph.

• 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.

Thank you for this remark. As done before, we want to add the backwards transform of the presented DFT subject to “zero-adding”. We are aware that we describe a different technique than zero-padding, but we now understand that we should explain this also to the reader and thus want to add a sentence like ”…not to be confused with zero-padding…”.

Considering the remark that “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.” We would like to argue that the concept of zero adding is very illustrative in showing how the effective sampling distance respectively the effective Nyquist frequency comes into being. Although the DFT presented in formula (16) has to be transformed to a more complex form using different running indices for the sine and cosine transform part of DFT as presented in Bretthorst (2001) - we would like to keep this formula as is. Bretthorst himself introduces the very explanatory concept of zero adding (although he does not use this terminology) to explain the effective sampling distances (in his work largest effective dwell time) for the DFT. It is also mentioned in VanderPlas (2018) that the concepts shown in Eyer & Bartoldi (1999) can be understood as an additional window with zeros at some points added to a uniform dataset. While he approaches the problem from the other side, the interpretation is synonymous to the concept of zero-adding. Thus we think it is sufficient to rewrite the paragraph in question emphasizing that we use the DFT formula here just as an illustrative example, mentioning  that the presented formula can’t be seen as a transform of DFT to non-uniform sampling as this is more complex (and refer to Bretthorst (2001)).

### References

Souvignier, International Tables for Crystallography (2016). Vol. A, Section 1.3.2, pp. 22–28.

Jacob T. VanderPlas (2018), Understanding the Lomb–Scargle Periodogram, ApJS 236 16 DOI 10.3847/1538-4365/aab766

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