Estimating Gradients of Physical Fields in Space

by Yufei Zhou and Chao Shen

General comments

This paper discusses the solution of the least-squares system that stems from a multi-dimensional Taylor series approximation of a scalar field. The authors foresee the inclusion of the time dimension in their analysis. The paper does a nice job of pointing out the opportunities and some of the difficulties of applying such least-squares gradient computation techniques to situations where a larger number of spacecraft is available and/or where one attempts to assess the higher-than-linear gradients. The existence conditions are interpreted in a geometric way, which is helpful. The paper contains new ideas and is at the forefront of research.

The paper is well-structured. There is a good overview of the relevant literature. The paper also puts the work properly into context. The conclusions are clear. The paper length is appropriate.

I have a few suggestions for improving the presentation of the material and I also have a few questions, see specific comments below.

There are several language and typographical issues in the manuscript; see technical corrections listed below. The paper would benefit from being thoroughly reread once more.

The paper will likely be suitable for publication after minor revision.

Specific comments

The authors discuss the weighted least-squares problem that arises when trying to fit a polynomial approximation to the observations of a scalar field at multiple points. This is, in general, an overdetermined problem. The solution of such a problem is well-known standard numerical mathematics: There is the theory of the “generalized inverse” of an overdetermined system and the use of the singular value decomposition to solve the system. None of that math, however, is referred to explicitly in the paper, while the paper reflects that math in the specific context of gradient computation. For instance, matrix (26) shows the singular value decomposition of the original matrix, which in the case discussed here is rank-deficient, reflecting the existence conditions for a solution in terms of the singular values as formulated by De Keyser et al. 2007. It would be extremely useful to highlight throughout the entire manuscript how the findings reflect this standard mathematical approach which many readers have as background knowledge.

The authors focus on the basic ALQG system. They argue that the approach utilizing gradients avoids having to make specific physical assumptions. However, in many practical cases there are physical constraints such as the divergence-free nature of the magnetic field. This is briefly touched upon at the beginning of section 2. As the authors suggest one can apply the scalar field approach to each of the vector field components, but that does not incorporate the constraint yet. In section 4.2 the authors propose that one can drop some gradient components, which is a (restricted) form of geometric constraints. How can the proposed technique incorporate more generic physical and geometric constraints? How do the conclusions of the present paper generalize to the situation where such constraints can be applied?

The numerical conditioning of the ALQG system depends on the scaling of the variables. The authors use the spatial coordinates as such, and the time coordinate multiplied by a characteristic speed. What if the spatial variations are very anisotropic, as is often the case in magnetic field dominated situations? Note that this is related to the question of geometric constraints and the “homogeneity scales” introduced by De Keyser et al. 2007.

The authors mention the problem of error estimation. How do they view/compare their approach with the one proposed by De Keyser et al. 2008, where the effects of measurement errors and approximation errors (in space and time) are combined?

Matrix (20) is of a form that is known to be likely ill-conditioned (a Vandermonde matrix) – admittedly, this ill-conditioning is more pronounced as the polynomial approximation degree becomes higher; for a degree 2 the situation is not so bad yet. But perhaps this deserves a word of caution: the technique is in principle applicable to higher-degree approximations, but in practice there are clear limitations also from the numerical point of view.

On line 267: For constellations with small spacecraft separation distances, the positioning error may become considerable. A word of caution would be welcome here.

It is not clear to why the authors introduce the wave field w. No specific conclusions are drawn in the error analysis of section 5 about this field. I therefore believe that the authors could just think of the wave field as part of the scalar field that is to be modelled. The end result would simply be to remove w from the formalism and thus simplifying it. I leave this to the authors to judge, but when keeping w, then an explicit discussion of its role in the error analysis in section 5 would be welcome.

Technical corrections

Throughout the text: replace “A” by “Appendix A” to refer to the appendix.

Throughout the text: “Keyser” -> “De Keyser”; the references on line 400-404 should read as follows: “De Keyser, J.: …”

30: “of reconnections” -> “the reconnection region”

35: “The algorithm” -> “An algorithm”

39: “As if” -> “If”

46: “point distribution” -> “the point distribution”

48 and later: notation f_’α is strange; shouldn’t this be f^’_α ?

150: “Similar result has” -> “A similar result has” or “Similar results have”

152: “be on a surface” -> better “lie on a surface”?

163: I do not think “great” has the connotation that you want in this sentence. Better “important”?

163: “desgins” -> “design”

172: “sphere” -> “the sphere” or “a sphere”

204: “exists” -> “exist”

235: “To the ease of” -> better “To facilitate the”?

245: “an unique” -> “a unique”

310: “have” -> “has”

The authors explore future multipoint techniques for constellation missions to estimate gradients of physical quantities. The analytical theory is well developed and comprehensive. I only have minor comments on the text as presented. The manuscript, however, contains no figures which this reviewer feels would greatly aid interpretation by readers who are less mathematical in their thinking and more visual. A few simple diagrams demonstrating the concepts and findings would greatly complement the existing text. If some figures are added and the minor points below are addressed, I would recommend publication.

Line 30: Change "of reconnections" to "in reconnection"

Lines 31-32: This states the reconstruction avoids assumptions, however, the underlying assumptions about the forms of gradients are omitted, e.g. that they are relatively consistent over the scales of the spacecraft separation.

Lines 56-59: It would be good to explicitly mention that in practical applications measurements include noise which may then affect estimates of gradients.

Line 70: change to "dipole (and higher-order moments)"

Lines 71-72: The magnetosheath is highly non-uniform over the scale of its thickness, so please be specific over what sorts of distances you are referring to.

Line 74: Are the wave fields really waves or just residuals? You mention they must have smaller scales, referring to their physical size, but do they not also need to have smaller amplitude fluctuations?

Line 77: It would be good to mention if the speed v needs to be chosen to be the same for all measurement points or if it can be allowed to vary.

Line 80 (and throughout): "A" needs to change to "Appendix A"

Line 147: "a algebraic" change to "an algebraic"

Lines 197-203: This is almost identical to the previous paragraph, remove.