Review of  egusphere-2024-1330 (ANGEO)

Quadratic Magnetic Gradients from 7-SC and 9-SC Constellations

by Chao Shen et al.

This paper describes a least-squares gradient computation technique for linear and quadratic magnetic gradients. The technique is applied to two test cases to show its performance. One of the goals is to demonstrate that 7- and 9-spacecraft constellations provide enough measurements to infer those gradients. The paper starts with an introduction that properly references earlier work on gradient computation. It then presents the technique, the test cases, and it ends with a conclusion.

The introduction could be better structured. This can probably be remedied by shifting some material from the description of the technique to the introduction, so that the characteristics of the technique are put in contrast with the earlier work on the subject (see details below). The actual contents of the paper is sound and will undoubtedly be useful for the community. I do have a number of questions/suggestions regarding the method, the test cases, and the presentation (see comments below).

The manuscript would benefit seriously from language editing. I have listed just a few language suggestions (see below). 

Major comments

In the abstract and at various places in the text, the authors say that 4 measurements are needed for computing the linear gradient and 10 measurements are needed for the quadratic gradient. This statement is somewhat imprecise. It would be more correct to state instead that 4 simultaneous measurements are needed for the linear spatial gradient and 10 simultaneous measurements for the quadratic spatial gradient components of a scalar field. Perhaps it would also be useful to mention from the start that, when using the least-squares approach, one adds the time derivatives and the mixed space-time derivatives, so that at least 5 measurements are needed for the linear and 15 for the non-linear gradients of a scalar field in general. 

In the description of the method, I was expecting that somewhere the condition div B = 0 would have been incorporated. If I understand well, that is not the case; rather that condtion is used for evaluating the precision of the technique. Still, inclusion of a div B = 0 constraint would make the technique more precise and robust, as it can remove a possible ill-posedness of the problem for certain spacecraft constellation geometries. Can the authors comment on whether and how such a condition can be included?

For the reader it is confusing that the time derivative is used (line 109) in the explanation of the technique, while time derivatives or mixed space-time derivatives do not appear in the variable count on lines 122ff. 

I think having the paragraph from line 122ff in the introductory section would help in setting the broader problem of balancing the number of unknowns versus the number of available observations.

The discussion of the volume tensor states that its determinant should be nonzero. At this point, no mention is made of the condition number, which is – practically speaking – more important than the tensor being non-singular. The statement that “This algorithm requires that the constellation be composed of at least seven spacecraft and that its configuration is non-planar. Because both the 9S/C HelioSwarm and 7S/C Plasma Observatory satisfy these requirements, the linear and quadratic magnetic gradients can be readily obtained” is therefore perhaps a bit optimistic. It is appreciated that in the examples the eigenvalues of the volume tensor are given. Still, that only partially describes the conditioning of the problem.

Figure 1 presents a very specific shape of the 7 S/C constellation. Such a constellation is nice for conceptually presenting the idea of “nested tetrahedra”, but cannot be easily maintained in space in practice. This figure is nowhere referenced nor discussed.

The effect of measurement errors is not included in the calculation. This is assuming a homogeneous set of instruments, but that may not be the case for Plasma Observatory, for instance, where there are different instruments on mother and daughter spacecraft.

Nothing is said about error estimates on the results (in the case where you do not know the exact solution). Does the technique allow you to produce such error estimates? If so, it would be useful to compare these estimates to the actual errors for the two test cases.

Minor comments

-    Title: Personally, I would try to avoid the “SC” abbreviation in the title. Better change into: “Quadratic Magnetic Gradients from 7- and 9-Spacecraft Constellations”

-    line 12: remove “therefore”

-    line 13: from -> from the

-    line 17: The tests -> Tests

-    line 18: verifies -> verified

-    line 23: iteration algorithm -> iterative algorithm

-    line 38: gradient -> gradients

-    line 43: tetrahedral -> a tetrahedral

-    line 43: such the missions -> such missions

-    line 62: consisting -> consisting of

-    line 66: an ESA’s new mission -> a new ESA mission

-    line 68: drawn -> inferred

-    line 74ff: I suggest to change punctuation into: “a description of the tests conducted for two typical magnetic structures (a cylindrical force-free flux rope and a dipole magnetic field), which were utilized to check the validity and accuracy of the new algorithm, is given …”

-    line 76: error -> accuracy

-    line 84: references -> reference frames

-    line 84: … of the magnetic field

-    caption of Figure 1: relative to the constellations -> relative to the constellation

-    line 95, 97, 195 and elsewhere: no capital needed at the beginning of the line

-    line 99: draw -> infer

-    line 225: “The characteristic size of the S/C is twice the square root of the maximum eigenvalue” makes no sense. Size of the S/C constellation?

-    Fig 4, 6, 8: light yellow lines are hardly visible

-    explain abbreviations when first used: NASA, ESA

Comments on the manuscript entitled

Quadratic Magnetic Gradients from 7-SC and 9-SC Constellations

submitted by Chao Shen, Gang Zeng, and Rungployphan Kieokaew.

General comments

The manuscript is concerned with a novel method to estimate the first and second spatial derivatives of stationary magnetic structures from multi-point measurements in constellations consisting of N spacecraft where N=7 (Plasma Observatory) or N=9 (HelioSwarm). In addition to the set of 3N simultaneous magnetic field measurements, also discrete representations of the 3N first time derivatives are utilised in the method, amounting to an effective number of 6N input data that are used for estimating the 33 model parameters (30 parameters in the second-order Taylor expansion, and 3 parameters for the velocity of the stationary structure). The paper presents the model equations and an iterative algorithm for estimating the parameters. The method is demonstrated using two magnetic field models. Deviations of the model predictions from their analytical counterparts are discussed.

While the study presented here can be considered a proof of concept that introduces the general framework and demonstrates the processing flow of the proposed method, a number of open issues and limitations need to be addressed and critically discussed, e.g., the concept of stationarity in the context of magnetohydrodynamics, the different types of errors, and the numerical stability of the inversion/reconstruction method.

Stationarity in the context of magnetohydrodynamics:

The method utilises discrete time derivative measurements through advection-type equations (3): $\frac{\partial \mathbf{B}}{\partial t} = - \mathbf{V} \cdot \nabla \mathbf{B}$. In magnetohydrodynamics, however, the local time derivative of the magnetic field $\mathbf{B}$ is connected to the velocity $\mathbf{V}$ through Faraday's law and an appropriate Ohm's law, which in the ideal case (collision-free plasmas in geospace and the heliosphere) equates the local time derivative with the curl of the cross product of velocity and magnetic field: $\frac{\partial \mathbf{B}}{\partial t} = \nabla \times ( \mathbf{V} \times \mathbf{B} )$ (hydromagnetic theorem), implying the invariance of magnetic flux through a surface transported with the plasma flow. The authors are asked to explain in which sense their notion of stationarity differs from the canonical interpretation (frozen-in magnetic flux) in space plasma physics.

Discretisation errors:

As pointed out by the authors in lines 223-227, the separation of spacecraft in the array introduces up to three different spatial discretisation scales. It should be added that a fourth spatial scale comes into play through the finite difference representation of local time derivatives, namely, the product of the intrinsic time scale (time resolution) with the velocity of the magnetic structure in the spacecraft frame.

Iteration errors:

When in Section 3 the convergence properties of the iterative method are discussed, a particular type of error considered there is the mismatch of the actual limit of the procedure and the approximation reached after a finite number of iterations. This error may be termed iteration error. It is not associated with the finite resolution of the spacecraft array or the time series and thus needs to be considered separately.

Random errors:

Due to imperfect (noisy) input data (measurement inaccuracies), the estimated parameters (first and second derivatives) will be subject to random errors, in addition to the discretisation errors and iteration errors mentioned above. In the current version of the manuscript, with demonstrations using noise-free model magnetic fields only, neither random errors are considered, nor the stability of the estimation (inversion) procedure (parameter reconstruction from noisy input data) which is likely to be associated with the set of different spatial discretisation scales. Since the inverse problem is weakly nonlinear, a condition number could be constructed for the linearised problem in the iterative procedure, or Monte Carlo simulations could be utilised to assess the impact of random errors. If such an approach is considered beyond the scope of this paper, the authors should at least critically discuss the implications of random errors, and outline the directions for future work.

Magnetic field divergence:

To quantitatively assess the limitations of this high-dimensional reconstruction problem with 33 model parameters, it is not sufficient to consider only a scalar quantity such as an estimate of the divergence of the magnetic field. Furthermore, in its original form, the divergence is normalised by the curl of the magnetic field (lines 201-203), while the latter quantity is zero for one of the two test cases (dipole field) in Section 3. To see if a dimensionless version of the divergence differs significantly from zero, meaningful reference values need to be chosen.

Terminology:

It is very unusual to refer to the tensor of second partial derivatives as the "quadratic gradient". It is strongly recommended to adjust the terminology. Canonical options are: "Hessian" or "Hessian matrix" (2nd derivatives of a scalar field) or "Hessian tensor".

Specific comments

Abstract and Key Points:

- The statements

"The tests for the situations of magnetic flux ropes and dipole magnetic field have verifies the validity and accuracy of this approach."

and 

"Magnetic flux ropes and dipole magnetic field testing verifies the validity and accuracy of the approach."

are too strong (and also difficult to understand in the first place). A proof of concept is presented, but a complete assessment of the accuracy would require studying all error types and the stability of the model inversion procedure.

Introduction:

- Line 54: The statement "To obtain high-order gradients in the magnetic field ..." is ambiguous as it could also refer to different orders of accuracy in discrete representations of the gradient. Instead, one could write "To estimate second derivatives of the magnetic field ..."

Method:

- Line 81: The statement "Calculation of the linear and quadratic gradients of a magnetic field generally requires magnetic measurements from at least ten spacecraft" should be made precise and briefly explained (3+9+18=30 parameters in the Taylor expansion up to second order, 3N magnetic field measurements in an array with N spacecraft).

- Lines 105/106: The statement "The errors in formula (3) are on the order V/c." is unclear. What kind of errors? Meaning of the variable c? Non-relativistic limit?

- Line 114: first-order or zero-order?

- Line 121: In the statement "The iterations are performed repeatedly until satisfactory results are achieved.", quantify what is meant by "satisfactory results" (which error measure/threshold).

- Line 142: In the statement "The temporal variation rate ... is readily obtained using time-series magnetic observation.", explain how the temporal variation rate is approximated (finite differencing? time resolution?).

Comparison of new method with analytical modelling:

- General comment: With spacecraft separations on the order 0.01 RE, and model magnetic fields varying on spatial scales on the order RE, the magnetic configurations vary only gradually on the spacecraft array scale, so these are not particularly challenging tests of the proposed method. In geospace, magnetic field structures can vary on much smaller scales. Furthermore, the model magnetic field configurations are simplified and highly symmetrical structures with a very small number of parameters so that only a minor subset of the 33 degrees of freedom can be assessed. The specifics and the limitations of the chosen test cases should thus be critically discussed.

- Lines 226/227: With the given value of the first eigenvalue $w_1 0.1643 R_E^2$, the characteristic size L should be $L = 2 \sqrt{w_1} = 0.8106 R_E$.

- Lines 254-258: Only total errors after a given number of iterations are discussed. It would be more interesting to get separate assessments of iteration errors and discretisation errors.

- Lines 277/278 and line 321: In the statement "The relative error approaches 50%; however, the absolute error is low." it is not clear which reference is used (low/small compared to what?)

Errors:

- General comment: As explained above, this section is very incomplete regarding the various types of errors. In particular, the current version of the manuscript lacks a critical discussion of random errors and the stability of the parameter estimation (inversion) procedure.

- Figures 12 and 13: It may be worth mentioning that the errors of the first derivative decrease quadratically with the scale L (second-order accuracy with regard to discretisation errors) whereas the errors of the second derivatives decrease linearly with L (first-order accuracy with regard to discretisation errors).

Conclusions:

- General comment: In line with the previous comments, this section should be rewritten to reflect the actual limitations of this study and the method, and explain where further work is required.