A technique for volumetric incoherent scatter radar analysis
 Institute for physics and technology, University of Tromsø, Tromsø, Norway
 Institute for physics and technology, University of Tromsø, Tromsø, Norway
Abstract. Volumetric measurements of the ionosphere are important for investigating spatial variations of ionospheric features, like auroral arcs and energy deposition in the ionosphere. In addition, such measurements make it possible to distinguish between variations in space and time. While spatial variations in scalar quantities such as electron density or temperature have been investigated with ISR before, spatial variation in the ion velocity, which is a vector quantity, has been hard to measure. The upcoming EISCAT3D radar will be able to do volumetric measurements of ion velocity regularly for the first time. In this article, we present a technique for relating volumetric measurements of ion velocity to neutral wind and electric field. To regularize the estimates, we use Maxwell's equations and fluiddynamic constraints. The study shows that accurate volumetric estimates of electric field can be achieved. Electric fields can be resolved at altitudes above 120 km which is the altitude range where auroral current closure occurs. Neutral wind can be resolved at altitudes below 120 km.
Johann Stamm et al.
Status: final response (author comments only)

RC1: 'Comment on angeo202211', Anonymous Referee #1, 03 May 2022
The paper is expanding on a technique published by Nicolls et al. (2014) by using physical constraints to measure the electric field from ion velocities measured from ISR. The technique seems reasonable and the authors show how the method will work through simulation examples.
The authors deal with the affects of the sensor e.g. finite beam width and range extent of the measurments within equation 10. There can be a time component to this as the plasma moves through the FOV it can move to different resolution voxels. The closest cross beam voxels at 100 km along the north south track are about 5 km. This can be similar to a bluring of the data. Will this result in a major change in the algorithm? I think for the most part it will just be an adjustment to the forward model but it may not be neccesary as you're just measuring velocities and not intrinsic plasma parameters with this technique. Plus the physics based regularization might help mitigate this impact.

AC1: 'Reply on RC1', Johann Stamm, 30 May 2022
Thank you for the valuable feedback
The model in the article does not consider time variations in other ways than that the neutral wind is assumed to vary slowly. There could be cases where a feature in electric field or neutral wind moves with a velocity large enough to be inside several voxels within one loop of transmit beams. For example the auroral arc could move over the field of view over this time. The electric field changes would then appear blurred in the output data.
Including a time component that considers changes in the measured plasma within one integration time into the model would be possible. It could be implemented similar to the dependency on the volume.
Another way of handling rapid timevariations could be to shorten the integration time. The price to pay for avoiding motionblurring this way is increased uncertainties. One could then apply a Kalman filter to the time domain. This is outside the scope of this study.
For Figure 11, we removed three beams completely to see how missing beams effect estimates and uncertainties. This could also be interpreted as an extreme valley in electron density. The result was that these are little affected. We expect that the technique could be applied in cases where faster scan times are used with beams in more sparse pointing directions leading to slightly larger uncertainties.
We will add to our manuscript a discussion of this topic:
The presented framework assumes that the ionosphere does not change faster than the integration time, which is 70 s for the presented example. Spatial and temporal variations occurring faster that the integration time will thus be blurred out. One way to mitigate this is to take into account in which direction the beam points at every point in time such that the model connects the time the measurement is taken to the results. Another possible mitigation procedure, is to use a shorter integration time. The latter will have increased uncertainty which may be compensated to some extent by a Kalman filter. A third option would be to use fewer beams as this needs shorter integration time. The regularization will then try to fill the gaps as best as possible as illustrated in the example above.
In general we can state that an improved timeresolution requires measurements in fewer pointingdirections, that is either covering a smaller volume with a compact set of beams or a more sparse set of beams covering a large volume.

AC1: 'Reply on RC1', Johann Stamm, 30 May 2022

RC2: 'Comment on angeo202211', Anonymous Referee #2, 04 Aug 2022
The comment was uploaded in the form of a supplement: https://angeo.copernicus.org/preprints/angeo202211/angeo202211RC2supplement.pdf

AC2: 'Reply on RC2', Johann Stamm, 25 Sep 2022
We thank the referee for the detailed review and comments. We will answer the comments in the order they appear.
We have included most of the referee comments before our answer such that it is easier to see what we are referring to. The referee comments are written in italics.
Major comments
1.
> The manuscript does not handle the Fregion parallel ion velocities correctly. The momentum equation in Eq. 3 is a vector equation, and it is approximately valid for the two perpendicular components. Nonetheless, this equation is not valid for the parallel component at Fregion altitudes. Using that fact that v_ × B = 0 and assuming that E_ is small, the parallel component of Eq. 3 reduces to v_ = u_ , implying that the parallel ion velocities are always equal to the parallel neutral velocities. This is generally not true in the Fregion. A proper treatment of ion parallel velocity in the Fregion requires the inclusion of gravity, ion pressure gradients, and ambipolar electric fields. The ion inertia terms can also become important during times of rapidly varying ion upflow.
> In principle, EISCAT 3D measurements could be used to volumetrically reconstruct all three components of the Fregion ion velocities, including the spatial variations of the ion upflow velocities. The algorithm presented in this manuscript, however, would fail to do that. This manuscript is not solving for v_, but instead solving for E_ and u_ assuming the two quantities are related to v_ through an invalid parallel momentum equation.
> Figure 8 shows low uncertainties in the vertical neutral wind estimates extending all the way up to 200 km altitude. This is unreasonable since the ion velocities that the radar measures become collisionally decoupled from the neutral velocities at high altitudes, meaning the radar data cannot actually be giving meaningful information on neutral velocities at those altitudes. This unreasonable result is a direct consequence of the invalid parallel momentum equation.
This study is for showing that it is possible to use multibeam multistatic ISR measurements to obtain volumetric measurements of electric field and neutral wind in the ionosphere. In that sense, we did some simplifications of the momentum equation where we neglected the smallest terms, that is advection, gravity and pressure gradients. It is possible to extend the model to include these terms. This will make the model more correct, but also requires even more calculations.
The shown uncertainties do not include those introduced by assumptions or simplifications made. The neglected influence of advection, gravity and pressure gradient terms on the velocities are therefore not reflected as increased uncertainties. Typically, these terms are small, but in the uppermost part of our volume, they can be large enough to become significant. If future work uses fieldaligned/vertical estimates of neutral wind or electric field, they will have to consider these terms.
In the revised manuscript, we will state the momentum equation including the terms mentioned above explicitly, before we simplify it to obtain the equation we use in the analysis. We will also clarify that the assumptions and simplifications we did are not included in the shown uncertainties .
2.
> For vector basis functions, the weights should generally be arrays not scalars. [...][It is implied that] all three components of the basis function get the same weight, which is an unusual restriction. To allow the three components to vary independently, the coefficients should be allowed to be different for the different vector components […]
> To be general, the three different components of η should be treated as three separate unknowns.
It appears that the text was confusing since the the three components of the basis functions are allowed to vary independently in the calculations and simulations.
This will be clarified in the revised manuscript.
3.
> The manuscript does not assume equipotential field lines and does not explain the rationale for allowing large variations in electric fields along a field line. Past Eregion neutral wind estimation techniques such as Thayer [1998] and Heinselman and Nicolls [2008] have always asserted that electric fields are invariant along field lines such that Fregion measurements of the electric fields can be mapped into the Eregion. The mapping of Fregion electric fields into the Eregion is crucial for all of these past studies of Eregion neutral winds using ISR; without that assumption the ion momentum equation is unsolveable in the Eregion. Past sounding rocket studies have demonstrated the reality of field line mapping using payloads that can measure electric fields independently of ion velocity [Sangalli et al., 2009].
> In this manuscript the a priori standard deviation of the electric field gradient is allowed to be 20 mV/m per 2.5 km in all three directions, including along the field lines. This is equivalent to asserting that fieldaligned mapping of the electric fields does not function between the F and Eregions; fields of 50 mV/m in the Fregion at 300 km can change by more that 100% over the distance to the Eregion at 100 km.
> Ignoring fieldaligned mapping of electric fields between the E and Fregion makes the problem of estimating Eregion neutral winds substantially more difficult, and it is clearly leading to unreasonable results in the examples presented. Figure 10 shows the algorithm estimates nonzero neutral winds at 125135 km altitude in a truth model simulation where the true neutral wind is zero. This behavior is pathological and unreasonable. The results assert that the variance of the estimated electric fields at low altitudes is nearly infinitely large, when in reality electric field mapping should guarantee that electric fields at low altitudes should nearly match the fields at high altitudes.
It is accurately noticed that we do not assume that the electric potential maps perfectly along the magnetic field. This allows our method to be used also in cases where this assumption can not be done.
The mapping of electric field along the magnetic field lines is not perfect. This is especially true on smaller scale structures or in the lower E region, as shown by Reid (1965) and Park and Dejnakarintra (1974) (see also Brekke (2013)). This is also seen in Sangalli et al. (2009) where the electric field in the E region fluctuates.
The allowed variances for the electric field gradients is not considered in itself, but through Maxwells equations. However, there are no neighbouring voxels at the borders, which means that these gradients can not be implemeted directly with differences between the neighbours. We thought about several techniques to handle this issue, which are shown in Figure 2 of the paper. Three of the techniques to handle the borders give bordercrossing derivatives a larger uncertainty. This uncertainty is 20 mV/m per 2.5 km (1 m/s per km for neutral wind).
Also, as described in Section 4.3, the setup we use for the borders give the same uncertainty to gradients over the border as to those within the borders.
Apparently, our manuscript could be understood as we would use these constraints throughout the volume, possibly in all directions. We will try to clarify the manuscript.
4.7.
We gratefully thank the referee for suggesting more reasonable strictnesses of the constraints we use. In general, those are stricter than those we use in our calculations. We performed new simulations using the constraints suggested by the referee in points 47 and letting the zeroth order Tikhonov regularization have an a priori standard deviation of 300 m/s. As a result, the electric field remains unchanged, but the neutral wind estimates become more varying and uncertain. To compare, we also tested how the model would react to the same situation if the wind blows with 300 m/s. Also here, the different strictnesses of regularization give small differences.
In general, one should not regularize an inverse problem more than necessary. Therefore the least strict assumptions are preferable because they then give more room for unforeseen variations in the ionosphere. We therefore do not want to tighten the constraints since it does not seem necessary. We will include these reasonings and references into the manuscript.
7.
> The use of a zerothorder Tikhonov regularization is going to bias the neutral wind estimates low. The assumed a priori variance is 200 m/s, but auroral neutral wind jets over 300 m/s have been observed, for example in the JETS rocket mission.
It is correct that zerothorder Tikhonov regularization biases the solution towards the a priori expected value, here zero. However, this bias is not as strong as the referee seems to suggest. The assumed standard deviation is 200 m/s  this means that winds of 300 m/s still would be at 1.5sigma. A standard deviation for the wind components of 200 m/s would lead to a small underestimate of these components of a few tens of metres per second.
Minor comments
1.
> The figure quality is generally low, with the text in the axis labels being highly pixelated.
We will increase the figure resolution in the next version of the manuscript.
2.
> An azimuthelevation plot of the beam geometry would substantially clarify the beam geometry. Figures 3 and 4 have so many lines on them that the 3D geometry is hard to see.
We will include a plot showing the azimuthelevation distribution of transmit beams.
3.
> Lines 209 and 210 should specify the interpulse period assumed for this experiment and specify how many independent estimates of the ACF/Spectra are obtained in 2 s of integration.
We will include this information to the manuscript
References
Brekke, Asgeir (2013): Physics of the upper polar atmosphere. 2. edistion, Springer, Heidelberg.
Park, C. G. and Dejnakarintra, M. (1974): Paper presented at Fifth International Conference on Atmospheric Electricity, GarmischPartenkirchen, West Germany, September 2–7, 1974.
Reid, G. C. (1965): Ionospheric effects of electrostatic fields generated in the outer magnetosphere. Radio Science, 69D, pp. 827–837.
Sangalli, L., D. J. Knudsen, M. F. Larsen, T. Zhan, R. F. Pfaff, and D. Rowland (2009): Rocketbased measurements of ion velocity, neutral wind, and electric field in the collisional transition region of the auroral ionosphere, Journal of Geophysical Research: Space Physics, vol. 114 number A4, doi: 10.1029/2008JA013757.

AC2: 'Reply on RC2', Johann Stamm, 25 Sep 2022
Johann Stamm et al.
Johann Stamm et al.
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