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 incoherent scatter radar (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 paper, 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 fluid-dynamic 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.

It would be of huge importance to measure the way in which electric fields in and around auroral arcs vary in time and space. This would allow us to gain new knowledge on the evolution of currents in Cowling channels, the closure of Birkeland currents and ultimately the dynamics of magnetosphere–ionosphere coupling in the auroral regions.
To investigate the spatial variation of the ionospheric electrical fields and currents, it is necessary to measure how physical quantities vary over a volume in the ionosphere (e.g.,

Investigating the spatial variation of the ionosphere can be done in two different ways: multi-beam scanning or aperture synthesis radar imaging (ASRI).
With multi-beam scanning, also known as volumetric imaging

A phased array is an array of (dipole) antennas where the beam can be steered by changing the phase of the transmitted or received signals. Combined with electronic control of the phases at every antenna, the beam steering can be performed between two consecutive pulses (e.g.,

With the first three sites of E3D, volumetric measurements of ion velocity will become possible. The core site with combined transmitter and receiver is going to be in Skibotn, Norway, and two remote receiver sites are built in Kaaresuvanto, Finland, and Kaiseniemi, Sweden. Each site will have a phased array, which will be built with up to 109 hexagonal subarrays consisting of 91 crossed dipole antennas each. In Skibotn, 10 additional outrigger subarrays will be built for interferometry

The technique for estimating electric field and neutral wind from ion velocity has been based on determining the electric field at high altitudes where the ion drift is dominated by

In this work, we present a technique to estimate the 3D variation of electric fields and neutral winds from multi-static ISR measurements of ion velocities.
A volumetric model makes it possible to use Maxwell's equations and the continuity equation for the neutral wind to constrain the estimates.
The work is a 3D generalization of the work of

This paper is organized as follows: the general technique to obtain neutral wind and electric field from ion velocity measurements is described in Sect.

The estimation of neutral wind and electric field consists of three steps: (i) measuring Doppler shifts, (ii) finding the ion velocity vectors, and (iii) estimating neutral wind and electric field.

Incoherent scatter radar (ISR) measurements are performed by transmitting a powerful radio wave and measuring the spectrum of the scattered signal, which at frequencies much larger than the plasma frequency, contains information about the plasma that scatters the radio waves. Due to the collective motion of the ions, the spectra are Doppler shifted. This shift is used to obtain the ion velocity component parallel to the Bragg scattering vector

The figure shows geometry and assumptions on E3D volumetric measurements. The figure is not to scale or angle.

The relationship between a measurement of the Doppler shift

Ion velocity is determined by the ion momentum equation:

To simplify the algebra, we rewrite the cross product with a matrix multiplication. We introduce the matrix:

This section defines the model that will be used to estimate electric field and neutral wind from multi-beam multistatic ISR observations of ion velocity. The electric field and neutral wind each have three components which have to be found from a discrete set of three components of ion wind. This gives six unknowns for three measurements. In addition to relating the ion velocity with the electric field and neutral wind, constraints are therefore also applied to find a more stable solution.

The discretization of the problem should keep most of its important features. The volume unknown is represented by discrete basis functions where we use a discretization corresponding to boxcars (voxels) in a desired coordinate system. This simplifies the search for discretization to find one coordinate system for each unknown. It is an advantage for computation speed to let the discretization be as coarse as possible because fewer parameters have to be estimated.

The electric field is strongly affected by the electric conductivities. This means that the fields are stronger in directions where the conductivity is low. Since the conductivity is much higher along the magnetic field than perpendicular to it

The neutral wind is expected to vary predominantly perpendicular to gravity and therefore following the surface of Earth. A geographic-oriented coordinate system is therefore an advantage for the neutral wind.

This means that the preferred coordinate systems for the discretization of electric field and neutral wind are different. We now introduce the discretization. We start with the measurements of the ion velocity. Here, for measurement

This converts the continuous vector field to a discrete form where the coefficients

The nature of the problem is underdetermined as shown by the earlier works (e.g.,

The main objective of regularization is to reduce the impact of noise amplification caused by the very smallest eigenvalues of the theory matrix. This can be achieved by a range of regularizing functions. It is preferable to choose regularization that biases the solution towards some sensible properties. In this paper, we suggest forcing the solution to adhere to plasma physics conservation equations: the continuity and momentum equations as well as Maxwell's equations. This gives a physical justification for regularization. However, it is still worth keeping the regularization as weak as possible to not impact the solution too much.

Here, we will show that for the electric field and neutral wind, we can use fundamental physical laws to obtain regularization terms similar to Tikhonov regularization. The outcome is twofold: (i) it gives a less noisy solution and (ii) it forces it to be physically reasonable.

By using Gauss' law

For the neutral wind, we use the continuity equation

With small neutral wind accelerations, one can also argue the use of previous neutral wind estimates as prior assumption of the next neutral wind estimate. This corresponds to a zeroth-order Tikhonov regularization and would then be similar to a Kalman filter, or to the approach introduced by

Many of the regularization terms we introduce contain spatial derivatives in multiple dimensions at the same time. For example, each component of Faraday's law uses derivatives in two directions, as illustrated in Fig.

Problems that arise at the borders of the grid. When using the definition of the derivative, at the one side, the derivative over the border cannot be included directly (black arrows). Possible solutions to the border problem for symmetric derivatives are also shown in the figure (cyan, blue, brown arrows).

Additionally, when differentiating in different dimensions, border issues appear in some cases since the derivatives can only be found in certain directions, see Fig.

Another possibility is to take the border-passing derivatives as stochastic variables, e.g.,

The problems described above do not apply to the 1D derivatives in the first-order Tikhonov regularization for the neutral wind. In this case, we simply use the definition of the derivative.

These regularizing constraints add several terms to our inverse problem. The physics-based regularized function we are minimizing is

Here, the covariance matrices in the different regularization terms fulfill the same role as the regularization parameter in a standard Tikhonov regularization. They balance how tightly the solution fits the constraints relative to how well they fit the observations.

It is possible to rewrite this in matrix form as

To analyze the resolution and accuracy that the proposed estimation technique provides, we perform a simulation of the system. Here we use different grids for ion drifts, electric field and neutral wind.

For the simulated measurements, we use an experiment consisting of

Longitude–height view of experimental layout. The radar beams are shown in blue, the grid for neutral wind in black and the grid for electric field in red.

Latitude–height view of experimental layout.

Azimuth–elevation distribution of transmit beams. The orange dot shows the direction of the magnetic field in 2022 as calculated with the IGRF model.

We model the measurements using a Gaussian beam pattern perpendicular to the range direction and triangular weights along the range.
The vertices of the triangle are placed in the center of the next range gate. At the nearest and furthest ranges, the triangles are symmetric. The Gaussian functions are centered around the line of sight with a standard deviation of 1

The grid for the neutral wind uses geographic coordinates, as shown in Figs.

For the electric field, we choose a special coordinate system. One axis is field-aligned and therefore slightly curved, as the magnetic field is not completely straight. However, in a short height range, as in Figs.

In this section, we will calculate the estimation uncertainties of the electric field and neutral wind for the example setup outlined in Sect.

When we calculate the uncertainties, we have neglected the effects of cases where transmit and receive beams only overlap partially, decreased transmit/receive gains for tilted beams and scattering angles below 90

The next step is to select suitable weights for the regularization terms, i.e., Maxwell's laws, the continuity equation and the assumption of low neutral wind acceleration. This can be interpreted as estimating the uncertainty in uncovered terms or the additional constraints they impose.
The equations for Gauss's law are equivalent to saying that the expected ionospheric charge density is zero with a variance that corresponds to some value of

In Faraday's law, the uncovered term is the time derivative of the magnetic field. In general, time variations in the magnetic field are mostly quite slow, but sometimes it changes very rapidly, for instance during substorms. To also include these conditions, we will use a rapid-changing magnetic field as a measure.
For example, wave-like structures in the magnetic field with amplitude up to 100 nT and a frequency of 0.5 Hz have been observed in situ

The continuity equation for neutrals is

In sum, with these variances, we assume that 67 % of the time, the net charge density in the plasma volume
is lower than 10

In addition, we need to have some estimate for the cases where we consider the derivative of electric field or neutral wind across the edges of our grids and for the constraint of small neutral wind accelerations. We implement both of these in the same way where we let the gradient be a stochastic variable with a variance as in Eq. (

For the variance of the neutral wind gradients, we use
approximate variations in measurements taken with a scanning Doppler imager as shown by

In addition, we constrain the magnitude of neutral wind components. For the horizontal wind, we assume that the estimates follow a normal distribution of mean zero and uncertainty of 200 m s

With these statements, we can proceed with finding the uncertainties in estimated electric field and neutral wind. The different solutions for handling the boundary problems also impose some properties of the neutral wind and electric field estimates. We did a short investigation of the different solutions as shown in Fig.

The resulting uncertainties in the estimates of electric field for the coordinate system, measurements and regularization described in this section are shown in Figs.

Uncertainty in electric field in local magnetic east direction.

Uncertainty in electric field in local magnetic north direction.

Uncertainty in electric field in field-aligned direction.

Like in the 1D case investigated by

The uncertainties in neutral wind estimates are shown in Fig.

Uncertainty in neutral wind estimates. Because the uncertainties vary little horizontally, the values are averaged for every altitude.

The same effect is also observed here, the neutral wind can be estimated with a high accuracy at low altitudes with a variance that increases rapidly above 110 km. The lowest estimates for the neutral wind have an accuracy of lower than 20 m s

In order to illustrate the results, we performed a vector field simulation
of neutral wind and electric field. We generated a vector field where the electric field in the N–S direction points inward to a certain latitude, thereby simulating an auroral arc, similar to

Electric field (blue) and neutral wind (red) used for simulations. Simulated ion wind measurements (green) are also shown. Because the neutral wind is set to zero, it is not seen in the plot. The vertical spacing in the plot is chosen so that the first plot covers our model and measurements between the 100 and 110 km range along the magnetic field, the second between 110 and 120 km, and so on. Since there are measurements every 5 km, each subplot contains two sets of measurements. For example, the 105 km plot contains the measurements from the line-of-sight ranges of 100 and 105 km. The plots for the uppermost and lowermost ranges look similar to their neighboring range and are not plotted.

Estimated neutral wind (blue) and electric field (red) together with ion wind measurements (green). The plots for the uppermost and lowermost ranges look similar to their neighboring range and are not plotted. Electric field vectors where at least one component has an uncertainty larger than 10 mV m

We used the generated fields to simulate the ion velocities in the coordinate system example described in Sect.

The generated vector fields for electric field and neutral wind are shown in Fig.

First of all, we note that the simulated ion velocity at the highest altitudes is perpendicular to the generated electric field. This is expected because at these altitudes, it is mainly influenced by the

The shown estimate of the electric field in Fig.

Moreover, the neutral wind estimates can be described as somewhat correct below 125 km altitude. Those estimates above this become increasingly worse, as in the 1D study.

This study introduces a method to estimate electric fields and neutral winds from multistatic multi-beam ISR measurements of ion velocity. We show that electric field uncertainties of a few millivolts per meter can be achieved at altitudes above 120 km. Estimation uncertainties of neutral wind should be small below 120 km.
It is the extension into 3D that marks the difference between this study and

For the presented estimates from the simulated ion drifts, the advantage of using the previous neutral wind estimate is not used. By using the previous neutral wind estimates as a prior knowledge of the state of the neutral wind, the time-variation of the neutral wind estimates will be smoothed. This is similar to a Kalman-filtering approach. This approach allows us to take into account that the neutral wind changes slowly with time.

The inverse problem in this study contains a number of regularization parameters that can be adjusted. When possible, we have tried to use weights for the regularization terms taken from measurements of related parameters. Elsewhere, physical models or reasoning were used.

The uncertainty in Gauss's law (

However, the ideal set of regularization parameters will have to be adjusted to the real observations on a per-case basis, at least initially.

Estimates of electric field with measurement gap. Three central measurement beams have been removed. Panel

As a performance test of the technique, we removed three of the central measurement beams and estimated electric field and neutral wind from the remaining measurements. The estimates with measurements between 180 and 190 km altitude are shown in Fig.

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 than the integration time will thus be blurred out. One way to mitigate this is to take into account the direction in which 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 by a Kalman filter to some extent. 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.

The code is available at

JV came up with the idea and programmed programs for the ISR spectrum and geographic calculations. JS programmed the model, carried out the calculations and prepared the article draft. AS, BG, JS and JV participated in developing the technique, the scientific discussions and the writing.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research has been supported by the Tromsø Forskningsstiftelse (Radar science with EISCAT_3D) and the Norges Forskningsråd (grant no. 326039). The publication charges for this paper have been funded by a grant from the publication fund of UiT, The Arctic University of Norway. EISCAT is an international association supported by research organizations in China (CRIRP), Finland (SA), Japan (NIPR and ISEE), Norway (NFR), Sweden (VR), and the United Kingdom (UKRI).

This paper was edited by Igo Paulino and reviewed by two anonymous referees.