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 , the radar beam is pointed in different directions to measure the local states in the ionosphere. Multi-beam scanning covers a large region in the ionosphere and is thereby useful for investigating large-scale structures. With ASRI, the phase difference in a received signal between receivers is used to investigate small-scale structures inside the radar beam (see e.g., ). In this paper, we investigate the multi-beam scanning with EISCAT3D (E3D). For ASRI with E3D, we refer to .

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., ). The advanced modular incoherent scatter radars (AMISRs) were the first incoherent scatter radars (ISRs) that combined these two, making it possible to perform measurements of scalar ionospheric parameters, such as electron density ne, electron temperature Te and ion temperature Ti in some tens of seconds . By assuming that the electric field along magnetic field lines is constant and that the field-aligned ion flow is completely constant, the variation in Doppler shift can be used to estimate horizontal variations in the electric field . However, full volumetric measurements of vector parameters require multiple receivers. At least one receiver for every component of the ion velocity vector is needed. This will be possible with E3D when it is finished .

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 E×B drift. Then, the electric field has been assumed to be constant along the magnetic field line so the neutral wind could be estimated at lower altitudes. This technique was introduced by and has been used in many studies of the neutral wind . However, for analyzing a vector field, the method has to be adjusted because only one beam will be field-aligned.

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 that investigated the possibility of using a field-aligned profile with E3D measurements of ion velocity to find estimates of electric field and neutral wind. When generalizing, one has to take into account that most of the measurements are not aligned with the magnetic field. With the improvements of , , and , we will develop a model that can be used to analyze the 3D vector fields of neutral wind and electric field.

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 framework for volumetric measurements and estimates is described in Sect. . Our chosen setup of the measurements and discretization of the neutral wind and electric field estimates is shown in Sect. . Section  discusses the uncertainties in the measurements, applicability of the assumptions and uncertainties of the estimates. A simulation of ion drift measurements is given in Sect. , followed by a discussion 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 kB which is equal to the difference between wave vectors of the scattered and transmitted wave (see also ). Figure  illustrates the characteristic geometry of E3D together with the wave vectors along which the ion velocity is measured.

The relationship between a measurement of the Doppler shift w and the ion velocity vector v for a transmitter–receiver pair p is 1wp=kpkp⋅v. A set of Doppler-shift measurements w⊤=[w1,…,wP] of the same volume from P pairs can be combined to the system 2w=Kv+ξw, where K⊤=k1/k1,…,kP/kP is the theory matrix and ξw is a vector containing the noise terms. If the measurements are sufficiently linearly independent, the ion velocity can be found with the method of least squares (see ).

Ion velocity is determined by the ion momentum equation: 3nimidvdt=-∇Pi+nimig+qiniE+v×B-∑knimiνikv-vk. In the equation, ni is the ion mass, mi is the average mass of ions, Pi is the pressure tensor for ions, g is the gravitational acceleration, qi is the ion charge, E is the electric field vector, B is the magnetic field, νik is the momentum-transfer collision frequency between ions and particle species k, and vk is the velocity of particle species k. At ionospheric altitudes, the dominant terms are the Lorentz force and collision with neutrals. Gravity and pressure gradients may influence the ions from the upper E region and upwards. The gravity force is constant and the pressure can be included when we have actual measurements. For the sake of clarity and without loss of generality, we can simplify the problem by neglecting these terms. We assume singly charged ions qi=qe, plasma quasi-neutrality ni=ne and only consider collisions with neutrals. When also assuming steady-state conditions, the ion momentum equation can be written as 40=qeneE+v×B-nemiνinv-u.

To simplify the algebra, we rewrite the cross product with a matrix multiplication. We introduce the matrix: 5Bg=0Bz-By-Bz0BxBy-Bx0, where x,y and z are the axes of the geographic coordinate system, i.e., east, north and up, respectively. This allows us to rewrite the cross product as v×B=Bgvg, where the subscript g shows that the matrix and vector are in geographic coordinates. Now, the momentum equation can be rewritten as 6I-κBBgvg=κBEg+ug, where 7κ=qeBmiνin is the ion mobility and I is the identity matrix. Inverting the matrix on the left-hand side is simplified by transforming it into local magnetic coordinates perpendicular to the magnetic field towards the east and antiparallel. The third component completes the right-handed system and will be referred to as northward. The transformation matrix from local geomagnetic to geographic coordinates is 8R=cos⁡δsin⁡Isin⁡δ-cos⁡Isin⁡δ-sin⁡δsin⁡Icos⁡δcos⁡Icos⁡δ0cos⁡Isin⁡I, where δ is declination and I is the magnetic dip angle . The matrix R is a rotation matrix, which means that R-1=R⊤. The matrix on the left-hand side of Eq. () can then be written as RCm-1R⊤, where 9Cm=11+κ21-κ0κ10001+κ2. The momentum equation can now be written as 10vg=R⊤CmRug+κBR⊤CmEm, indicating that we will estimate the electric field in local magnetic coordinates.

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 , electric fields and their variations are expected to mainly be in the perpendicular direction for higher altitudes. To avoid aliasing-type problems, it is preferable to use a discretization that is aligned with the magnetic field.

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 ℓ, the measured ion velocity υℓ is considered as an integral over the probed volume indicated with the function βℓ. The measurement can be written as 11υℓ=∭VvrβℓrdV+ξℓ, where ξℓ is a vector which contains the errors of the ion velocity vector measurements, i.e., the errors of the solution of Eq. (). Equation () can be expanded using the momentum equation, Eq. (). This gives 12υℓ=∭VR⊤CmRugrdetJuβℓrdV+∭VκBR⊤CmEmdetJEβℓrdV+ξℓ, where J is the Jacobian from the coordinate system of the ion velocity to that one indicated by the subscript, u for neutral wind and E for electric field. Then, the unknown continuous vector fields are discretized by replacing them with sums of basis functions Φj and Ψj: 13Ex≈∑j=1NEηjxΦjx and 14ux≈∑j=1NuΓjxΨjx, and correspondingly for the other dimensions.

This converts the continuous vector field to a discrete form where the coefficients ηj and Γj are our new set of unknowns. They are constant over the integrated volume and can therefore be taken out of the integral. We will now define the variables 15aℓjE=∭VκBR⊤CmΦjdetJEβℓrdV and 16aℓju=∭VR⊤CmRΨjdetJuβℓrdV. Equations () and () let us write Eq. () as 17υℓ=∑j=1NuaℓjuΓj+∑j=1NEaℓjEηj+ξℓ, which can be recognized as a matrix equation υ = AEη+AuΓ+ξ. If we define the unknowns as one single vector x⊤=Γ⊤,η⊤ and stack the matrices A⊤=Au⊤,AE⊤, the equation relating the measurements to the unknowns becomes 18υ=Ax+ξ. The equation can be recognized as a standard linear inverse problem for which we develop a general physics-based solution in this paper.

The nature of the problem is underdetermined as shown by the earlier works (e.g., ). We therefore have to use regularization.

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 ∇⋅E=0 for a charge-neutral plasma and Faraday's law ∇×E=0 for a time-stationary magnetic field, we are adding four equations for every unknown vector of the electric field.

For the neutral wind, we use the continuity equation ∇⋅ρu=0, where ρ is the mass density of neutral particles. Also, we assume that the acceleration of the neutral wind is small. This means that when the same particles have moved for some time, and thereby distance, they have the same velocity. Furthermore, this implies that the spatial variation of the neutral wind vector field is small. We implement this approximation by assuming that the first-order differences of the neutral wind components in all directions are smaller than some parameter 1/α. These constraints are mathematically equivalent to first-order Tikhonov regularization .

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. . Since these derivatives are not symmetrical in this case, we use a weighting of the derivatives in both directions. They are approximated by 19dExdy(y)≈W1Ex(y+Δy1)-Ex(y)Δy1+W2Ex(y)-Ex(y-Δy2)Δy2 for the example of electric field in the x direction. In the equation, W1 and W2 are weights. We note that the separation in the grid is varying because the grid may be curved and stretched. Therefore, we have to take into account that Δy1≠Δy2.

Additionally, when differentiating in different dimensions, border issues appear in some cases since the derivatives can only be found in certain directions, see Fig. . Mathematically, the solutions to this problem differ depending on the weights (W1 and W2) that are used. We are aware of three possible solutions. The first is to ignore the derivatives passing the border. Then, one of the weights is zero, which is shown as the blue line in Fig. .

Another possibility is to take the border-passing derivatives as stochastic variables, e.g., 20Ex(y)-Ex(y-Δy2)Δy2∼N0,σΔE2. A third possibility is to weigh the two derivatives in another way, e.g., by focusing on those inside the borders. An example is illustrated by the cyan arrows in Fig. .

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 21m-Ax⊤Σm-1m-Ax+∇×E⊤ΣF-1∇×E+∇⋅E⊤ΣG-1∇⋅E+∇⋅(ρu)⊤ΣK-1∇⋅(ρu)+dudr⊤Σdudr-1dudr+u-uprev⊤Σdudt-1u-uprev.

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 22υR=ARx+ξR, where the extended theory matrix is AR⊤=A⊤,L⊤. Here, the matrix L is the regularization matrix which contains all the regularization terms constraining the problem.

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 5×7 beams, as illustrated in Figs. , and . The beams are pointed evenly as a fan with zenith angles from 13∘ S to 5∘ N, with a spacing of 3∘. In the E–W direction, we have beams every 2.5∘ between 5∘ W and 5∘ E. According to the International Geomagnetic Reference Field (IGRF) model , in 2022, the magnetic field at 69∘ N, 20∘ E has a declination of 10∘ and an inclination of 78∘. The beams and the magnetic field are pointed out in Fig. . In every beam, we measure with ranges every 5 km range resolution from 90 to 210 km range.

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∘ corresponding to the half-power beamwidth (HPBW). The Gaussian is truncated at 2 standard deviations and normalized such that it still integrates to 1.

The grid for the neutral wind uses geographic coordinates, as shown in Figs.  and . The grid centers are placed every 0.15∘ between 68.9–69.5∘ latitude and every 0.3∘ between 19.8–20.7∘ longitude. In altitude, we place the centers every 10th kilometer between 90 and 210 km.

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.  and , the curvature is not visible. The other axes consist of geographic latitude and longitude at the surface of Earth. We place the horizontal grid centers for the electric field every 0.1∘ within 69.3–69.9∘ in latitude and every 0.2∘ within 20.0–21.0∘ in longitude on the surface of Earth. The grid contains 7 voxels in latitude and 6 voxels in longitude. Along the magnetic field axis, the centers are placed every 10th kilometer between 90 and 210 km.

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 . Inside the arc, the field is zero. Also the other components of the electric field are set to zero. This can be compared to the Cowling channel model by . The neutral wind is set to zero everywhere.

We used the generated fields to simulate the ion velocities in the coordinate system example described in Sect. . Then, normally distributed noise is added with a standard deviation of 20 m s-1 in the horizontal direction and 5 m s-1 in the vertical direction. Finally, the simulated ion velocities are used to find estimates of neutral wind and electric field. Here, we use the same grids as for the generated fields and the regularizations as described in Sect. .

The generated vector fields for electric field and neutral wind are shown in Fig.  along with the ion wind measurements simulated from these. The estimated vector fields are shown in Fig. . The estimates where the uncertainty in at least one electric field component is above 10 mV m-1, are not plotted, neither are those of neutral wind where at least one component has uncertainty above 30 m s-1.

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 E×B drift which was used by to find electric field estimates. At lower altitudes, the ion drift becomes increasingly more dependent on the neutral wind.

The shown estimate of the electric field in Fig.  is quite close to the starting point at 125 km and upwards, but only inside the measured volume. This is the same result as found in the one-dimensional case by . We note that in the eastern boundary region of Fig. , there is a small curving artifact that is caused by Faraday's law.

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 . The estimates from this 3D technique give a more stable solution than in the 1D case. Even if the study is more sophisticated in 3D, the approaches give similar results which depend on how the regularization is performed. In both cases, the results indicate that even with adding regularization, electric field and neutral wind cannot be estimated well at the same altitudes without further assumptions.

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 (10-3 V m-2) is a very large value. A stricter value can be found by considering the current continuity equation from and ignoring the conductance gradients. Then, 24∇⟂⋅E≈J∥ΣP, where J∥ is the field-aligned current density and ΣP is the Pedersen conductivity. If one assumes a field-aligned current density of 5×10-5 A m-2 and a Pedersen conductivity of 5 Ω-1 and that field-aligned variation is small, the allowed divergence of electric field decreases with some orders of magnitude. Considering the variability around dynamic auroral arcs , we do not want to use stricter regularizations than necessary. In our model tests, it appears that the used regularization is strict enough.

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

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. a, and the Ex uncertainties in Fig. b. The deviations relative to the estimates using the full set of measurements (see Fig. ) are small. Maybe more importantly, the uncertainties do not increase by much. This shows that this type of Tikhonov regularization leads to solutions that degrade gracefully while satisfying Maxwell's equations. A consequence of this is that it should be possible to use sparser beams to estimate electric field and neutral wind. This can be used to either improve the time resolution or to expand the observed volume. However, the removal of beams comes with a cost of slightly increased uncertainties, which can be seen by comparing Figs. b and .

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.