For a known radar system and radar operation mode, the signal power dP from a plasma volume dV at the point rt is 1dP=Neξ0sin⁡2χPT1+Te/TiGT4πrT2GR4πrR2λ24πWrdV, where Ne is electron number density, ξ0=10-28 m² is the single-electron radar cross-section, χ is the polarization angle (angle between the electric field vector of the incident wave and wave vector of the scattered wave), PT is the transmitted power, Te and Ti are electron and ion temperatures, respectively, GT and GR are gains of the receiver and transmitter antennas at rt, rT and rR are distances from the target to the transmitter and the receiver, respectively, and λ is wavelength of the radar carrier signal. Wr is the range ambiguity function, which includes the effects of phase-coding and decoding in the signal processing chain, which narrows down the plasma volume in the range direction according to the applied coding and decoding scheme. The radar is assumed to transmit circular polarization and the receiver polarisation is assumed to be matched with the polarization ellipse of the scattered wave. Total power received from a scattering volume V with known Ne,Te, and Ti located around rt can be integrated using Eq. (), 2P=∭VdP, if technical specifications of the radar system and range resolution of the measurement are known. A detailed discussion of determination of the scattering volume V is provided in Appendix B of . To simplify and increase the computational speed of the calculations, the beam patterns and pulse shapes are assumed Gaussian.

Noise contributions from thermal noise in the radar receiver and sky-noise are described by the system noise temperature Tsys. For a known receiver bandwidth δν the background noise power is 3PB=kBTsysδν, where kB is the Boltzmann constant. Another noise contribution, the so-called self-noise, arises from the phase-code decoding, which effectively converts the signal from unwanted ranges into zero-mean noise. The self-noise power PS can be integrated from Eq. () if the range ambiguity function Wr, corresponding to one bit in the phase-coded pulse, is replaced with Wp, matched to the length of the whole pulse. To get a reasonable and computationally light-weight approximation of the self-noise power, the self-noise calculation assumes that the electron density altitude profile has a Gaussian shape that peaks at rt and has a user-defined width. This approximation is used only in the self-noise calculation, not for the signal powers.

These approximations allow us to calculate the total noise power PN=PB+PS and the signal-to-noise ratio as 4SNR=PPB+PS. One should note that the self-noise contribution PS will often be significant in E3D due to its high transmitted power and large antenna gains. Ignoring the self-noise contribution would thus lead to unrealistically high SNR especially in the E region, where high electron densities are often present at a short distance from the radar.

The uncertainty estimation procedure relies on linearization of the non-linear incoherent scatter theory, which yields a relation between the incoherent scatter autocorrelation function (ACF) and the plasma parameters. We denote the theoretical ACF by f(k,τ;x). Via the Wiener-Khinchin theorem the ACF is the Fourier transform of the incoherent scatter spectrum S(k,ω) given in Appendix . The three arguments of the ACF are the scattering wave vector k, the time lag τ and the model vector of (normalized) plasma parameters 5x=ne/n0,Ti/T0,Te/Ti,νin/ν0,p,vs/vs0T, where ne, Ti, Te, νin, p, and vs are respectively the plasma density, ion temperature, electron temperature, ion-neutral collision frequency for momentum exchange, the fractional abundance of oxygen ions (assuming the plasma is composed of a mixture of NO⁺, O2+, and O⁺), and the line-of-sight ion drift measured from site s. The variables n0, T0, ν0, and vs0 are normalization constants, and NO⁺ and O2+ are represented as one molecular ion with mass 30.5 u. Thus the set of N measurements of the ACF at time lag τ with scattering vector k is represented by the measurement vector 6m=d0,d1,…,dNT, which is related to the ACF via 7m=f(k,τ;x)+ϵ. Here ϵ is taken to be Gaussian zero-mean noise. For non-zero vs the measurement vector m is complex.

In linearizing the ACF, 8f(k,τ;x0+Δx)≈f(k,τ;x0)+AΔx, we define a theory matrix A that consists of the partial derivatives of f with respect to each plasma parameter xk. As with the measurement vector m, this matrix is complex for non-zero vs; it is nevertheless possible to write the real and imaginary parts of Eq. () separately such that it may be represented as a purely real system of equations. Provided an estimate of the measurement noise covariance Σm, one can also calculate the posterior model covariance Σx, or covariance of plasma parameter estimates, using the standard result from linear inverse theory: 9Σx=Σx,pr-1+ATΣm-1A-1, where Σx,pr is the prior model covariance matrix that represents any prior information that we may have about the distributions and interrelationships of the plasma parameters x. (In this work we use Σx,pr-1=0.) Equation (), which is the basis for uncertainty estimation, only requires specification of the theory matrix A and does not require that Δx be nonzero.

At the highest level, our uncertainty estimation procedure consists of two steps: estimation of the ACF measurement error covariance matrix Σm, and plasma parameter uncertainty estimation via Eq. ().

In the first step we estimate the ACF measurement covariance matrix Σm as follows. We assume that measurements of the ACF at different lags are uncorrelated and have the same variance σ2, so that Σm is an identity matrix multiplied by σ2. With these assumptions, the ACF noise level may be defined as 10γ2=σ2f(k,0;x)2Nτ=NaNτ1SNR2, with Na the number of averaged ACF samples, Nτ≈2π/(ω0Δτ) the approximate number of lags to the first zero of the ACF assuming zero plasma bulk flow, ω0=k2kBTi/mi the “ion angular Doppler frequency” for scattering wave number k, and Δτ=2Δr/c the lag spacing (sampling interval) of ACF measurements. The lag spacing is matched with the user-specified range resolution Δr. The number Na accounts for the number of samples averaged in phase-code decoding, as well as post-integration in range and time.

With this definition, we calculate the SNR and the ACF noise level at each site as described in Sect.  using the radar equation assuming a particular set of radar system parameters, and assuming Gaussian beam shapes and pulse shapes. (See Sects. B.3–B.6 in , as well as the routine “multistaticNoiseLevels” in .) The outputs from this step are the separate ACF noise levels γs for each transmitter-receiver pair s.

In the second step (plasma parameter uncertainty estimation) we use the fact that uncertainty of the line-of-sight velocity vi is practically independent of the errors of the other parameters, and assume that all receivers see the same ne, Te, and Ti. The latter assumption is equivalent to assuming that Te and Ti are isotropic. In reality Te and Ti may be neither isotropic nor equal; see discussion in, e.g.,and references therein. The uncertainty estimation itself can therefore be divided into two parts: calculation of (i) the variance of estimates of scalar parameters, which do not have a direction, and (ii) the covariance of ion drift velocity vector estimates.

To calculate (i) we simply sum information from all receivers (since information or precision is the inverse of variance) to get a combined ACF noise level for the full multistatic system: γ=∑sγs-2-1/2. We then calculate the theoretical ACF f(k,τ=0;x), and solve for the ACF variance σ2 in Eq. () (see routines “parameterFitErrors” and “ISspectrumSimple” in ). We thus have an estimate of the ACF covariance matrix Σm. We then calculate the theory matrix A, and finally calculate the posterior model covariance matrix in Eq. (), with the scalar plasma parameter uncertainties taken to be the square roots of its diagonal elements.

To calculate (ii) one must consider the system geometry, and in general the estimate of the line-of-sight ion drift for each transmitter-receiver pair has a different uncertainty. We combine the different uncertainties following the methodology outlined in Appendix B. For a monostatic system the ion drift velocity covariance matrix reduces to a scalar representing the variance of the line-of-sight ion velocity.

A simple way to constrain the otherwise overwhelmingly large parameter space that comprises E3D experiment design is to begin with identifying a goal or set of goals that one wishes to achieve via a particular experiment, as is done in Sect. . These goals will often be related to a scientific hypothesis or question. In such cases, some general considerations for experiment design can be stated:

Gather information about the desired study targets or conditions. Where do they occur in terms of latitude, longitude, local time, and/or altitude? When do they occur in terms of time of day, season, or relevant solar wind or geomagnetic conditions? What are the typical spatial and temporal scales? Over what altitude range? Which plasma parameters are most important?

Regarding the first point, the field of view of the Skibotn transceiver is confined to approximately 65–68° magnetic latitude (MLat) at 100 km altitude (indicated by the innermost blue ring in Fig. 1, 68–71° geodetic latitude). Meanwhile, various auroral forms exist over different ranges of local times and MLats e.g., illustration in Fig. 1 of. Thus whether phenomena located in the polar cap, auroral zone, or at sub-auroral latitudes can be observed by E3D is a question of local time and geomagnetic activity. A detailed list of phenomena that may be observable with E3D has been given by .

Auroral arcs and aurora-like features can appear at all local times within the latitudinal extent of the auroral oval, where the latter depends on the level of geomagnetic activity. A well known auroral form is the standard discrete arc that occurs during geomagnetically quiet periods – quiet, discrete aurora (hereafter QDA). In spite of its apparent simplicity, basic facts about the generation, evolution, and lifetime of QDA remain unknown and references therein.

One interesting suggestion found in the literature is that “arc proper motion” (motion of an auroral arc relative to background plasma convection) u′=u-v, where u and v are respectively arc velocity and plasma convection, may contain information about energy and momentum transfer between the magnetosphere and ionosphere e.g.,. The meaning of arc proper motion is nonetheless debated and references therein, at least partly due to a lack of estimates of u′ with sufficiently low uncertainty.

Here we envision using all-sky cameras to estimate u and E3D to estimate v at spatial and temporal resolutions exceeding those of previous studies . Typical in these studies were integration times of 10–60 s for one beam, and uncertainty in plasma drift velocity of 20–100 m s⁻¹ depending on conditions, but mostly above 40 m s⁻¹, and displacement of the radar beam relative to the location of the arc by a few kilometers or tens of kilometers. also notes that the total displacement of the plasma during the observation window has an uncertainty of ± 7–14 km. It could also be possible to obtain an additional, independent estimate of u by for example tracking enhancements of E3D plasma parameter estimates.

In answer to the questions posed in the guidelines presented in Sect. 4.1 we arrive at the following.

QDA occurs over 60–80° MLat, with a peak occurrence rate at 70° MLat. It generally occurs between 14:00–08:00 MLT and is most frequently observed between 22:00 and 23:00 MLT , but can occur at virtually all MLTs.

QDA arcs are generally aligned with geomagnetic east-west, with tilts of as much as ±8° locally . Typical arc widths are 10–20 km , with lifetimes of up to tens of minutes . QDA arcs have an emission height range of 80–400 km, with an emission peak at 110 km altitude .

QDA is typically associated with Kp ≤ 4 . Given the range of MLats visible to E3D (Fig. a) and the statistical MLat distribution of the auroral oval for Kp ≃ 2–4 generally between 65–70° MLat; see Figs. 2 and 4 in, QDA should be visible to E3D over ∼ 22:00–05:00 MLT during both winter and summer. Note, however, that the probability of occurrence of QDA during summer relative to winter is decreased e.g.,.

Figure  illustrates an example “keogram” experiment designed to conform to the requirements given immediately above. Figure a shows that the experiment consists of 40 beams arranged in a rectangular (10 × 4) grid. This beam pattern was selected via trial and error from several arbitrarily chosen beam patterns with varying numbers of beams. Our experimentation with different numbers of beams and beam patterns given a fixed observational area over E3D indicates that, for this particular experiment, the exact number of beams and the beam pattern itself is less important than selection of the overhead area and the total amount of time allotted for integrating over all beams, as the latter two are closely related to the overall level of uncertainty. This process also revealed that convection velocity uncertainty generally increases rapidly poleward of 68° MLat.

For a wide variety of arbitrarily chosen beam patterns with total numbers of beams numbering between 20 and 120 we tested (not shown), we found that a total integration time of 300 s, or 7.5 s per beam, was necessary to achieve an acceptable level of plasma convection speed uncertainty (∼ 10–25 m s⁻¹) within the desired horizontal area of a few thousand square kilometers at 110 km altitude. For reference,found that a per-beam integration time of 5 s was suitable for achieving uncertainties of 1–10 m s⁻¹, although it is critical to note that their model did not account for radar self-noise. When we exclude self-noise effects the uncertainties are reduced by factors of 4–6, not shown here A per-beam integration time of a few seconds is generally less than the integration times used in previous EISCAT systems, typically a minute or more. E3D allows for slightly shorter integration times as a result of the planned relatively higher transmission power of E3D (3.5 MW versus for example ∼ 1 MW for the EISCAT Svalbard radar) since the ACF noise level γ is inversely proportional to the transmitter power PT (cf. Eqs. 1–2, 4, 10).

We also found ideal range resolutions of 5 and 2.5 km (corresponding to modulation bit lengths Δτ= 16.7 µs and Δτ= 33.4 µs) for statistical ionospheric conditions, as given by IRI, during winter and summer respectively. Figure S1 in the Supplement shows a typical diagnostic figure used for making this assessment.

The plasma ACF is assumed sampled at regular intervals of 5 km (bit length Δτ= 33.4 µs) between 180 and 400 km altitude (blue triangles and box, Fig. a), with ionospheric conditions specified using the International Reference Ionosphere-2016 (IRI2016) and the Naval Research Laboratory-MSIS models for either “winter” (00:00 UT, 23 December 2020) or “summer” (00:00 UT, 23 June 2020) conditions. Results for winter and summer are respectively shown in the second and third columns of Fig. . Our use of IRI2016 to specify the F-region ionospheric parameters that are the basis for the uncertainty calculations (including the uncertainty of ion drift velocity) is somewhat artificial, since IRI2016 does not account for the temperature and density enhancements that typically accompany auroral activity. We nonetheless deem the uncertainties we obtain from using IRI2016 to be a reasonable reference point.

At each measurement point, the resulting convection covariance matrix is then mapped down to 110 km altitude following the methodology of (their Eqs. 3 and 4). After mapping to a reference altitude of 110 km, the measurement points extend over ∼ 66–68° MLat (black triangles and gray box in Fig. a).

The convection covariance matrices are then used together with the grid of curl-free SECS shown in Fig. b to obtain estimates of east-west and north-south convection velocity uncertainties (square root of the diagonal of the posterior convection covariance for each velocity component), as shown respectively in panels (c1)–(c2) and (d1)–(d2) of Fig. . This is done by calculating the posterior model covariance matrix Σx via Eq. (), where the theory matrix A now describes the relationship between the amplitudes of the SECS basis functions and the ionospheric convection pattern that they are used to reconstruct see, e.g.,, Σm is the data covariance matrix (here the covariance of convection measurements mapped to 110 km altitude), and Σx,pr is the prior model covariance. (We use Σx,pr-1=0, i.e., no regularization.) The posterior convection covariance is then obtained as Σv=AvΣxAvT, where Av is the theory matrix that maps SECS amplitudes to convection velocity components equation 4 in.

From these panels it is evident that the uncertainties are overall higher during winter than during summer within the experiment observation volume: typical uncertainties within the volume during winter conditions vary from 80 m s⁻¹ to over 100 m s⁻¹ but some places as low as 55 m s⁻¹, while during summer conditions the uncertainties are in some places lower than ∼ 20 m s⁻¹.

Heuristically, for approximately equally spaced beams numbering between 20 and 120 within the area indicated by the blue box in Fig. a, we find an order-of-magnitude average convection uncertainty δve∼δvn∼O0.4B/Ttot m s⁻¹, where B is the area covered by the beam pattern at 110 km altitude in km² and Ttot is the total integration time for the entire beam pattern in seconds. For example, for B=2×103 km² and Ttot=300 s, we have δve∼δvn∼O50 m s⁻¹.

Results in Fig. c1–d2 also indicate that the convection uncertainty is very high at latitudes poleward of ∼ 67.3, regardless of season. This reflects the fact that the geometry of the three E3D sites is increasingly unfavorable for uniquely resolving all three convection velocity components as the radar beam is moved to higher latitudes.

Figure e1–e2 show the resolution as defined in Sect. 2.2 of using point spread functions of the resolution matrix R=ΣxATΣm-1A their equation 5. The resolution may be thought of as indicating the smallest resolvable scale size as a function of the locations and covariances of the convection measurements, and the locations of the SECS poles. It is clear that the smallest resolvable scale size within the gray box that defines the measurement volume is generally close to the spacing of the SECS poles (10 and 5 km in the latitudinal and longitudinal directions, respectively). A spatial resolution of 5–10 km is acceptable for the plasma convection in producing estimates of arc proper motion u′.

Rather than targeting a specific phenomenon or set of conditions, the goal of this experiment is to provide routine estimates of the high-latitude ionospheric convection pattern in the vicinity of E3D. As demonstrated by , measurements from such an experiment may enable a three-dimensional, volumetric reconstruction of the ionospheric current system, or estimation of the neutral wind over altitudes at which ion motion is dominated by collisions (typically below about 130 km altitude), or both. In answer to the questions posed in the guidelines presented in Sect. 4.1 we arrive at the following.

This experiment focuses on routine measurement of ionospheric convection and subsequent reconstruction of ionospheric electrodynamics rather than a particular ionospheric phenomenon. At the small-scale end, ionospheric convection can vary over distances of meters and times of seconds and references therein, while on the largest scales temporal and spatial variations occur over tens of minutes or hours and hundreds of kilometers, respectively e.g.,. Here we choose to focus on what might be termed mesoscales, corresponding to temporal variations of order minutes and spatial variations of order a few kilometers. As have shown via the E3D-based OSSE they present, it will likely be reasonable for an appropriately designed E3D experiment to resolve convection on these scales over an area of ∼ 10⁵ km².

Following , we use output from the Geospace Environmental Model of Ion‐Neutral Interactions (GEMINI) as ground truth in the E3D example experiment presented in this section. We use the same modified E3D station locations that used: each site is artificially moved 200 km south of its actual location “to probe a more relevant part of the GEMINI simulation output, covering the transition between the up and down FAC regions”.

Figure  shows metrics for three different beam patterns, with each column corresponding to one beam pattern. The beam patterns at 120 km altitude are indicated with orange dots in panels (a)–(c). The beam pattern shown in panel a is the pattern used for the reconstruction performed by that is shown in their Figs. 2 and 3. Each of the three reconstructions uses the same input potential pattern from GEMINI (blue contour lines in panels a–c) and system of spherical elementary currents. The beam pattern in the center column consists of 25 beams spaced by 40 km on a cubed sphere grid (area of ∼ 160 km²). The beam pattern in the right column consists of 47 beams spaced by 30 km on a cubed sphere grid (area of ∼ 180 km²).

Figure a–i show that differences in the three beam patterns play little role overall in reconstruction of the potential pattern (gray lines in panels a–c), and the plasma convection (black arrows in panels d–f and scatter points in panels g–i). (Two-dimensional histograms for each of the original and reconstructed convection components are also shown in Figure S2 of the Supplement.) Particularly in the vicinity of measurement-dense areas there appears to be little to no difference in the reconstruction. Correspondingly, Fig. j–o show that the uncertainty (square root of diagonal elements of the covariance matrix) of the reconstructed eastward (j–l) and northward (m–o) components of convection tend to be lowest in the vicinity of measurements regardless of the overall beam pattern. That is, there is a clear connection between the density of measurements and the overall reconstructed convection uncertainty. There is also an overall tendency for uncertainties of the eastward convection to be slightly lower than those of the northward convection at corresponding locations.

Figure p–r, which show the resolution as defined by , indicate that the resolvable scale size at each location is closely tied to the density of measurements in the vicinity. It is clear that the third beam pattern (right column), in which the beam spacing is tightest, produces the most uniform resolution within the observational area. More variation in the resolution is seen in the first and second beam pattern; the second pattern in particular exhibits large gradients in resolution over much of the observational area, where resolutions in neighboring grid cells can in some locations differ by nearly 50 km.

We therefore conclude that for this particular experiment, which focuses on reconstruction of the ionospheric potential and the plasma convection over E3D, differences in beam patterns have somewhat surprisingly little influence on the overall reconstruction. Unless more specific information were given about the scientific requirements for this experiment, from the evidence in Fig.  there is no clear reason to prefer one beam pattern over another. One exception is that it might in some cases be preferable to optimize the experiment for uniformity in the resolution and plasma convection uncertainties; in such cases the third, tightly spaced beam pattern with more uniform coverage may have some advantage.