MAARSY is an active phased antenna array operating at 53 MHz, located in Andoya, Norway (69.30∘ N, 16.04∘ E). The array consists of 433 antenna elements, each with its own transceiver module that allow us to modulate the antennas in phase and amplitude independently. Using this capability, the transmitting or receiving beam can be steered in a desired direction up to 30∘ off zenith, with an angular resolution of 3.6∘ e.g.,. In addition to its multibeam capability, MAARSY can be used for in-beam imaging experiments. In this case, the signals from a selected number of receiving antennas are stored, and later a digital beamforming algorithm (imaging) is applied to the data. Unlike the multibeam experiment, imaging allows a 2-D image to be obtained at once, avoiding the interleave from beam to beam. Currently, only 16 receivers are available at MAARSY. These 16 receive signals can be selected from groups of seven antennas, each called “hexagons”, or from a group of seven hexagons called “anemones” see, e.g., for further technical details. For this campaign, we conducted an imaging experiment using 15 hexagons on reception similar to 's experiment. One receiver is always connected to the full antenna array and it is used as in the standard multiple experiments. The radar parameters of this experiment are summarized in Table .

In order to improve the performance of our imaging experiment, we applied a coherent MIMO technique . The technique employs multiple independent transmitting antennas and multiple receiving antennas, both spatially separated, to take advantage of the transmit–receive geometry and to increase the angular resolution of the radar. If the antennas are closely separated or collocated, the signals from each transmitting–receiving path are coherent and can be combined to form a larger virtual receiving array. The resulting number of virtual receivers is equal to the number of transmitters times the number of receivers.

Depending on the transmitting and receiving antenna configuration, some virtual receivers can be redundant. In our experiment, we carefully selected the transmitting and receiving antenna configuration to get three special redundant virtual receivers. These three redundant virtual receivers were used for phase calibration of the transmitters as was done by . Figure a shows the 15 hexagons used in reception and the three anemones used in transmission (B, D, F). Figure d shows the resulting virtual receiving antennas, whereby three of them are redundant and located at the origin.

In order to separate the contribution of each transmitter, a form of transmit diversity was needed. In , three types of transmit diversity were proposed: code, time, and polarization. Code diversity is recommended for atmospheric observations given that this is not sensitive to the temporal correlation or polarization of the target of interest. Unfortunately, code diversity cannot be currently used in MAARSY. For targets for which the temporal correlation is less than the time separation between transmitters, time diversity can be applied. Given that PMSEs have a relative long correlation time (a few hundreds of milliseconds), we applied time diversity to enhance the spatiotemporal features of PMSE. The effective time separation between transmitters was 4, 4, and 8 ms between pairs BD, DF, and BF, respectively.

As explained by , in a monostatic coherent MIMO radar, the relationship between the normalized spatial cross-correlation of signals from two different transmitting-receiving paths and the angular distribution of scattered power for a given range and frequency bin can be described by 〈vm,p⋅vn,q*〉〈vm,p2〉〈vn,q2〉=VkΔrm,n+Δrp,q=ej2πfdτ+ϕm,n+ϕp,q⋅∫B(θ)⋅e-jkΔrm,n+Δrp,qdθ, where vm,p is the signal from the transmitting-receiving path (m,p), m being the receiver and p the transmitter; vn,q* is the complex conjugate of the signal from the transmitting-receiving path (n,q), n being the receiver and q the transmitter; 〈vm,p*vn,q*〉 is the cross-correlation of two signals from antennas spatially separated; VkΔrm,n+Δrp,q is the visibility sample at Δrm,n+Δrp,q; k is the wave number vector equal to 2π/λθ. And λ is the radar wavelength; θ is the angle of arrival equal to (θx,θy,θz) which are the direction cosines in the (x, y, z) direction; B(θ) is the angular scattered power distribution, also known as brightness; Δrm,n is the spatial separation between receivers m and n; Δrp,q is the spatial separation between transmitters p and q; 2πfdτ is the phase difference due to the Doppler shift of the target, fd, where τ is the time separation between transmitters; ϕp,q is the phase difference between transmitters; ϕm,n is the phase difference between receivers.

A quick comparison between the visibility (sampling domain) for SIMO and MIMO, shown in Fig. b and e, indicates that the antenna aperture for MIMO is larger than the SIMO by ∼50 %. The difference lies in that the MIMO antenna aperture is defined as the maximum separation between two virtual receiving antennas, i.e., MaxΔrm,n+Δrp,q, whereas for SIMO, Δrp,q=0, and the antenna aperture is only defined by the maximum spatial separation between two receiving antennas. Figures c and f show the resulting instrument function or point spread function for SIMO and MIMO, respectively. As expected the half-power beamwidth (HPBW) for MIMO is ∼50 % smaller than for SIMO, resulting in an angular resolution of 2.4∘ for MIMO compared to 3.6∘ for SIMO. Furthermore, the sidelobes in the MIMO configuration are strongly reduced, given that the visibility is larger and contains no gaps.

As described by , the angular resolution obtained from a direct inversion of Eq. () using the inverse Fourier transform is limited by the longest baseline and the unmeasured antenna separations (visibility gaps). proposed a new technique to improve the angular resolution based on the work of . Capon can be seen as an extension of the inverse Fourier transform. The difference lies in the fact that Capon chooses the antenna weights adaptively in order to minimize the sidelobe interference from signals outside of the direction of interest according to the data. Capon's technique provides an estimate of the brightness function given by B(θ)^=1MH⋅V-1⋅M, where M=e-jkrm0+rp0,e-jkrm0+rp1,…,e-jkrm1+rp0,…T is the Fourier kernel and V=VkΔrmi,mj+Δrpk,pl is the visibility due to the virtual receivers vmi,pk and vmj,pl, with i and j being the receiver indices and l and k being the transmitter indices.

Even when MIMO is used, the problem is still underdetermined. Thus, there are infinite possible image solutions, B, which agree with the data, V. Of all possibilities, MaxEnt chooses the solution with the maximum entropy or minimal information content e.g., as the one to be the most likely brightness distribution and the most consistent with the available visibility data and their statistical uncertainties. The entropy for a given frequency bin and range can be defined as S=∑θB(θ)^ln⁡B(θ)/F^,F=∑θB(θ)^, where F is the summation of the brightness distribution over the region of interest. The solution of Eq. () is defined by max⁡θSsubject toV-M⋅B(θ)^<ϵ, where ϵ is the noise amplitude associated with the visibility measurements. In this work, we have also considered the improvements of . Specifically, we have taken into account the transmitting beam pattern and the statistical uncertainties of all the visibility pairs.

Since the estimated brightness is expressed in polar coordinates (θx,θy,R), a cubic spline interpolation was applied to convert them to Cartesian coordinates, B(θx,θy,R) to B(x,y,z), with the radar being located at the center (x=0, y=0, z=0). Below we show the results of two selected events (Events 1 and 2) after performing such interpolation. For both events, we show x vs. y cuts for a given z, as well as x vs. z cuts for a given y, where x, y, and z represent the east–west (EW) direction, north–south direction (NS), and altitude, respectively.

Examples of EW–NS and EW–altitude 2-D images for Event 1 obtained by applying Capon and MaxEnt to two different antenna configurations, SIMO and MIMO, are shown in Figs.  and , respectively. Four different results are shown: (a) SIMO-Capon, (b) SIMO-MaxEnt, (c) MIMO-Capon, and (d) MIMO-MaxEnt. Having a quick look at the results, it is clearly observable that (1) MaxEnt outperforms Capon when the same antenna configuration is used, either SIMO or MIMO. This was already pointed out by previous works e.g.,. (2) As expected, MIMO shows a cleaner and more defined image compared to SIMO, when either Capon or MaxEnt is employed. (3) The improvement of using MIMO instead of SIMO is much better in MaxEnt than Capon. The improvement of MIMO-Capon with respect to SIMO-Capon is about 50 % due to the larger virtual antenna array, whereas the improvement of MIMO-MaxEnt with respect to SIMO-MaxEnt is much better than 50 %. This difference lies in the fact that Capon tries to reduce the sidelobes, adaptively steering them to echo-free zones. Unfortunately, for the two events shown, most of the illuminated area is filled with PMSE scattering, and thus the performance of Capon is expected to be comparable to the conventional beamforming (inverse Fourier transform). In the case of MaxEnt, the improvement is mainly due to the larger virtual antenna array and the use of statistical uncertainties as described by . Unlike Capon, our MaxEnt implementation takes advantage of the redundant visibility pairs, giving more weight to pairs with less uncertainty, i.e. more redundancy. Figure b and e show the visibility pairs and their redundancy for SIMO and MIMO, respectively.

Coming back to our comparison of SIMO vs. MIMO, with MIMO-MaxEnt, small wave-like structures of 2 km wavelength can be clearly observed, which are invisible in SIMO implementations or MIMO-Capon. For example, observe the two wavefronts at x=10 km in Fig. d, right beside the larger meridionally oriented wavefronts of 7 km wavelength. This indicates that wave-like structures of different wavelengths coexist within PMSEs as previously seen in NLCs e.g.,. In addition, Fig. d shows that the ascending structures (red color) have a higher signal-to-noise ratio (SNR) than the descending structures (blue color).

We show similar 2-D cuts for Event 2 in Figs.  and  for z=82.7 km and y=-6 km, respectively. In this case, the observed wavelength is 12 km. Unlike the first event, the SNR is similar for targets with negative and positive Doppler. Figure d shows two very interesting points: (a) a very well-defined wave-like structure between 82 and 84 km and (b) a quasi-uniform structure between 84 and 86 km, which apparently has been modulated by the first wave. In this case, the wave-like structure is easily discernible even with SIMO-Capon, given that the wavelengths are larger than in Event 1 (see Fig. a).

Having shown the better qualitative performance of MIMO-MaxEnt with respect to the other three implementations for the two selected events above, next we present extended results using just MIMO-MaxEnt.

Figure  shows the evolution in time of two selected events, i.e., Event 1 (panels a, b, and c), and Event 2 (panels d, e, and f). Figure a and d show the time evolution vs. altitude for selected EW and NS coordinates. In these plots, we can appreciate how variable PMSE structures are, showing different altitudinal extensions. Note that the effective horizontal area is less than 1 km2 in both cases.

The second and third row of Fig.  show the time evolution vs. EW direction and the time evolution vs. NS direction, for EW and NS keograms, respectively. We have included the zonal (u0) and meridional (v0) wind velocity estimated from combining a couple of specular meteor radars (SMRs) (pink arrow) and from MAARSY based on PMSE Doppler velocities (yellow arrow). The wind values are shown in Table . Since this is a “time” vs. “distance” plot, the zonal and meridional wind are represented by arrows for which their slopes indicate the wind magnitude, i.e, how long a target is displaced in the y axis for a certain time in the x axis. The SMR winds were obtained from combining SMR detections from Andenes and Tromsø in northern Norway see, e.g.,for details. In order to estimate the winds from PMSEs we used the following formula: vradθx,θy,θz=u⋅θx+v⋅θy+w⋅θz, where vrad is the radial wind, (θx,θy,θz) are the direction cosines, and (u, v, w) are the zonal, meridional, and vertical wind direction, respectively. Assuming a constant u, v, and w for a given altitude bin and time bin, and taking all the measurements with a SNR higher than -5 dB, we invert Eq. () and get u0, v0, and w0, mean values of u, v, and w respectively.

The keograms for Event 1, i.e., from 00:50 to 01:05 UTC, show that the meridionally oriented wavefronts have a limited vertical extent centered at 85 km (Fig. a), and since this wave has a finite wavelength in the EW direction, the zonal wave propagation can clearly be observed in Fig. b, which shows that the elongated meridionally oriented wavefronts are zonally drifting with the same direction and with the same speed as the wind. In the NS direction, the meridional drifting of the wave is not clearly observed due to the elongated structure. Mesospheric wave-like features observed with airglow imagers (ripples) have also been noticed to drift with the background wind e.g.,. These ripples have been associated with gravity wave breaking and are a clear signature of atmospheric instability.

Figure d shows another interesting wave-like example (Event 2). Unlike the first case, this wave does not keep its amplitude in the vertical direction; see Fig. d. It grows and then disappears. Its direction of propagation in the zonal and meridional direction is also interesting. As shown in Fig. e, the direction of propagation in the zonal direction is completely opposite to the background wind. Whereas the wind is going from east to west, the wave propagates from west to east. In the NS direction (Fig. f), the wind is close to zero and we do not expect changes in this direction. Since its wavelength is relatively small, this structure might be classified as an instability; however, the opposite direction of propagation suggests that it could be a propagating gravity wave. Further investigation of these events is needed to understand the physical mechanisms behind them, including lidar and airglow imager observations.

PMSEs have been used as a neutral wind tracer, assuming that u, v, and w are constant and homogeneous during the analyzed time e.g.,. Therefore, those works assumed that scatters from PMSEs are moving with the neutral wind at the same velocity and in the same direction. Unlike winds obtained from SMR, winds from PMSEs are affected by local disturbances, as shown in Fig. e and f. When the dynamics of local structures are not in agreement with the wind dynamics, a bias could be introduced in the wind estimation (as shown in the Event 2). However, when these local disturbances are moving with the wind, the estimated wind is not affected (Event 1). Note that the PMSE winds are in good agreement with the SMR winds in Event 1, but they are not for Event 2, particularly for the meridional component.

An animated sequence of the two events has been included in the Video supplement, i.e., Movies S1 and S2. For both events, the sequence includes selected cuts of EW–NS, EW–altitude, and NS–altitude. In Movie S1, we identify at least four examples of monochromatic waves with different wavelengths drifting with the wind in the northwest direction (at 23:57:37, 00:02:24, 00:10:57, and 00:55:33 UTC). Interestingly, in this case, longitudinal and transverse waves both drift with the background wind. In Movie S2, we show the complete evolution in time of Event 2. In the EW–altitude cut, the wave structure between 82 and 85 km drifts against the wind, whereas a layer at 87 km between 05:20 and 05:30 UTC follows the background wind. Note the projected radial wind (from red to blue) indicates a westward wind. These events are good examples of the complicated dynamics within PMSEs. Further analysis and interpretation of these high-resolution spatiotemporal structures will be done in a future work.

Figures  and  show 3-D maps of (a) the signal-to-noise ratio (SNR), (b) radial velocity, (c) locally enhanced SNR, and (d) residual radial velocity (i.e., vres), for Events 1 and 2, respectively. In addition contours of locally enhanced SNR are overplotted on both the radial velocities. The SNR and radial velocity were obtained from the first and second spectral moments e.g.,. The locally enhanced SNR has been obtained using a 2-D Gaussian function kernel with a width of 6 pixels. The local enhancements allow us to observe weak structures within the strong one. For example, wave fronts are distinguishable in Fig. c which were not visible in Fig. a. On the other hand, the residual radial velocity was estimated by removing the contributions of the estimated mean horizontal velocities in the measured radial velocities; i.e., vresθx,θy,θz=vradθx,θy,θz-u0⋅θx+v0⋅θy.

Assuming that the vres is mainly due to the vertical motion, we can clearly see in Fig. d how up (red) and down (blue) structures drift across the illuminated area, maybe due to KHI. Similarly, Fig.  shows animated images of Event 2. In this case, the horizontal wind was small and most of the radial velocity was due to the vertical motion; i.e., radial velocity and residual velocities are almost the same. As mentioned above, in this event, the waves propagate horizontally against the weak horizontal wind.

The animated versions of Figs.  and  are shown in Movies S3 and S4, respectively. Although the information might be redundant when compared to Movies S1 and S2, we have decided to include them to provide a more standard view of typical spectral parameters of a multibeam radar.

Making a quantitative comparison between SIMO and MIMO for real targets is not an easy task. We need a prior knowledge of the brightness to make a good analysis. This is not the case for PMSE. Fortunately, our observations include echoes from specular meteors; see the bright echoes located at (-10.5, -12.5) in Fig. d. Indeed, meteor echoes can be observed in the PMSE region, but the great majority of them occur outside this window. When a meteor echo occurs in the PMSE altitude it will be short-lived (less than a few hundred milliseconds). In previous studies meteor echoes were treated as outliers and were removed from the measurements e.g.,. For our benefit they can also be used to evaluate the angular resolution that can be achieved with our implementations quantitatively. A specular meteor echo could be considered to be a point target. Along with its trajectory, the trail is long (hundreds of meters to a few kilometers), but its angular response is narrow. In the transverse direction to the trail, it is very narrow and its angular response is also narrow.

In Fig.  we show the normalized angular scattered power distribution for a specular meteor using SIMO and MIMO in combination with Capon and MaxEnt. As expected, the range resolution does not change for SIMO or MIMO (see Fig. a). We see a peak at 89.1 km and low power at other ranges. However, when comparing Capon and MaxEnt, MaxEnt shows us a clean power distribution along all the ranges, while Capon shows us a remaining sidelobe contamination in other ranges, coming from other angles. This indicates that, even with MIMO, Capon does not suppress the sidelobes as well as MaxEnt. Figure b and c show us the angular power distribution for θx and θy, respectively, for which the points are the samples for a given angle and the continuous line is a fitted Gaussian function. Using the fitted function, we estimated the half-power beamwidth (HPBW) for each implementation. Table  summarizes the angular resolution and the improvement factor for each method compared to the theoretical angular resolution of the full array MAARSY radar. As we expected the improvement between SIMO and MIMO is about 1.5, given that we increased the antenna aperture for MIMO by ∼50 %. When combining MIMO and MaxEnt, surprisingly, we got an angular resolution of ∼0.6∘, i.e., more than 5 times better than MAARSY's HPBW.