Every day the Earth's atmosphere is bombarded by billions of dust-sized particles and larger pieces of material from space. In doing so these objects, called meteoroids, ablate and produce phenomena called meteors . These phenomena are commonly seen as visible streaks of light in the night sky.

Meteor phenomena cause ionized plasmas that can be roughly divided into two distinctly different regimes: a dense and transient plasma region co-moving with the ablating meteoroid and a trail of diffusing plasma left in the atmosphere and moving with the neutral wind. Both of these plasmas reflect radio waves . When measured with a radar, they cause so-called meteor head and meteor trail echoes . To determine the position of these radar targets, interferometric or multi-static radar systems must be used.

Observing this incoming material is important for several reasons. To mention a few, it gives us a unique opportunity to examine the motion and population of small bodies in the solar system , it provides information about the extraterrestrial input of material into our atmosphere , and it provides a possibility to assess the neutral wind at an altitude otherwise difficult to probe . The typical ablation altitude where meteor phenomena occur lies between 70 and 130 km and references therein. This region is characterized by variability driven by atmospheric tides as well as planetary and smaller-scale gravity waves. As this region is difficult to observe with other methods due to the low atmospheric density and high altitude, specular meteor trail radars have become widespread scientific instruments to study atmospheric dynamics. The extraterrestrial input of matter also affects various physical and chemical processes important for a wide range of phenomena, such as the formation of clouds at 15–25 km altitude responsible for ozone destruction in the polar regions, midlatitude ice clouds at 75–85 km, which are possible tracers of global climate change, and metallic ion layers in the atmosphere .

The diffusing meteor trail is an elongated plasma and not a point target. Therefore, specular reflection dominates. This makes interferometric all-sky radar systems efficient at observing the meteor trail phenomena with relatively inexpensive hardware. This has made such systems widespread and there are currently systems deployed at locations covering latitudes from Antarctica to the Arctic . When determining the position of an object by interferometry, there may be an ambiguity problem . The position is determined by finding the direction of arrival (DOA) of the incoming echo onto the radar. Depending on the spatial configuration of the receiving antennas and their individual gain patterns, the voltage responses can be the same for several different plane wave DOAs, thereby making it impossible to determine which one is correct. Noise can create the same effect even if the system contains no theoretical ambiguities. These ambiguities then appear with a probability that is a function of the signal-to-noise ratio (SNR) . This problem is general to all DOA determinations made by interferometric radar systems.

Due to its simplicity and theoretically unambiguous meteor trail position determination capability, the so-called Jones 2.5λ radar design has become the standard for specular meteor trail radars . investigated the Jones antenna configuration and found that the use of 2.5, 3, and 5.5λ spacings could produce a more accurate echo DOA. developed the concept further and presented a solution which utilizes all possible antenna pairs of a meteor radar antenna configuration. However, it has been noted by several authors that if the SNR is low enough, the position determination is still ambiguous . For this reason, the widespread SKiYMET system has implemented a variable in their database referred to as “ambig” which gives several possible position solutions for the same event. A relatively small part of the total number of detected echoes have these angular ambiguities, generally around 10 %–20 % Fig. 3a. Practically, these events cannot be used for wind determination as the correct location is needed in order to determine the line-of-sight direction for the wind component estimation. Resolving the ambiguity issue is important also to, e.g., facilitate detailed analysis of meteor head echoes and long-lasting trails.

As the SNR is the deciding factor for whether a detection is ambiguous or not, standard temporal integration methods can be applied to increase the SNR of the signal. The amount of time that can be integrated depends on the coherence time of the phenomena. A meteor trail drifts with the local wind speed, and the echo phase change over time can be reliably modelled during its entire existence. The range rates and Doppler frequencies of head echoes change fast. In practice, unpredictable effects from, e.g., fragmentation limits the usable coherence time to single radar pulse sequence transmissions.

used direct Monte Carlo (DMC) simulations to theoretically characterize the ambiguities of several radar systems commonly used for both head and trail echo meteor measurements, the Jones 2.5λ radar being one of them. However, the study did not explore to what degree such simulations are practically useful in validating analysis of real measurement data. The simulation results were not applied in the context of comparisons with measurement data or to remove measurement ambiguities.

The standard SKiYMET data products contain ambiguities that can theoretically be resolved by temporal integration but also validated and resolved by DMC simulations. Here, we test the methods presented in the previous study by implementing two different temporal integration methods to compare with the DMC results and resolve angular ambiguities.

examined the morphology of ambiguities for the Jones 2.5λ design and other radar systems using the beam-forming point spread function (PSF). In the case of radars with identical antenna elements, each having a separate signal channel, the PSF is identical for all input DOAs. The identification done in thereby applies for all input DOA. The PSF reported in Fig. 1 matches with the morphology of the Monte Carlo simulations of DOA determination performed here and in . Given a Bayesian method to assign probability distributions to the ambiguities, many of the previously unusable data may again be usable. Events with high certainty in inference of the true DOA provide a validation for methods to resolve the ambiguity in the analysis itself without simulations.

implemented a coherent detection algorithm where linear regression was applied to the measured cross correlation angles. In essence, the sample zero-lag cross correlation described there is the same as the temporal integration of sample cross correlations we have implemented here. Henceforth we drop the word “sample” in the expression “sample spatial cross correlations” for brevity. When we temporally integrate spatial cross correlations of a virtually stationary plane wave, the integration is coherent. However, a more effective coherent integration is to apply a matched filter to each channel and coherently integrate prior to calculating the cross correlation. implemented a coherent deconvolution on a coded continuous-wave meteor radar. This is practically the same as the matched-filter temporal integration implemented here. A simple derivation of the effectiveness and coherence of these two temporal integration techniques is given in Appendix .

We have applied two separate methods for inferring the true location of an ambiguous DOA measurement. The first method is based on DOA determination direct Monte Carlo simulation and Bayesian inference and has been presented in . This method is complex and costly in terms of computation resources but yields a probability distribution over the ambiguities. The second method is temporal integration of the spatial correlation matrix, described in detail in Sect. . This second method is easily implemented in data analysis pipelines as it can rely on previous DOA determination implementations and is only analysing a single integrated pulse rather then all pulses individually.

The DOA determination performance investigation of the Jones 2.5λ radar presented by successfully made use of the multiple signal classification (MUSIC) algorithm developed by . We have applied the same implementation of MUSIC as described by to analyse the SKiYMET data in this study.

We have four independent methods of determining the DOA: the SKiYMET standard data product, temporal integration of the spatial correlation matrix (R¯), the spatial correlation matrix of matched-filter integrated channel signals (R(xm)), and Bayesian inference based on DMC simulations of DOA determinations (P(j)). We have compared all four methods in order to test and validate them.

In total 2222 events were automatically analysed. To select authentic specular trail echoes we used the time for which the amplitude of the backscattered signal falls to one half of its maximum value τ. We did not include events with undetermined τ (these may be ground echoes or long-lived non-specular echoes pp. 356), τ less than 0.001 s (these may be ionosonde interference), or radial velocities exceeding 100 m/s (these may be due to Farley–Buneman instabilities ) as such events are unlikely to be genuine specular trail echoes. Some of the events still appear likely to have been range-spread non-specular trail echoes (e.g. the “straight line” of black dots in Fig.  around kx=-0.15,ky=0.25 all occurred within 1.5 s of each other), but as this does not impact the DOA evaluation, we did not attempt to remove such events from the analysis. By contrast, the methods presented here can be used to successfully analyse such events. A summary of the analysis results is given in Table .

Out of the total of 2222 events 294 were listed as having angular ambiguities. These were analysed with both the DMC-simulation-based Bayesian inference and the temporal integration versions of MUSIC. An example of such an event is illustrated in Fig. . The upper middle panel illustrates SNR versus radar pulse, and the two vertical black lines denote the region within which the trail event was identified by the SKiYMET analysis. These are the pulses that were also reanalysed using the MUSIC algorithm. The DOA output from these IPPs is illustrated as blue dots in the left and right column of panels. The upper left panel shows azimuth as a function of IPP and the lower one elevation. Here, the transparent grey lines denote the azimuth and elevation given in the MPD file. In this case, the standard SKiYMET analysis produced two angular ambiguities. The orange transparent line denotes the DOA output from the R¯-based MUSIC and the cyan transparent line denotes the DOA output from the R(xm)-based MUSIC. The same information is given in wave vector ground-projected space, kx,ky in the lower right panel with a zoomed-in version in the upper right panel. Here the MPD-file results are marked by grey crosses, the R¯-based MUSIC by an orange circle, the R(xm)-based MUSIC by a cyan circle, and the regular MUSIC output for each IPP is marked by the blue dots. Finally, the distribution of the MUSIC response F is illustrated in the lower middle panel as a function of IPP. Here, the solid lines denotes the MUSIC response for the temporally integrated versions.

For each of the 294 events with angular ambiguities we also performed a series of DMC DOA determination simulations. The noise in the simulations was set to sample from the distribution of SNRs measured for the events themselves so as to reproduce the multinomial probabilities. In Fig. , the simulations and the Bayesian inference results are illustrated alongside the measurement data for the event, also illustrated in Fig. . The upper left panel shows the MUSIC DOA output in wave vector ground-projected space, i.e. kx and ky. The black circles denote the possible ambiguities and their inclusion regions, i.e. the Ai sets from Eq. (). The upper middle panel shows the measured distribution of the SNR versus MUSIC response compared to the simulated distribution. The remaining panels show DMC DOA determination simulations with different true inputs. In each of these panels, the input DOA is marked by the large red circle. For reference, the MPD-file results are also marked by transparent grey crosses in each panel. The title of these figures give the Bayesian inference results P(j) in percent rounded to one decimal. The Bayesian inference can also be evaluated manually by comparing the simulated distribution of DOA outputs with the measurements given in the upper left panel. In this case, the Bayesian inference and the temporally integrated MUSIC, as well as a manual inspection, all agree and identify the same DOA as the true location.

Of the 294 analysed events, 256 (87 %) were identified as containing a nominal echo, ideally scattered from a single meteor trail. The remaining 38 events (13 %) contained anomalous signals in the sense that the target model does not match with the real target(s). The SNR of individual pulses in each event typically span several orders of magnitude. The mismatch of model and reality was detected automatically by using the fact that the MUSIC F value did not increase with increasing SNR as a criterion. This indicates that regardless of the measured signal strength, the sensor response model could not be matched to the detected signal. Further examination to identify the sources or causes of these anomalous events was considered outside the scope of this study. The events were marked as anomalous signals in Table  and were not processed further.

In 95 % of the nominal ambiguous echoes, all three non-SKiYMET methods found one and the same output DOA, and this DOA was listed as one of the possible ambiguous DOAs in the original SKiYMET analysis. Similarly, in all unambiguous cases with a nominal signal, the results from all three methods also concurred. For the unambiguous cases a comparison between the temporally integrated MUSIC algorithms and the SKiYMET standard data product is illustrated in Fig. . This further validates the robustness of the implementation as their difference is typically less (root mean square differences are 0.68∘ for the temporal integration results and 0.79∘ for the matched-filter results) than the reported expected angular measurement accuracy of the system (≈1∘, ).

Upon manual examination of the remaining 5 % where all three methods did not find one and the same DOA, the following observations were made:

In five of the cases where the Bayesian inference indicated a solution not listed in the MPD file, the event had too few IPPs to be well determined by any method. In the other two events, both the R¯ and the R(xm) solutions concurred with P(j) and manual inspection indicated they were correct.

The DOA distribution for all analysed events is illustrated in Fig. . The black dots are the 1928 unambiguous meteor trail events, the blue stars are the original ambiguous nominal signal locations that could be discarded, and the red crosses are the 249 resolved ambiguities for which the Bayes solution matched with one of the listed ambiguous locations in the original SKiYMET analysis. There are multiple blue stars per resolved ambiguous event, i.e. per red cross.

The DOA distribution is concentrated towards low elevation in the north. These data were recorded during the Geminid meteor shower. The solid line shows the locations where specular reflection could occur assuming that the meteor originated from the centre of the Geminid meteor shower radiant region (right ascension 112∘, declination 33∘) at the time of the measurement.

At around 00:00 UTC, the Geminid radiant as well as most sporadic meteor source regions were located towards the south since Sodankylä is located at a high northern latitude. Therefore, most meteors with trajectories fulfilling a specular condition with respect to the radar appeared towards the north. This explains the concentration of DOAs towards the north at low elevation in Fig. .

When the remaining resolved direction only has one ambiguous range possibility within the meteor zone, the resolved angular ambiguity also solves the range ambiguity problem, but this is not always the case. The problem with range ambiguities is independent from the angular ambiguity problem and outside of the scope of this study. However, the range ambiguity would be easily avoided with an update of the system to use coded transmission sequences. An advanced example is give by , who used coded transmission sequences with a multiple-input–multiple-output bi-static and continuous-wave setup. In the case of the mono-static Sodankylä SKiYMET system, it would be enough to use a small set of binary-phase shift key codes on the pulsed transmitter to enable distinguishing between the pulse trains that come from subsequent echoes within the meteor zone.

The reason why range ambiguities appear in the standardized SKiYMET analysis is the combination of uncoded radar pulses and high PRF. At low-elevation angles from the radar, this means that two or more consecutively transmitted pulses may simultaneously give rise to echoes of meteors in the standardized acceptable altitude range 70–110 km . The commonly used PRF 2144 Hz of the SKiYMET radar systems corresponds to a range aliasing of ≃70 km.

We have shown that DMC simulations can characterize the DOA determination behaviour of a system and validate the performance of other analysis methods. Together with a Bayesian inference approach, these simulations can systematically be used to determine the true location of weak meteor radar trail echoes with ambiguous DOA. To perform the Bayesian inference, many simultaneous DMC simulations are required, and this method is as such computationally expensive. However, as the results can be used to validate other methods and characterize the behaviour of a radar system, they are a valuable tool for development work.

We have implemented two versions of standard temporal integration techniques used to increase the SNR of the spatial correlation matrix and subsequent application of the MUSIC algorithm. This is computationally inexpensive and implementation-wise a very simple method able to resolve ambiguous DOAs for SKiYMET meteor radar systems. However, even though the concept itself is not new (cf. ), it does not seem to have been implemented in the SKiYMET standard data analysis. Both the temporally integrated spatial correlation matrix version of MUSIC and the matched-filter version provided the correct output DOA according to the Bayesian inference in ∼ 96 % of the ambiguous cases containing a nominal signal. In the cases when they did not agree, there were either not enough sample points to temporally integrate for the method to be effective or we could not manually conform a specular meteor trail echo in the raw data. For all unambiguous cases both methods coincide with the SKiYMET standard data product.

Standard meteor trail radar systems, such as SKiYMET generally produce enough sample points from each registered meteor for the temporally integrated MUSIC to work. The results indicate that this method will solve the angular ambiguity problem in almost all cases. The problem with range ambiguities is independent from the angular ambiguity problem and outside the scope of this study, but we note that it could be avoided with an update of the system to use coded transmission sequences.

The presented methods do not depend on the MUSIC algorithm per se: for example, the complex signal amplitudes that are used in the DOA determination algorithm by to calculate the ϕ angles can be temporally integrated in the same way as the spatial correlation matrix R was temporally integrated here.

Finally, one can pose the question whether and at what point the temporally integrated MUSIC becomes ambiguous. This question as well as the validation of a pipeline implementing temporal integration techniques can be addressed by the same type of DMC simulations.