Meteors and hard targets produce coherent radar echoes. If measured with an interferometric radar system, these echoes can be used to determine the position of the target through finding the direction of arrival (DOA) of the incoming echo onto the radar. Depending on the spatial configuration of radar-receiving antennas and their individual gain patterns, there may be an ambiguity problem when determining the DOA of an echo. Radars that are theoretically ambiguity-free are known to still have ambiguities that depend on the total radar signal-to-noise ratio (SNR). In this study, we investigate robust methods which are easy to implement to determine the effect of ambiguities on any hard target DOA determination by interferometric radar systems. We apply these methods specifically to simulate four different radar systems measuring meteor head and trail echoes, using the multiple signal classification (MUSIC) DOA determination algorithm. The four radar systems are the Middle And Upper Atmosphere (MU) radar in Japan, a generic Jones

Radar systems are a vital part of current research infrastructure. They are used for a wide variety of novel

When determining the position of an object by interferometry, there is an ambiguity problem

Every day the Earth's atmosphere is bombarded by billions of dust-sized particles and larger pieces of material from space. This incoming material gives us a unique opportunity to examine the motion and population of small bodies in the solar system

Meteor trail plasma drifts with the ambient atmosphere. The drift velocity is therefore a measure of the neutral wind at the observation altitude. The typical ablation altitude where meteor phenomena occur lies between 70 and 130 km. This region is characterised by variability driven by atmospheric tides and planetary and smaller-scale gravity waves. Specular meteor trail radars have become widespread scientific instruments for studying atmospheric dynamics deployed at locations covering latitudes from Antarctica to the Arctic Svalbard

Due to the altitude distribution of meteor phenomena, the far field approximation is almost always valid, which means that an incoming echo can be modelled as a plane wave

There are analytical methods for determining all the ambiguities present in a radar system, although these scale poorly and have several restrictions

The existence of head echo events, such as the one illustrated in Fig.

Example meteor head event measured with the MU radar which seems ambiguous. Panel

There are no methods, to our knowledge, for resolving noise-induced ambiguities in DOA determinations or for determining the probability of misclassification. We have therefore extended upon the study performed in

In Sect.

Section

We have focused on radars measuring the meteor phenomena. However, the analysis methods applied are usable on any kind of interferometric hard-target detections made by radars. The radars that we have applied these techniques on are described in Sect.

To find the ambiguities present when determining the DOA, we need a radar sensor response model

Usually, the wave amplitude

In the pursuit of an analytical solution,

The normalised sensor response model is written as follows:

Ambiguities are formed when the following occurs:

We call an ambiguity perfect, i.e. unambiguous DOA determination is impossible even at infinite SNR, if

We define a set of ambiguities to

Following the definition in Eq. (

There are several other ways to define ambiguities. The requirement is that the definition is invariant of the sensor response norm and is to a constant phase offset in all dimensions; which definition to use depends on the behaviour of the DOA determination algorithm. One important property of algorithms is whether or not they preserve orientation with respect to

One of several possible other formulations is using a distance function. The definition would then be

If the set

Define a source wave direction

Generate a set of

Do a gradient ascent search, using the gradient of Eq. (

Collect the peak locations

Remove duplicate results yielding the set of all ambiguities and their peaks

This method can be used with any sensor response model to find ambiguities. After the set

An important factor to note is that these ambiguities are not necessarily transitive relations. They are only transitive when the indicator function is 1, i.e. they occupy the same point in sensor response space. The intransitive relation means that, for

All the radar systems considered in this study have operating frequencies in the very high frequency (30–300 MHz) range. In this range, the galactic background radiation dominates the noise

In pseudocode, the noise can now be simulated as

In order to relate results from simulations to measured data, the noise-controlling variable

Given a sensor response model and a noise model, we can perform a direct Monte Carlo (MC) on any DOA determination algorithm. Given a true direction

Example Monte Carlo (MC) direction of arrival (DOA) determination simulation with 500 samples. The generic Jones

The example MC DOA determination simulation in Fig.

To account for a limited DOA determination accuracy, we choose an inclusion distance

Practically, in the following, if:

We are interested in the misclassification and algorithm failure probability. We examine this by regarding the source as a variable

Even though ambiguities are not necessarily transitive relations, as mentioned in Sect.

Assuming that we have chosen an echo from the target and that our models are representative, then either

Overview of the relation between ambiguity sets.

To provide a set of input and output wave vectors that accounts for all possible true

Practically, considering several sets of ambiguous measurements from independent events, i.e.

There are some practical consideration when implementing the construction of

Using the definitions for

The columns of the matrix

The error of the probability matrix

It is generally not advisable to use data that are ambiguous. Quantitatively describing when data are too ambiguous for further usage is one of the goals of this study. For example, as illustrated in Fig.

If there is a need to analyse ambiguous data, we suggest a Bayesian approach. As an example, let us return again to the simulation presented in Fig.

Given a model with parameters

The

This approach is compatible with the problem at hand. Assuming the matrix of probabilities

This method may be able to infer the true direction even in very ambiguous data. At the very least, this method provides a probability distribution over the possible directions, as exemplified in Sect

A method for investigating whether the Bayesian approach makes a significant improvement on analysis is measurement simulation. The matrix

Effects like mutual coupling, errors in cable lengths and other hardware-related issues can introduce phase errors in radars

There has been extensive work done to determine phase offsets as a whole on radar channels

To examine the impact of phase errors on the antenna level within subgroups, we performed a pair of MC simulations for the MU radar subgroup model. As input, the

It should be noted that in this test the phase offsets that generated the simulated noisy signal were also included in the MUSIC analysis model.
Finally, we ran two MC simulations at 5 dB SNR for the MU subgroup model, using

For the purposes of consistency and simplicity, we have used the same DOA determination method for all radar systems examined, namely the MUSIC algorithm

We define a measured sensor response as the complex vector

The correlation matrix consists of coherently integrated channel-to-channel phase differences over the temporal samples. The eigenvalues of the correlation matrix correspond to signal powers, and the eigenvectors corresponding to the largest eigenvalues span the signal subspace

We write the projection function in terms of matrix operations as follows:

As Eq. (

We have chosen to apply a two-step maximisation method. First, a finite grid search over all possible

However, there is no guarantee that the initial grid search will always be able to identify the correct slope as an initial condition for the gradient ascent. If the peak width is smaller, then the grid size of any slope may be found instead. To solve this problem, we also implemented an option of running multiple gradient ascents in parallel. When this option is enabled, instead of using only the maximum point from the grid search as a starting value, the

The sensor response for all radars covered in this study was modelled using two different models, with a simplified model as follows:

In this study, we have assumed that the antennas have omnidirectional gain. This is, of course, not the case as mentioned in Sect.

We hereafter refer to the model in Eq. (

In Fig.

The different radars considered in the DOA determination study. For the radars which consist of subgroups of antennas, the subgroup centres have radar-specific colours, while antennas and subgroup borders are always grey and black. The Jones

Radar systems designed for studying meteor trail echoes commonly consist of a wide angle (all-sky) transmitter system and an interferometric receiver system

The receiver system design is beset by two problems

As described by

The Jones antenna configuration has remained predominant in meteor radar installations and is often referred to as removing (in principle) any angular ambiguities

To our knowledge, there are no further quantitative investigations of the Jones 2.5

The 46.5 MHz Middle And Upper Atmosphere (MU) radar near
Shigaraki, Japan (34.85

Early meteor head echo measurements, using the original setup with four receiver channels

The new Middle Atmosphere Alomar Radar System (MAARSY) was constructed in 2009–2010 on the Norwegian island of Andøya (69.30

The smallest MAARSY sub-array unit consists of seven antennas distributed in a hexagonal pattern, as illustrated in Fig.

An alternative configuration of MAARSY subgroups used as radar channels to the one illustrated in Fig.

The radiation pattern of MAARSY has been studied and validated through observations of cosmic radio sources

The Antarctic Syowa Mesosphere Stratosphere Troposphere Incoherent Scatter (PANSY) is a mesosphere–stratosphere–troposphere/incoherent scatter (MST/IS) radar located at the Japanese Syowa Station (69.01

PANSY operates on a centre frequency of 47 MHz and with a peak power of 500 kW and 5 % duty cycle. The radar is a challenge for DOA determinations as the subgroups are located at different altitudes and partially disjointed and have to be moved or intermittently be disconnected from the system, depending on snow accumulation conditions. Even the antennas within subgroups are elevated non-symmetrically. Currently the antennas are distributed in altitudes ranging between

In 2017, a peripheral antenna array for detecting field-aligned irregularities (FAI) was installed

The PANSY radar has recently been complemented by a meteor trail echo interferometric receiver system (Taishi Hashimoto, personal communication, 2020). The antenna configuration is displayed in Fig.

To demonstrate the above methods, we present results from numerical simulations. The next step will be applying them on measurement data. We aim to implement these methods in our data analysis pipelines for meteor head echoes measured by the MU radar and the PANSY radar in the future and to classify the location probability of ambiguous meteor radar trail echoes using Bayesian inference. However, the current study allows us to quantitatively evaluate how DOA determination behaves with respect to SNR and to qualitatively evaluate if ambiguities are relevant or not. Such results are useful in the configuration and construction of pipelines.

For each of the radar systems described in Sect.

For each of these chosen sources, the following steps were performed:

Determine all ambiguities using 1000 starting conditions according to the method outlined in Sect.

Run an MC simulation of 500 samples for each input direction in

Discretise the MC results into probability matrices

Simulate measurements according to Sect.

The ambiguity dynamics change as a function of input DOA for all systems but the planar Jones 2.5

This kind of map shows in which source directions the radar is able to resolve well and in which directions it cannot uniquely determine. While it does not illustrate the morphology of ambiguities, it does show the qualitative connection between input DOA and limiting SNR. The white areas are regions where no ambiguities were found using the selected algorithm settings.

Map of worst ambiguity as a function of input DOA for the MU radar subgroup model

The analysis results for each of the radar systems (except the planar Jones 2.5

Additionally, on the maps for the MU radar and the MAARSY 15 channel radar, two elevation limits are shown as two concentric circles. The inner circle represents the elevation above which DOA determination is practically unambiguous. The outer circle illustrates the elevation below which unambiguous DOA determination is practically impossible.

Before application on measurement data, one should validate that the sets

First, we report results for the Jones 2.5

Following the steps outlined above, the resulting DOA sets

The discretised output DOA distribution as a function of SNR and input DOA for the Jones 2.5

Summary results for all input DOAs used to perform an MC DOA determination simulation. These are the

A series of MC simulations were performed using the set

As expected, the results for

Figure

Following the method outlined in Sect.

In contrast to the Jones

Given the MU asymmetric subgroups, one could consider the model in Eq. (

As the ambiguity results for the phase centre model are close to trivial (see Appendix

Ambiguity analysis summary illustration for the MU radar using the subgroup model. The three columns represent the source DOAs

The MU radar subgroup model is expected to have an ambiguity map that varies as a function of input DOA. We present maps for

Comparing the phase centre model with the subgroup model, the DOA determination situation improves slightly for

The phase output of subgroup 1, the outer subgroup to the west in Fig.

Comparison of the signal phase measured using the subgroup model and the phase centre model of MU radar channel 1 (the outer asymmetric subgroup to the west, illustrated in Fig.

The discretised output DOA distribution as a function of SNR and input DOA for the MU radar, using the subgroup model and

The discretised output DOA distribution as a function of SNR and input DOA for the MU radar, using the subgroup model where input 1 is

The discretised output DOA distribution as a function of SNR and input DOA for the MU radar, using the subgroup model and

As the phase centre model and subgroup model converge towards the zenith, we only present MC results from the subgroup model. The summary of the MC MUSIC DOA determinations for that model is illustrated for

From Fig.

To test if a narrow peak was causing problems, we applied the parallel gradient ascent technique described in Sect.

As in Fig.

The different input locations differ significantly in the SNR needed for stable DOA determination. This SNR limit also differs with respect to the used sensor response model. As such, the MUSIC peak value is a more stable quality indicator than SNR for DOA determination. The MUSIC peak value directly describes how well the used sensor response model matches the measured signal. The MUSIC peak distribution for the MU sub-array model simulations is given in Fig.

Distribution of MUSIC peak values as a function of SNR compiled from all MU subgroup model MC simulations, with the zenith as the input DOA (see Fig.

As the MU radar is a more complex system than the Jones 2.5

The MAARSY radar system is limited to 16 output channels but with a flexible subgroup configuration. Studies looking at the interferometry of meteor echoes with MAARSY have predominantly used two different configurations. These configurations are illustrated in Figs.

The phase centre model of the MAARSY eight channel configuration contains many perfect ambiguities. In this antenna configuration, all subgroups contain reflection symmetry lines, which means that their individual subgroup gains do not resolve ambiguities. However, the MAARSY eight channel configuration also contains the entire array as one of the channels. This channel has a significantly different gain pattern compared to the other channels. This creates a small shift in the sensor response between different directions. We have neither included an illustration of the ambiguity analysis of the phase centre model nor the subgroup model for the MAARSY eight channel configuration as they are trivial and ambiguous.

The discretised output DOA distribution as a function of SNR and input DOA for the MAARSY 15 channel radar, using the subgroup model and

For the phase centre model of the MAARSY 15 channel configuration, there are many close-to-perfect ambiguities and a few perfect ones. The distribution of ambiguities is close to identical to the MAARSY eight channel one. The addition of the smaller hexagonal subgroups, illustrated in Fig.

In the MAARSY 15 channel configuration, half of the channels have vastly different antenna gain patterns. This fact makes the configuration better when the subgroup model is applied. We do not present any MC simulations for the phase centre model of the MAARSY 15 channel configuration but focus on the subgroup model.

In Fig.

A significant complication was discovered regarding the application of the MUSIC algorithm on the MAARSY 15 channel subgroup model in that the vast differences in gain between channels narrow down the peaks in the MUSIC spectrum significantly. This is usually a desirable property as it allows more precise DOA determination. However, if the narrowing is extreme, a simple grid search for a peak will become unreasonably costly in terms of computations. As the difference in gain between the channels increases with zenith angle, the narrowing is a function of elevation, thus making low-elevation sources harder to determine with grid methods.

A peak at 45

As such, we applied the multiple gradient ascent method described in Sect.

The most prominent problem for DOA determination of echoes in the zenith with the MAARSY 15 channel configuration is that they are ambiguous with many DOAs below 57

We also simulated a case in which an elevation restriction was added to the DOA determination algorithm. It was found that if the algorithm could be restricted to only accept matches above 70

For

The PANSY radar is a special and interesting case when it comes to DOA determination. Firstly, all antennas are distributed vertically, ranging between

The discretised output DOA distribution as a function of SNR and input DOA for the PANSY radar, using the subgroup model where input 1 is

The MC DOA determination simulation summary for

We only present summary results for the PANSY

To summarise the comparison between the standard Jones

The main purpose of the ambiguity analysis and the MC DOA determination simulations was to provide improved understanding of DOA determination dynamics. These results and methods provide simulated theoretical references that are useful when analysing real measurement data.

We compared the phase centre models and subgroup models for the MU, PANSY and MAARSY radars. For the MU radar, even though the subgroup model has ambiguities at low elevations, this is the expected behaviour in real data as well. Its performance in terms of limiting SNR is also better than the phase centre model. As such, the subgroup model is overall the better choice. For the PANSY radar the situation is similar as the phase centre model is outright nonphysical and should not be used. In the case of MAARSY, if the DOA search is restricted to high elevations, either model is sufficient.

The simulations also provided insight into the construction of DOA determination algorithms. It was shown for the MAARSY, MU and PANSY systems that an additional step of a scattered gradient ascent had to be implemented due to the topology of the MUSIC function. The success of this method suggests that there may be other optimisation algorithms that could further improve performance, such as the bird swarm algorithm

The comparison between a standard Jones

Considering the application of these methods and results on measurement data, they provide a reference, not only for SNR limits but also for model validation. If measurements do not follow the dynamics simulated by these methods, assuming the pipeline itself is validated and stable, it points towards the models not representing reality. This makes such simulations a good validation tool for analysis pipelines. For example, it has been frequently shown that multiple-receiver radar systems are in need of phase calibrations

We have explored a Bayesian approach to determine the most probable DOA of a target given several measurements distributed among noise-induced ambiguities. Such an approach can be applied if it is not possible to increase the SNR using coherent integration. The results indicate that this is a suitable method for providing a quantitative probability for which DOA is correct. Using the Bayesian method, it appears possible to analyse echoes down to 4 dB SNR for both the standard Jones 2.5

Lastly, the MC simulations in this paper demonstrated quantitatively that ambiguities are more or less relevant, depending on radar system configurations. In systems where ambiguities are not prevalent, the DOA determination failure onset is the important variable to determine. In systems where noise-induced ambiguities are relevant, it is important to determine the SNR range in which they emerge. Our results show that the PANSY system is not affected by noise-induced ambiguities, while the MU radar has a small region of SNRs where they could be relevant. The Jones-type systems and MAARSY all have relevant noise-induced ambiguities.

Table

Using interferometric radar systems to perform meteor head echo measurements, the trajectory can be directly determined

Adding a phase offset to each element in the subgroups defined in Eq. (

Inserting the phase offset subgroup model from Eq. (

Here,

If

Additionally, if

The results are the same if the ambiguity indicator

What follows is a derivation of the fact that planar arrays with single-antenna channels are uniquely identified by a single translated ambiguity map.

Inserting Eq. (

Equation (

The results are the same if the ambiguity indicator

All the data used in this manuscript were generated by the authors and are available from the Swedish National Data (SND) service findable, accessible, interoperable and reusable (FAIR) repository at

An animation of the simulated MC DOA determination at different SNRs for the MU radar subgroup model (with array SNR in the range from

DK developed the model code and performed the simulations. DK prepared the paper with contributions from JK.

The authors declare that they have no conflict of interest.

We thank Koji Nishimura for providing the PANSY radar antenna configuration data, Taishi Hashimoto for providing the PANSY meteor radar antenna configuration data and Carsten Schult for providing the MAARSY radar antenna configuration data.

This paper was edited by Jorge Luis Chau and reviewed by two anonymous referees.