Corresponding author: J. P. Younger, School of Chemistry and Physics, University of Adelaide, Adelaide, SA, 5005, Australia. (joel.younger@adelaide.edu.au)

Abstract

[1] Meteor radars have become common and important tools in the study of the climate and dynamics of the mesosphere/lower thermosphere (MLT) region. These systems depend on accurate angle-of-arrival measurements to locate the positions of meteor trails in the atmosphere. Mutual coupling between antennas, although small, produces a measurable error in the antenna pair phase differences used to deduce the angle of arrival of incident radiation. Measurements of the scattering parameter matrix for antennas in an interferometric meteor radar array have been made and applied to the existing angle-of-arrival calculation algorithm. The results indicate that mutual coupling of antennas in the array produces errors in the zenith angle estimate of less than ± 0.5°. This error is primarily in the form of a gradient across the field of view of the radar, which can be removed using existing phase calibration methods. The remaining error is small but will produce small systematic variations in the height estimates for detected meteors.

[2] The accuracy of directional information is crucial to the scientific application of meteor radar data. Meteor radars use the trails of ionization left behind by ablating meteoroids as scattering targets to measure winds and diffusion coefficients in the upper mesosphere/lower thermosphere region of the atmosphere around 80–100 km. In order to use meteor radar detections to construct a three-dimensional wind field from the radial wind components of individual meteor trails, to estimate a vertical temperature profile from echo decay times [Tsutsumi et al.1994], or to determine orbital data of incident meteors from directional data [Jones and Jones, 2006], the location of the meteor trail relative to the radar must be accurately known.

[3] As meteor radars have become a ubiquitous tool in the study of the middle atmosphere, there is an imperative to account for all possible systematic errors in the angle-of-arrival measurements. Zenith angles derived from angle-of-arrival estimates determine the height of the meteor, so the ability of a meteor radar system to assign meteor detections to a geospatial coordinate system depends on accurate angle-of-arrival information. Fixed phase biases through the receive signal chain can be corrected for by a number of methods [Holdsworth et al.2004], but the effect of mutual coupling between antennas has thus far been limited to purely theoretical numerical modeling, the results of which have not been published in detail.

[4] The most common design of dedicated meteor radar consists of two interferometric baselines consisting of two antenna pairs each, usually arrayed in a cross pattern with a common central antenna [Jones et al.1998], an example of which is shown in Figure 1. Each baseline has two antenna pairs, with separations of 2.0 and 2.5 times the operating wavelength, λ. This particular set of spacings has been found to be the best compromise between array size and accuracy, but any set of antenna pairs with separations that differ by 0.5 λ would function in the same manner.

2 Angle-of-Arrival Estimation

[5] Angle-of-arrival estimation using interferometers relies on the difference in the phase of an incident wave between antennas at different locations in the array. For a wave incident at angle ψ to the line between two antennas separated by distance d, the difference in phase observed between the two antennas is given by ϕ = 2πdλ^{ − 1} cosψ, where λ is the radar wavelength.

[6] Normally, a separation of greater than 0.5 λ would produce an ambiguous set of possible angles, but the difference in phase across the two pairs can be combined to produce a virtual 0.5 λ baseline. Using the 2 and 2.5 λ phase differences, the phase differences for the 0.5 and 4.5 λ separations can be constructed as follows:

ϕ0.5λ=ϕ2.5λ+ϕ2λϕ4.5λ=ϕ2.5λ−ϕ2λ.(1)(1)

These phase differences can then be used to progressively select the best angle-of-arrival candidates for each ambiguous antenna separation up to the 4.5 λ antenna pair, as illustrated in Figure 2. Initially, the 0.5 λ phase difference provides an unambiguous angle-of-arrival estimate. Ambiguity in the 2.0 or 2.5 λ antenna pair is resolved by selecting the value of ψ from a set of possible values with the smallest deviation from the 0.5 λ estimate. The process is then repeated for the 4.5 λ antenna pair, using the unambiguous estimate from the 2.0 or 2.5 λ antenna pair as a reference.

[7]Holdsworth [2005] pointed out that, due to the linearity of ϕ with respect to sinψ, the best candidate is actually that which minimizes Δϕ = | sinψ_{2λ} − sinψ_{0.5λ} | . This antenna arrangement and the progressive selection process eliminates ambiguity in the angle-of-arrival estimate, reduces the effect of mutual coupling through increased inter-antenna spacing, and allows for the use of the longest, and therefore most accurate, antenna separation.

3 Scattering Parameters

[8] It is usually assumed that each antenna in the interferometric array behaves as if in isolation. In reality though, each antenna is part of an array. The current on one antenna gives rise to an electric field that induces an electric field on all other antennas in the array, producing a coupled system. Therefore, the field on each antenna results not only from the applied field of the incident scatter from the target alone but is also the result of the combined field of the incident backscatter and all other antennas in the array. Under these conditions, the measured voltage V_{i} on antenna i is given by

VO=ZI(2)

where Z is the matrix of mutual impedances between the antennas in the array and I is the current in the antennas [see, e.g., Balanis, 2005].

[9] The matrix of mutual impedances can be found through the scattering parameters, beginning with the s-parameters [see, e.g., Anderson et al., 1997]. These terms, usually found in discussions of network port theory, describe the magnitude and relative phase of an induced voltage on one antenna due to a voltage on another antenna. Using the s-parameters, the mutual impedances between antennas is given by the following:

Z=ZL(E−S)−1(E+S),(3)

where E is the identity matrix. Z_{L} is the load impedance of the antennas, which is 50 Ω for the system considered in this paper.

[10] If the matrix of mutual impedances between antennas in an array is known, then the observed voltage, V_{O}, can be predicted for an applied voltage, V_{A}. In the case of a radar, the applied voltage is the electric field of the incident returned scatter. First, the admittance parameters, given by

Y=(ZLE+Z)−1,(4)

must be calculated. Y is analogous to 1 / R in Ohm's law, which leads to the expression for current, I = YV_{A}.

[11] Using the above definitions, the observed voltage on the antennas of an array can then be calculated as

VO=ZLEYVA.(5)

Hence, the observed phase of an incident wave can be predicted for any incident wave on all of the antennas in an array. This allows the error in angle-of-arrival estimates due to mutual coupling to be determined using the measured s-parameter matrix for an array by comparing supplied angles of arrival with those that would be observed after the application of the known mutual impedances. This process can also be reversed in order to correct for the effects of mutual coupling in angle-of-arrival estimates.

4 Measurement of the Buckland Park Array

[12] The Buckland Park meteor radar is operated by the University of Adelaide at the Buckland Park field site 45 km north of Adelaide, South Australia. It is a typical example of interferometric meteor radars operating around the world. The radar operates at 55 MHz and is configured in a cross pattern with a common antenna in the center of the receiving array. The receive antennas are gamma-matched crossed dipoles with a single reflecting element placed at λ / 4 under the main element. The all-sky transmit antenna is located approximately 3 λ to northeast of the receiving array and utilizes a folded cross-dipole design. A λ / 4 delay cable produces circular polarization in the transmitted radiation.

[13] The current radar configuration is powered by a second-generation 40 kW solid-state transmitter-receiver (STX II) that is shared with a 144-element stratospheric/tropospheric (ST) array located to the north of the meteor array. The receive array baselines are aligned with the cardinal directions, as illustrated in Figure 1.

[14] The s-parameters of the antennas in the receive array were measured using an Agilent E5061A vector impedance meter. Antennas not part of each measurement were left connected to the receivers, which is equivalent to termination in matched loads. It was not possible to include the transmit antenna due to hardware incompatibilities, but the contribution should be small, given the outlying position relative to the five receive antennas.

[15] The results shown in Table 1 consist of a matrix of the magnitude and phase of the s-parameter for each antenna pair combination. These matrices are symmetric about the diagonal, due to the reciprocal equivalence of antennas in a specific pairing. The bidirectional equivalence of antenna pair s-parameters was confirmed during the measurement process. This means that for an arbitrary array of n elements, the matrix S has n(n + 1) / 2 unique values.

Table 1. S-Parameters Measured for the Receive Array Antennas of the Buckland Park Meteor Radara

Antenna

Magnitude (dB)

0

1

2

3

4

^{a}The array is configured with antenna 0 in the center, crossed by antenna pairs 1/2 (W-E) and 3/4 (S-N), with antennas 2 and 3 having 2.5 λ separation from antenna 0 and the transmit antenna located in the northeast corner nearest antennas 2 and 4.

0

− 20.9

− 35.8

− 38.2

− 39.3

− 36.7

1

− 35.8

− 23.6

− 45.8

− 44.0

− 43.0

2

− 38.2

− 45.8

− 29.6

− 46.7

− 44.6

3

− 39.3

− 44.0

− 46.7

− 22.4

− 47.9

4

− 36.7

− 43.0

− 44.6

− 47.9

− 23.2

Antenna

Phase (deg)

0

1

2

3

4

0

92.9

− 32.6

149.7

− 29.2

142.0

1

− 32.6

88.1

157.2

9.3

− 43.0

2

149.7

157.2

63.4

56.7

− 44.6

3

− 29.2

9.3

56.7

61.5

− 24.4

4

142.0

− 43.0

− 44.6

− 24.4

87.0

[16] The measured s-parameters were used to numerically simulate the observed angles of arrival for incident waves across the radar field of view. The skymap was divided into bins of 1° in both azimuth and zenith. An incident wave was then applied for each of these directions, and the observed voltages were then calculated using equation (5). The angles of arrival were calculated using the process outlined by Holdsworth [2005] and compared to the original values to determine the error in azimuth and zenith.

[17] The zenith is of particular importance due to the role it plays in the height calculation for meteor detection position. The construction of vertical wind and temperature profiles depend on accurate height estimates, so zenith error was chosen as the primary metric for assessing the impact of mutual coupling. The results of the simulation, shown in Figure 3, show that mutual coupling produces less than 0.5° of error in the zenith estimate for observations from the Buckland Park meteor radar. This is consistent with the simulated coupling effect of less than 0.5° given by Jones et al. [1998] for a pair of side-by-side dipole antennas spaced at 2.0–2.5 λ.

[18] A closer look at Figure 3 allows us to make several general statements about the effect of mutual coupling. The predicted mutual coupling errors display a gradient across the radar field of view, increasing mostly towards the north/northwest. Absent from the predicted coupling errors is a strong symmetry about the NW-SE axis of symmetry of the array. It may be possible that the predicted gradient is slightly skewed by the presence of the transmit antenna to the east-northeast, the ST array to the north, or reflections from the equipment hut to the northeast, but the contribution of all of these should be small. This is reinforced by the fact that the transmit antenna and the ST array are oriented such as to present a shallow polarization angle relative to the meteor receive array antennas, which will further reduce the possibility of coupling. Given the magnitudes of the s-parameters, it is more likely that differences in individual receive antenna properties between adjacent antennas are more significant contributing factors to the observed gradient than the 0.5 λ difference in baseline lengths.

[19] The gradient behavior is fortuitous in this case, as a gradient in the angle of arrival is equivalent to fixed phase biases across the interferometer baselines. Such biases are easily removed using existing calibration methods, as described by Holdsworth et al. [2004]. This corrects for the most severe mutual coupling errors, which are concentrated at the edges of the radar field of view, where zenith angle has the greatest effect on the calculation of height.

[20] The remaining error that is not part of a consistent gradient across the radar field of view displays a complex structure, which is the result of the 5 × 5 element mutual impedance matrix mapping onto the two-dimensional radar skymap. These small variations have magnitudes less than about 0.2° and are the dominant feature in the center of the radar field of view. Fortunately, zenith angle is much less significant to height calculations near zenith, so the errors introduced by mutual coupling here are small compared to the uncertainty stemming from uncertainty in the estimates of detection range and random error in the antenna phase measurements.

5 Conclusions

[21] This research has shown that mutual coupling, although reduced by the 2.0 and 2.5 λ separations used in most meteor radar designs, does play a small part in contributing to the error in zenith angle measurements. This error is not sufficient to cause a breakdown in the 0.5/2.0/4.5 λ method for progressive removal of ambiguity in angle-of-arrival measurements but does introduce small systematic errors in the zenith angle estimate that are a function of the true angle of arrival. The measurements of mutual coupling of an operational meteor radar give insight into the effect of mutual coupling on positional estimates of individual meteors.

[22] Simulations based on measurements of mutual coupling in the Buckland Park meteor radar receive array predict that the effect of mutual coupling on zenith angle estimates is below 0.5° across the entire radar field of view. When applied to meteor detections around 90 km, this corresponds to an error in height estimates of about 1 km.

[23] The presence of a gradient in the predicted error map is important, as this corresponds to a constant phase bias along each interferometric baseline segment. Phase biases are already corrected for through calibration techniques that encompass all phase biases along the entire receive signal chain. Thus, the dominant error contribution of mutual coupling is already corrected for by existing methods. The remaining small variations in zenith angle error are well within the uncertainty due to noise in the phase estimate, so there is little systematic contribution to the overall uncertainty in height estimates. In theory, it is possible to correct the small fluctuating errors if the matrix of mutual impedances between antennas is known. Further work should be conducted to determine the coupling behavior of arrays as individual antenna tuning degrades.

[24] While this work has focused on a specific array, the broad conclusions should be true for all radars sharing the same design. Mutual coupling, although small, does produce small systematic errors in the angle-of-arrival estimate of less than 0.5°. The structure of the errors introduced by mutual coupling may be complex, but the process of measuring and accounting for the errors is straightforward.

Acknowledgments

[25] This study has been supported by ARC grants DP0878144 and DP1096901. The authors would like to thank Richard Mayo of ATRAD Pty. Ltd. for his assistance in making measurements.