A relatively simple design for a meteor radar interferometer with 4 receiving antennas is described. A basic 3 element interferometer provides a set of possible directions for each radio echo detected. The ambiguity is resolved using a 4th antenna, placed according to the results of a numerical simulation. Details of the procedure are provided. The resolution in echo direction is between ∼1° and ∼2° for altitudes (elevations) above ∼30° at all azimuths. A general purpose meteor radar based on this principle has been in use since 1986 at Grahamstown, South Africa.
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 Interferometric methods have been used for echo-direction finding since early in the history of meteor radars [e.g., Robertson et al., 1953, and references therein]. The technique relies on the comparison of the phase of an echo signal at each of a number of spatially separated receiving antennas. The design of the receiving arrays is complicated by conflicting requirements that the antennas be sufficiently close to allow unambiguous determination of echo direction, while being sufficiently well separated to avoid undesirable coupling effects and provide good direction resolution. Various systems which effectively satisfy all these criteria have been devised. A successful 5 antenna array, capable of good all-sky coverage and a precision of ∼1.5° in measurement of echo directions was introduced by Hocking et al.  and has been described in detail by Jones et al. . It is now incorporated in the widely deployed SKiYMET meteor radar systems [Hocking et al., 2001a]. Similar antenna configurations have been used in other applications (e.g., lightning studies [Rhodes et al., 1994]). An interferometer for meteor work involving only 4 elements is embedded in a much larger multipurpose VHF radar array [Hocking, 1997; Hocking et al., 2001b], but removal of direction ambiguity is generally restricted to a comparatively limited range of zenith angles unless additional information (e.g., echo range) is available.
 The form of the antenna layout is shown in Figure 1. Antennas at the corners of right-angled isosceles triangle ABC, with AC = BC = ℓ, form a simple interferometer as explained in section 3. Such an arrangement is suitable for ionospheric work since echo directions from near the zenith can be obtained unambiguously, even for antenna separations ℓ of several wavelengths. To cater for all-sky operation, a fourth antenna (D, the Discriminator) is required in a position that allows a unique resolution of the direction ambiguities that generally arise for ℓ greater than half a wavelength (see section 3). Such antenna separations are needed to maximise precision in echo directions and minimise phase errors due to impedance coupling between the antennas.
3. Echo Directions
 In the equations below and subsequent discussion, phase angles ϕ as well as distances are expressed in units of wavelength, λ. One unit is thus equivalent to a phase angle of 2π radians. This convention has the effect of simplifying the equations.
 For any pair of antennas, X and Y, the angle of arrival θ is defined as shown in Figure 2. If the measured phase of the signal received at Y relative to X is ϕYX then
This condition defines a conical surface of half-angle (90 − θ) about the axis XY. In general more than one value of θ may be consistent with a given ϕYX if ℓ > λ/2. Two values of θ will yield indistinguishable measured phases ϕYX if the corresponding values of ℓ sin θ (physical path delay) differ by an integral number (n) of wavelengths (or, equivalently, 2nπ radians).
 Direction-finding using the configurations shown in Figure 1 involves the determination, for each echo, of ϕCA, ϕCB and ϕCD, the signal phase at C referred, respectively, to A, B and D. The intersection of the conic surfaces for AC and BC defines a possible echo direction. If the corresponding angles of arrival are θCA and θCB, then the azimuth A of the echo referred to a meridian along the direction AC (i.e., north-east; the conversion to true azimuth is trivial) is given by
and the altitude (elevation) α by
The phase difference ϕCD expected for a given A and α is
where β is the angle shown in Figure 1 (∠BCD + 90°) and d is the distance DC.
 To resolve the direction ambiguity for any echo, ϕ*CD is found for all the echo directions that could have resulted in the measured ϕCA and ϕCB, and compared with the measured ϕCD. The differences ϕn = ϕ*CD − ϕCD are then ranked such that ∣ϕn∣ < ∣ϕn+1∣. Any ∣ϕn∣, or difference ∣ϕn+1∣ − ∣ϕn∣, is considered not to differ significantly from zero unless it exceeds δϕT, a ‘discrimination threshold’ derived from considerations of uncertainty in phase measurement (see below). The values of A and α corresponding to n = 1 are then selected as defining the correct echo direction with the proviso that if ∣ϕ1∣ > δϕT (ϕ*CD and ϕCD not in sufficiently good agreement) or ∣ϕ2∣ − ∣ϕ1∣ < δϕT (strictly < δϕT, insufficient discrimination between directions), the echo is classed as ‘ambiguous’. Such echoes are recorded for rate purposes, but direction information is discarded. ϕ1 has also been monitored as a check on phase calibration; for a correctly calibrated system ϕ1 is expected to be distributed about a mean of zero. A representative histogram is shown in Figure 3.
 The overall performance of a radar interferometer depends largely on the precision with which signal phase can be measured. In general, this might be assessed before the system becomes operational through direct experimentation or modeling (section 5), and verified later from the data as in the following example. The observed scatter in ϕ1 (Figure 3), represented quantitatively by the rms deviation σ, can be directly related to uncertainties in measured phases (ϕCA, ϕCB, ϕCD) through the above set of equations. For the typical distribution of Figure 3 σ = 0.045λ, corresponding to a phase uncertainty of ∼0.03λ (∼10°), in broad agreement with initial modeling predictions. σ also serves as a guide for the adoption of an appropriate discrimination threshold δϕT, which was arbitrarily set at 0.05λ (18°). Over the years of operation σ seldom exceeded this value, lying mostly in the range ∼0.04λ < σ < 0.05λ depending on noise conditions. With the above criterion, between ∼20% and ∼25% of echoes (in Figure 3, all those outside the range −0.05λ < ϕ1 < +0.05λ) were adjudged ‘ambiguous’.
4. Position of D
 The discriminating ability of antenna D depends critically on its position relative to the other antennas. There is more than one way in which suitable coordinates for point D may be determined, but we focus here on the procedure finally adopted for the Mk II system.
 A rectangular coordinate grid is set up with the origin at O (refer to Figure 1), extending at least as far as ±ℓ along the x and y axes, with grid spacing ≪λ. Each grid point is then examined as a potential position for antenna D. To test a particular grid point we find the set of echo directions (i.e., A and α) that would be compatible with a randomly chosen combination of ϕCA and ϕCB, and for each of these directions we find the associated ϕ*CD (the expected ϕCD) and hence the (positive) differences Δϕ*CD between all possible pairs of ϕ*CD values. This process is repeated for a large number (at least 100) of random ϕCA and ϕCB combinations to find the overall probability P that Δϕ*CD exceeds the discrimination threshold δϕT (section 3). In the context of normal operation of the radar, P may be interpreted as the expected probability that two arbitrary ϕ*CD values from the set computed for any echo direction will differ significantly. A high P would imply good discrimination on the part of antenna D since it means that during normal operation of the radar the different ϕ*CD values computed for any meteor echo will generally be well separated, enabling the correct echo direction to be clearly identified.
 The above exercise is repeated for all the grid points, and we then seek regions for which P ≈ 100%. Figure 4 is a contour plot showing the distribution of P for the Mark II array, in the region to the west of antenna B. A larger scale plot covering the region immediately surrounding the coordinates finally chosen for antenna D is presented in Figure 5. (Many of the contours in Figure 5 should actually coincide, indicating discontinuities in P, but for clarity have been slightly separated here by suitable choice of grid spacing.) To provide a safety margin when implementing this procedure it was required that there should be an area with a radius of at least 0.05λ(= δϕT) surrounding the chosen grid point (‘circle of tolerance’, also featured in Figures 4 and 5) within which discrimination, if not optimum, was acceptable.
 Referring to Figure 4, several satisfactory solutions (regions where P ≈ 100%) can be identified, and the final choice of coordinates for antenna D (Table 1) was governed by convenience and the requirement that coupling effects be avoided. Antenna impedance measurements indicated that these effects rapidly become negligible as antenna separations approach ∼1λ [see also Jones et al., 1998, Figure 7]. In both the antenna configurations used here the smallest separation is the distance BD; 0.90λ and 1.46λ for Mk I and Mk II, respectively.
Table 1. Antenna Configuration
(Coordinates of D)/λ
 It might also be noted that the required coordinates are generally irrational and result in an asymmetric positioning of D. Clearly, identical performance would have been obtained with the sign of the y coordinate reversed. The mean ‘ambiguity level’ for triangle ABC (the number of possible echo directions for a given ϕCA and ϕCB) was found to be 6.3 for the Mark I system and 12.6 for Mark II.
 For the purposes of this report the feasibility of systems with ℓ > 2λ has been briefly investigated. The ambiguity level increases rapidly with increasing ℓ (approximately as ℓ2) with a consequent decrease in P. For ℓ > ∼2.2λ P nowhere reaches 100% and islands of high P decrease rapidly in size with increasing ℓ, making the placing of antenna D more critical. For ℓ = 3λ P does reach maxima of ≈97%, but gradients are steep in all directions and P rapidly drops below 90% (well within the circle of tolerance). It does nevertheless seem possible that usable systems could be devised if large ℓ is a priority, but the positioning of D would probably have to involve some trial and error.
 In summary, the steps involved in setting up the 4 antenna array are as follows:
 1. Decide on the dimensions and orientation of the right-angled isosceles triangle that defines the basic 3 element interferometer (antennas A, B, C). Antenna separations should be at least ∼1λ.
 2. Assess uncertainty in phase measurement by experiment or modeling, and hence deduce a value for the discrimination threshold, δϕT. (These quantities can be reviewed later in the light of data acquired and updated if necessary.)
 3. Set up a rectangular grid covering the area surrounding triangle ABC. For each grid point find P, the probability that the quantity Δϕ*CD, computed for a random distribution of ϕCA and ϕCB, exceeds δϕT.
 4. Locate circular regions of radius at least δϕT (0.05λ in the present case) within which P ≈ 100%. The 4th antenna (D) can be placed at the center of any of these circles, subject to the restriction that it is no closer than ∼1λ to A, B or C.
5. Phase Uncertainty and ‘Pointing Error’
 The precision of the direction finding process is limited by the precision with which the phase of an echo signal at each of the antennas can be measured. A direct assessment of phase uncertainty (by observing the response of the receiving system to a known signal of terrestrial origin with a signal-to-noise ratio typical of meteor echoes) was contemplated, but in the event this process was modeled using an early version of the procedure described later in this section. Phase uncertainties continued to be monitored during normal operation of the radar by noting the spread of ϕ1 about its mean (see section 3). On the strength of these data and the modeling predictions, a value of 10° was adopted as representative of the uncertainty in the measurement of the phase of a typical meteor echo signal detected by the Grahamstown radar under average noise conditions.
 For any point in the sky, the uncertainty in echo direction arising from measured phase error can be represented in the form of an ellipse with its major axis on a great circle passing through the zenith. The magnitudes of the major and minor axes (a and b, respectively), deduced by propagating a phase error of 10° through the equations of section 3, are given in Table 2 for various altitudes α. a increases rapidly for α < 20°, but in the case of the Grahamstown radar echoes at these altitudes are in any case discarded as they are subject to range aliasing. Over the range of α shown in Table 2, a also varies with azimuth by a factor of ∼1.6, with maxima in the four cardinal directions (N, E, S, W); the values shown are averaged over azimuth. b is constant everywhere.
 In order to investigate the dependence of phase uncertainty and pointing error on echo shape and signal-to-noise ratio S we have also carried out a numerical modeling exercise similar to that of Jones et al. , whereby gaussian noise is added to a range of simulated base-band echo signals assigned various amplitudes, envelope shapes and phases. As this simulation was geared specifically to the Grahamstown Mark II radar, the following brief discussion will not generally be of relevance to other systems, and many operational details have been omitted. In the Grahamstown system signal phase is derived from the complex spectra of contiguous time cells containing receiver output, rather than from time-domain signals in quadrature as in Jones et al. The presence of a genuine echo in a given cell is assumed if the frequencies of the associated spectral maxima (lines) in each of the four receiving channels agree to within an adopted limit. A feature of this method is that echo detectability depends on duration as well as amplitude. The model confirms the observation during operation that a low level echo with S < 0 dB can be ‘picked out’ of the noise if sufficiently long-lived (∼0.5s, the length of a time cell). The rms phase deviations and associated pointing errors for such echoes are of the same order as those presented in Table 2, as might be expected since a significant proportion of echoes recorded at Grahamstown are of the ‘transition type’ [Poole, 1997] with comparatively long durations and small amplitudes. Minimum detectable echo duration and phase error both decrease with increasing echo amplitude. To a good approximation, it is also found that the phase error of an echo with a given amplitude is inversely proportional to its duration. Because of the differences in analytical approach, these modeling results are not directly comparable with those reported by Jones et al. , but appear similar in that, for a given echo duration, the modeled errors decrease by a factor of ∼2 for a 6 dB increase in S.
 The direction-finding capability and sky coverage of the 4 element interferometer described here is similar to that achieved by the 5 element system of Jones et al. . Although the present discussion has been specific to a particular system, the application of the principles to other types of meteor radar should not pose any serious problems. The method does involve some initial effort in the placing of the fourth antenna, as well as additional steps in the data processing, but it is a workable option when the expense or availability of hardware is a consideration. In the case of the Grahamstown radar the 4 antennas were served by just two receivers [Poole, 1988]; any addition to the array would have been very difficult to accommodate.
 The author is indebted to the Rhodes University Joint Research Committee for ongoing support over the years, and to Professor J. Jones for suggesting that attention be drawn to this radar design, and for other useful comments. Source codes for the routines which implement the procedures discussed here are available from the author.