This article is a companion to *Custovic et al*. [2013] doi:10.1002/2013RS005156.

Regular Article

# Elevation angle-of-arrival determination for a standard and a modified superDARN HF radar layout

Article first published online: 12 DEC 2013

DOI: 10.1002/2013RS005157

©2013. American Geophysical Union. All Rights Reserved.

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#### How to Cite

2013), Elevation angle-of-arrival determination for a standard and a modified superDARN HF radar layout, Radio Sci., 48, 709–721, doi:10.1002/2013RS005157.

, , , , , and (#### Publication History

- Issue published online: 16 JAN 2014
- Article first published online: 12 DEC 2013
- Accepted manuscript online: 25 NOV 2013 09:20PM EST
- Manuscript Accepted: 13 NOV 2013
- Manuscript Revised: 11 NOV 2013
- Manuscript Received: 14 JAN 2013

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- Cited By

### Keywords:

- superDARN;
- interferometer;
- elevation angle;
- layout

### Abstract

- Top of page
- Abstract
- Introduction
- Elevation Angle-of-Arrival Calculation With a Single-Interferometer Array
- Elevation Angle-of-Arrival Calculations for a Dual-Interferometer Antenna Layout
- Simulation of Elevation Angle-of-Arrival Calculations for a Dual-Interferometer Layout
- Discussion
- Conclusions
- Acknowledgments
- References

[1] Calculations have been developed for the determination of elevation angle of arrival for a modified Super Dual Auroral Radar Network (SuperDARN) HF radar antenna layout consisting of dual auxiliary interferometer arrays: one behind and one in front of the main array. These calculations show that such a layout removes the 2*π* ambiguity or angle-of-arrival *aliasing* effect observed in existing SuperDARN HF radars. Ray tracing and simulation results are presented which show that there is significant potential for aliasing with existing SuperDARN radars and the standard interferometer algorithm under routine operating conditions.

### Introduction

- Top of page
- Abstract
- Introduction
- Elevation Angle-of-Arrival Calculation With a Single-Interferometer Array
- Elevation Angle-of-Arrival Calculations for a Dual-Interferometer Antenna Layout
- Simulation of Elevation Angle-of-Arrival Calculations for a Dual-Interferometer Layout
- Discussion
- Conclusions
- Acknowledgments
- References

[2] The Super Dual Auroral Radar Network (SuperDARN) is a global network of over 26 coherent scatter HF radars designed to detect backscatter from plasma density irregularities in the high latitude and midlatitude ionosphere [*Greenwald et al.*, 1983, 1995; *Chisham et al.*, 2007] and backscatter from the ground propagating via specular reflection from the ionosphere [*Andre et al.*, 1998]. The SuperDARN radars detect scatter from periodic electron density fluctuations in the ionosphere with wavelength equal to half the radar wavelength. Backscatter from the land surface is also observed due to natural surface roughness and suitably oriented mountain slopes [*Ponomarenko et al.*, 2010], and from the sea surface due to surface roughness and coherently from successive ocean wave fronts of half the radar wavelength [*Greenwood et al.*, 2011; *Wyatt et al.*, 2011].

[3] At middle to high latitudes, ionospheric irregularities are most commonly observed in regions of high-velocity plasma convection such as the auroral electrojet at *E* region altitudes [*Greenwald et al.*, 1975] and plasma flows in the *F* region driven by electric fields generated at the magnetospheric boundary mapping to the ionosphere along geomagnetic field lines [*Ruohoniemi and Baker*, 1998]. The irregularities tend to drift with the electron flow in the background plasma, providing an image of plasma convection in the ionosphere [*Greenwald et al.*, 1978; *Kelley et al.*, 1982]. The formation of ionospheric irregularities is thought to be associated with plasma instabilities, in particular the gradient drift instability in the ionospheric *F* region [*Simon*, 1963], and the cross-field two-stream Farely-Buneman instability [*Buneman*, 1963; *Farley*, 1963] and gradient drift instability [*Reid*, 1968] in the ionospheric *E*region.

[4] The SuperDARN HF radars operate in the frequency range 8–20 MHz, employing a linear phased array of 16 antennas to create a narrow, azimuthal, steerable beam [*Greenwald et al.*, 1995]. The radar measures the average backscatter power from each range cell for a number of discrete azimuthal beam directions. A multipulse scheme, such as the seven-pulse scheme [see, for example, *Farley*, 1972], is adopted to determine range information unambiguously and provide sufficient frequency resolution for the determination of line-of-sight Doppler velocities of the order of 1000 m/s without aliasing [*Greenwald et al.*, 1983, 1985]. The single-pulse width determines the range resolution; typically 300 μs for a resolution of 45 km. A longer pulse width would improve SNR at the expense of range resolution. The interpulse lag unit of around 2400 μs determines the frequency resolution. Quadrature samples are processed using autocorrelation techniques developed for use with incoherent scatter and VHF auroral radars [*Farley*, 1972; *Greenwald et al.*, 1978] in order to rapidly calculate Doppler velocities. Simultaneous measurement of the autocorrelation function (ACF) at multiple ranges is performed with up to 21 lagged products formed. The longest time lag is assumed to be comparable to the correlation time of the ionospheric scattering medium. Thus, there is an implicit assumption of spatial and temporal uniformity of backscatter targets within a given range/azimuth cell over the time taken to complete the multipulse sequence, of the order of 100 ms. Clutter from uncorrelated scatter from differing ranges, and noise, can be averaged out with integration over a sufficient time, typically 3 s [*Farley*, 1972]. The Fourier transform of the autocorrelation function gives the Doppler power spectrum. In practice, however, for statistical reasons, rather than perform a Fourier transform, the power and spectral characteristics are obtained from a functional fit to the decorrelation envelope of the ACF based on collective wave-scattering theory [*Villain et al.*, 1987; *Hanuise et al.*, 1993]. The basic unit of lag, typically around 3 ms, is a compromise between accurate determination of the decorrelation time and frequency resolution. The line-of-sight Doppler velocity is calculated from the rate-of-change of the phase of the complex ACF with lag.

[5] A second receive-only phased linear array with fewer antennas (typically four), displaced from the main array, acts as an interferometer for the calculation of elevation angle of arrival. The phase difference between the signals arriving at the main and interferometer arrays can be determined from the cross-correlation function of the combined signals from the main and interferometer arrays [*Farley et al.*, 1981; *Baker and Greenwald*, 1988]. The phase difference between the two planar arrays is then used to obtain the vertical (elevation) angle of arrival. The number of antennas in the auxiliary interferometer array can be considerably less than in the main array since only the correlated portion of the signals incident on the main and auxiliary arrays are used for the determination of the main-auxiliary phase lag.

[6] There is considerable variation in the displacement between the main and auxiliary (interferometer) arrays within the SuperDARN network, which is typically around 100 m. The main-auxiliary array spacing needs to be of the order of 100 m for accuracy of elevation angle measurements and to reduce shadowing effects between the two planar arrays [see, for example, *Custovic et al.*, 2013]. The interferometer spacing is, in all cases, larger than the wavelength of the HF signals, which is a maximum of 37 m at 8 MHz. This results in a 2*π* ambiguity or vertical angle-of-arrival *aliasing* effect where the *measured* phase difference in the range −*π* to *π*, between signals arriving at the main and auxiliary arrays, differs from the *total* phase difference by an unknown multiple of 2*π*[*Milan et al.*, 1997; *Ponomarenko et al.*, 2011]. The total phase difference or total *phase lag*between the main and auxiliary arrays is required for the unambiguous estimation of the angle of arrival.

[7] Elevation angle of arrival is critical for determining the propagation modes of received backscatter, the type of scatter observed (ionospheric scatter or ground scatter) and the altitude in the ionosphere from which scatter or ionospheric propagation originates [*Milan et al.*, 1997; *Uspensky et al.*, 2001]. However, only a small number of studies have utilized elevation angle-of-arrival information from SuperDARN radars. *Andre et al.* [1998] used ground scatter analysis of elevation angles versus group range to determine ionospheric parameters, *Milan and Lester* [2001] used elevation angles to determine the altitudes of particular classes of backscatter in the auroral electrojet *E* region, *Chisham et al.* [2008] explored the statistical relationship between elevation angle and group range for ionospheric scatter, *Gillies et al.* [2009] used elevation data to approximate the index of refraction in the ionospheric scattering region to refine Doppler velocity measurements, while *Ponomarenko et al.* [2011] used elevation to estimate the *F* region peak electron density directly. The determination of ground surface parameters such as sea state conditions [*Greenwood et al.*, 2011] using HF SuperDARN radars is also greatly enhanced by having accurate elevation angle-of-arrival data. Elevation data from many of the SuperDARN radars, however, is either unavailable or considered to be unreliable, despite nearly all the SuperDARN radars being equipped with interferometer arrays. It is likely that many more studies requiring information on scattering altitude, ionospheric propagation, and angle of incidence at the ionosphere and the ground surface would be possible with improvements to the performance of SuperDARN elevation interferometry.

[8] This paper aims to give a detailed description of the extent of the interferometer 2*π* phase ambiguity problem for HF radars in the SuperDARN network and investigate the application of a second interferometer array to a SuperDARN radar as a solution to this problem.

### Elevation Angle-of-Arrival Calculation With a Single-Interferometer Array

- Top of page
- Abstract
- Introduction
- Elevation Angle-of-Arrival Calculation With a Single-Interferometer Array
- Elevation Angle-of-Arrival Calculations for a Dual-Interferometer Antenna Layout
- Simulation of Elevation Angle-of-Arrival Calculations for a Dual-Interferometer Layout
- Discussion
- Conclusions
- Acknowledgments
- References

[9] In this paper we use *elevation angle* to refer to the vertical angle of arrival taken from the horizontal.

[10] In SuperDARN radars the measurement of the elevation angle of arrival is determined from the phase lag between scattered signals arriving at the main array and the interferometer array after beam phase delays have been corrected for and the signals received on each array summed. In the case of a single auxiliary interferometer array, it is assumed that the total main-auxiliary interferometer phase lag lies between the maximum phase lag possible, corresponding to zero elevation angle, and the first 2*π* ambiguity, corresponding to an elevation angle Δ_{max} [*Milan et al.*, 1997]. The measured phase lag, between −*π*and *π*, is translated to this range by adding an appropriate integer multiple of 2*π*. It is assumed that elevation angles below Δ_{max} are not aliased.

[11] Another way of viewing this is to consider each planar array acting as a single antenna with the spacing between the two greater than the wavelength of the received signal. As such, the problem is related to ambiguities in linear array manifolds when used for azimuth-only direction finding [see, for example, *Manikas et al.*, 1997; *Manikas and Proukakis*, 1998]. With SuperDARN radars, there are effectively just two antennas and the aim is not to identify or resolve hard targets, but to distinguish between direct, single, and multihop ionospheric backscatter returns from soft targets (electron density waves).

[12] Figure 1 shows simulated mapping of elevation angle to main-auxiliary array phase lag using the standard interferometer expressions developed for SuperDARN radars [see, for example, *Andre et al.*, 1998; *Milan et al.*, 1997]. Here the simulated measured phase lag (Figure 1, top) is the total phase lag (Figure 1, bottom) modulo 2*π*, shifted to the range −*π* to *π*. Note, that for elevation angles greater than Δ_{max} (dashed vertical lines), a one-to-one mapping does not exist between measured phase lag and elevation angle.

[13] In practice, the total phase lag, required for calculation of angle of arrival, is to be determined from the measured phase lag. Figure 2 shows the mapping of measured phase lags corresponding to incident elevation angles 0° to 90° (horizontal axis), to calculated elevation angles using the standard SuperDARN interferometer algorithm. For elevation angles less than Δ_{max}, the elevation angle can be determined correctly. Elevation angles greater than Δ_{max}, however, are interpreted incorrectly as an elevation angle between 0 and Δ_{max}.

[14] The simulations of Figures 1 and 2 (left) are for a typical SuperDARN radar layout with main-auxiliary array displacement of 100 m, beam swinging to a maximum of 24° from the boresight and frequency equal to 10.2 MHz, the most common operating frequency. As the beam direction moves away from the boresight, Δ_{max} decreases. Recent SuperDARN installations utilizing twin terminated folded dipole (TTFD) antennas [see, for example, *Sterne et al.*, 2011] routinely perform beam swinging up to 37° from the boresight, since TTFD antennas can be placed slightly closer together than log periodic dipole array antennas. Furthermore, with an increase in main-auxiliary array displacement or operation at higher frequencies, Δ_{max} also decreases significantly. Figure 3 shows values of Δ_{max}, the maximum elevation angle that can be measured before elevation aliasing occurs, for a number of SuperDARN radars. Solid lines correspond to the extreme azimuthal beam (beam 0) and dashed lines the beam closest to the boresight. For TIGER (Tasman International Geospace Environmental Radar)-Bruny and Hankasalmi, beam 0 is 24° from the boresight, while Blackstone and Wallops Island have wider fields of view with beam 0 at 37° from the boresight. Main-auxiliary array displacements are 58.9 m for Blackstone, 100 m for TIGER-Bruny and Wallops Island and 185 m for Hankasalmi. Reducing the main-auxiliary array displacement *increases*Δ_{max}. However, this also increases RF interaction and shadowing between the main and auxiliary arrays and decreases the accuracy of elevation angle measurements [*Custovic et al.*, 2013].

[15] The radars selected in Figure 3 are a representative sample of the SuperDARN radar network, covering the range of interferometer displacements and extremes of azimuthal field of view which exist across the network. Δ_{max} is observed to fall approximately into the range 25° to 45° under routine operating conditions for many existing SuperDARN radars. The geomagnetic latitude, interferometer spacing, and maximum azimuth from the boresight for the current radars of the SuperDARN network are shown in Table 1. The radars corresponding to Figure 3 are in bold.

Geomag. | |||
---|---|---|---|

Radar Name | Lat. (°) | d (m) | Φ_{max} (°) |

^{a}A negative value of *d*indicates an auxiliary array behind the main array.^{b}The radars corresponding to Figure 3 are in bold.
| |||

Adak East | 47.6 | −69.8 | 37.3 |

Adak West | 51.9 | −69.8 | 37.3 |

Blackstone^{b} | 48.2 | −58.9 | 37.3 |

Christmas Valley East | 49.5 | −80.0 | 37.3 |

Christmas Valley West | 49.5 | −80.0 | 37.3 |

Clyde River | 78.8 | 100.0 | 24.3 |

Fort Hays East | 48.9 | −80.0 | 34.0 |

Fort Hays West | 48.9 | −80.0 | 34.0 |

Goose Bay | 61.1 | 100.0 | 24.3 |

Hankasalmib | 59.1 | 185.0 | 24.3 |

Hokkaido | 37.1 | −100.0 | 24.3 |

Inuvik | 71.5 | 100.0 | 24.3 |

Kapuskasing | 60.2 | 100.0 | 24.3 |

King Salmon | 57.5 | 100.0 | 24.3 |

Kodiak | 57.2 | −100.0 | 24.3 |

McMurdo | −80.0 | 70.1 | 24.3 |

Pykkvibaer | 64.6 | 100.0 | 24.3 |

Prince George | 59.6 | −100.0 | 24.3 |

Rankin Inlet | 72.6 | −100.0 | 24.3 |

SANAE | −61.8 | 100.0 | 24.3 |

Saskatoon | 60.9 | −100.0 | 24.3 |

Stokkseyri | 64.9 | 100.0 | 24.3 |

Syowa East | −66.5 | 92.0 | 24.3 |

Syowa South | −66.5 | 92.0 | 24.3 |

TIGER-Bruny^{b} | −54.8 | −100.0 | 24.3 |

TIGER Unwin | −54.4 | −100.0 | 24.3 |

Wallops^{b} | 48.7 | 100.0 | 37.3 |

Zhongshan | −74.9 | 100.1 | 24.3 |

[16] Significant echo returns with elevation angles up to 45° or more are routinely observed by existing SuperDARN radars, where elevation angles are available [see, for example, *Chisham et al.*, 2008; *Andre et al.*, 1998]. To support this observation, the range of potential elevation angles has been evaluated using a ray-tracing model coupled with the International Reference Ionosphere (IRI-2012) [*Bilitza et al.*, 2011]. The ray-tracing code is based on a two-dimensional formulation of Fermat's principle in the propagation plane [*Coleman*, 1998] and implemented specifically for SuperDARN backscatter analysis (http://vt.superdarn.org/Ray-tracing). Ionospheric scatter predictions are based on the relative orientation of the background magnetic field with each ray, that is, aspect conditions [*Ponomarenko et al.*, 2009; *Carter et al.*, 2012]. Ionospheric irregularities tend to align along the direction of the local geomagnetic field, with wave vectors close to orthogonal to the field lines [*Ruohoniemi and Baker*, 1998]. Thus, for coherent backscatter to be detected by the radar, the incident radar signal must be nearly orthogonal to the local geomagnetic field, which places severe geometric constraints on the detection of coherent backscatter from a particular radar site. Refraction of the HF radar signal within the ionosphere is critical for achieving the orthogonality condition [*Greenwald et al.*, 1983]. For a particular geographic location, it is thus possible to make a good prediction of the expected angle of arrival of received ionospheric backscatter, based on refraction by a model ionosphere and achievement of the orthogonality condition. The range from the radar and the background electron density at the potential scatter point are also taken into account in the simulations.

[17] Figures 4 and 5 show simulated time series plots of ionospheric backscatter returns, color coded in elevation angle, for several SuperDARN radars over a range of geomagnetic latitudes. The operating frequency of 11 MHz is very typical for the standard SuperDARN operating mode called “common,” used about 70% of the time. Figure 4 shows simulations corresponding to solar maximum (31 August 2012) for the (a) Rankin Inlet, (b) Hankasalmi, and (c) Wallops radars having geomagnetic latitudes 72.7°, 59.1°, and 48.7°, respectively. Close to solar maximum, increased refraction of HF signals within the ionosphere leads to higher-elevation angle rays satisfying the orthogonality condition described above. Ground scatter returns via specular reflection from the ionosphere at high-elevation angles are also expected to be enhanced close to solar maximum, due to the increased ionospheric refraction of HF signals. Radars located at lower geomagnetic latitudes such as Wallops (Figure 4c), observe ionospheric backscatter returns at particularly high-elevation angles due to the inclination of the geomagnetic field being further from the vertical. Maximum elevation angles at solar maximum range from approximately 35° for the highest-latitude radars such as Rankin Inlet to at least 50° for the lower-latitude radars such as Wallops. Simulations at the higher frequency of 14 MHz at solar maximum, and also at solar minimum at 11 MHz for the Wallops radar (Figure 5) indicate that elevation angles up to 35° can be expected for midlatitude radars over the full range of typical operating conditions. Within the SuperDARN network there are currently 11 radars with geomagnetic latitudes between 35° and 55°, 10 radars with geomagnetic latitudes between 55° and 65°, and seven radars with geomagnetic latitudes greater than 65°.

[18] Comparing Figures 4 and 5 with Figure 3, we see that for the radars with large interferometer spacing and moderate azimuthal field of view (such as Hankasalmi) or an enhanced azimuthal field of view and moderate interferometer spacing (such as Wallops), Δ_{max} falls well within the expected elevation angle range for routine radar operation. For example, for Wallops, Δ_{max} ranges from 34° to 50° along the boresight and from 29° to 42° for the extreme azimuth. The predictions of Figures 4c and 5 show that elevation angles within these ranges are typical at this latitude under a wide range of operating frequencies and solar conditions. For the TIGER-Bruny radar with the most common interferometer spacing of 100 m and moderate azimuthal field of view, Δ_{max} ranges from 33° to 51° along the boresight and from 32° to 47° for the extreme azimuth. With a geomagnetic latitude midway between that of Figures 4b and 4c, similar ray-tracing simulations show that, close to solar maximum, TIGER-Bruny can expect elevation angles up to 50° at 11 MHz and 40° at 14 MHz, reducing to around 30° at solar minimum (results not shown).

[19] A detailed experimental study of elevation angle of arrival by *Baker and Greenwald* [1988] using the Goose Bay HF radar found that elevation angles are often higher than those expected from the regular stratified ionosphere used in the present simulations, due to the presence of unstratified structures such as ionospheric tilts.

[20] Thus, for SuperDARN radars, it is often the case that a significant proportion of backscatter echo returns exist with elevation angles greater than Δ_{max}. As shown in Figure 2, such echo returns will be incorrectly interpreted or *aliased* and can contaminate the entire elevation angle data. This has been observed in previous studies and shown to cause problems with data interpretation [*Milan et al.*, 1997]. It is also apparent from Figure 2 that elevation angles greater than Δ_{max} by just a few degrees will be mapped to between 10° and 20°. Since this falls in the range where nonaliased angles would be expected, aliased elevation angles can be difficult to recognize and filter out.

[21] It should be noted that the typical antenna gain pattern of SuperDARN radars favors the receiving of elevation angles in the range 10° to 45°. However, with the increased sensitivity of the latest generation of SuperDARN radars, it will be possible to receive signals at even higher-elevation angles [*Custovic et al.*, 2011].

### Elevation Angle-of-Arrival Calculations for a Dual-Interferometer Antenna Layout

- Top of page
- Abstract
- Introduction
- Elevation Angle-of-Arrival Calculation With a Single-Interferometer Array
- Elevation Angle-of-Arrival Calculations for a Dual-Interferometer Antenna Layout
- Simulation of Elevation Angle-of-Arrival Calculations for a Dual-Interferometer Layout
- Discussion
- Conclusions
- Acknowledgments
- References

[22] A dual-interferometer antenna layout is shown in Figure 6, which has been developed in conjunction with the third TIGER radar at Buckland Park, South Australia [*Custovic et al.*, 2011]. This layout provides two different main-auxiliary array displacements normal to the boresight, permitting resolution of the 2*π* ambiguity, while maintaining some beam steering capability for the rear auxiliary array.

[23] Figure 7 shows a view in elevation of a signal being received at the main array and one of the auxiliary arrays. The total phase lag Ψbetween the signal reaching the main and auxiliary arrays can be expressed in terms of the wave vector **k** and the path length difference *δ**P*;

- (1)

where |**k**|=2*π*/*λ*, Δ is the elevation angle and *d*^{′} is the main-auxiliary array displacement for beam swinging along an azimuth *φ* relative to the radar boresight. *d* is the fixed main-auxiliary array displacement along the boresight (see Figure 8).

[24] The total phase lag Ψ in (1) is typically greater than 2*π* since the path difference *δ**P* is greater than the signal wavelength *λ*, for all but the highest-elevation angles. *d* is of the order of 100 m while *λ* is a maximum of 37 m. The phase lag measured by the radar, however, is modulo 2*π* such that

- (2)

where *ψ* is the measured phase lag and *m* is some unknown integer.

[25] The azimuth for transmit and receive is set by a phasing matrix applied to a large number of collinear antennas normal to the boresight. The azimuth is not constant with elevation angle, with the maximum of the antenna pattern lying on the surface of a cone with its axis aligned along the antenna array [*Milan et al.*, 1997]. Following *Andre et al.* [1998], the azimuth *φ* at elevation Δ is related to the azimuth at zero elevation angle *φ*_{0} by the expression

- (3)

[26] For the dual-interferometer layout shown in Figure 6, we form two expressions, one for each auxiliary array, for the elevation angle Δ in terms of the measured main-auxiliary array phase lag *ψ*_{1} or *ψ*_{2}. From equations (1), (2), and (3),

- (4)

- (5)

where *ψ*_{1} and *ψ*_{2} are the measured main-auxiliary array phase lags for the front and rear arrays, respectively. *a*=*d*_{1}/*λ* and *b*=*d*_{2}/*λ*, where *d*_{1} and *d*_{2} are the main-auxiliary array displacements (see Figure 6) and *λ* is the signal wavelength. *m* and *n* are unknown integers resulting from the 2*π* ambiguity in the total phase lag. The determination of the elevation angle Δ for given measured phase lags *ψ*_{1} and *ψ*_{2} depends on the determination of *m* and *n*. Note, that in the standard single-interferometer SuperDARN layout, just one of these expressions is formed.

[27] It should be noted that the phase lags measured by SuperDARN radars are always with respect to the main array: for an interferometer array, in front of the main array *d*>0 and the total phase lag Ψ>0; for a rear interferometer array, *d*<0 and the total phase lag Ψ<0. Expressions (4) and (5) are valid regardless of the value or sign of *d*_{1} and *d*_{2}. However, in the calculations which follow, it is simplest to treat *d*_{1} and *d*_{2} both as positive, such that the total phase lag Ψis always positive. In practice, this is valid as long as we make the substitution *ψ*^{′}=−*ψ* for any measured phase lags *ψ*associated with a rear interferometer array.

[28] For any dual-interferometer layout, *d*_{1} and *d*_{2} should be chosen such that

- (6)

where *λ*_{min} is the minimum wavelength used by the radar, equal to around 15 m at 20 MHz. For example, for the layout shown in Figure 6, *d*_{1}=67 m and *d*_{2}=−80 m, satisfying the conditions (6). We then have 0<|*b*|−|*a*|<1.

[29] The direct solution for *m*, *n*, and Δ is found in the following way. Since 0≤*C*≤1, subtracting (4) from (5) yields the inequalities

- (7)

- (8)

[30] For measured phase lags *ψ*_{1},*ψ*_{2}∈[−*π*,*π*), we identify two distinct cases; *ψ*_{2}≥*ψ*_{1}or *ψ*_{2}<*ψ*_{1}.

[31] In case one, where *ψ*_{2}≥*ψ*_{1}, (*ψ*_{2}−*ψ*_{1})∈[0,2*π*) so that (*ψ*_{2}−*ψ*_{1})/2*π*∈[0,1). Then from (7), *n*−*m*≥0 and from (8), *n*−*m*≤0. So *n*=*m*.

[32] In case two, *ψ*_{2}<*ψ*_{1}, (*ψ*_{2}−*ψ*_{1})∈[−2*π*,0) so that (*ψ*_{2}−*ψ*_{1})/2*π*∈[−1,0). Then from (7), *n*−*m*≥1 and from (8), *n*−*m*≤1. So *n*=*m*+1.

[33] In practice, we measure *ψ*_{1} and *ψ*_{2}, express *n* in terms of *m* according to either case one or case two above, and then solve (4) and (5) uniquely for *m* and Δ. The algorithm above will work for any dual-interferometer layout which satisfies the conditions (6), where *ψ*_{2} should be taken as the measured phase lag associated with the larger of the two interferometer spacings. Thus, in principle, layouts with two front arrays, two rear arrays, or one front and one rear array are equally valid. For measured phase lags associated with rear interferometer arrays, the substitution *ψ*^{′}=−*ψ* should be made before applying the algorithm.

[34] A graphical representation of the solution to (4) and (5) is shown in Figure 9. Measured phase lags *ψ*_{1} and *ψ*_{2} have been simulated for particular elevation angles by using equations (1), (2), and (3) to find the total phase lag Ψ, then taking modulo 2*π* (see also Figure 10). The azimuth comes from the fixed azimuthal beam we are transmitting and receiving on; in this case beam 7. The values of *ψ*_{1} and *ψ*_{2} are then used in expressions (4) and (5) to simulate all possible solutions for Δ for the unknown *m* and *n*. Solutions have been generated for negative as well as positive integers to demonstrate that the algorithm works regardless of whether the interferometers are in front of or behind the main array. It can be seen that there is only one value of *m* and *n* for which the elevation angle prediction from interferometer array 1 and 2 is identical. In Figure 9, this occurs for 18° and 58° respectively, which we can verify as the elevation angles chosen to simulate the values of *ψ*_{1} and *ψ*_{2}. The corresponding values of *m* and *n* are consistent with the direct solution as described above.

### Simulation of Elevation Angle-of-Arrival Calculations for a Dual-Interferometer Layout

- Top of page
- Abstract
- Introduction
- Elevation Angle-of-Arrival Calculation With a Single-Interferometer Array
- Elevation Angle-of-Arrival Calculations for a Dual-Interferometer Antenna Layout
- Simulation of Elevation Angle-of-Arrival Calculations for a Dual-Interferometer Layout
- Discussion
- Conclusions
- Acknowledgments
- References

[36] Simulations of the dual-interferometer algorithm have been performed corresponding to *d*_{1}=67 m and *d*_{2}=−80 m (see Figure 6).

[37] Figure 10 shows the simulated mapping of elevation angle to phase lag for the front auxiliary array (Figure 10, left) and the rear auxiliary array (Figure 10, right) for the modified antenna layout shown in Figure 6 using the expressions (1), (2), and (3). Shown are the total phase lag (Figure 10, bottom) and the expected measured phase lag (Figure 10, top) in the range −*π* to *π*.

[38] Figure 11 shows the mapping of the measured phase lags of Figure 10, corresponding to incident elevation angles 0° to 90° (horizontal axis), to calculated elevation angles using the dual-interferometer algorithm described in section 3. All elevation angles are determined unambiguously, and there is no chance of aliasing occurring. The only limit on measurable elevation angles is imposed by the coupling between azimuth and elevation described by equation (3). This limit is independent of frequency or interferometer spacing and requires only that cos2*φ*_{0}> sin2Δ. This is shown by the dashed vertical lines in Figure 11 for various azimuthal beams. The upper limit of measurable elevation angle of arrival decreases as angle from the boresight increases, varying from 90° along the boresight to 66° for the outermost beam 0. At the greatest azimuthal angle from boresight currently used in SuperDARN radars of 37°, the upper limit of measurable elevation would be 53°.

[39] The dual-interferometer algorithm described above has been tested in simulations for all frequencies in the HF range 8–20 MHz and a large range of interferometer spacings *d*_{2} and *d*_{1} which satisfy the condition (6) described in section 3. Within these conditions, there is always a one-to-one mapping between measured phase and elevation angle identical to Figure 11. For different values of *d*_{1} and *d*_{2}, positive or negative, the phase lags shown in Figure 10 will be quite different, but the resultant resolution of elevation angle of arrival remains the same. To mitigate against phase-measurement errors, |*d*_{2}|−|*d*_{1}| should be as large as possible without exceeding the minimum radar signal wavelength. Note that for the purposes of the calculation, *d*_{2} is by definition the larger of the two interferometer spacings.

[40] Further simulations have been performed for a set of much larger interferometer spacings, which still fulfill the conditions (6). Figure 12 (left) shows the simulated total and measured phase lags for a dual interferometer with *d*_{1}=172 m and *d*_{2}=−185 m. A much larger range of elevation angle ambiguities are possible now due to the larger interferometer spacing. Figure 12 (right) shows a graphical solution for 40° actual elevation angle, the black circle indicating the correct solution as found by the direct algorithm in section 3. With the standard SuperDARN algorithm for an interferometer spacing |*d*|=185 m, 40° maps to 22.4° (see Figure (2, right). This is shown by the red circle in Figure 12 (right). The mapping between actual elevation angle and measured elevation angle for these values of *d*_{1} and *d*_{2} is again identical to Figure 11.

[41] The equivalent of Δ_{max} for the proposed dual-interferometer system, the maximum elevation angle of arrival detectable by the system, are the vertical dashed lines shown in Figure 11. These values only depend on the azimuth from the boresight (beam number) and do not depend on the interferometer spacing, provided the conditions (6) are met.

### Discussion

- Top of page
- Abstract
- Introduction
- Elevation Angle-of-Arrival Calculation With a Single-Interferometer Array
- Elevation Angle-of-Arrival Calculations for a Dual-Interferometer Antenna Layout
- Simulation of Elevation Angle-of-Arrival Calculations for a Dual-Interferometer Layout
- Discussion
- Conclusions
- Acknowledgments
- References

[42] It is important to acknowledge that the signal-to-noise ratio (SNR) of a single-antenna subarray, as shown in Figure 6, will be significantly lower than that of a traditional four-antenna array. A single antenna will have around 3 dB lower SNR than a four-antenna array and in addition, the more isotropic radiation pattern will amount to approximately 6 dB less gain along the desired azimuth. However, the new antenna array has improved output filtering, antenna impedance matching and lower loss cabling, which accounts for at least a 2 dB increase in system SNR [see *Custovic et al.*, 2013]. Also, the higher transmit power will result in around a 6 dB gain in signal power. Overall, the single-antenna front subarray is expected to have SNR performance reduced by just 1 or 2 dB compared to the original four-antenna array. However, depending on the noise and interference levels at a specific site, it may be necessary to use two or more antennas for the front subarray to get sufficient SNR, particularly for current generation SuperDARN radar hardware.

[43] In the case where the dual-interferometer technique failed due to insufficient SNR at the second interferometer array, we would expect this to occur for backscatter coming from the greatest ranges, where the received power is lowest. The longest range echo returns, however, tend to arrive at the lowest elevation angles. Thus, it is likely that the standard SuperDARN single-interferometer algorithm could still be used in these circumstances.

[44] There is an additional problem associated with the 2*π* ambiguity in SuperDARN radars that is not discussed in this paper. This is the frequent aliasing of small elevation angles to large elevation angles which occurs when the mean elevation angle is smaller than the noise value for the elevation measurement resulting from the measurement uncertainty of the interferometer phase measurement. This has been observed previously by *Uspensky et al.* [2001] and is the focus of a study currently being conducted. We note here that the use of dual-interferometer arrays is more robust against this kind of aliasing also, due to the degree of redundancy inherent to the method.

[45] Given the complex propagation environment for HF signals in the ionosphere, there is always a possibility that multiple scattering targets exist with a similar group range at different elevation angles, such that they fall into a common range gate. This, however, is not the typical scenario. Well-correlated coherent scatter at all lags leading to a well-defined ACF typically occurs for just a few range bins, for a given azimuth. If backscatter from multiple targets did occur in the same range bin, however, the received signals from the multiple targets would be expected to be less well correlated since they are likely to have slightly different group delays. Also, the phase lags of the complex cross-correlation function between the main and auxiliary arrays will be uncorrelated for backscatter from different elevation angles so the main-auxiliary phase-delay information would be discarded. Thus, multipath propagation is not expected to lead to false results for elevation angle of arrival using the interferometry techniques described in this study. This assumption also applies to the existing SuperDARN radar interferometer design.

[46] The dual-interferometer methods described here will be implemented at a new digital HF radar facility in Buckland Park, South Australia. An analysis of experimental results and the operation of the dual-interferometer system will be the subject of future studies.

### Conclusions

- Top of page
- Abstract
- Introduction
- Elevation Angle-of-Arrival Calculation With a Single-Interferometer Array
- Elevation Angle-of-Arrival Calculations for a Dual-Interferometer Antenna Layout
- Simulation of Elevation Angle-of-Arrival Calculations for a Dual-Interferometer Layout
- Discussion
- Conclusions
- Acknowledgments
- References

[47] The standard SuperDARN interferometer layout and elevation angle algorithm has performed well over the years in many cases. However, the 2*π* ambiguity has been shown to cause significant aliasing of elevation angles. Elevation angles greater than Δ_{max} are interpreted incorrectly as angles somewhere in the range zero to Δ_{max}, causing difficulties with data interpretation. Δ_{max} is typically between 25° and 45°, depending on interferometer spacing and azimuth from the boresight. For all SuperDARN radars, in particular for those with large interferometer spacing or an enhanced azimuthal field of view, Δ_{max} falls within the expected elevation angle-of-arrival range for routine radar operation. Elevation angle-of-arrival aliasing is expected to occur most often away from solar minimum and for the SuperDARN radars with the lowest geomagnetic latitudes.

[48] It has been shown that the addition of a second interferometer array, such as a single-antenna subarray, with a well-chosen displacement from the main array, has the potential to completely remove the 2*π* ambiguity or elevation angle aliasing effect. It is suggested that the retrofitting of existing SuperDARN radars with an additional interferometer array could greatly improve the quality of elevation angle-of-arrival data for the SuperDARN network.

[49] With dual-interferometer phase lag measurements, the only limit on measurable elevation angles is imposed by the coupling between azimuth and elevation; the “cone angle” effect. This limit is independent of frequency or interferometer spacing and requires only that cos2*φ*_{0}> sin2Δ, where *φ*_{0} is the azimuth at zero elevation and Δis the elevation angle of arrival. The upper limit of measurable elevation angle of arrival decreases as angle from the boresight increases, varying from 90° along the boresight, to 66° for the outermost beam of a typical SuperDARN radar or to 53° for the recent TTFD radars with a wider field of view.

### Acknowledgments

- Top of page
- Abstract
- Introduction
- Elevation Angle-of-Arrival Calculation With a Single-Interferometer Array
- Elevation Angle-of-Arrival Calculations for a Dual-Interferometer Antenna Layout
- Simulation of Elevation Angle-of-Arrival Calculations for a Dual-Interferometer Layout
- Discussion
- Conclusions
- Acknowledgments
- References

[50] Andrew McDonald is supported by a post-doctoral fellowship through the Department of Defence, Defence Science Technology Organisation and by an ARC LIEF grant in partnership with IPS Radio and Space Services.

### References

- Top of page
- Abstract
- Introduction
- Elevation Angle-of-Arrival Calculation With a Single-Interferometer Array
- Elevation Angle-of-Arrival Calculations for a Dual-Interferometer Antenna Layout
- Simulation of Elevation Angle-of-Arrival Calculations for a Dual-Interferometer Layout
- Discussion
- Conclusions
- Acknowledgments
- References

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