### SuperDARN interferometry

The main array of SuperDARN radars is used both for radio wave emission and reception of echoes. It contains a line of 16 elements (either log-periodic director antennas or broad-band wire dipoles), which are separated by ≈15 m. The horizontal alignment of the array elements produces a knife-like diagram which is relatively narrow in the azimuthal plane and broad in the vertical plane. Scanning (beam-forming) in azimuth is achieved by applying a linear phase shift between array elements. Usually, SuperDARN arrays scan consecutively through 16 azimuthal directions (beams) within ±27° from the boresight direction. Elevation selection is based on measuring phase shift *Ψ* between the echoes received by the main array and an auxiliary (interferometer) array. The latter consists of four elements and is usually located at 100 m in front of or behind the main array (Fig. 1). Both arrays form beams pointing in the same azimuthal direction so that the narrower beam from the main array is “embedded” into the wider interferometer beam. The elevation angle is calculated as *Δ* = cos^{− 1}[*Ψ*/*kd* cos *ϕ*], where *k* = 2*π*/*λ* is the radar wave vector magnitude, *d* is the interferometer base and *φ* is azimuth measured from the boresight direction (Milan et al. 1997).

There is an intrinsic ambiguity arising from the fact that the interferometer base of *d* = 100 m is larger than the radar wavelength, *λ* ≈ 20–30 m. As a result, the maximum phase shift (observed at zero elevation), *Ψ*
_{max} = *kd* cos *ϕ*, lies between 6π and 10π, while instrumentally the phase can only be measured between −π and +π. Figure 2 shows the relationship between elevation and phase shift for central beams (*φ* = 0) at the typical Hokkaido East frequency of 11 MHz. The black line shows the total phase shift while the blue lines correspond to the actually measured phase shift which is confined to within ±π range. The latter illustrates the above-mentioned ambiguity with the same phase shift corresponding to multiple elevation values.

The ionospheric scatter echoes are expected to come from anisotropic plasma irregularities aligned with the geomagnetic field lines, and the maximum backscatter power is observed when the radio wave propagates in the direction orthogonal to the field lines. At high latitudes, the field lines are nearly vertical so that the backscatter elevation is expected to be closer to the horizontal direction, somewhere between 0 and 30°–40°. As a result, in SuperDARN software, the assumption made that the echoes come from within the first segment, i.e. between *Ψ*
_{max} − 2π and *Ψ*
_{max}
*,* and the resulting elevation estimates lie between zero and some maximum value *Δ*
_{2π
} = cos^{− 1}[(*Ψ*
_{max} − 2*π*)/*Ψ*
_{max}] (Milan et al. 1997). This phase shift range is highlighted by red in Fig. 2.

### Theoretical dependence of elevation vs range for ground scatter

In order to estimate a phase offset, first we need to identify an expected pattern for a known propagation mode of the radio wave. In order to produce plasma circulation maps, SuperDARN utilises HF backscatter returns from small-scale ionospheric irregularities (ionospheric scatter, IS). Importantly, spatio-temporal characteristics of IS are subject to different kinds of plasma instabilities which may or may not be present in the radar’s field-of-view, in contrast to the regular ground scatter (GS) echoes, which represent radio waves “reflected” by the regular ionospheric layer before they are scattered back by the rugged ground surface. While GS echoes are treated as interference by SuperDARN data processing procedures, their regular character makes this propagation mode very useful as a reference in analysing interferometer phase patterns.

Figure 3 represents a ray tracing simulation of HF propagation in a simplified situation of a single Chapman layer with a maximum density located at 300-km altitude and a scale height of 30 km. The ray tracing code was developed by Ponomarenko et al. (2009) and utilises Snell’s law and the simplest form of the Appleton-Hartree equation, \( {n}^2=1-{f}_p^2/{f}_0^2 \), where *f*
_{0} and *f*
_{p} are the wave frequency and plasma frequency, respectively. The ray trajectories were simulated between 0° and 85° separated by 0.1° in elevation at the radar location. The physical range resolution is 1 km (in Fig. 3, the rays are plotted at 1° intervals only). The blue shading shows the spatial distribution of the ionospheric refractive index. The black dots show group range in 250-km steps. There are three major propagation modes: (i) low-angle rays (green), (ii) high-angle (Pedersen) rays (red), and (iii) escaping rays (orange). While all three propagation modes can contribute to IS, only the first two reach the ground and are capable of generating GS. The yellow contour corresponds to the turning (“reflection”) points of the rays with the top and bottom branches produced by the Pedersen and low-angle rays, respectively, while their convergence point at close ranges corresponds to the skip zone boundary on the ground. The Pedersen mode covers a narrow angular range (≈3° in this case) and corresponds to divergent rays “gliding” along the ionospheric maximum. This contrasts with the low-angle mechanism covering a comparatively large angular range and producing significantly larger power density due to convergent rays “reflected” from the bottom part of the ionospheric layer.

As a consequence of the larger ray/power density, the low-ray mode is expected to dominate GS returns. This assumption is supported by analysing simulated GS elevation at each 45-km range gate. The median value for all trajectories reaching the ground was estimated within each 45-km range of group delays. The result is presented in the middle panel of Fig. 4. Here, different colours correspond to different maximum plasma frequencies increasing from 5 to 10 MHz (dark blue to red) in 1-MHz steps. The elevation values gradually decrease with increasing range until they approach zero level, as would be expected from the low-angle mode. The horizontal dash line shows the maximum measured elevation, Δ_{2π}. The top panel in Fig. 4 shows respective phase shift values. For convenience, the phase was shifted by the maximum possible phase value, namely (*Ψ-Ψ*
_{max}), so that a smaller phase shift corresponds to a lower elevation and *vice versa*. The bottom panel shows the respective values of the virtual height calculated from the group range *r* and elevation Δ accounting for the spherical geometry, \( {h}_v=\sqrt{R_E^2+{r}^2+2{R}_Er \sin \varDelta }-{R}_E \), where *R*
_{
E
} is the Earth’s radius (Andre et al. 1998). The virtual height decreases with distance at close ranges and then stays at an almost constant level. These “saturation” altitudes are lower for higher critical frequencies. The near-range cut-off represents an ionospheric projection of the skip zone boundary. This boundary moves away from the radar with decreasing *f*
_{mF2} because the radio wave gets “reflected” at a progressively lower elevation, subject to the secant law. The closest data point at the highest critical frequency (10 MHz, red) has an elevation value that exceeds the maximum measured value of Δ_{2π} ≈ 43°, so its phase was adjusted by the radar software producing an artificially low value of Δ ≈ 30°.

In order to estimate sensitivity of the obtained elevation patterns to the presence of the horizontal ionospheric gradients, we performed an additional simulation with an overall ionospheric density increase or decrease with range. For a realistic density gradient value of 30 % per 1000 km, there were only minor changes in the elevation patterns (not shown) which did not alter the overall decrease of elevation with range.

At the next stage, we investigated distortions to the elevation and virtual height patterns caused by an arbitrary phase offset, Δ*Ψ*. In Fig. 5, the unperturbed values (Δ*Ψ* = 0) for *f*
_{mF2} = 9 MHz are plotted by a solid black line, while the coloured diamonds correspond to different offset values ranging from 0 to 360° in 60° steps. The offset causes some phase values to go outside of the “allowed” range so these values are automatically brought inside the range by either adding or subtracting 2π. As a result, most of the phase curves in Fig. 5 exhibit a 2π discontinuity which shifts to longer ranges with increasing Δ*Ψ*. The general effect on elevation is that the retrieved angle, instead of following a monotonous dependence on range, is split into two seemingly unrelated populations. To the left from the phase discontinuity (closer ranges), elevation decreases faster compared to the situation with no offset, while to the right (farther ranges) the situation is the opposite. The most noticeable effect on the virtual height is that at farther ranges, instead of becoming nearly constant, the derived virtual height increases monotonically.

### Proposed calibration method

Based on the simulation results, the first step would be to analyse if the experimental elevation-range dependences for GS match the expected dependence on range. If they show patterns similar to those in Fig. 5, i.e. with sharp phase jumps and increasing virtual height at larger ranges, then one would need to introduce an extra phase/time offset so that the patterns would look as those in Fig. 4. While this task seems to be simple, there are several critical points to consider.

The most important factor is a correct interpretation of the data uncertainties. Previous studies by Ponomarenko et al. (2011a) have demonstrated that the accuracy of the phase estimates is mainly limited by variations arising from the statistical nature of the radar returns. The same phase fluctuation levels lead to larger elevation errors at lower angles due to the strongly non-linear phase-elevation transfer function near *Ψ*
_{max} (Fig. 2). At low elevation levels, the bulk of measured phase shift values will lie just below the maximum possible value, *Ψ*
_{max}, but the statistical spread in the measured phase will shift some of the “true” phase values beyond *Ψ*
_{max}. The data processing software automatically subtracts 2π from these values to shift them inside the “allowed” phase range so that the adjusted phase values become close to *Ψ*
_{max} − 2π, i.e. the respective elevation values are shifted to the maximum value corresponding to the first 2π “wrap”, Δ_{2π}. As a result, at far ranges (low elevation), even properly calibrated data should contain some sporadic discontinuities caused by the statistical variability of the phase measurements. However, on the range-time map, these discontinuities should present just isolated pixels of high elevation values on the predominantly low elevation background.

The above “statistical” discontinuities should not be confused with those observed when echo elevation truly exceeds Δ_{2π}. In contrast with the statistical error in the phase, the latter are automatically shifted to the lower elevation values. This sort of discontinuity is easy to recognise because it is generally located at closer ranges and represents a regular feature which is observed across consecutive scans. Most importantly, in this case, there will be no increase in the virtual height at progressively larger ranges.

Based on the above observations, we propose a new method to detect and measure the phase offset using visual analysis of the range-time elevation maps for ground scatter. The technique is based on consecutive adjustments of the interferometer phase until the elevation data show the expected pattern, i.e. a general decrease of elevation with range which shows at large distances a near-zero background accompanied by sporadic isolated “jumps” to values close to Δ_{2π}. Conveniently, in the hardware radar profile, there is a parameter *tdiff* which allows for phase adjustment in terms of time delay expressed in nanoseconds. At the time of publication, the Virginia Tech SuperDARN group (2015) provided an opportunity to perform the adjustment procedure by tuning *tdiff* and visually analysing the resulting elevation patterns.