Phase calibration of a radar interferometer by means of incoherent scattering



[1] In this paper it is suggested that it is possible to phase calibrate a radar interferometer using a beam-filling scattering process, and the technique is demonstrated using incoherent scattering data from the European Incoherent Scatter Svalbard Radar. Although the coherence due to the beam-filling scattering is very small, its associated phase can be measured to great accuracy with enough integration. The technique can be used by any radar interferometer with sufficient sensitivity in the receiver elements to detect incoherent scattering, possibly after several minutes of integration.

1. Introduction

[2] Radio-frequency interferometers typically use two or more receivers to position a signal source somewhere in the field of view of the instrument, the position being a function of the observed phase difference of the signal between the two receivers. Hence, it is of vital importance to separate the sources of phase difference due to instrumental causes from those due to the source itself. This process is known as phase calibration, and is necessary for beam forming methods and aperture synthesis imaging as well as for simpler interferometers.

[3] Over the years, phase calibration has been achieved in many ways. For instance, Chen et al. [2002] used radar echoes off aircraft together with TV images from a colocated camera to phase calibrate a VHF spaced antenna setup for atmospheric studies, while Sullivan et al. [2006] used radar reflections from satellites together with optical observations of the same satellites to calibrate the European Incoherent Scatter (EISCAT) Svalbard Radar (ESR) two-antenna interferometer. Earlier, Sahr [1996] introduced the concepts of closure phase and self-calibration from radio astronomy to the field of radar imaging. The 2-D imaging setup that has been used with great success at Jicamarca Radio Observatory [e.g., Hysell, 1996] can be calibrated in several ways: by means of a reference beacon on a nearby hilltop; using known astronomical radio sources; or simply by centering the distribution of scatterers in each baseline observation [Chau et al., 2008].

[4] In this paper, a distinction is made between having the data from different interferometric baselines agree on the position of a given scatterer (called relative calibration), and placing said scatterer at its correct location in the field of view (called absolute calibration). Furthermore, it is shown that absolute calibration of the ESR two-antenna interferometer can be achieved using the stable phase associated with the tiny coherence produced by beam-filling incoherent scatter. This means that for a radar interferometer with receivers capable of detecting incoherent scatter, phase calibration can be performed easily and continuously, independent of dark skies or conditions suitable for optical observations. The application of this calibration technique has acquired renewed interest in connection with the EISCAT_3D project which has produced the design of a new generation radar with built-in full 3-D imaging capabilities based on aperture synthesis imaging radar.

[5] For the first interferometric observations made with the ESR, described in previous papers [Grydeland et al., 2003, 2004], phase was used only to determine whether two scattering structures at different Doppler shifts were likely to coexist in the same volume or not. For this purpose, it was only necessary to compare the phases of the two scattering structures, absolute calibration was not required, but relative calibration is still required, otherwise coherence will be underestimated.

2. Theory

[6] In the two-element radar interferometer, the index 1 is used for the transmit/receive antenna and index 2 for the receive-only antenna. From Farley et al. [1981] and Grydeland et al. [2004], the visibility function is:

equation image

where A is the baseline length in wavelength units; and f1 and f2 are the signals received in the transmit/receive and the receive-only antenna, respectively. When g1 and g2 are the amplitude antenna patterns of these antennas, these signals are given by

equation image
equation image

where x and y are the cross-beam directions along and perpendicular to the baseline, respectively, and Zo is the distance to the observed volume. The complex cross correlation between these signals is given by

equation image

In this final expression, n(x, y; Zo) is nominally electron density, but any quantity which modifies the scattering cross section (e.g., coherent structures) will be included in this term. If, for simplicity, identical Gaussian antenna patterns are used for the two antennas, then

equation image


equation image

where G is the antenna (power) pattern. Using the condition that scattering from disjoint volumes are uncorrelated results in

equation image

Assuming that the electron density is homogeneous within the scattering volume, which is generally the case for ordinary incoherent scattering, these expressions can be evaluated as follows:

equation image
equation image
equation image
equation image
equation image
equation image

where the expression for the Fourier Transform of exp{−x2/σ2} was used for the integral in the numerator.

[7] The visibility function for the beam-filling scatterer is then

equation image

The phase of this visibility function is zero whenever the product of the backscatter intensity (electron density) and the antenna pattern is symmetric, and specifically when the electron density is constant inside the beam and the antenna pattern is symmetric.

[8] In the case of the ESR, the operating frequency is 500 MHz, which means a wavelength of 60 cm. The radar has two parabolic dish antennas, one fully steerable with a diameter of 32 m and one fixed along the geomagnetic field with a diameter of 42 m. The distance between their center points is 128 m, or 213.5 λ. Using the nominal beam width of the steerable 32 m antenna, 0.6°, as the width of the Gaussian patterns in the expressions used to derive (14), the constant visibility becomes

equation image

Using the values for both 32 and 42 m antennas, the visibility becomes slightly larger.

3. Experimental Setup

[9] The ESR data acquisition system was originally described by Wannberg et al. [1997]. The system has been upgraded since then, but the upgraded system still contain “channel boards” with digital numerical oscillators and mixers for every digital channel, so for our purposes this description is sufficient.

[10] The numerical oscillators are free-running and unless they are explicitly reset, they introduce a random phase, coherent over time but in the general case different for every digital channel.

[11] The signal from antenna j in digital channel a can be expressed as

equation image

where f is the signal from the source itself, ψj,a is the phase of the free-running numerical oscillator for channel a from antenna j, ϕj is the phase introduced by the analog part of the instrument that delivers the signal from antenna j, while θj is the phase introduced by the distance from the scatterer to the projection of the focal point of the antenna into the aperture plane of the interferometer along the line of sight to the scatterer. The random noise is represented by εj.

[12] For any signal processing involving only correlations within a single antenna/frequency channel, the random phase components are of no consequence, as long as they remain coherent for times much longer than the coherence time of the target. Hence, phase calibration is not necessary for any of these classical modes.

[13] On the other hand, cross-correlation function estimates are influenced by phases of two fundamentally different origins: First we have the systematic phases in the analog system. This is caused by things like cable lengths, delays in analog components, temperature variations in same, etc; this is the phase usually sought to correct for in phase calibration procedures. In addition, in the ESR receiver system, every cross-correlation function estimate is also influenced by two random phases, one from each digital receiver.

equation image

where ϕjk = ϕj − ϕk, etc.

[14] It is θjk = θj − θk, the difference between the θj factors for multiple antennas, that are of interest here. The other phase factors are what we aim to correct for through phase calibration.

[15] When a radar is capable of switching between different transmit frequencies in a few milliseconds, one way of increasing the duty cycle (and hence signal to noise ratio) of a radar mode is to transmit several compatible modulations on multiple frequencies, sample the scattered signal in independent channels and integrate the results together. All of the EISCAT radars have this capability.

[16] In the examples shown here, a radar mode was used which exploits this capability by transmitting identical long pulses on two frequencies. The signals at these two frequencies are then received simultaneously and integrated together wherever the observed ranges overlap. The mode is called iSlopes (interferometric Slopes, where Slopes is Simple Long-Pulse Experiment for SPEAR). This experiment was adapted for interferometry by modifying it to take raw data simultaneously on both antennas of the ESR. The data were processed off-line to produce autocorrelation and cross-correlation function estimates, using components developed for the Software Radar project [Grydeland et al., 2005b].

[17] The mode has some similarity to the GUP0 experiment, an early version of which was described by Wannberg et al. [1997], and which was used for the observations reported by Grydeland et al. [2003, 2004].

[18] For interferometry, it is then necessary that data obtained from the different frequencies for a given scatterer all have the same relative phase, i.e., relative calibration. That is, unless the factors exp{jk,a} are the same for all channels a, b, …, then integrating over multiple channels will in general not contribute at the same phase, and coherence is underestimated.

[19] In previously reported interferometric observations from the ESR [Grydeland et al., 2003, 2004], a MIDAS-W type data acquisition system was used [Holt et al., 2000], where the digital mixing and downconversion was performed using Software Radar components [Grydeland et al., 2005b] on the band-pass sampled voltage-level data. With this processing chain, all digital signal processing is performed with the same phase contribution in all antennas for a given channel, so that the factors ψjk,a are all zero, and relative calibration (in the sense defined above) was achieved automatically.

[20] As mentioned in the introduction, the phase of each coherent scattering structure is a measure of its position within the illuminated volume, up to a certain ambiguity given by the baseline. Relative calibration then means that the position indicated by all baselines for a given scatterer all indicate the same location in the instrument's field of view. In imaging, this corresponds to an image which is focused and free of “ghost” images.

[21] In order to combine data from multiple baselines to obtain an image which is both focused and corresponding to the true position of the scattering structure, it is necessary for all baselines to be calibrated to a common (true) reference, i.e., absolute calibration.

[22] That is, all factors ϕjk should be corrected for, such that only the factors θjk contribute to the cross-correlation phase. Even with absolute calibration (in the sense defined above), there is still a phase ambiguity of 2π for the signal seen in each of the interferometric elements. As long as all signals are shifted by the same phase value, the resulting image is the same. This correspond to not knowing the actual phase of the scatterer f in the expression above, but this phase does not influence the measured phase difference.

4. Observations

[23] Figure 1 shows an example of the type of observations used in the calibration procedure, this one using data from the iSlopes mode. Figure 1 (top) shows the power spectra of the signals seen in the two antennas used in this mode, in arbitrary units on a logarithmic scale, and where data from both pulses in every IPP have been summed. These data are from a 2.5 s integration centered at a range of 205.5 km. The double-humped spectrum between −5 and 5 kHz is typical incoherent scattering, with no sign of instability of any kind. Figure 1 (middle) shows the coherence, or normalized cross-spectrum, between the same two signals, computed as described by Grydeland et al. [2004]. Separate curves are shown for coherence computed using only the first pulse in every IPP; using only the second pulse; combining the two without regard to calibration; and combining the two using the calibration method described herein. Figure 1 (bottom) shows the cross-spectrum phase between the two signals, for the same four cases as used for coherence.

Figure 1.

Example of the observations. Shown is a 2.5 s integration from the iSlopes mode, taken on 23 January 2006, at 0601:59 UT. (top) Power spectra from the 32 and 42 m antennas, and (middle and bottom) coherence and cross-spectrum phase of these two signals. In Figure 1 (middle and bottom), coherence and phase are plotted separately for the two pulses (indicated by “p0” and “p1” in the legend), merged without calibration (indicated by “m(u)”) and with the calibration discussed in this paper (indicated by “m(c)”).

[24] The phase plotted in Figure 1 (bottom) is only expected to be meaningful when the coherence is significant, roughly between −5 and 5 kHz. In this interval, we see that the cross-spectrum computed using only the first pulse has a phase value close to π, while using only the second pulse results in phase values close to −π/2. In Figure 1 (middle) we see that these two sets result in different coherences. When combining the data from the two pulses without phase calibration (indicated by “m(u)” in the legend), an ordered phase in the vicinity of −3π/4 is obtained, and the resulting coherence is lower than that seen in either of the two pulses separately.

[25] The phase calibration involves multiplying the complex cross-spectra with a complex exponential such that the phase of each averages to zero in the region where it takes on a constant value. The phases of these complex exponentials are given in Figure 1, separately for each pulse. Combining the data calibrated in this way results in the lines indicated by the legend “m(c)” in Figure 1 (middle and bottom). The coherence of the combined data is now between those seen in the two pulses separately and its phase is close to zero, both of which are as expected.

[26] Notice also that the coherence seen in Figure 1 appears to be significantly higher than the background level, and much higher than what is expected from the theoretical expression (15). Using Figure 3 of Grydeland et al. [2003], a coherence of 0.3 at 200 km range corresponds to a horizontal size of a scattering structure of about 250 m, while the nominal width of the beam (full width at half maximum) of the 32 m antenna at this range is about 2 km. Using a better representation of the antenna patterns might help alleviate this discrepancy.

[27] The radar signal processing produces estimates of the autocorrelation and cross-correlation functions of the signals, while the spectral domain data shown in Figure 1 are obtained through a Fourier transform. Therefore, the phases used for calibration were computed in the frequency domain. In the cross-correlation function estimates, the phases were averaged for all short lags (∣τ∣ ≤ 50 μs) and moderate ranges (<350 km). In Figure 2, this average is plotted against time for about 20 min of observations, including a couple of noteworthy events. The phase values used for calibration in Figure 1 are the averages of these values in the interval from zero to 750 s (excluding the 125–250 s interval). Figure 2 shows both that the cross-spectrum phase seen in the data from each of the two pulses is constant over time, and that when there are deviations from this average due to strong off-axis targets, the phase deviates in the same direction in the data from both pulses. Examples are seen around 200 s, just before 800 s and around 1150 s.

Figure 2.

Measured cross-correlation phase. The phase values close to π are from the cross correlations computed using only the first pulse in every IPP, while the other values are computed using only the second pulse. The first vertical dashed line indicates the time of the data shown in Figure 1.

5. Discussion

[28] As shown by Woodman and Hagfors [1969], the cross-correlation phase estimate is approximately Gaussian and bias free, as long as

equation image

where N is the number of independent estimates and ρ is the coherence, computed using formula (1). The variance is then

equation image

Using this expression, the necessary number of estimates can be calculated for a given observed coherence and desired phase accuracy. A coherence of 0.3 and a desired phase accuracy of 0.05 means approximately 2000 independent estimates are required.

6. Conclusions

[29] This paper shows that it is possible to perform phase calibration of a radar interferometer using a beam-filling scattering process without knowledge of the various instrumental phases that contribute to the signal in the digital channels used. The technique uses the simple observation that the phase of the cross correlation of a beam-filling target should be zero, and any computed cross-correlation function for such a target can be phase calibrated by rotating its observed phase to zero. The variance of the observed phase can be reduced as much as desired through long integrations. This phase calibration can be monitored continuously whenever the scattering volume is free from hard targets or coherent structures. This technique has the great advantage that it is independent of other instruments and their limitations, such as, e.g., the need for dark skies and no clouds for optical satellite observations.

[30] The technique is applicable for any radar interferometer with receiver elements sufficiently sensitive to detect incoherent scattering, possibly after minutes of integration. The 32 m antenna of the ESR is demonstrably sensitive enough. Preliminary results (not shown) indicate that the same might be the case for the receive antennas of the EASI project [Grydeland et al., 2005a]. We suspect that the same will be the case for the modules used in the Jicamarca imaging setup, and the modules of the planned EISCAT_3D radar system will certainly have sufficient sensitivity for this technique to be used.


[31] T.G. thanks E. M. Blixt, V. Belyey, and B. Gustavsson for stimulating discussions; and special thanks to J. Sullivan for starting the work on calibrating the ESR interferometer. It was while preparing data for her studies that key observations underlying the current paper were made. The Slopes experiment was developed by Paul Gallop. This work was supported by the Department of Physics and Technology, by grants from the Norwegian Research Council (NFR), and the European Union's Sixth Framework program funds of the EISCAT_3D project. The EISCAT Scientific Organization is supported by research councils and institutes of Norway (NFR), Sweden (VR), Finland (SA), United Kingdom (STFC), Germany (DFG), Japan (NIPR and STEL), and China (CRIRP).