High-precision measurements of ionospheric TEC gradients with the Very Large Array VHF system



[1] We have used a relatively long, contiguous VHF observation of a bright cosmic radio source (Cygnus A) with the Very Large Array (VLA) to demonstrate the capability of this instrument to study the ionosphere. This interferometer, and others like it, can observe ionospheric total electron content (TEC) fluctuations on a much wider range of scales than is possible with many other instruments. We have shown that with a bright source, the VLA can measure differential TEC values between pairs of antennas (δTEC) with a precision of 3 × 10−4 TECU. Here, we detail the data reduction and processing techniques used to achieve this level of precision. In addition, we demonstrate techniques for exploiting these high-precision δTEC measurements to compute the TEC gradient observed by the array as well as small-scale fluctuations within the TEC gradient surface. A companion paper details specialized spectral analysis techniques used to characterize the properties of wave-like fluctuations within this data.

1. Introduction

[2] The effects of the ionosphere have always been an obstacle for ground-based radio frequency observations of astronomical sources. This is especially true for interferometric observations used to make images of objects with relatively small angular sizes. VHF and UHF interferometers such as the Very Large Array (VLA) in New Mexico, the Westerbork Synthesis Radio Telescope (WSRT) in the Netherlands, the Giant Metrewave Radio Telescope (GMRT) in India, and the Australia Telescope Compact Array (ATCA), among others, are all affected by the ionosphere in the same way.

[3] The fundamental principles for the operation of an interferometer are well described in the literature [e.g., Thompson et al., 1991]. Briefly, an interferometer measures the time-averaged correlation of the complex electric fields measured at pairs of antennas pointed at a particular object. These correlations, or “visibilities” provide a measure of the spectrum of the sky brightness distribution at different spatial frequencies. These frequencies are essentially the difference between the position vectors of the two antennas normalized by the wavelength of the observation in a coordinate system based on the position of the object in the sky. These spatial frequencies are commonly referred to as u, v, and w, and the coordinate system used is defined such that u is the spatial frequency in the east-west direction, v corresponds to the north-south direction, and w gives the spatial frequency along the line of sight to the object. Thus, as the object moves through the sky, the u, v, w coordinates of a pair of antennas, or “baseline,” changes. The visibility measured for a specific baseline at a particular time is given by

display math

where Ω denotes solid angle, ν the frequency of the observed signal, and I is the intensity on the sky at a position given by the direction cosines l, m, and inline image which are measured relative to the position of the observed object on the sky.

[4] Generally, interferometers are “fringe-stopped,” that is, the measured visibilities are multiplied by a factor of exp(−2πiw) so that visibilities from a source at the center of the field of view will have a phase of zero (i.e., the source will not produce fringes). This is mainly done because with fringe-stopping and for small fields of view (i.e., n ≃ 1), equation (1) becomes a simple two-dimensional Fourier transform such that the observed visibilities, measured as a function of u and v, may be converted to maps of intensity on the sky using standard numerical methods.

[5] As signals from astronomical sources pass through the ionosphere, a phase term is added given by

display math


display math

where Ne is the electron density and x denotes the path-length through the ionosphere.

[6] Thus, the phase of the observed visibility for a baseline is altered by the difference between the ionospheric phase terms observed by the two antennas, which is proportional to the difference in the total electron content (TEC) observed along the lines of sight from the two antennas to the observed object. To first order, if these phase terms are not removed, then during the Fourier inversion process involved in making an image of the sky, the ionospheric phase terms have the effect of changing the apparent positions of objects in the image plane. When higher order ionospheric effects begin to dominate, objects begin to appear distorted in the image plane, and in extreme cases, almost disappear.

[7] Therefore, to make an image, one must remove these phase terms through a calibration process which estimates the required phase corrections. Typically, after initial calibrations for instrumental effects are performed, one usually uses some form of a procedure referred to as “self-calibration” [Cornwell and Fomalont, 1999]. This involves dividing the visibilities by an assumed sky model for the observed field of view which removes any contribution by the sky brightness distribution to the observed visibility phases. Following this, a linear fit is used to determine the complex gain for each antenna. Since the interferometer only provides phase differences between antenna pairs, the absolute phase of the complex gain for each antenna cannot be determined by this fitting process. The general practice is to choose one reference antenna for the interferometer and to set the phase for its complex gain to zero.

[8] Since for N antennas, there are N(N − 1)/2 baselines, this is generally an over-determined problem and can be done over relatively short time periods, depending on the brightness of the source. One may also use this calibration to make an image, deconvolve the image to produce a better sky model, and then repeat the process until it converges. This has been shown to be a rather robust procedure for determining the complex antenna gains [see Cornwell and Fomalont, 1999, and references therein].

[9] According to equations (2) and (3), the phase corrections obtained from the determined antenna gains are essentially measurements of the difference between the TEC along the lines of sight of a particular antenna and that of the reference antenna. These equations also demonstrate that the effect of the ionosphere will be more substantial for VHF observations. Given the size of available astronomical VHF interferometers (baselines ranging from <1 km to as large as ∼30 km), the robustness of the self-calibration procedure on relatively small time scales (typically ∼1 minute, but as small as a few seconds for extremely bright sources), and the sensitivity of such interferometers to relatively small TEC variations (fluctuations in differential TEC of as small as 0.001 TECU), these instruments are capable of studying TEC fluctuations on substantially finer scales than many others available. Subsequently, previous work has been done using astronomically motivated observations [e.g., Cohen and Röttgering, 2009; Intema, 2009] as well as observations geared toward studying the ionosphere [e.g., Jacobson and Erickson, 1992a, 1992b] to explore the phenomenology of the ionosphere on these fine scales.

[10] Much of this work has been performed with the VLA (latitude = 34° 04′ 43.497″ N and longitude =107° 37′ 05.819″ W). This is in part due to the fact that the VLA is relatively well suited to the study of the ionosphere because its 27 antennas are distributed in a “Y”-shape which allows it to probe structures along three different directions. Arrays such as the WSRT and ATCA are so-called “east-west” arrays because the antennas are aligned along a single east-west axis and can therefore only observe ionospheric fluctuations along one dimension. The VLA is also unique in that the antennas are moved from time to time among four different configurations, referred to A, B, C, and D. The configurations go from larger and more spread out to smaller and more compact. At its largest in the A configuration, each of the array's three arms is about 20 km long with the shortest antenna separations being about 1.5 km. In the most compact configuration, D, the arms are at most 0.6 km long and the shortest spacings are about 0.04 km. This allows the VLA the ability to study the ionosphere over a wider range of physical scales than other similar interferometers.

[11] Finally, the VLA had a somewhat unique VHF system in place that allowed observations to be made simultaneously at 74 MHz and 327 MHz using a pair of dipole antennas (one for each band) mounted near the prime focus of each antenna. Recently, the VLA electronics and receivers were upgraded to establish the new Expanded VLA (EVLA) which does not include the old VLA VHF system. However, a new and improved VHF system is now being developed and will be available in the near future.

[12] Past work using the VLA and other similar instruments has led to interesting results. These include discoveries such as the new class of magnetic-eastward-directed (MED) waves, predominantly found at night, discovered by Jacobson and Erickson [1992b] as well as larger statistical studies such as the measurements by Cohen and Röttgering [2009] of the dependence of differential ionospheric refraction on relatively large angular scales (>10°) using data from a 74 MHz all-sky survey which showed large dependences on time of day. However, there is much more to be learned from these types of data, especially at smaller time and amplitude scales. Therefore, we are embarking on a program utilizing the VLA data archives (https://archive.nrao.edu) that seeks to push this type of analysis to even finer scales. We will use previously unexplored data sets of the brightest VHF objects. We will apply new techniques for calibration and the mitigation of radio frequency interference (RFI) to other data sets to significantly improve their sensitivity to small amplitude TEC fluctuations as well as fluctuations occurring on smaller time and spatial scales than could be explored previously.

[13] Here, we describe the first step in this program, a thorough evaluation of the ionospheric information contained within a single, relatively long VHF observation of one of the brightest radio sources in the sky with the VLA. This data set provides the opportunity to develop and establish techniques for processing, analyzing, and interpreting similar data in future segments of our program. This paper focuses on the data selection and calibration as well as the post-processing done on the phase information extracted from this exemplar data set to measure TEC gradients with the array. In a companion paper [Helmboldt et al., 2012], we detail new techniques for spectral analysis of these data.

2. Data Acquisition and Processing

2.1. Phase Correction Determination

[14] In the study presented in this paper, we have sought to explore the kinds of ionospheric phenomena capable of being observed with the VLA at the lowest fluctuation levels and the smallest times scales possible. To ensure that we were able to determine ionospheric phase terms to the the level of accuracy needed at the smallest available time intervals, including the resolution of any 2π ambiguities, we chose to use a data set focused on Cygnus A (or “Cyg A”; also known as 3C405). With a total intensity of more than 17000 Jy (= 1.7 × 10−22 W m−2 Hz−1) at 74 MHz, Cyg A is one of the two brightest sources in the sky at frequencies below 100 MHz. Its brightness distribution on the sky is also known relatively well, implying that it can be used within the self-calibration procedure described briefly in section 1 to solve for the phase corrections (both instrumental and ionospheric) for each antenna over relatively short time intervals.

[15] The data set we have selected consists of simultaneous 74 MHz and 327 MHz dual polarization observations of Cyg A with the VLA over a period of more than 12 hours on 12–13 August, 2003 (VLA program number AK570). During this period, there was a moderate amount of geomagnetic activity (Kp index ≈ 2–4) and low solar activity (F10.7 = 123 SFU; 1 SFU = 10−22 W m2 Hz−1). For this data, the VLA was in the A configuration with the VHF dipole system available on all but three antennas (antenna numbers 11, 13, and 15; see section 1 and Figure 1). The exact layout of the antennas is shown in Figure 1. The data set contains a 35 minute scan (i.e., a contiguous block of observing time), and a longer, nearly 13 hour scan which, taken together, cover a range of local time from roughly 17h00m on 12 August to 06h30m the next morning. For the 74 MHz band, the total bandwidth was 1.5 MHz; it was 3 MHz for the 327 MHz band. For both bands, the time-averaged visibilities were measured over intervals of 6.67 seconds simultaneously in both RR and LL polarizations.

Figure 1.

The layout of the VLA antennas during the observations of Cyg A. The reference antenna (see section 2) is highlighted in black.

[16] The processing of VHF interferometric observations from the VLA or other similar interferometers is, in the average case, a lengthy and difficult process. One usually must deal with significant sources of RFI, which typically appear much stronger on shorter baselines. For the VLA, the field of view at 74 MHz is more than ten degrees in diameter, and the ionospheric conditions may vary significantly from one part of the field of view to another, which requires special calibration techniques. One must also typically produce an image for the entire field of view, regardless of which object(s) one is interested in because the removal of sidelobes (i.e., secondary peaks of the impulse response in the image plane) produced by other objects is crucial to minimizing noise in the final image. Fortunately, the unusually high intensity of Cyg A makes it substantially brighter than any source of RFI or any other object in the field of view. This makes the calibration process much simpler and more straight forward.

[17] The calibration of the data was performed using standard tasks within the Astronomical Image Processing System (AIPS [Bride and Greisen, 1994]) (http://www.aips.nrao.edu). The first step in the calibration was to determine the bandpass response of each band (AIPS task BPASS). This was done within one minute time intervals by dividing the data by the visibilities measure within a few central channels where the response typically peaks. The relative amplitude and phase responses across each bandpass were then measured and interpolated onto the full set of time steps. After trimming the first and last few channels from each band where the response drops substantially, the data were corrected for the bandpass responses.

[18] Following this, at 6.67 second intervals (i.e., the shortest possible for these data), the data for each band and polarization were used with model visibilities computed using images of Cyg A presented by Lazio et al. [2006] (currently publicly available at http://lwa.nrl.navy.mil/tutorial/) to compute the phase corrections for each antenna at each time step. This was done with the AIPS task CALIB which does a series of consistency and sanity checks as it determines the solutions needed to fit the measured visibilities to the model and flags antennas and intervals that appear spurious or of poor quality. We relaxed some of the criteria for these checks, namely the minimum number of antennas (we used four) required and the minimum signal to noise ratio (we used three) because of the relatively short time intervals used. As noted above, an interferometer like the VLA can only measure relative phases, implying that for this calibration to work, a reference antenna must be chosen whose phase is arbitrary and is subsequently set to zero. Typically, the antenna closest to the center of the VLA is not used because the effects of RFI tend to be the worst for this antenna. However, as noted above, Cyg A is bright enough that this is not a consideration for our data set, and we therefore used this antenna as our reference (antenna 9; see Figure 1).

2.2. Processing of Phase Corrections

[19] Following the determination of phase corrections using standard AIPS routines, several steps were taken to extract ionospheric information from the phase data. This was done using ad hoc, python-based software.

[20] The phase corrections measured by the calibration process contain contributions from several effects, the ionosphere being the largest, especially at 74 MHz. In particular, the phase difference between two antennas within a single baseline is given by

display math

where Δϕion denotes the difference in the ionospheric phases along the lines-of-sight of the two antennas given by equation (3), Δϕinstr represents difference in the instrumental effects of the two antennas, Δϕsour is the contribution to the phase difference from the structure of the observed source, and Δϕamb is the contribution from 2π ambiguities. We can remove Δϕsour by dividing the observed visibilities by a model of Cyg A and Δϕamb by having short enough time sampling to “unwrap” the phases (see below). The dual 74 and 327 MHz observing mode is very useful for removing instrumental effects since the ionospheric phase simply scales with wavelength (see equations (2) and (3)), whereas instrumental effects do not necessarily. The instrumental components of the phase corrections include errors in the delays added to individual antenna signals used to fringe-stop the visibilities (see section 1) and offsets between the antenna pointing and the actual source position. These two effects and 2π ambiguities were dealt with in three separate steps.

[21] First, the AIPS task CALIB flags antennas that are spurious at each time step so that the phase corrections are “missing” for some antennas at a relatively small fraction (∼2%) of time steps. Generally, a small amount of missing data is not problematic. However, to facilitate the use of fast Fourier transforms (FFTs) within our spectral analysis of the data presented in a companion paper which work best with evenly sampled data, these missing time steps were filled in. We have done this by linearly interpolating the real and imaginary parts of the complex antenna gains, gA, computed by CALIB for all antennas onto a common time-step grid consisting of 7299 steps spaced by 6.67 seconds, and then recomputing the phase corrections {= tan−1[Im(gA)/Re(gA)]}.

[22] Next, the time sampling of the data (6.67 seconds) was sufficiently short that the phase, ϕ, as a function of time for each antenna could be “unwrapped” in the conventional way, i.e., by correcting phase jumps of more than π radians by adding or subtracting 2π. There was one caveat to this process, however. For each antenna, there were a few (ranging from zero to five) times steps where, for one reason or another, the phase correction was either spurious or represented a real and very short jump in the instrumental phase, appearing as sharp spikes in the unwrapped phases. Since there were a total of 7299 time steps, a few short jumps in phase would not be an issue except for their effect on the unwrapping process. Any of these spikes can cause an artificial large phase jump if it is included in the unwrapping process, which we have illustrated in Figure 2 (top).

Figure 2.

(top) The unwrapped phase, ϕU for antenna 4, RR polarization, as a function of time (relative to midnight, 13 August) when no attempt was made to “de-spike” the phase data (see section 2.2). (middle) The difference between the cosine of the wrapped phase at a time step i, cosϕi, and the next time step as a function of time used to find spikes in the phase data. Flagged spikes are highlighted with circles. (bottom) The phase as a function of time, unwrapped with the flagged spikes excluded.

[23] To combat this, we wrote a simple algorithm that computes the difference between cos(ϕ) at each time step and the value for the next time step, where ϕ is the wrapped phase. Any time step where the absolute value of this difference was more than ten times the standard deviation among all time steps for a give antenna was flagged and not included in the unwrapping process. These empty time steps were then filled by linearly interpolating the unwrapped phase data for the un-flagged time steps. It should be noted that these spikes only occupy 1–2 time steps (6.67–13.34 seconds) and that the surrounding data are otherwise well-behaved, making interpolation a reasonable and straightforward solution to eliminating instances of such spikes. An example of how the data were flagged is illustrated in Figure 2 (middle), while the resulting unwrapped phases are shown in Figure 2 (bottom). From this result, one can see that the spikes are not always completely removed from the data, but the goal of eliminating their effect on the unwrapping process has been achieved.

[24] A number of instrumental effects can contribute to the phase corrections, including errors in the fringe-stopping process (see section 1) and offsets between the position of the source and that of the observed field center (i.e., pointing errors). These effects are generally stable in time, changing insignificantly over periods of days [see, e.g., Cohen et al., 2007]. However, for the VLA, the instrumental phase is known to occasionally have short jumps that one must be wary of. We have found such a jump in our data occurring at a local time of about 23h06m on 12 August. The jump is most obvious when one plots the difference between the 74 MHz phase and the 327 MHz phase scaled by a factor of (327/74) so that the ionospheric phases cancel out (note, this only gives us the scaled difference between the 327 and 74 MHz instrumental phases, not the instrumental phases themselves). We have plotted this difference as a function of time for antenna 14 for both polarizations in Figure 3 to show the location of what we will refer to as the “phase jump region.” While being fairly subtle in the LL polarization, it is quite obvious in the RR polarization. The jump lasted about 6 minutes and can be seen in the data for several antennas. To deal with this, we have treated the phase jump region, as well as the time periods before and after it, as separate scans, assuming that each scan has its own instrumental phase. This basically resulted in us treating the data as if it contained four scans instead of two.

Figure 3.

As a function of time, the difference between the unwrapped 74 MHz phase, ϕ74, and the 327 MHz phase, ϕ327, with the 327 MHz phase scaled such that the ionospheric components of the two phases canceled out for the RR (upper) and LL (lower) polarizations for antenna 14. The region where the instrumental phases have changed relatively abruptly (i.e, the “phase jump” region first referred to in section 2.2) is shaded in gray.

[25] To remove instrumental effects, we have used a kind of continuum subtraction process. Within this process, we have treated ionospheric fluctuations as features superimposed on a smooth continuum consisting of the instrumental phases, which vary relatively slowly with time as well as any slowly varying component of the ionospheric phase. With the current data, we unfortunately do not have the means to separate the slowly varying component of the ionospheric phase from the instrumental effects and can therefore only measure fluctuations in TEC on relatively short (<1 hour; see below) time scales. In the future, with instruments with larger bandwidths, it will be possible to use the wavelength dependence of the phase corrections to separate these effects since the instrumental phases are ∝ ν and the ionospheric phase is ∝ ν−1.

[26] We have chosen to perform our continuum subtraction process for each antenna, band, and polarization by smoothing the unwrapped phases with a one-hour-wide boxcar which appeared to preserve any apparent fluctuations while giving a good representation of the continuum. For the first scan and the phase jump region, we simply subtracted a single mean value from all the phases since they are each shorter than an hour. We did the same for the first and last hour of each of the remaining two scans so that the same filter width would be used for all times steps.

[27] Following this, we found that the position offset component of the instrumental phases presented a problem for this process near the edges of each scan. This is because, depending on the antenna, these phases can vary significantly over one hour, especially at 327 MHz. From the fringe-stopped (see section 1) version of equation (1), one can see that a position offset in the direction cosines l and m of Δl and Δm will produce an additional phase of −2π(uΔl + vΔm). Since u and v are normalized by the observed wavelength, any such phase will be 4.4 times larger for the 327 MHz band. In addition, the offsets can be different for each band and this difference can vary with baseline. This is due to a number of factors including the fact that different model images were used for the bands which may not be exactly aligned and that Cyg A has a significant amount of resolved structure which larger baselines are more sensitive to, especially at 327 MHz where the angular resolution in the image plane is 4.4 times better than that at 74 MHz.

[28] We have demonstrated this in Figure 4 (top) where we have plotted ϕ74 − ϕ327(327/74) for antenna 3, LL polarization as a function of time. With ϕ327 scaled by (327/74), the ionospheric phases are removed and all that is left is the difference between the instrumental phases for the two bands. The data follow a smooth curve which is inconsistent with the known behavior of VLA instrumental phases, especially at 74 MHz. Furthermore, the curve that the data follows is easily fit by a linear combination of the un-normalized versions of the u and v coordinates (see the red curve in Figure 4, top). A single baseline (antenna 3 with the reference antenna) will sweep out an ellipse in the u, v-plane because of the rotation of the earth [see, e.g., Thompson, 1999]. Therefore, this is exactly what one would expect for a scenario where there is a single position offset for each of the two bands during each scan.

Figure 4.

(top) As a function of time, the difference between the unwrapped 74 MHz phase, ϕ74, and the 327 MHz phase, ϕ327, with the 327 MHz phase scaled such that the ionospheric components of the two phases canceled out for antenna 3, LL polarization. A fit to this data of a linear combination of the spatial frequencies u and v (see equation (1)) which assumes a single position offset (see section 2.2) is also plotted in red. (middle) The same as in Figure 4 (top), but for the phases with the smoothed (with a one-hour wide box) phases subtracted from each band. (bottom) The same as Figure 4 (middle), but with a fit of a linear combination of u and v used in the continuum subtraction of the 327 MHz phases at the edges of each scan (see the discussion in section 2.2).

[29] To show the effect of the time dependence of the position offset phase on our continuum determination process, we have plotted the continuum-subtracted version of ϕ74 − ϕ327(327/74) versus time in Figure 4 (middle). One can see from this plot that within the first scan and within the last hour of the second and fourth scans where a single mean continuum value was subtracted from each, the gradient of the position offset phases has introduced an artificial difference which increases/decreases with time. Since the position offset phase is much larger for the 327 MHz band, we have introduced the following additional step for the continuum subtraction of the 327 MHz data. Within the first scan, the phase jump region, and the first and last hours of the other two scans, we have fit a linear combination of the un-normalized versions of u and v to ϕ74 − ϕ327(327/74) separately for each time range. Within each of these time periods, we used the mean values for ϕ74, ϕ372, and the u, v fit to construct a time-variable continuum for the 327 MHz data. The benefits of this are illustrated in Figure 4 (bottom) where we have plotted the continuum-subtracted version of ϕ74 − ϕ327(327/74), this time, including the additional computation for the time-variable 327 MHz continuum with the first/last hour of each scan. One can see that the roughly linear features seen in Figure 4 (middle) have been removed and that the remaining difference between the 74 MHz and 327 MHz continuum-subtracted phases is essentially noise.

[30] Following the application of the corrections detailed above, we used equation (3) to convert the continuum subtracted phases for each antenna, band, and polarization to values of differential TEC, or δTEC. Then, at each time step and antenna, we computed the median δTEC among the four values (i.e., two bands and two polarizations) as well as the median absolute deviation (MAD) as an estimate of the uncertainty in the median. To increase the reliability of the MAD computations, we included with each time step the four nearest time steps (using their own individual median values) for a total of 20 data points per time step. In both computations (median and MAD), the median was used to minimize the effects of any spurious data which remained.

[31] The resulting δTEC values are plotted as functions of time for each antenna in the northern arm in Figure 5, the southeastern arm in Figure 6, and the southwestern arm in Figure 7 along with the MAD values to illustrate the relatively high precision to which δTEC has been measured. The typical δTEC uncertainty, represented by the MAD computations, is about 3 × 10−4 TECU, demonstrating the remarkable ability of the VLA to detect extremely small TEC fluctuations, even on time scales <10 seconds when an object as bright as Cyg A is used.

Figure 5.

For each antenna in the northern arm (see Figure 1) of the VLA, the the difference between the TEC fluctuation (i.e., above or below the mean TEC) along its line of sight and that measured along the reference antenna's line of sight, δTEC. The estimated uncertainty (see section 2.2) is plotted in red in each panel.

Figure 6.

For each antenna in the southeastern arm (see Figure 1) of the VLA, the the difference between the TEC fluctuation (i.e., above or below the mean TEC) along its line of sight and that measured along the reference antenna's line of sight, δTEC. The estimated uncertainty (see section 2.2) is plotted in red in each panel.

Figure 7.

For each antenna in the southwestern arm (see Figure 1) of the VLA, the the difference between the TEC fluctuation (i.e., above or below the mean TEC) along its line of sight and that measured along the reference antenna's line of sight, δTEC. The estimated uncertainty (see section 2.2) is plotted in red in each panel.

3. Measuring TEC Gradients

3.1. General Approach

[32] With the fully reduced δTEC data, including robust estimates of the uncertainties, we are in a position to explore a wide range of ionospheric phenomena. First, we note that since the VLA measures differential TEC values between antenna pairs, it is essentially only sensitive to changes in the TEC gradient. Given the geometry of the array (see Figure 1), we cannot numerically compute the TEC gradient at each antenna location from our data, and measuring the full TEC gradient requires a somewhat ad hoc approach. Such measurements are crucial for any analysis of observed TEC fluctuations because without modification, the set of δTEC time series can only be spectrally analyzed for specific assumed pattern models (e.g., a single plane wave [see Jacobson and Erickson, 1992a]).

[33] This is different from the normal mode of operation for radio interferometers in which standard techniques are used to invert and de-convolve sparsely sampled visibility data to make an image. As equation (1) demonstrates, the observed visibilities are functions of the differential antenna positions, u, v, and w. For small fields of view, the contribution of the w term is negligible if fringe-stopping is applied (see section 1). Thus, even for a Y-shaped array, reasonably good u, v-coverage can be obtained. This is improved further by the rotation of the earth which causes each baseline to sweep out an ellipse in the u, v-plane (see section 2.2 and Thompson [1999]). Using this fact to obtain better u, v coverage is sometimes referred to as “earth rotation synthesis.”

[34] In contrast, the gradient of an arbitrary set of TEC fluctuations varies over the array as a function of the actual antenna positions projected onto the ionosphere. Improved spatial coverage can be obtained by exploiting the change in the apparent position of the observed source (i.e., rotation of the earth) and the movement of the fluctuations themselves, effectively converting temporal baselines into spatial ones. However, since the TEC fluctuations presumably have a distribution of speeds and directions, this is not as straightforward as in earth rotation synthesis. One must decompose the time series into temporal spectral modes and then analyze how the properties of each mode vary across the array to extract the size, speed, and direction of the dominant pattern(s) for that mode (such spectral techniques are detailed in a subsequent paper). Therefore, we must still contend with data that has been sampled in a Y-pattern which cannot be inverted in a straightforward manner.

[35] We have consequently developed two ad hoc techniques designed to provide measurements of the TEC gradient time series over the full array and along each of the VLA arms. Before implementing either technique, we first had to perform two basic geometric corrections to the data so that the measured TEC gradients would correspond to vertical TEC gradients as closely as possible. First, to ensure that our characterization of the shape of the observed TEC surface is physically meaningful, we needed to project the antenna pattern displayed in Figure 1 onto the locations where the lines of sight of the antennas pass through the ionosphere, or “pierce-points.” Second, we needed to compute the slant-to-vertical TEC corrections for the line of sight to Cyg A as its apparent position on the sky changed throughout the observation. For a plane parallel approximation, both of these corrections are relatively straightforward. However, since Cyg A was as low as 12° above the horizon during the observing run, a plane parallel approximation was far from valid at all time steps. We have therefore computed the two required geometric corrections using a spherical model detailed in Appendix A.

[36] Within this model, the ionosphere was approximated with a thin shell located at the height of the maximum electron density, or “peak height” [see Lanyi and Roth, 1988; Ma and Maruyama, 2003]. We obtained estimates of the peak height as a function of time by using the International Reference Ionosphere (IRI [Bilitza, 2001]) software, inputting the date and time of our observations and the latitude and longitude of the VLA. We then re-determined the peak heights using the latitudes and longitudes of the pierce-points. We found that additional iterations of this process only marginally changed the results and chose to use one iteration only. The final peak heights used are plotted in Figure 8 (top) along with the projected separations from the array center for the farthest antennas of each arm (antennas 1, 7, and 22; see Figure 1) and the corresponding slant-TEC corrections.

3.2. Polynomial Fits

[37] After applying the geometric corrections to the antenna positions and the δTEC measurements, we sought to characterize the full two-dimensional TEC gradient observed by each antenna at each time step. Rather than assume a particular dominant structure (e.g., a plane wave), we simply assumed that since the array is smaller than many transient ionospheric phenomena, the observed TEC surface at any time step could be approximated with a low-order Taylor series. We examined many time steps and found that a second order, two-dimensional Taylor series adequately approximated the amount of curvature in the TEC surface detected by the VLA. This Taylor series has the following form

display math

where x and y are the north-south and east-west antenna positions, respectively, projected onto the surface of the ionosphere at the estimated peak height. To maximize the amount of data used to constrain the parameters of each fit, we used the difference between δTEC for each of the 300 unique antenna pairs at each time step. Thus, the form of equation (5) actually fit to the data was

display math

where the i and j subscripts denote the values for the ith and jth antennas, respectively. We also utilized some standard sigma-clipping during the fitting process for each time step by computing the rms of the fit residuals, rejecting all antenna pairs with absolute residuals >3 rms, and repeating 50 times. As many as about 10 and as few as zero were rejected for any given time step. We note that each time step was fit independently to preserve the presence of any small-scale spatial/temporal fluctuations.

Figure 8.

(top) The height of the maximum electron density at the latitudes and longitudes of the ionosphere pierce-points during the observations as computed by the International Reference Ionosphere (IRI) software as a function of local time. (middle) For each of the three furthest antennas in the array, the distance of the ionosphere pierce-point of the antenna from that of the array center assuming a thin-shell model at the heights plotted in Figure 8 (top) as a function of time. (bottom) The multiplicative factor used to correct the observed slant-δTEC values assuming a thin-shell model at the heights plotted in Figure 8 (top).

[38] The fitted coefficients as a function of time are plotted in Figure 9. From these, one can see the same large amplitude and period fluctuations at the beginning of the observing run that are visible in the individual antenna data plotted in Figures 5–7. Note that they are not quite as large here because of the applied slant-TEC correction discussed above. Here, we can see that they are most visible in the p0 coefficient which is the partial derivative of the TEC surface at the center of the array along the north-south direction. With the plots in Figure 9, one can also see the same thing beginning to happen near the end of the run toward dawn. This is qualitatively consistent with the known behavior of medium-scale traveling ionospheric disturbances (MSTIDs) which are prevalent near sunrise and sunset [e.g., Hernández-Pajares et al., 2006]. During the middle of the night, the second-order terms become more significant.

Figure 9.

The fitted values of the coefficients from equation (6) as a function of local time relative to midnight for 13 August 2003.

3.3. Arm-Based Approach

[39] While the polynomial-based measurements provide useful information about the variation of the full two-dimensional TEC gradient, they neglect the ability of the VLA in its A configuration to detect fluctuations on scales as small as a few kilometers. In principle, one could do this by simply increasing the order of the polynomials used. However, it is likely that the small-scale structures observed do not span the array. This implies that such fits would not yield accurate representations of the full TEC gradient at each antenna, especially those near the ends of the arms (see Figure 1). Therefore, instead of using higher order polynomial fits, we have opted for an alternative approach to make full use of the data.

[40] This complementary method computes the projection of the TEC gradient (or, the spatial derivative of δTEC) along each VLA arm. The projected gradient was computed at each time step separately for the antennas of each arm using simple three-point Lagrangian interpolation. Given the typical δTEC precision of 3 × 10−4 TECU and the mean separation between antennas of 2.5 km, the precision of these projected TEC gradient measurements is typically about 2 × 10−4 TECU km−1. The time series of the projected gradient at each antenna is plotted in Figure 10. We have also plotted in red the projected gradient computed using the polynomial coefficients plotted in Figure 9. One can see that for the larger amplitude, longer period disturbances, the polynomial fits largely recover the structure observed using the data for individual antennas. However, during the middle of the night, there appears to be a significant amount of smaller-scale structure missed by the polynomial fits that can only be observed using the individual antenna gradients, especially for the shortest baselines near the center of the array.

Figure 10.

The projection of the TEC gradient along each VLA arm (see Figure 1) at each antenna as a function of local time (black curves). Plotted in red are the projected TEC gradients computed using the fitted polynomial coefficients plotted in Figure 9. In each panel, the antenna number and the mean separation among the nearest antennas (Δr) are printed.

4. Discussion

[41] Our exploration of a long, VHF observation of Cyg A with the VLA has successfully demonstrated the power of this instrument to characterize a variety of transient ionospheric phenomena. For this observation, the typical 1σ uncertainty in the δTEC measurements was 3 × 10−4 TECU, yielding more than an order of magnitude better sensitivity to TEC fluctuations than can be achieved with GPS-based relative TEC measurements [see, e.g., Hernández-Pajares et al., 2006].

[42] Large amplitude, long period waves reminiscent of MSTIDs are visible within the δTEC data near dusk and dawn as well as other times intermittently throughout the night. The polynomial-based approach we have detailed in section 3.2 appears to be able to recover the properties of the full two-dimensional TEC gradients associated with these relatively large disturbances as they passed over the array. This information can be used to estimate the size, speed and directions of such patterns down to scales of roughly half the size of the array (∼20 km). This is demonstrated in more detail in a subsequent paper describing the spectral analysis of these data.

[43] In addition, our approach of measuring the projected gradient at each antenna along each arm has shown that there are smaller-scale TEC fluctuations observed throughout the night, most prominently after midnight local time. Thus, the VLA can be used to simultaneously study fine-scale ionospheric dynamics. This may include a host of phenomena such as the small-scale distortions/structure within MSTID wavefronts, turbulent fluctuations from ion-neutral coupling within the lower ionosphere/thermosphere, and sporadic-E (Es) layers. In the case of Es, Coker et al. [2009] demonstrated with a combination of VLA data and optical observations that many of the small-scale fluctuations seen by the VLA during summer nighttime are likely associated with Es layers. Coker et al. [2009] showed that the TEC gradients caused by these layers are typically ≈0.001 TECU km−1 which is easily detectable using the arm-based gradient method. A specialized spectral analysis technique has also been developed for these measurements and will likewise be detailed in the companion manuscript to this paper.

Appendix A:: Geometric Corrections

[44] Two basic geometric corrections must be applied to the antenna positions and δTEC measurements so that they more accurately represent the actual conditions within the ionosphere. Since we have used observations of Cyg A that include times when it is relatively close to the horizon, we cannot use a plane-parallel approximation. Instead, we have used a thin shell approximation for the ionosphere where the shell is located at the height of maximum electron density, zion, as computed by the IRI software for the dates and times of the observations (see section 3.1 and Figure 8). The full spherical corrections used are detailed below.

[45] First, the positions of the antennas on the ground must be converted to projected positions within the ionosphere which, for a non-plane-parallel atmosphere, change with the elevation of the observed source. For a spherical shell, we may define a “pierce-point” for each antenna where its line of sight to the source passes through the ionosphere. The positions of these pierce-points relative to that for the center of the array can then be used as their projected ionosphere positions. Figure A1 provides a schematic illustration (not to scale) of how these positions are determined. We first define a set three position vectors, RA, RPP, and RS, which define the positions of the array center/antenna, the ionosphere pierce-point, and the observed source, respectively, with the center of the earth as the origin of the coordinate system.

Figure A1.

A schematic representation (not to scale) of the procedure detailed here for computing the required geometric corrections. See the text of Appendix A for a detailed discussion of the different components of this schematic. Note, the image used for Cyg A is a false-color image made with the VLA at 327 MHz [see Lazio et al., 2006].

[46] Next, we note that the vast majority of astronomical sources, including Cyg A, are essentially infinitely far away, which implies that the line of sight from the array center/antenna location to the source is essentially parallel to that from the center of the earth to the sources, or RPP − RARS. If we define a “left-handed” coordinate system such that RA points along the z-axis, then the source position is given by

display math

where h is the angular elevation of the source, a is the azimuthal angle measured from north though east, and the inline image and inline image unit vectors point toward the north and east, respectively, as viewed from the array/antenna. Combining this with the assumption of parallel lines of sight to the source and the fact that the length of RPP is set to Rearth + zion yields the following expression

display math

where xA, yA and zA are the coordinates of the antenna relative to the array center and r = ∣RPP − RA∣. Since the antenna positions are known, r is the only unknown variable. Equation (A2) can then be rewritten as a quadratic equation and solved for r keeping in mind that 0 ≤ r < Rearth + zion which allows one to compute the x, y, and z coordinates of RPP for the array center and each antenna in the current coordinate system. Following this, a coordinate rotation was performed such that RPP for the array center pointed along the z-axis and the x and y axes pointed toward north and east, respectively, as viewed from the location on the earth directly below the array center pierce-point. These rotated coordinates were then taken to be the x and y antenna positions projected onto the ionosphere thin shell for each time step. Figure A1 shows a graphical representation of these computations for the array center (in black) and for an exemplar antenna (in gray).

[47] The second correction deals with the fact that the path length through the ionosphere is increased when the observed source is closer to the horizon. For a thin spherical shell, it is increased by a factor of sec(ϵ) where ϵ is the angle between the line of sight from the VLA to the source and a line from the ionosphere pierce-point to the location on the earth directly below it. In the schematic in Figure A1, ϵ is the angle between the position vectors RPP and RPP − RA. Therefore, to compute the factor needed to correct our δTEC measurements, cos(ϵ), we simply computed the dot product between these two vectors and divided by the product of their lengths, r(Rearth + zion).

[48] Finally, while computing the above geometric corrections, we also computed estimates of the apparent motion of Cyg A within the coordinate system of each time step. This was done to estimate the degree of Doppler shifting of the temporal/spatial frequencies of any detected wave phenomena. We did this for each time step by recomputing the position of the array center pierce-point for the two nearest time steps within the coordinate system of the current time step. These positions were then used to numerically compute the time derivatives of the x and y coordinates of the array center pierce-point to obtain the north-south and east-west components of the sidereal velocity. These are plotted in Figure A2 as functions of time along with a histogram for the azimuth angle (measured north through east) of the sidereal velocity vector for the entire observing run. One can see from this figure that the velocities were sometimes significant, especially when the source was at lower elevations. In addition, while the motion is generally from east to west, as one would naively assume, there is a significant spread in position angle of more than 100°.

Figure A2.

(top) The estimated apparent velocity of Cyg A at the location within the thin-shell ionosphere of the array center pierce-point in both the north-south (black) and east-west (gray) directions as functions of time. (bottom) A histogram for the distribution of azimuth angles measured from north through east for the velocity vectors plotted in Figure A2 (top).


[49] The authors would like to thank the referees for useful comments and suggestions. Basic research in astronomy at the Naval Research Laboratory is supported by 6.1 base funding. The VLA was operated by the National Radio Astronomy Observatory which is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.