## 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

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 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(−2*πiw*) 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

or,

where *N*_{e} 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.