We present complementary methods for calibrating the dual-polarization feed of a ground-based tracking antenna used for ionospheric measurements. Several calibration parameters are defined which describe the state of the measurement equipment. These key parameters then form a transformational Mueller matrix which may be used to remove system bias in data received from known beacons. Several strategies are developed to quantify these time-dependent distortions for the Green Bank 140′ diameter antenna and receiver, although these methods could be applied to other similar systems. One approach quantifies the receiver biases in terms of amplitude and gain differentials. This is accomplished by measuring polarimetric differences between channels using a test signal with a known amplitude and phase. Due to variations over time periods on the scale of a beacon track, this procedure is most effective when the calibration signal is injected concurrently with the beacon measurements. To determine antenna feed distortion, including cross-polarization and ellipticity, data is recorded from an external celestial source. An example shows the effectiveness of this calibration technique by comparing calibrated data to data without correction. Error estimates of the calibration parameters in the example establish an upper bound of 0.22 TEC units for measurements at 150 MHz along track.
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 Observations of beacon signals emitted from satellites have been performed for many years. These investigations have used a variety of transmission frequencies, polarizations, and platforms to characterize and measure trans-ionospheric propagation. Signals from satellite beacons can provide information on the propagation and scattering characteristics of the ionospheric path between the satellite borne transmitter and the receiving ground station. Near energy sources created by sharp density gradients and externally imposed electric fields, the propagation can become quite complex including significant changes due to diffraction [Yeh and Liu, 1982], multipath [Evans and Wand, 1983], and magnetoionic/depolarization effects [Lee et al., 1982; Lee and Klobuchar, 1988]. These responses have both direct and indirect dependence on the characteristic length and time scale of the ionospheric plasma. Fundamental processes which can be remotely sensed include ionospheric irregularities, coherent plasma waves, and traveling ionospheric disturbances.
 Both radio frequency applications systems and science observations depend on a detailed knowledge of ionospheric propagation effects. In particular, propagation and scattering anomalies have the potential to corrupt the information content of trans-ionospheric communications and radio-astronomical signals by broadening spectral distributions and causing amplitude and phase fluctuations. Emerging radio astronomy large aperture, low frequency arrays such as the Murchison Widefield Array require the active tracking and removal of ionospheric polarization effects for successful astronomical observations of weak sources and for measurement of the background heliospheric environment. For instance, ionospheric Faraday rotation effects can vary from a rotation measure of ∼0.5 radians/m2 at solar minimum to 5 radians/m2 at solar maximum [Salah et al., 2005]. These correspond to TEC values between ∼5 and ∼50 TEC units (=10^16 electrons/m^2) and corresponding Faraday angular rotations of ∼115 to ∼1150 degrees at 150 MHz frequency using typical midlatitude background magnetic field values. By comparison, coronal mass ejection simulations constrained by Pioneer 6, Pioneer 9, and Helios spacecraft data collected during a wide range of disturbances show typical heliospheric Faraday rotation levels of only ∼0.05 radians/m2 (11 degrees of rotation at 150 MHz) or less, or ∼0.5 TEC units [Jensen et al., 2010]. With a well known satellite beacon and calibrated antenna, it is possible to determine this polarization rotation to high accuracy.
 Satellite beacon measurements with high spatial resolution and signal-to-noise ratio using tracking capable high gain antenna systems provide an attractive technique for precisely quantifying first and second order amplitude, phase, and polarization effects. Measurements in these configurations provide a means of observing these signals with near horizon-to-horizon visibility, without the relatively complex analysis techniques and potentially incorrect assumptions necessary when using conventional low gain receiving systems and relatively simplistic calibration models [e.g., van Velthoven et al., 1990]. However, experiments of this nature are rare. One of the only published accounts is found in the works of Evans and Wand  and Evans et al. , who describe UHF experiments conducted from 1969 to 1973 with the 84 foot Millstone tracking antenna as part of a precision propagation study. The work observed the angular bias refraction, amplitude scintillation, and less common angular scintillation of Navy TRANSIT UHF beacon signals from geophysical effects such as traveling ionospheric disturbances. These previous attempts were conducted in an era where little information was available on the mesoscale ionospheric plasma state. However, this contextual information can now be collected in a relatively straightforward manner using modern distributed space environment sensors. In particular, the combined techniques of total electron content (TEC) mapping using the existing GPS receiver network [Rideout and Coster, 2006], high cadence line of sight GPS TEC, and incoherent scatter radar detailed probing of the plasma state [Evans, 1969] provide a spatial and temporal view of generative structures such as large scale electron density gradients and perturbations. These efforts are a natural complement to the precise observations afforded by trans-ionospheric beacon data.
 The midlatitude location of the National Radio Astronomy Observatory facilities at Green Bank, West Virginia (38.43 degrees N latitude/79.83 degrees W longitude) provides an excellent opportunity to observe both disturbed and quiet time ionospheric variations on trans-ionospheric signals within the inner plasmasphere in a region characterized by strong magnetosphere-ionosphere coupling. Several satellites with orbiting dual-frequency CW coherent beacon transmitters are available, including DMSP F15 (840 km nominal altitude; 98.7 deg. inclination) and RADCAL (749–882 km nominal altitude; 89.5 deg. inclination). Transmitting at VHF (150.012 MHz) and UHF (400.032 MHz), the characteristics of these beacons are similar to the geodetic U.S. Navy TRANSIT and ATS series geosynchronous beacon satellites used for previous work [e.g., Evans and Wand, 1983]. Recently, several current satellites such as the FORMOSAT-3/COSMIC constellation and the Communication/Navigation Outage Forecasting System (CNOFS) platform have been equipped with beacon transmitters using the Coherent Electromagnetic Radio Tomography (CERTO) dual VHF/UHF highly phase coherent beacon at these same frequencies [Bernhardt and Siefring, 2006]. The CERTO instrument is due to fly as well on upcoming sounding rocket and satellite launches such as the Canadian CASSIOPE/Enhanced Polar Outflow Platform (ePOP) satellite. The CERTO system uses several different versions of transmission antennas with different radiated polarization characteristics, including helical, crossed-dipole, and monopole designs.
 In general, making precise ionospheric measurements using satellite beacon transmissions along with a tracked, high gain polarimetric receiving system is a challenging task, as a very detailed time dependent system knowledge is required. Maintaining high sensitivity levels capable of detecting very small fluctuations while retaining precision is also difficult. Proper instrument calibration becomes critical since inherent biases in the system can distort the data and skew results. Channel-to-channel amplitude and phase variations alter the received polarization. Compensating for these variations requires temporally sensitive characterization of the receiving system. Furthermore, since this experiment is measuring polarization rotation using discrete fixed elements, the relational positions of the transmitting and receiving components must be established for the duration of the track. The common frame of reference is defined with time-varying coordinate transformations, since the source on the orbiting platform is moving (often rapidly) relative to the measurement device. The feed elements of the moving antenna are also changing orientation during beacon tracking, adding the issue of feed alignment to the complexity of the problem.
 We report in this paper on results of recent efforts to obtain calibration parameters for high gain, dual polarization antenna systems as a crucial step in a sensitive satellite beacon based ionospheric diagnostic with full polarization, amplitude, and phase capability. This work was done as a joint collaboration between MIT Lincoln Laboratory (MIT LL), the Atmospheric Sciences Group at MIT Haystack Observatory, and the National Radio Astronomy Observatory (NRAO). In the following sections, the high precision instruments used in the experiments will be discussed, along with a consideration of their time-dependent relational motion. The methods used to calibrate these instruments and remove bias will then be outlined. The calibrations parameters are then generated by using a collection of beacon data with a test signal, and by measuring broadband radiation emitted from a celestial source. The derived parameters are confirmed by comparison with other related measurements of the instruments and components under investigation. An example is presented using calibration parameters to collected beacon data which illustrates the effectiveness of the technique.
 NRAO′s 140′ diameter antenna is part of an extensive assortment of radio telescopes in Green Bank, West Virginia. The largest equatorial mount telescope in the world, it has been instrumental in enabling a variety of major discoveries in astronomy [e.g., Lovas et al., 1979; Snyder, 1997]. The site was vacated as a user facility in 1999, but was re-commissioned in 2004 by MIT LL, and the 140′ parabolic dish was restored to operation. During the refurbishing phase, the controls were automated, including the antenna drives, the RF switching and monitoring, and the data recording, such that much of the operation of the dish could be executed remotely over a network. This extensive modernization effort is further detailed in (G. I. Langston, manuscript in preparation, 2011).
 The feed attached at the front end of the receiver is a wide-band dual-polarized feed developed at Chalmers University [Olsson et al., 2006] (Figure 1). This unique arrangement of dipole elements provides the desired reflector illumination over a range of frequencies while maintaining a frequency invariant phase center. It has been characterized at several frequencies by measuring radiation patterns at a compact test range at MIT LL and NRAO.
 The received vertical and horizontal signals are amplified in the receiver feed box attached to the feed by a set of low noise amplifiers (LNAs) in a dewar cooled to 77 K. Approximately 330 feet of low loss RF cable is required to traverse the distance from the feed box to the base of the antenna. An additional 200 feet of cable connects this line to the adjacent trailer which houses the receiver equipment and digital recording system.
 The basic receiver architecture, illustrated in Figure 2, provides high dynamic range while also allowing a wide frequency selection. After splitting and channelization, four data streams are digitized and recorded, two vertical and two horizontal. These individual channels may be tuned independently, but for this study the vertical and horizontal channels were paired in frequency. One pair was typically tuned near a particular beacon frequency, 150 MHz; the other near 400 MHz. Phase commonality is maintained for each vertical/horizontal pair by using a single local oscillator for the necessary frequency upconversions and downconversions. One additional feature essential to the characterization of the system is a test source which may be introduced to the receiver channels at the feed preceding the LNAs.
 In this paper we focus on multiple techniques used to calibrate this equipment to high precision, while a future article (P. J. Erickson et al., Satellite beacon based ionospheric diagnostics of the solar minimum mid-latitude ionosphere, manuscript in preparation, 2011) will detail results of application of the methods here for high sensitivity determination of ionospheric variations at midlatitudes.
3. Geometry and Receiver Orientation
 There are several challenges in characterizing the Faraday rotation of a polarized beacon signal from a ground-based receiver. From the point of view of the receiver, the signal is a distant but rapidly moving source which fluctuates due to the earth's changing atmosphere. One crucial step is establishing an appropriate frame of reference and coordinate system common to all the essential elements of the problem. Multiple geometric transformations are necessary (Figure 3) to move from the polarized radiator (orbiting source) to the polarized feed (ground receiver). The radiator's position and orientation change relative to the receiver during a single collection. Therefore the transformation matrices are inherently time-variant.
 To simplify, we can take advantage of several known factors: first, the satellite is 3-axis stable within its orbit, as is the case for the functioning RADCAL and DMSP satellites. Furthermore, the beacon's radiating antenna is stationary with respect to the satellite's body and with a given orientation. For example, on the DMSP F15 satellite, the monopole is mounted in the forward−looking (ram) direction along the orbital path, while the RADCAL helical antenna is at nadir. We point out furthermore that, although the RADCAL antenna is a right hand circularly polarized antenna when viewed at boresight, at any other observation angle the effective polarization is elliptical and therefore contains a significant linearly polarized component which will experience Faraday rotation.
 Regarding the 140′ antenna and receiving system at the Green Bank site, we carefully orient the feed's dipole elements during installation such that they lie along the east-west/north-south spars of the antenna running from the edges of the dish to the feed box (Figure 4). Additionally, we ensure that the parallactic angle of the feed [Heiles et al., 2001] does not change by fixing the feed box to its housing, preventing any mechanical rotation with respect to the antenna. Therefore an additional rotation matrix (used for alt-az telescopes) is unnecessary. The axes of rotation established by the equatorial mount, then, make it possible to relate the known dipole orientation directly to an earth-centered frame of reference (Figure 5), allowing us to bypass the site-specific coordinate frame. An orbit propagation routine, such as SGP4 [Hoots and Roehrich, 1980], is then used to generate earth-centered orbital positions for the object of interest, and therefore link the radiation source antenna orientation to the receiving dipole orientation.
4. Description of Calibration Parameters
 While the receiver system installed at the Green Bank site allows flexibility in terms of frequency selection and provides high dynamic range, its complexity leads to a certain amount of signal distortion present in all channels. For the most part, the sources of these non-idealities are unavoidable and may not be easily calibrated. For example, the various system components, such as filters, mixers, amplifiers and attenuators, do not have identical frequency responses. Furthermore, aside from the fact that the cable lengths for parallel channels are never precisely equal, the amount of distortion varies channel-to-channel, preventing the establishment of a baseline from differential measurements. The environment and local conditions can also play a significant role in the biases of the system. The local temperature often changes over the recording time, leading to variations in cable electrical lengths and component response. Likewise, antenna motion can translate to stresses on the cables linking the feed to the recording system, creating some additional signal distortion.
 The sources for error are combined into several common parameters which are then used to form a system transformation matrix. Commonly referred to as the Mueller matrix, M, this transfer function is typically expressed in terms of the four Stokes' parameters (I, Q, U, V):
 For a dual-linearly polarized feed, the Mueller matrix may be simplified to [Heiles et al., 2001]:
where ΔG is the gain amplitude differential between the vertical/horizontal channels, and ψ is the phase differential. The cross-polarization coupling between the two dipole elements at the feed are described by ɛ (amplitude) and ϕ (phase). The parameter δα represents the ellipticity angle of the beam formed by the feed. It may be expressed as a ratio of the semi-major axis to semi-minor axis of the polarization ellipse describing a field's propagation.
 While cross-polarization and ellipticity are caused solely by the feed, gain and phase differences may be attributed to the distortions in the cables and components of the receiver system. This separation of parameters to individual sub-systems simplifies the calibration measurement approach. In this particular case, it is possible to use two different methods for determining the three parameters which characterize the feed (ɛ, ϕ, and δα), and the two parameters arising from the receiver system (ΔG and ψ).
4.1. Gain and Phase Difference (ΔG, ψ)
 The gain and phase differences may be determined by injecting a calibrated signal of known frequency and amplitude into the receiver system immediately following the feed (Figure 6). This test signal is common to both the vertical and horizontal channels. The recorded gain differential (ΔG = GV − GH) between the two channels is then directly measured at or near the frequencies of interest.
 The phase differential between the two channels is determined using a test signal swept over a series of frequencies. The electrical path length difference, dV – dH, is:
where f is frequency, c is the propagation velocity in the cables, and the measured phase difference is ψV − ψH. Note that for this particular measurement, the frequency step size must be small enough to avoid phase-wrapping. The relation in (5) assumes the cables are without dispersion, which is valid since the signal is narrowband, and Doppler shift variations are small (less than 50 kHz).
4.2. Cross-Polarization and Beam Ellipticity (ɛ, ϕ, δα)
 The Mueller matrix calibration parameters for the antenna's feed include the cross-polarization (amplitude and phase) and ellipticity. It is challenging to use a calibrated test source to determine the installed feed since local external signals are difficult to introduce within reasonable proximity of the feed at the site. It may be possible to place a signal source at the center of the antenna, which then directly illuminates the feed, but several sources of error limit the effectiveness of this technique. One issue is that the source will not be in the far field, so quantifying the fields at the feed will be difficult. Even more challenging to overcome is the effect of multipath, which can be significant for ground-based sources near a sensitive receiver such as the one at Green Bank.
 One way to avoid these issues is to measure the feed at a range in an isolated test chamber. This has in fact been done for the Chalmers University feed at MIT Lincoln Laboratory. However, these measurements, while useful in providing a rough indication of cross-polarization and ellipticity, are taken in an ideal environment, one that is not an accurate representation of the changing external conditions at site. A more physical representation of the calibration parameters will be time-varying, dependent on conditions at Green Bank.
 A better method for determining the cross-polarization parameters at site is to take advantage of the simplifications which occur when measuring radiation generated from a randomly polarized astronomical source (Figure 7). In this case, the input signal, , is represented as:
 From (1) and (4), the received signal then isolates the calibration parameters:
 The cross-polarization parameters of the feed may be extracted from the two Stokes' parameters Uout and Vout, which are 90° out of phase. The amplitude, ɛ, of cross-coupled polarization is simply:
while the combined phase ϕ + ψ is
 The cross-polarization phase between the vertical and horizontal dipole elements at the feed (ϕ) may be separated from the system phase distortion by directly measuring ψ using test signals at the feed as described earlier.
 The remaining feed parameter, beam ellipticity angle (δα), is a measure of the inequality in response between the two polarization elements of the feed. An ideal linear feed has an ellipticity angle of 0 or 90 degrees. If the receive strength of one element is different from the other, the resulting bias creates an additional ellipticity, making the feed slightly nonlinear. This parameter is not measured using the above approach. However, for this particular feed the ellipticity angle is assumed to be very small (<1 degree). Range measurements of the feed's ellipticity taken at NRAO (Figure 8) show this to be a valid assumption for the main portion of the beam.
5. Calibration Results
 As an example of the detailed calibration techniques, recent measurements of RADCAL and DMSP F15 beacons were collected using the Green Bank 140′ antenna and receiver system. We present in this section a calibration for precise removal of system biases through determination of the Mueller matrix parameters.
5.1. Gain and Phase Difference (ΔG, ψ)
 The gain and phase difference calibration parameters were determined initially using test signals near the frequencies of interest which, for the CERTO beacons, are 150.012 MHz and 400.032 MHz. A single test signal generator and a splitter fed common inputs to both the vertical and horizontal channels. Phase differences were measured on several different days (Figure 9), where the slopes represent the physical delay lengths between the two channels. Generally, the slopes near 400 MHz averaged to around 0.04 radians/MHz (dV − dH ∼2 m). It is interesting to note that the same quantity was measured at Arecibo (a system significantly different from the one at Green Bank) as approximately 0.1 radians/MHz, or dV − dH ∼5 m [Heiles, 2001].
 The results demonstrated that the day-to-day variations in phase and phase slopes were large, implying that the local environment changed the bias of the system. Due to the magnitude of these variations, it was decided to quantify the phase differences on the day of the collection. However, as the results in Figure 10 demonstrate, the variations over this smaller time scale are still significant, with electrical length variations as much as 0.004 radians/MHz (0.2 m). Quantifying the geophysical effects of interest at a scale equivalent to Jensen et al.'s  heliospheric data would require Faraday rotation measurement accuracies within 0.05 radians/m2, meaning the phase difference errors must be limited to below 0.2 radians (11 degrees) near 150 MHz, or less than 0.03 radians (1.6 degrees) at 400 MHz. Since the measurements on longer time scales did not reach this goal, the calibration process was modified to improve accuracy by recording the parameters concurrently with the beacon signal collections at a rate of 10 Hz.
 The system hardware was modified to couple in a test signal after the feed with the reception of a signal from the feed's dipoles on the same cable. By taking advantage of the wide-band architecture of the recording system, it was possible to record a beacon signal and a test signal at the same time, where the test signal is offset slightly from the frequency of interest. This offset is set in the range of +6 to +12 MHz from the beacon center frequency, far enough to avoid interference from any amount of Doppler shift in the beacon's received signal, but close enough to minimize possible dispersion effects in the transmission line. This method was then applied for a series of data collections of the course of a year, focusing on two particular beacons, DMSP F15 and RADCAL. The test signal was applied at a single frequency, although a swept frequency scheme over a small bandwidth was also used. An example of a beacon track using a calibration frequency at a single offset frequency is shown in Figure 11.
 The hardware modification enabling the simultaneous injection of a test signal allowed on-time generation of calibration parameters while beacon data is collected. Figure 12 shows the gain and phase differentials of the system measured during a beacon track. The test signal was frequency swept over 1 MHz centered at 412 MHz. The gain difference did not show repeatability for each sweep, suggesting the two channels were changing over time. The phase differential measurement exhibited a more stable characteristic over each frequency sweep within the collection time (Figure 13). In the case of the calibrations near 150 MHz, the phase difference was often measured with variations less than ± 0.33 degrees, an improvement by almost a factor of 10 compared to the phase difference measurements in Figure 10. Furthermore, the frequency dependency of this phase difference variation matched the frequency response of the surface acoustic wave (SAW) filters in the receiver system as reported by the manufacturer [Sawtek Company, 2003]. This lends a high degree of confidence in the technique's ability to measure phase variation within the required precision.
5.2. Cross-Polarization and Beam Ellipticity (ɛ, ϕ, δα)
 As mentioned previously, the feed calibration parameters, namely cross-polarization amplitude (ɛ), phase (ϕ), and beam ellipticity angle (δα), are difficult to determine using an external test source. One approach is to use a randomly polarized celestial source, such as produced by Cygnus, Taurus, or Cassiopeia. Solar radiation also has mixed polarization, although this possibility was ultimately rejected due to the fact it is not a point source but distributed over the receiving beam produced by the Green Bank 140′ antenna.
 The feed parameters, ɛ and ϕ, which represent the amplitude and phase cross-polarization coupling between the vertical and horizontal dipole elements, were determined using Cassiopeia A (Figure 14). For this particular measurement, the antenna was pointed on-source for three minutes. For comparison, data was also collected during times when the antenna was off-source. The tuning of the receiving system was set at 150 MHz for one vertical/horizontal channel pair and 400 MHz for the other, with a single sub-band frequency bandwidth of around 1.5 MHz. The results averaged over the sub-band showed a cross-polarization of −17 dB to −18 dB ± 0.2 dB for each channel pair. The amount of cross-polarization of the feed determined independently by measurements at NRAO's external range was as high as −20 dB near both frequencies, although the degree of precision was limited due to multipath. It should be noted that the off-source data yielded similar results as the unpolarized content collected from the source, but showed more variation. The corresponding phase measurement, representing the combined feed and system phase difference ϕ + ψ, was also calculated with high precision (±5°). The parameters can be de-coupled since an averaged system phase difference has been determined using a test signal (Figure 13).
 An important caveat in determining calibration parameters from unpolarized external sources is that, unlike the method using a directly coupled test signal, the calibration parameters cannot be measured simultaneous to the beacon signal. Therefore, to minimize the temporality of these biases, the measurements must be taken as close to the beacon signal collection as possible.
5.3. Calibration Example
 The previous sections derived calibration parameters for the Green Bank 140′ system necessary for determining ionospheric variations from observing Faraday rotation. The application of those system corrections to the data is demonstrated in this section. For this example, data was collected for DMSP F15's CERTO beacon for approximately 400 s. One vertical/horizontal channel pair of the receiver system was tuned near 150 MHz, and the other near 400 MHz. Concurrently, a calibration signal was injected at 156 MHz, and one at 412 MHz, as illustrated in Figure 11. The receiver gain and phase difference calibration parameters (ΔG and ψ) were generated every 0.1 s during the continuous collection. The feed calibration parameters describing cross-polarization amplitude and phase (ɛ and ϕ) were determined based on previously recorded observations of a randomly polarized celestial source, in this case Cassiopeia A. The final feed calibration parameter concerning beam ellipticity bias (δα) is, as discussed earlier, assumed to be zero in this case.
 Assuming it is fixed in the forward-looking (ramming) direction of the satellite, the CERTO beacon's orientation can be determined relative to the orientation of Green Bank's vertical and horizontal dipole elements. Figure 15 shows the direction of the beacon at a particular time during the data collection using the vertical and horizontal dipole orientations as the axes in two dimensions.
 The beacon data may be expressed as either vertical and horizontal field amplitudes and phases or the four Stokes' parameters. In the most general case, the fields are elliptically polarized. The orientation of the received field's ellipse, Ψe, is the angle of the semi-major axis measured counter-clockwise from the horizontal axis. The ellipticity angle α is defined as the arctangent of the semi-minor axis to semi-major axis ratio. These parameters are determined directly from the Stokes' parameters [Goldstein and Collett, 2003]:
 The polarization parameters of the ellipse are typically different for the calibrated data from the uncorrected data (out ≠ in). An illustration of the received beacon data expressed in terms of the electric field and polarization ellipse is shown in Figure 15 for both the uncorrected case and the case using the system calibration parameters applied at a particular time during the collection period.
 The Faraday rotation angle is defined here as the angle between beacon orientation and received polarization. Figure 16 shows an example of how that angle changes over time during a single beacon track. It should be noted that this particular track was at a relatively low elevation (∼25 degrees). Applying the calibration parameters to the raw data reduces the amount of variation (representing precision) to a large degree. The parameter with the largest impact in terms of data improvement is the phase difference between vertical/horizontal channels, ψ.
 The sources of error in the Faraday rotation angle measurement come from error in the system calibration parameters and error in the raw data. The errors from the calibration parameters can be calculated, and are represented in the figure as black error bars. Note these errors are generally very small, indicating the receiving system has been well characterized. The remaining variation of the corrected data is shown by the much larger blue error bars. This total variation (minus the small calibration error) represents other uncharacterized sources of error in the data, as well as real geophysical effects. Using an appropriate geometric projection onto the terrestrial magnetic field as provided by the International Geophysical Reference Field model [International Association of Geomagnetism and Aeronomy, Working Group V-MOD, 2010], we estimate that the total geophysically driven rotation angle variation along track has an upper bound of approximately 0.22 TEC units at 150 MHz.
 It is useful to compare this result with earlier midlatitude measurements made by the Los Alamos VHF interferometer system at 136 MHz by Hoogeveen and Jacobson . That work estimates a solar minimum quiet mean carrier phase fluctuation which corresponds to approximately 0.08–0.16 TEC units out to plasmaspheric altitude (∼20,000 km). These values depend on the exact path through the ionosphere along with seasonal and solar cycle effects, and are likely lower than our upper bound estimate due to sources of uncharacterized error in the Green Bank system. Finally, the angular variability is at or less than the 11 degree rotation level, approximately 10 percent of total ionospheric Faraday rotation contribution and a value comparable to Jensen et al.'s  heliospheric data. Thus, although the results presented here are from beacon paths which are ionospheric in nature, the Green Bank system calibration provides a measurement sensitive enough to determine large scale heliospheric variations for radioastronomical applications using polarized celestial sources.
 The calibration parameters of the Green Bank 140′ dual-polarized feed and receiver were determined using several methods. For the gain and phase differentials, a common test signal was used which was introduced at the feed. The calibration data was taken concurrently with the beacon data at an offset frequency, measuring the system's vertical/horizontal amplitude differences to ± 0.26 dB and phase difference variations less than ± 0.33 degrees. The non-idealities at the feed were determined by collecting data from a celestial source emitting broadband randomly polarized radiation. The amplitude of cross-polarization was between −17 dB and −18 dB (± 0.2 dB), a result which has been confirmed by range measurements. The remaining system distortion mechanism, namely the feed's ellipticity, δα, was shown to be even smaller, and therefore elided from the calibration process. The current approach would require modification to quantify this effect.
 The resulting Mueller matrix formed by these calibration parameters can be used to remove the biases of the system. An example using calibration parameters derived during collection data from the CERTO beacon on DMSP F15 shows a significant increase in measurement precision. This example was useful in that it allowed us to place an upper bound of 0.22 TEC units at 150 MHz. To achieve even greater precision, other approaches would likely be required, such as one that attempts to describe all system biases simultaneously at each time interval of recorded data.
 By applying these calibration techniques, the removal of measurement errors will allow us to quantify geophysical parameters, in this case propagation effects due to the ionosphere, to very high precision. This ability to obtain accurate measurements of ionospheric effects is important for future solar and heliospheric measurements, such as those taken for the MWA project. Finally, although the results shown in this paper demonstrate successful calibration for the Green Bank 140′ system specifically, the approach is general enough that it should work for other high precision instruments in radioastronomy as well.
 The authors would like to thank Pete Chestnut and Gary Anderson at NRAO for providing invaluable operational and site support. The system's hardware and software were maintained by Bill Beavers, Nick Mosher, and Sarah Welch at MIT LL. Michael Shields at MIT LL also provided essential information on the Chalmers University wideband feed, including range measurements. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Incorporated. Analysis at MIT Haystack Observatory was supported under a subcontract to MIT Lincoln Laboratory. Disclaimer: This work was sponsored by the Department of the Air Force under Air Force contract FA8721-05-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the United States Government.