A detector interferometric calibration experiment for high precision astrometry
A. Crouzier, F. Malbet, F. Henault, A. Leger, C. Cara, J. M. LeDuigou, O. Preis, P. Kern, A. Delboulbe, G. Martin, P. Feautrier, E. Stadler, S. Lafrasse, S. Roc… 展开作者
A. Crouzier, F. Malbet, F. Henault, A. Leger, C. Cara, J. M. LeDuigou, O. Preis, P. Kern, A. Delboulbe, G. Martin, P. Feautrier, E. Stadler, S. Lafrasse, S. Rochat, C. Ketchazo, M. Donati, E. Doumayrou, P. O. Lagage, M. Shao, R. Goullioud, B. Nemati, C. Zhai, E. Behar, S. Potin, M. Saint-Pe, J. Dupont
Context: Exoplanet science has made staggering progress in the last two decades, due to the relentless exploration of new detection methods and refinement of existing ones. Yet astrometry offers a unique and untapped potential of discovery of habitable-zone low-mass planets around all the solar-like stars of the solar neighborhood. To fulfill this goal, astrometry must be paired with high precision calibration of the detector.
Aims: We present a way to calibrate a detector for high accuracy astrometry. An experimental testbed combining an astrometric simulator and an interferometric calibration system is used to validate both the hardware needed for the calibration and the signal processing methods. The objective is an accuracy of 5e-6 pixel on the location of a Nyquist sampled polychromatic point spread function.
Methods: The interferometric calibration system produced modulated Young fringes on the detector. The Young fringes were parametrized as products of time and space dependent functions, based on various pixel parameters. The minimization of func- tion parameters was done iteratively, until convergence was obtained, revealing the pixel information needed for the calibration of astrometric measurements.
Results: The calibration system yielded the pixel positions to an accuracy estimated at 4e-4 pixel. After including the pixel position information, an astrometric accuracy of 6e-5 pixel was obtained, for a PSF motion over more than five pixels. In the static mode (small jitter motion of less than 1e-3 pixel), a photon noise limited precision of 3e-5 pixel was reached.
GRAVITY is the second generation Very Large Telescope Interferometer instrument for precision narrow-angle astrometry and interferometric imaging in the Near Infra-Red (NIR). It shall provide precision astrometry of order 10 microarcseconds, and imaging capability at a few milliarcsecond resolution, and hence will revolutionise dynamical measurements of celestial objects. GRAVITY is currently in the last stages of its integration and tests in Garching at MPE, and will be delivered to the VLT Interferometer (VLTI) in 2015. We present here the instrument, with a particular focus on the component…
Pulsars are special objects whose positions can be determined independently from timing, radio interferometric, and Gaia astrometry at sub-milliarcsecond (mas) precision; thus, they provide a unique way to monitor the link between dynamical and kinematic reference frames. We aimed to assess the orientation consistency between the dynamical reference frame represented by the planetary ephemeris and the kinematic reference frames constructed by Gaia and VLBI through pulsar positions. We identified 49 pulsars in Gaia Data Release 3 and 62 pulsars with very long baseline interferometry (VLBI) posi…
The achievement of $μ$arcsec relative astrometry with ground-based, near infrared, extremely large telescopes requires a significant endeavour of calibration strategies. In this paper we address the removal of instrument optical distortions coming from the ELT first light instrument MICADO and its adaptive optics system MAORY by means of an astrometric calibration mask. The results of the test campaign on a prototype mask (scale 1:2) has probed the manufacturing precision down to $\sim$ 50nm/1mm scale, leading to a relative precision $δσ\sim 5e-5$. The assessed manufacturing precision indicate…
The main objective of this work is to accelerate the Maximum-Likelihood (ML) estimation procedure in radio interferometric calibration. We introduce the OS-LS and the OS-SAGE radio interferometric calibration methods, as a combination of the Ordered-Subsets (OS) method with the Least-Squares (LS) and Space Alternating Generalized Expectation maximization (SAGE) calibration techniques, respectively. The OS algorithm speeds up the ML estimation and achieves nearly the same level of accuracy of solutions as the one obtained by the non-OS methods. We apply the OS-LS and OS-SAGE calibration methods…
During the Hipparcos mission in September 1992, I presented a concept for using direct imaging on CCDs in scanning mode in a new and very powerful astrometric satellite, Roemer. The Roemer concept with larger aperture telescopes for higher accuracy was developed by ESA and a mission was approved in 2000, expected to be a million times better than Hipparcos. The present name Gaia for the mission reminds of an interferometric option also studied in the period 1993-97, and the evolution of optics and detection in this period is the main subject of the present report. The transition from an interf…
We describe a flagging and calibration pipeline intended for making quick look images from GMRT data. The package identifies and flags corrupted visibilities, computes calibration solutions and interpolates these onto the target source. These flagged calibrated visibilities can be directly imaged using any standard imaging package. The pipeline is written in "C" with the most compute intensive algorithms being parallelized using OpenMP.
Radio interferometric gain calibration can be biased by incomplete sky models and radio frequency interference, resulting in calibration artefacts that can restrict the dynamic range of the resulting images. It has been suggested that calibration algorithms employing heavy-tailed likelihood functions are less susceptible to this due to their robustness against outliers in the data. We present an algorithm based on a Student's t-distribution which leverages the framework of complex optimisation and Wirtinger calculus for efficient and robust interferometric gain calibration. We integrate this a…
This paper introduces an amendment to radio interferometric calibration of sources below the noise level. The main idea is to employ the information of the stronger sources' measured signals as a plug-in criterion to solve for the weaker ones. For this purpose, we construct a number of source clusters, with centroids mainly near the strongest sources, assuming that the signals of the sources belonging to a single cluster are corrupted by almost the same errors. Due to this characteristic of clusters, each cluster is calibrated as a single source, using all the coherencies of its sources simult…
Observations of the redshifted 21-cm line from the epoch of reionization have recently motivated the construction of low frequency radio arrays with highly redundant configurations. These configurations provide an alternative calibration strategy - "redundant calibration" - and boosts sensitivity on specific spatial scales. In this paper, we formulate calibration of redundant interferometric arrays as a complex optimization problem. We solve this optimization problem via the Levenberg-Marquardt algorithm. This calibration approach is more robust to initial conditions than current algorithms an…
Residual calibration errors are difficult to predict in interferometric radio polarimetry because they depend on the employed observational calibration strategy, encompassing the Stokes vector of the calibrator and parallactic angle coverage. This work presents analytic derivations and simulations that enable examination of residual on-axis instrumental leakage and position angle errors for a suite of calibration strategies. The focus is on arrays comprising alt-az antennas with common feeds over which parallactic angle is approximately uniform. The results indicate that calibration schemes re…
The data reduction procedure for radio interferometers can be viewed as a combined calibration and imaging problem. We present an algorithm that unifies cross-calibration, self-calibration, and imaging. Being a Bayesian method, that algorithm does not only calculate an estimate of the sky brightness distribution, but also provides an estimate of the joint uncertainty which entails both the uncertainty of the calibration and the one of the actual observation. The algorithm is formulated in the language of information field theory and uses Metric Gaussian Variational Inference (MGVI) as the unde…
Recent developments in optimization theory have extended some traditional algorithms for least-squares optimization of real-valued functions (Gauss-Newton, Levenberg-Marquardt, etc.) into the domain of complex functions of a complex variable. This employs a formalism called the Wirtinger derivative, and derives a full-complex Jacobian counterpart to the conventional real Jacobian. We apply these developments to the problem of radio interferometric gain calibration, and show how the general complex Jacobian formalism, when combined with conventional optimization approaches, yields a whole new f…
In this paper we present DMC, a model and associated tool for polarimetric imaging of very long baseline interferometry datasets that simultaneously reconstructs the full-Stokes emission structure along with the station-based gain and leakage calibration terms. DMC formulates the imaging problem in terms of posterior exploration, which is achieved using Hamiltonian Monte Carlo sampling. The resulting posterior distribution provides a natural quantification of uncertainty in both the image structure and in the data calibration. We run DMC on both synthetic and real datasets, the results of whic…
Heterodyne receivers register the sky signal on either a circular polarization basis (where it is split into left-hand and right-hand circular polarization) or a linear polarization basis (where it is split into horizontal and vertical linear polarization). We study the problem of interferometric observations performed with telescopes that observe on different polarization bases, hence producing visibilities that we call "mixed basis" (i.e., linear in one telescope and circular in the other). We present novel algorithms for the proper calibration and treatment of such interferometric observati…
Pixel-Level Calibration for Space-Based High Precision Astrometry Using Young's Fringes
Institut de Planétologie et d’Astrophysique（行星与天体物理研究所）；PYXALIS；Centre de Recherche Astrophysique de Lyon（里昂天体物理研究中心）；Centre National d’Etudes Spatiales（法国国家空间研究中心）
It has recently been shown that radio interferometric gain calibration can be expressed succinctly in the language of complex optimisation. In addition to providing an elegant framework for further development, it exposes properties of the calibration problem which can be exploited to accelerate traditional non-linear least squares solvers such as Gauss-Newton and Levenberg-Marquardt. We extend existing derivations to chains of Jones terms: products of several gains which model different aberrant effects. In doing so, we find that the useful properties found in the single term case still hold.…
Combination of more than two fields provides constraints on the systematic error of simultaneous observations. The concept is investigated in the context of the Gravitation Astrometric Measurement Experiment (GAME), which aims at measurement of the PPN parameter $γ$ at the $10^{-7}-10^{-8}$ level. Robust self-calibration and control of systematic error is crucial to the achievement of the precision goal. The present work is focused on the concept investigation and practical implementation strategy of systematic error control over four simultaneously observed fields, implementing a "double diff…
Solution intervals are often used to improve the signal-to-noise ratio during radio interferometric gain calibration. This work investigates how factors such as the noise level, intrinsic gain variability, degree of model incompleteness, and the presence of radio frequency interference impact the selection of solution intervals for calibration. We perform different interferometric simulations to demonstrate how these factors, in combination with the choice of solution intervals, affect calibration and imaging outputs and discuss practical guidelines for choosing optimal solution intervals. Fur…
For a ground-based radio interferometer observing at low frequencies, the ionosphere causes propagation delays and refraction of cosmic radio waves which result in phase errors in the received signal. These phase errors can be corrected using a calibration method that assumes a two-dimensional phase screen at a fixed altitude above the surface of the Earth, known as the thin-layer model. Here we investigate the validity of the thin-layer model and provide a simple equation with which users can check when this approximation can be applied to observations for varying time of day, zenith angle, i…
In order to cope with the increased data volumes generated by modern radio interferometers such as LOFAR (Low Frequency Array) or SKA (Square Kilometre Array), fast and efficient calibration algorithms are essential. Traditional radio interferometric calibration is performed using nonlinear optimization techniques such as the Levenberg-Marquardt algorithm in Euclidean space. In this paper, we reformulate radio interferometric calibration as a nonlinear optimization problem on a Riemannian manifold. The reformulated calibration problem is solved using the Riemannian trust-region method. We show…
We have built a reliable and robust system that takes as input an astronomical image, and returns as output the pointing, scale, and orientation of that image (the astrometric calibration or WCS information). The system requires no first guess, and works with the information in the image pixels alone; that is, the problem is a generalization of the "lost in space" problem in which nothing--not even the image scale--is known. After robust source detection is performed in the input image, asterisms (sets of four or five stars) are geometrically hashed and compared to pre-indexed hashes to genera…
Adaptive optics (AO) systems deliver high-resolution images that may be ideal for precisely measuring positions of stars (i.e. astrometry) if the system has stable and well-calibrated geometric optical distortions. A calibration unit, equipped with back-illuminated pinhole mask, can be utilized to measure instrumental optical distortions. AO systems on the largest ground-based telescopes, such as the W. M. Keck Observatory and the Thirty Meter Telescope require pinhole positions known to 20 nm to achieve an astrometric precision of 0.001 of a resolution element. We characterize a photo-lithogr…
The aim of the new generation of radio synthesis arrays such as LOFAR and SKA is to achieve much higher sensitivity, resolution and frequency coverage than what is available now, especially at low frequencies. To accomplish this goal, the accuracy of the calibration techniques used is of considerable importance. Moreover, since these telescopes produce huge amounts of data, speed of convergence of calibration is a major bottleneck. The errors in calibration are due to system noise (sky and instrumental) as well as the estimation errors introduced by the calibration technique itself, which we c…
Calibration of radio interferometric observations becomes increasingly difficult towards lower frequencies. Below ~300 MHz, spatially variant refractions and propagation delays of radio waves traveling through the ionosphere cause phase rotations that can vary significantly with time, viewing direction and antenna location. In this article we present a description and first results of SPAM (Source Peeling and Atmospheric Modeling), a new calibration method that attempts to iteratively solve and correct for ionospheric phase errors. To model the ionosphere, we construct a time-variant, 2-dimens…
Calibration is a key step in the signal processing pipeline of any radio astronomical instrument. The required sky, ionospheric and instrumental models for this step can suffer from various kinds of incompleteness. In this paper we analyze several important calibration methods, ignoring for now the ionosphere. The aim is to use established statistical and signal processing tools to provide a generic method to assess calibratability of an instrument. We show how currently popular calibration techniques differ in their assumptions and also discuss their theoretical commonalities. We also study t…
Astrometry is a powerful technique in astrophysics to measure three-dimensional positions of stars and other astrophysical objects, including exoplanets and the gravitational influence they have on each other. Interferometric astrometry is presented here as just one in a suite of powerful astrometric techniques, which include space-based, seeing-limited and wide-angle adaptive optics techniques. Fundamental limits are discussed, demonstrating that even ground-based techniques have the capability for astrometry at the single micro-arcsecond level, should sufficiently sophisticated instrumentati…
Calibration of a typical radio interferometric array yields thousands of parameters as solutions. These solutions contain valuable information about the systematic errors in the data (ionosphere and beam shape). This information could be reused in calibration to improve the accuracy and also can be fed into imaging to improve the fidelity. We propose a distributed optimization strategy to construct models for the systematic errors in the data using the calibration solutions. We formulate this as an elastic net regularized distributed optimization problem which we solve using the alternating di…
Radio interferometric imaging aims to estimate an unknown sky intensity image from degraded observations, acquired through an antenna array. In the theoretical case of a perfectly calibrated array, it has been shown that solving the corresponding imaging problem by iterative algorithms based on convex optimization and compressive sensing theory can be competitive with classical algorithms such as CLEAN. However, in practice, antenna-based gains are unknown and have to be calibrated. Future radio telescopes, such as the SKA, aim at improving imaging resolution and sensitivity by orders of magni…
We recently proposed the use of consensus optimization as a viable and effective way to improve the quality of calibration of radio interferometric data. We showed that it is possible to obtain far more accurate calibration solutions and also to distribute the compute load across a network of computers by using this technique. A crucial aspect in any consensus optimization problem is the selection of the penalty parameter used in the alternating direction method of multipliers (ADMM) iterations. This affects the convergence speed as well as the accuracy. In this paper, we use the Hessian of th…
Correlation Calibration: A Hybrid Calibration Technique for Radio Interferometric Arrays
McGill University（麦吉尔大学）；Dunlap Institute for Astronomy and Astrophysics, University of Toronto（多伦多大学邓普勒天文学与天体物理研究所）；Perimeter Institute for Theoretical Physics（基础物理研究所）
Distributed radio interferometric calibration based on consensus optimization has been shown to improve the estimation of systematic errors in radio astronomical observations. The intrinsic continuity of systematic errors across frequency is used by a consensus polynomial to penalize traditional calibration. Consensus is achieved via the use of alternating direction method of multipliers (ADMM) algorithm. In this paper, we extend the existing distributed calibration algorithms to use ADMM with an adaptive penalty parameter update. Compared to a fixed penalty, its adaptive update has been shown…
High fidelity radio interferometric data calibration that minimises spurious spectral structure in the calibrated data is essential in astrophysical applications, such as 21 cm cosmology, which rely on knowledge of the relative spectral smoothness of distinct astrophysical emission components to extract the signal of interest. Existing approaches to radio interferometric calibration have been shown to impart spurious spectral structure to the calibrated data if the sky model used to calibrate the data is incomplete. In this paper, we introduce BayesCal: a novel solution to the sky-model incomp…
Interferometric arrays seeking to measure the 21 cm signal from the Epoch of Reionization must contend with overwhelmingly bright emission from foreground sources. Accurate recovery of the 21 cm signal will require precise calibration of the array, and several new avenues for calibration have been pursued in recent years, including methods using redundancy in the antenna configuration. The newly upgraded Phase II of Murchison Widefield Array (MWA) is the first interferometer that has large numbers of redundant baselines while retaining good instantaneous UV-coverage. This array therefore provi…
MICADO is the Multi-AO Imaging Camera for Deep Observations, and it will be one of the first light instruments of the Extremely Large Telescope (ELT). Doing high precision multi-object differential astrometry behind ELT is particularly effective given the increased flux and small diffraction limit. Thanks to its robust design with fixed mirrors and a cryogenic environment, MICADO aims to provide 50 $μ$as absolute differential astrometry (measure star-to-star distances in absolute $μ$as units) over a 53" FoV in the range 1.2-2.5 $μ$m. Tackling high precision astrometry over large FoV requires M…
The arrival of the Square Kilometer Array (SKA) will revitalise all aspects of Very Long Baseline Interferometry (VLBI) astronomy at the lower frequencies. In the last decade there have been huge strides towards routinely achieving high precision VLBI astrometry at frequencies dominated by the tropospheric contributions, most notably at 22GHz, using advanced phase referencing techniques. Nevertheless to increase the capability for high precision astrometric measurements at low radio frequencies (<8GHz) an effective calibration strategy of the systematic ionospheric propagation effects that is …
Distributed calibration based on consensus optimization is a computationally efficient method to calibrate large radio interferometers such as LOFAR and SKA. Calibrating along multiple directions in the sky and removing the bright foreground signal is a crucial step in many science cases in radio interferometry. The residual data contain weak signals of huge scientific interest and of particular concern is the effect of incomplete sky models used in calibration on the residual. In order to study this, we consider the mapping between the input uncalibrated data and the output residual data. We …
Increasing data volumes delivered by a new generation of radio interferometers require computationally efficient and robust calibration algorithms. In this paper, we propose distributed calibration as a way of improving both computational cost as well as robustness in calibration. We exploit the data parallelism across frequency that is inherent in radio astronomical observations that are recorded as multiple channels at different frequencies. Moreover, we also exploit the smoothness of the variation of calibration parameters across frequency. Data parallelism enables us to distribute the comp…
TMT has defined the accuracy to be achieved for both absolute and differential astrometry in its top-level requirements documents. Because of the complexities of different types of astrometric observations, these requirements cannot be used to specify system design parameters directly. The TMT astrometry working group therefore developed detailed astrometry error budgets for a variety of science cases. These error budgets detail how astrometric errors propagate through the calibration, observing and data reduction processes. The budgets need to be condensed into sets of specific requirements t…
This paper investigates the possibility of improving radio interferometric images using an algorithm inspired by an optical method known as "lucky imaging", which would give more weight to the best-calibrated visibilities used to make a given image. A fundamental relationship between the statistics of interferometric calibration solution residuals and those of the image-plane pixels is derived in this paper. This relationship allows us to understand and describe the statistical properties of the residual image. In this framework, the noise-map can be described as the Fourier transform of the c…
Latest-generation spectro-interferometric instruments combine a milliarcsecond angular resolution with spectral capabilities, resulting in an immensely increased information content. Here, I present methodological work and results that illustrate the fundamentally new scientific insights provided by spectro-interferometry with very high spectral dispersion or in multiple line transitions (Brackett and Pfund lines). In addition, I discuss some pitfalls in the interpretation of spectro-interferometric data. In the context of our recent studies on the classical Be stars β CMi and ζ Tau, I present…
We describe a new method which achieves high precision Very Long Baseline Interferometry (VLBI) astrometry in observations at millimeter wavelengths. It combines fast frequency-switching observations, to correct for the dominant non-dispersive tropospheric fluctuations, with slow source-switching observations, for the remaining ionospheric dispersive terms. We call this method Source-Frequency Phase Referencing. Provided that the switching cycles match the properties of the propagation media, one can recover the source astrometry. We present an analytic description of the two-step calibration …
The stars orbiting the compact radio source Sgr A* in the Galactic Centre are precision probes of the gravitational field around the closest massive black hole. In addition to adaptive optics assisted astrometry (with NACO / VLT) and spectroscopy (with SINFONI / VLT, NIRC2 / Keck and GNIRS / Gemini) over three decades, since 2016/2017 we have obtained 30-100 mu-as astrometry with the four-telescope interferometric beam combiner GRAVITY / VLTI reaching a sensitivity of mK = 20 when combining data from one night. We present the simultaneous detection of several stars within the diffraction limit…
We report on a broader evaluation of statistical bootstrap resampling methods as a tool for pixel-level calibration and imaging fidelity assessment in radio interferometry. Pixel-level imaging fidelity assessment is a challenging problem, important for the value it holds in robust scientific interpretation of interferometric images, enhancement of automated pipeline reduction systems needed to broaden the user community for these instruments, and understanding leadingedge direction-dependent calibration and imaging challenges for future telescopes such as the Square Kilometer Array. This new c…
Image quality in mosaicked observations from interferometric radio telescopes is strongly dependent on the accuracy with which the antenna primary beam is calibrated. The next generation of radio telescope arrays such as the Allen Telescope Array (ATA) and the Square Kilometer Array (SKA) have key science goals that involve making large mosaicked observations filled with bright point sources. We present a new method for calibrating the shape of the telescope's mean primary beam that uses the multiple redundant observations of these bright sources in the mosaic. The method has an analytical sol…
In a companion paper, we presented BayesCal, a mathematical formalism for mitigating sky-model incompleteness in interferometric calibration. In this paper, we demonstrate the use of BayesCal to calibrate the degenerate gain parameters of full-Stokes simulated observations with a HERA-like hexagonal close-packed redundant array, for three assumed levels of completeness of the a priori known component of the calibration sky model. We compare the BayesCal calibration solutions to those recovered by calibrating the degenerate gain parameters with only the a priori known component of the calibrati…
Interferometric imaging now achieves angular resolutions as fine as 10 microarcsec, probing scales that are inaccessible to single telescopes. Traditional synthesis imaging methods require calibrated visibilities; however, interferometric calibration is challenging, especially at high frequencies. Nevertheless, most studies present only a single image of their data after a process of "self-calibration," an iterative procedure where the initial image and calibration assumptions can significantly influence the final image. We present a method for efficient interferometric imaging directly using …
Calibration is an essential step in radio interferometric data processing that corrects the data for systematic errors and in addition, subtracts bright foreground interference to reveal weak signals hidden in the residual. These weak and unknown signals are much sought after to reach many science goals but the effect of calibration on such signals is an ever present concern. The main reason for this is the incompleteness of the model used in calibration. Distributed calibration based on consensus optimization has been shown to mitigate the effect due to model incompleteness by calibrating dat…
Radio astronomical imaging using aperture synthesis telescopes requires deconvolution of the point spread function as well as calibration of the instrumental characteristics (primary beam) and foreground (ionospheric/atmospheric) effects. These effects vary in time and also across the field of view, resulting in directionally-dependent (DD), time-varying gains. The primary beam will deviate from the theoretical estimate in real cases at levels that will limit the dynamic range of images if left uncorrected. Ionospheric electron density variations cause time and position variable refraction of …
We report on the successful science verification phase of a new observing mode at the Keck interferometer, which provides a line-spread function width and sampling of 150km/s at K'-band, at a current limiting magnitude of K'~7mag with spatial resolution of lam/2B ~2.7mas and a measured differential phase stability of unprecedented precision (3mrad at K=5mag, which represents 3uas on sky or a centroiding precision of 10^-3). The scientific potential of this mode is demonstrated by the presented observations of the circumstellar disk of the evolved Be-star 48Lib. In addition to indirect methods …
Precise centroid estimation plays a critical role in accurate astrometry using telescope images. Conventional centroid estimation fits a template point spread function (PSF) to the image data. Because the PSF is typically not known to high accuracy due to wavefront aberrations and uncertainties in optical system, a simple Gaussian function is commonly used. PSF knowledge error leads to systematic errors in the conventional centroid estimation. In this paper, we present an accurate centroid estimation algorithm by reconstructing the PSF from well sampled (above Nyquist frequency) pixelated imag…
ASTRON, The Netherlands Institute for Radio Astronomy（荷兰射电天文研究所（ASTRON））；Netherlands eScience Center（荷兰电子科学中心）
We present a stochastic, limited-memory Broyden Fletcher Goldfarb Shanno (LBFGS) algorithm that is suitable for handling very large amounts of data. A direct application of this algorithm is radio interferometric calibration of raw data at fine time and frequency resolution. Almost all existing radio interferometric calibration algorithms assume that it is possible to fit the dataset being calibrated into memory. Therefore, the raw data is averaged in time and frequency to reduce its size by many orders of magnitude before calibration is performed. However, this averaging is detrimental for th…
A Formalism for Calibrating the Instrumental Polarization of Radio Interferometric Arrays at Meter Wavelengths using Unpolarized Sky: A Demonstration using the MWA Observations
