Generalized redundant calibration of radio interferometers

Redundant calibration is a technique in radio astronomy that allows calibration of radio arrays whose antennas lie on a lattice by exploiting the fact that redundant baselines should see the same sky signal. Because the number of measured visibilities scales quadratically with the number of antennas but the number of unknowns describing the individual antenna responses and the available information about the sky scales only linearly with the array size, the problem is always overconstrained as long as the array is big and dense enough. This is true even for nonlattice array configurations. In this work we study a generalized algorithm in which a per-antenna gain is replaced with a number of gains. We show that it can successfully fit data from an approximately redundant array on square lattice with pointing and geometry errors, but that the model’s parameters are difficult to link to the quantities of interest. We discuss the parametrization, limitations, and possible extensions of this algorithm.

Figure 1

Illustration of the pixelization scheme for $M=1$. Labeled points illustrate the lattice points on the $uv$-plane for a square close-packed array. For example, (1,0) corresponds to baselines one lattice spacing apart in the E-W direction and (0,1) is the same for N-S baselines. (0,0) is the origin of the $uv$-plane and corresponds to single-dish observations. Individual beam is described by $3\times 3$ pixelized grid, so $(\tilde{B}⊛{\tilde{B}}^{†})$ is the $5\times 5$ pixelized grid shown in blue around (1,0). Making prediction for a particular (1,0) baseline amounts to multiplying the values of the blue beam-convolved map on the blue grid and the corresponding points in the $uv$-plane and summing them up. See text for further discussion.

Figure 2

Redundancy ratio as a function of $N_a$ with varying $M$. Solid black line is $r=1$ and dashed black line is $r=5$ corresponding to significant redundancy.

Figure 3

Radius response function with $R=0.4L$ and $\mathrm{taper}=0.05L$.

Figure 4

2D visualization of the beam $\tilde{B}$ used on the left and its Fourier transform, the actual primary beam $B$, on the right. The left image spans the $uv$-plane while the right is in the x-y plane. In the limit of no taper, the beam on the right would be an Airy disk pattern.

Figure 5

Example image of a $4\times 4$ array in $uv$ space with a perfectly redundant system on the left and a perturbed version of the same systems on the right. The top images are generated using $M=30$ and bottom are the same system but downscaled to $M=1$ resolution. The complex gain of each pixel in the telescope dish is turned into an RGB value for this image with the color bar referring to the overall amplitude ranging from 0 to slightly greater than 1. The perturbations for the right-side graphs were generated with geometric errors at ${\sigma }_g=0.1L$ and pointing errors at ${\sigma }_p=0.5$. Pointing errors which lead to phase changes are modeled as a gradient on each beam and geometry errors are modeled as circle offsets.

Figure 6

Histogram comparing the magnitude of observed visibilities generated between the two methods. There are 100 discrete sources used to model the sky. The overall statistics of the two are comparable, alleviating concern about the sky simulation with $M$.

Figure 7

Comparison of our algorithm (green) against perfect knowledge of beam shape (orange) and also against an assumption of a perfectly redundant array (blue). Blue and orange have fixed beams, at the prior and truth, respectively, and can only vary the visibilities and overall gain factors. As such, these two methods are forms of standard redundant calibration. Our algorithm solves using $M=1$ and the data are generated with $M=1$, meaning that the model is a perfect description of the data. The top row corresponds to a weak departure from perfect redundancy with the geometric and pointing errors set to 0.01, while the bottom row is for relatively strong departure from redundancy with geometric and pointing errors set to 0.1. The plots in the first column show ${\chi }^2$ for each model to see if we can recreate the data. The purple dashed box is the 5% percentile to 95% percentile bound of the expected ${\chi }^2$ for generalized redundant calibration with no regularization. ${\chi }^2$ below this box is indicative of overfitting. The second column plots the variance between the solved and true visibilities, ${\sigma }_V$, to see if we are accurately recovering the true sky—since we are generating and solving using $M=1$ we can compare the solved to true values directly. The third column plots the variance between the solved and true beam shapes, labeled ${\mathrm{\Sigma }}_B$, to see if we recover the correct array. As the blue and orange lines use a fixed beam shape (the beam prior and true beams, respectively), their values for ${\mathrm{\Sigma }}_B$ are constant. The fourth column plots the variance between the solved and prior beam shape, ${\sigma }_B$ from Eq. (17).

Figure 8

Same as Fig. 8 but with data generated using $M=14$ which is essentially indistiguishable from the continuous model. The data were still fitted using $M=1$ model, which is now an approximate model. For ${\sigma }_v$, the central pixels from the fitted $uv$ data have been compared with the averaged $uv$-plane pixels from the corresponding area from the model representing the truth. See text and caption to Fig. 8 for complete description.