Redundant Baseline Calibration in CHIMEA First Implementation & Application as Beam ProbebyDeborah C. GoodBachelor of Science, Colorado School of Mines, 2014A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Astronomy)The University of British Columbia(Vancouver)August 2017c© Deborah C. Good, 2017AbstractThe nature of dark energy is one of the most intriguing scientific questions of thetwenty-first century. There are many ways to probe dark energy, but one methodinvolves detecting baryon acoustic oscillations (BAO) throughout the universe’shistory. BAO have a characteristic size scale and therefore act as a “standard ruler,”an advantageous property for a method of tracking the universe’s expansion history.While baryon acoustic oscillations can be probed in many ways, one of themost intriguing and promising methods is through twenty-one centimeter hydrogenintensity mapping. Several experiments devoted to twenty-one centimeter hydro-gen mapping will be coming on line in coming years, and these experiments havestringent calibration requirements due to the need to remove bright foregroundsignals. These calibration requirements necessitate new and improved methodsfor calibration. One proposed method is redundant baseline calibration, a self-calibration method which takes advantage of the massively redundant designs ofmany hydrogen intensity mapping experiments.With the Canadian Hydrogen Intensity Mapping Experiment as a test case,we demonstrate that the redundant baseline method is effective in even its sim-plest implementation for an idealized version of a real telescope. We then showthat redundant baseline calibration fails in real CHIME Pathfinder data in a waythat is consistent with deviations from redundancy observed in processed CHIMEPathfinder data. These deviations from redundancy are themselves consistent withthe effects of feed-to-feed beam pattern variations, a possibility not considered inthe conventional redundant baseline calibration algorithm.We simulate the CHIME Pathfinder including beam width perturbations andverify that similar failures in the redundant baseline calibration can be generatediiwith beam perturbations. We then use the principles of redundant baseline calibra-tion to solve for our simulated beam perturbations. Finally, we compare redundantbaseline calibration results to point source holography results and show that thetwo are equivalent probes of relative feed-to-feed beam variation.We conclude that redundant baseline calibration is a promising path forward incalibrating hydrogen intensity mapping experiments, both as a conventional cali-bration method and as a probe of beam structure.iiiLay SummaryOne goal for cosmology in the twenty-first century is to understand dark energy andthe accelerating expansion of the universe. One way to learn about dark energy isby detecting baryon acoustic oscillations (BAO) with 21 cm hydrogen intensitymapping. The Canadian Hydrogen Intensity Mapping Experiment (CHIME) is anexperiment designed for this purpose.Hydrogen Intensity Mapping is difficult, because the desired signal is muchdimmer than other sources of detected light. This means that experiments mustbe calibrated carefully to allow the removal of unwanted sources. One method forcalibration is redundant baseline calibration (RBC), which takes advantage of thedesign of telescopes like CHIME to precisely calibrate without detailed knowledgeof the sky or electronics.RBC is successful in idealized situations, but real telescopes break assumptionsunderlying the algorithm making RBC inaccurate. However, these inaccuraciespredict important properties of the telescope, so it is still a useful tool.ivPrefaceThis thesis is based on work conducted as part of the CHIME Collaboration andthe CHIME Pathfinder experiment specifically. None of the text of this thesis istaken directly from previously published articles. The analysis in this thesis is thework of D. Good with the supervision of Kris Sigurdson and J. Richard Shaw aswell as incidental input from other members of the CHIME Collaboration.Several CHIME software modules were used to create the simulated CHIMEPathfinder data in Chapters 2 and 4. These modules were caput, cora, ch pipeline,ch util, draco, driftscan, and were developed by members of the CHIME collabo-ration, largely Kiyoshi Masui and J. Richard Shaw.The holography data in Chapter 5 was collected by the CHIME collaboration atlarge and Philippe Berger specifically. Holographic beam mapping for the CHIMEPathfinder is discussed in more detail in [6].vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi1 A Brief Introduction to ΛCDM Cosmology & 21 cm Hydrogen In-tensity Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 An Introduction to Dark Energy . . . . . . . . . . . . . . . . . . 21.1.1 Vacuum Energy & the Cosmological Constant . . . . . . 21.1.2 Experimental Probes of Dark Energy . . . . . . . . . . . 41.2 21 cm Intensity Mapping . . . . . . . . . . . . . . . . . . . . . . 101.2.1 Understanding 21 cm Neutral Hydrogen . . . . . . . . . . 101.2.2 21 cm Intensity Mapping as a Probe of Dark Energy . . . 111.2.3 Designing 21 cm Experiments . . . . . . . . . . . . . . . 131.2.4 A Brief Census of 21 cm Experiments . . . . . . . . . . . 141.3 An Introduction to CHIME . . . . . . . . . . . . . . . . . . . . . 152 Implementing Redundant Baseline Calibration . . . . . . . . . . . . 192.1 Constructing Visibilities . . . . . . . . . . . . . . . . . . . . . . 20vi2.2 Amplitude Calibration . . . . . . . . . . . . . . . . . . . . . . . 232.2.1 Complications to the Amplitude Calibration . . . . . . . . 242.3 Phase Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4 Redundant Baseline Calibration on Simulated Data . . . . . . . . 272.4.1 Simulating the CHIME Pathfinder . . . . . . . . . . . . . 272.4.2 First implementation . . . . . . . . . . . . . . . . . . . . 302.4.3 Determining Ideal Noise Covariance . . . . . . . . . . . . 322.4.4 Conclusions from initial implementation . . . . . . . . . 373 Using Redundant Baseline Calibration in CHIME Pathfinder Data . 383.1 Redundant Baseline Calibration on CHIME Pathfinder Data . . . 393.1.1 Modifying redundant baseline calibration for use on real data 393.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 Examining Redundancy in the CHIME Pathfinder . . . . . . . . . 444 Redundant Baseline Calibration with Perturbed Beams . . . . . . . 524.1 Creating a Simulation with Beam Perturbations . . . . . . . . . . 524.1.1 Design of beam perturbation . . . . . . . . . . . . . . . . 524.2 Redundant Baseline Calibration Results . . . . . . . . . . . . . . 544.3 Solving for Beam Perturbation Values . . . . . . . . . . . . . . . 554.3.1 Solving with One Perturbation of Known Structure . . . . 584.3.2 Solving for Beam Perturbations with only Amplitude In-formation . . . . . . . . . . . . . . . . . . . . . . . . . . 594.3.3 Relaxing Assumptions about V 0i j and V1i j . . . . . . . . . . 614.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675 Holography and Redundant Baseline Calibration as Beam Probes . 695.1 Holography: A method for probing CHIME beams . . . . . . . . 695.2 Comparing redundant baseline beam measurements with hologra-phy data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.2.1 Comparing by eye . . . . . . . . . . . . . . . . . . . . . 715.2.2 Ratio analysis . . . . . . . . . . . . . . . . . . . . . . . . 715.2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 76vii6 Conclusions and Further Work . . . . . . . . . . . . . . . . . . . . . 796.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89A CHIME simulation pipeline . . . . . . . . . . . . . . . . . . . . . . . 89A.1 Modification to pipeline code . . . . . . . . . . . . . . . . . . . . 89A.1.1 Use of CylinderPerturbed Telescope Object . . . . . . . . 89A.1.2 ExpandPerturbedProducts . . . . . . . . . . . . . . . . . 91A.2 Example results . . . . . . . . . . . . . . . . . . . . . . . . . . . 92B Supplemental Figures . . . . . . . . . . . . . . . . . . . . . . . . . . 95viiiList of FiguresFigure 1.1 Achievable parameter space for BAO detection with 21 cm in-tensity mapping. The left exclusion arises from limited ob-servable volume, the top exclusion from nonlinearity obscur-ing the BAO wiggles, and the bottom exclusion from brightforegrounds exceeding the removable threshold. Image from[9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Figure 1.2 Constraints on the dark energy equation of state and its red-shift dependence from DETF stage I + Planck (outermost line),DETF stage I+ III + Planck (intermediate dotted line), DETFstage I + III + IV + Planck (inner dotted line), intensity map-ping + Planck (inner solid line for best case, outer solid linefor worst case, and all options combined (dashed lines for bestand worst - almost indistinguishable). Image from [9]. . . . . 13Figure 1.3 The CHIME Pathfinder at DRAO. It consists of two 20 m by36 m cylinders and 256 total inputs, and operates between 400-800 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Figure 1.4 CHIME under construction at DRAO. It consists of four 20 mby 100 m cylinders and 2028 total inputs. It will operate at thesame frequency range as the CHIME Pathfinder. . . . . . . . 16ixFigure 2.1 The simulated sky map used for simulations in Section 2.4. Itcontains both point sources and galaxy, but the point sourcesappear as isolated pixels and are therefore almost invisible atthis size. The colour bar has been scaled to show the galaxy,as it is by default saturated by the brightest point sources. Themap is created using a combination of the Haslam 408 MHzmap, known bright point sources in the CHIME frequencyband, and Gaussian realizations of dimmer point sources. . . 28Figure 2.2 We see here the complete gain results for redundant baselinecalibration implemented on a simulated CHIME Pathfinder dataset, assuming identity noise covariance. Notice the set of linesat exactly 1.0, showing feeds which are masked out and there-fore forcibly set to 0 (then exponentiated to become 1). Noticealso the improved precision of results at around 2000 secondsafter the beginning of the file, a result of improved signal-to-noise ratio during point source transit. The long-term wavystructure in the gains closely trace the input gain fluctuation. . 31Figure 2.3 We see here the underlying visibilities derived from the redun-dant baseline calibration algorithm. We notice that the algo-rithm has correctly located the point source transits and someunderlying features of the galaxy. The point source transits arethe visible parabolic structures located at the beginning andend of the file, which corresponds to about 6 hours of timeseries data. We notice very little structure at times when theinput simulated visibilities are predominantly noise, which isencouraging. . . . . . . . . . . . . . . . . . . . . . . . . . . 32xFigure 2.4 We show here the redundant baseline-derived gains, using anoise covariance calculated directly from the data. Compareto Figure 2.2 and Figure 2.5. We again see the congregation ofturned off or opposite polarization inputs at 1, improved resultsat higher signal to noise, and a general correspondence with theshape of the gain fluctuations. However, we also see decreasednoise in the solution relative to the identity noise covarianceresults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Figure 2.5 We show here the redundant baseline-derived gains, using anoise covariance calculated directly from the radiometer equa-tion. Compare to Figure 2.2 and Figure 2.4. We again seethe congregation of turned off or opposite polarization inputsat 1, improved results at higher signal to noise, and a generalcorrespondence with the shape of the gain fluctuations. How-ever, we also see decreased noise in the solution relative to theidentity noise covariance results. This decreased noise is verysimilar to the covariance from data results in Figure 2.4. . . . 34Figure 2.6 In this figure, we compare the input gains to the results for re-dundant baseline amplitude calibration for the identity noisecovariance (cyan), data-derived noise covariance (magenta),and radiometer noise covariance (blue) for a sampling of in-puts on the west cylinder. We see that the results trace thegeneral structure of the gain at all times, that they are gener-ally improved at times corresponding to point source transits,and that results for the data and radiometer noise covarianceshave less noisy solutions. Similar figures for other portions ofthe simulated array are shown in the Appendix. . . . . . . . . 35xiFigure 2.7 In this figure, we compare the percent deviation from inputgains for the identity noise covariance (cyan), data-derived noisecovariance (blue), and radiometer noise covariance(magenta)for a sampling of inputs on the west cylinder. The improvedprecision at higher signal to noise regions is less obvious here,but the smaller scatter in radiometer and data derived noise co-variances is present clear. . . . . . . . . . . . . . . . . . . . 36Figure 3.1 Redundant baseline true visibilities for a June 2015 CygA tran-sit. In this analysis, we use only inter-cylinder baselines andexclude “dead feeds” using our more complete algorithm.Thetwo polarizations are solved for separately. The recovered truevisibilities certainly recover the existence of a point sourcetransit and a reasonable shape estimate for it. We do observesome spikes through the solution - these may be attributable tothe varying level of redundancy for short vs. long baselines.The units on the y-axis are correlator units, as redundant base-line is a purely relative calibrator. We could normalize this plotto know values for CygA should we prefer a plot in Jansky. . 42Figure 3.2 Redundant baseline gains calculated for a June 2015 CygAtransit. In this analysis, we use only east-west baselines, andexclude “dead feeds.” As described previously, we solve forEast and South polarizations separately and recombine the re-sults after our calculations. Notice the definite slope in the gainvalues during the point source transit. This is contrary to ourexpectation that gains would not vary much on a short, pointsource transit time scale. . . . . . . . . . . . . . . . . . . . . 43xiiFigure 3.3 In this figure, we calculate a rough measure of the slope ofeach gain, obtained by taking the rise over run for an individ-ual feed’s redundant baseline gain results between samples 60and 100. The 40 sample range represents approximately onestandard deviation around transit, which is the period of timein which we are most confident in our results. This figure in-dicates that the slope effect observed by eye does appear to besignificant. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 3.4 We compare redundant baseline gain results for selected feedson the west cylinder calculated over two consecutive CygAtransits. We notice that there is very little deviation betweenthe two days and regard this as evidence that the cause of theslope in the gain solution is not strongly time dependent and islikely a property of the array. An identical figure showing theeast cylinder is included in the Appendix. . . . . . . . . . . . 46Figure 3.5 We compile redundancy comparisons for a short intercylinderbaseline during a CasA transit in pass 1p for four test frequen-cies. Nominally, these are instances of the same redundantbaseline, and we would therefore expect there to be little tono variation between curves in a given frequency. Instead weobserve significant deviation. We propose that this deviationis largely derived from variations in the beam pattern betweenfeeds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Figure 3.6 We compile slices at CasA transit peak for each nominally re-dundant instance of a short inter-cylinder baseline at each offour test frequencies. We notice that there is not a defined pat-tern in visibility value based on feed location, which indicatesthat the effect causing the deviation from redundancy does notvary in a systematic way along the CHIME Pathfnder cylinder. 48xiiiFigure 3.7 We compile redundancy comparisons for a short intercylinderbaseline during a TauA transit in pass 1p in the same manneras Figure 3.5. We note that while the exact magnitude of devi-ations from redundancy may differ, the general structure of thedeviation is similar to that present in the CasA data, indicatingthe existence of such deviations is not declination-dependentalthough the values may be. . . . . . . . . . . . . . . . . . . 49Figure 3.8 In this figure, we observe all instances of a nominally redun-dant, short inter-cylinder baseline at frequency 518 MHz, com-pared between CSD 693 and CSD694. Notice that both indi-vidual days are significantly non-redundant, lessening the like-lihood that the deviations from redundancy present in the side-real stack for pass1 p are caused by a deviant day included inthe pass. It appears further that deviation from redundancy isnot strongly time-dependent. . . . . . . . . . . . . . . . . . 50Figure 3.9 The right hand panel shows the ratio between the two pan-els of Figure 3.9 and the left hand panel shows the differencebetween them. Each is a short inter-cylinder baseline at fre-quency 518 MHz on CSD 693 and CSD694. The deviationbetween days is on the order of 100, while the spread withina day is on the order of 1000, meaning the deviation betweeninstances is much larger. Though the ratio is relatively largefor areas outside of the central transit, at the transit peak, it isapproximately 1. . . . . . . . . . . . . . . . . . . . . . . . . 51xivFigure 4.1 The left panel displays perturbed beam redundant baseline gainamplitude results, for selected feeds of a given polarization ina simulated perturbed beam telescope, compared with the in-put gains and a redundant baseline analysis conducted on datawithout gains added (i.e. an analysis that detects only beameffects). The right panel compares the beam only analysis tothe full analysis with redundant baseline gains subtracted. Thisfigures shows only a small sampling of inputs; more are shownin the Appendix. We notice that the full redundant baseline so-lutions deviate from the input gain solution near the peak ofthe beam only solutions. We infer that this deviation is causedby the beam perturbations, and the right hand panel confirmsthis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Figure 4.2 The left panel shows perturbed beam redundant baseline gainamplitude results, for selected feeds of a given polarization ina simulated perturbed beam telescope, compared with the in-put gains and a redundant baseline analysis conducted on datawithout gains added (i.e. an analysis that detects only beameffects. The right panel compares the ratio of the beam andgain to beam only analysis and the input gain variations. thefull analysis with redundant baseline gains subtracted, show-ing that the ratio recovers the correct input gain. As in Figure4.1, results for more inputs are shown in the appendix. . . . . 57xvFigure 4.3 This figure shows beam perturbation solution for a 16 feed to-tal telescope with random perturbations α applied in the beamwidth of all feeds has a few noticeable features. The actual in-put α values used to create the simulation are plotted in blue,but are almost exactly overplotted by the green values. Thegreen values represent the result when the output of the sim-ulation is used as the “recovered visibility,” and we thereforeexpect this close correspondence. The red curve represents theresults using the redundant baseline calibration results as the“recovered visibility” and is noticeably less accurate than thegreen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Figure 4.4 We see here the percent difference between the actual simu-lated α beam perturbation values and the recovered α valuesfor both the actual input visibility and the recovered redundantbaseline solution as input visibility. We observe that the inputvisibility has quite good agreement with the actual perturba-tion values, but the recovered redundant baseline solution isuseful only for order of magnitude approximations. . . . . . 62Figure 4.5 This figure examines the beam perturbation solution for a 16feed total telescope with all feed perturbed for a sidereal day.Each time point is solved independently, but time dependentfeatures are consistent with the redundant baseline solutionmore generally, e.g. that solution improves with improved sig-nal to noise ratio. The top panel shows the result for the solu-tion using the simulation output as the “recovered visibility,”the middle panel shows the result for a solution using the re-dundant baseline calibration results as the “recovered visibil-ity,” and the final panel shows a single visibility’s time seriesduring this sidereal day. . . . . . . . . . . . . . . . . . . . . 63xviFigure 4.6 We examine the ratio (α j +α0)/(α2+α0) for both the recov-ered (α j +α0) values and the actual (α j +α0) values for a 32feed one cylinder telescope. The solution values are close tothe expected values, except at noticeable outlier feed 16. Theimperfect correspondence to the correct answers, in spite ofthe absence of noise, is due to the underdetermined nature ofthe problem. . . . . . . . . . . . . . . . . . . . . . . . . . . 67Figure 4.7 We examine the ratio (α j−α0)/(α2−α0) for both the recov-ered (α j−α0) values and the actual (α j−α0) values for a 32feed one cylinder telescope. Unlike the (α j +α0)/(α2+α0)solution, the (α j−α0)/(α2−α0) solution exists only for ap-proximately every other input. As these results derive fromthe same underdetermined problem, they too deviate from theexpected values in ways that can be examined more carefullyusing the null space. . . . . . . . . . . . . . . . . . . . . . . 68Figure 5.1 Comparison between holography transit and calculated redun-dant baseline gain for selected feeds on the west cylinder, look-ing only at results from the east west polarization. The twogain traces represent solutions for CygA transits on consecu-tive days. The average transit peak time is marked by the bluevertical line, and the shift to before or after the average peaktime appears to correlate with the slope of the redundant base-line gain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Figure 5.2 Comparison between holography transit and calculated redun-dant baseline gain for the selected feeds on the east cylinder,EW polarization. The average transit peak time is marked bythe blue vertical line, and the shift to before or after the averagepeak time appears to correlate with the slope of the redundantbaseline gain. . . . . . . . . . . . . . . . . . . . . . . . . . 73xviiFigure 5.3 Redundant baseline gain amplitude results relative to a ref-erence redundant baseline gain solution (from feed 65). Thevariations in this plot are expected to trace feed-to-feed beamamplitude variations. These feed-to-feed beam amplitude vari-ations will then be compared to holography results to verifythe correspondence. . . . . . . . . . . . . . . . . . . . . . . 75Figure 5.4 In this figure, the points represent the average ratio differenceacross feeds at a given time point, while the shaded band repre-sents one standard deviation in the ratio difference. Notice thatthe deviation values (and especially the standard deviation) aresmall nearest the transit (which peaks at about sample 100) andlarger near the edges. Additionally, notice that the average de-viation at points nearest the transit (the center of the plot) aresignificantly smaller than the range in values within the set. . 77Figure 5.5 In this figure, we plot the average deviation between hologra-phy and redundant baseline gains for each feed in our sample,averaged over the 80 time samples closest to the transit peak(about 25 minutes of data). Notice that for all feeds, the devia-tion from zero is less than ±0.1, much smaller than the valuesof the redundant baseline gain ratios themselves. . . . . . . . 78Figure A.1 In this figure, we see the progression of simulated data throughthe simulation pipeline. First, a sidereal stream is generated foreach unique baseline. Then, the sidereal stream is expanded sothat there is a representation of each individual baseline. Atthis stage of the pipeline process, these products do not in-corporate complex gains or noise and are therefore perfectlyredundant. Third, we expand from sidereal streams to indi-vidual time streams, with “20 second” samples, mimicking theactual CHIME data. Fourth, we add a constant receiver tem-perature to our timestream. Fifth, we add complex gains to thetimestream, and finally ,we add sample noise. . . . . . . . . 90xviiiFigure A.2 Sample sidereal stream prior to expansion of redundant base-lines and re-combination of perturbed components. . . . . . . 92Figure A.3 The left hand panel shows sample sidereal stream after to ex-pansion of redundant baselines, for an identical, unperturbedsimulation. The right hand panel shows a sample perturbedsidereal stream after the expansion of redundant baselines andre-combination of perturbed components. There are now asmany different baselines as real CHIME Pathfinder data andredundancy has been broken by the addition of per-feed beamperturbations. However, at this stage, the simulated data doesnot include instrumental gains or any noise estimate and istherefore not generally suitable for analysis tasks. In this par-ticular figure, the perturbation is turned up to approximately0.1 (from approximately 0.01) to make its existence more ob-vious. Without the perturbation, the two panels would be iden-tical, as the underlying telescope configuration is the same asis the input sky map. . . . . . . . . . . . . . . . . . . . . . . 93Figure A.4 After the product expansion stage, the simulation pipeline shouldproceed as in a standard unperturbed version. First, we addthe receiver temperature, then time dependent complex gains,then we add sample noise. After adding the complex gainsand sample noise, would-be redundant baselines are no longerredundant in either the perturbed or unperturbed case, but doresemble raw CHIME Pathfinder data . . . . . . . . . . . . . 94Figure B.1 From Chapter 2. In this figure, we compare the percent devi-ation from input gains for the identity noise covariance, data-derived noise covariance, and radiometer noise covariance fora sampling of inputs on the east cylinder. . . . . . . . . . . . 96Figure B.2 From Chapter 2. In this figure, we compare the absolute devi-ation from input gains for the identity noise covariance, data-derived noise covariance, and radiometer noise covariance fora sampling of inputs on the east cylidner. . . . . . . . . . . . 97xixFigure B.3 From Chapter 2. Here, we view the average deviation betweenthe input and recovered gain. The pink band represents onestandard deviation above and one standard deviation below theaverage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Figure B.4 From Chapter 3. We compare redundant baseline gain resultsfor selected feeds on the east cylinder calculated over two con-secutive CygA transits. We notice that there is very little de-viation between the two days and regard this as evidence thatthe cause of the slope in the gain solution is not strongly timedependent and is likely a property of the array. . . . . . . . . 99Figure B.5 From Chapter 4. A continuation of Figure 4.1 . . . . . . . . . 100Figure B.6 From Chapter 4. A continuation of Figure 4.1 . . . . . . . . . 101Figure B.7 From Chapter 4. A continuation of Figure 4.1 . . . . . . . . . 102Figure B.8 From Chapter 4. A continuation of Figure 4.2 . . . . . . . . . 103Figure B.9 From Chapter 4. A continuation of Figure 4.2 . . . . . . . . . 104Figure B.10 From Chapter 4. A continuation of Figure 4.2 . . . . . . . . . 105Figure B.11 From Chapter 5. Differences between redundant baseline gainratios and ChIME-26 m cross correlation ratios; indicative ofdifference between beam estimates for a given feed on the westcylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Figure B.12 From Chapter 5. Differences between redundant baseline gainratios and CHIME-26 m cross correlation ratios; indicative ofdifference between beam estimates for a given feed on the eastcylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107xxAcknowledgmentsThis thesis would never have been possible without the help of a legion of people,both those who supported me scientifically, and those who supported me person-ally. While I could by no means list everyone who has had a role in my success todate, I will take a moment to acknowledge some of the most pertinent.First, no thesis is ever completed without the work of a dedicated supervisor.Thank you, Kris, for introducing me to the wonderful world of redundant baselinesand to my entire “theory of experiment” niche and for helping me develop as anindependent scientist.Thank you, Richard, for your continual willingness to help, both in figuring outthe science itself and in my never-ending stream of software issues. Also, thankyou for teaching me how to drive the CHIME car; sorry I left you at the DRAOgate.Thank you to the entire CHIME collaboration for both providing direction formy thesis and also pushing me to always be able to articulate what I’m workingon and why it’s important. In particular, thank you to Gary and Mark for servingas my “step-advisors,” and to my fellow UBC CHIME graduate students, whohave supported me both scientifically and emotionally. Also, thank you to thequadruped graduate student, Looney, for letting me scratch her ears and rant whenI was having a bad day.Finally, thank you to all of my wonderful family and friends, who supported mein a myriad of ways and never wavered in their belief that I would be an excellentphysicist.xxiChapter 1A Brief Introduction to ΛCDMCosmology & 21 cm HydrogenIntensity MappingBefore delving into specific discussions of redundant baseline calibration in theCanadian Hydrogen Intensity Mapping Experiment, it is worth discussing morebroadly the scientific motivation for and design of CHIME.We will begin by discussing the theoretical underpinnings of and experimentalevidence for dark energy. With this general background and motivation, we willshift our focus to twenty-one centimeter intensity mapping, the technique CHIMEwill use, and discuss the theory, potential applications, and a few current and up-coming experiments using twenty-one cm intensity mapping. Finally, we will dis-cuss the specific design of CHIME and its unique calibration requirements.This will provide a structure for the more specific discussion of redundant base-line calibration in Chapter 2 - 5.11.1 An Introduction to Dark Energy1.1.1 Vacuum Energy & the Cosmological ConstantThe “cosmological constant problem” looms over the past century’s attempts toexplain the large-scale universe. Shortly after Albert Einstein formulated generalrelativity in 1915-1916, he attempted to apply his theory to the universe at large,assuming as was commonly thought at the time, that the universe was static. Hewrote, “The most important fact that we draw from experience is that the relativevelocities of the stars are very small as compared with the velocity of light” [38].However, Einstein failed to discover a static solution to his equations and there-fore resorted to the introduction of a cosmological constant, altering the Einsteinequation to beRµν − 12gµνR−λgµν =−8piGTµν . (1.1)In the early 1920s, Alexander A. Friedmann introduced a metric which re-moved the need for the cosmological constant, the metric which would becomeknown as the FRW metric and which would become the fundamental metric forcosmology in the twentieth century [38]. George Gamow later wrote that Einsteinconsidered that “the introduction of the cosmological term was the biggest blunderhe ever made in his life” [1].However, though the cosmological advances of the 1920s indicated that thecosmological constant was not necessary for the reasons Einstein originally pro-posed, the idea never entirely left physics. Indeed, by the late 1980s, it was again atopic of serious discussion. Studying the standard model of particle physics led toincreased concern about vacuum energy, the state of least energy density in stan-dard model particle physics [1]. Such a vacuum energy is composed of a bare cos-mological constant, the value the cosmological constant would take without anymatter in the universe, and the energy density arising from quantum fluctuations[1].Particle physicists define vacuum as a ground state, and therefore concludethe vacuum must be Lorentz invariant. This in turn means the stress-energy-momentum tensor must be proportional to a Minkowski metric. Knowing thatthe stress-energy-momentum tensor of a perfect fluid has the diagonal (ρ,P,P,P),2we can conclude that the vacuum is a perfect fluid with the equation of statePvac = −ρvac. Assuming adiabatic compression and expansion, ρvac remains con-stant and is related to a cosmological constant Λ by Λ= 8piGρvac [8].This vacuum energy is irrelevant in classical, non-gravitational physics. How-ever, it becomes relevant in quantum theory. We can generalize our description ofthe vacuum energy to a quantum field theory formulation by modeling a relativis-tic field as a collection of harmonic oscillators. This allows us to write the vacuumenergy asE0 =∑j12h¯ω j, (1.2)summing over possible modes of the field. Suppose the system is in a box withside L and that L goes to infinity. We can impose periodic boundary conditions andreformulate Equation 1.2 asE0 =12h¯ L3∫ d3k(2pi)3ωk. (1.3)To obtain ρvac, we allow L→ ∞ and divide by L3. We set ω2k = k2+m/h¯2 and setkmax m/h¯. Then we can write an expression for ρvacρvac ≡ limL→∞E0L3= h¯k4max16pi2(1.4)As k approaches ∞, ρvac diverges. This is not uncommon for low-energy theo-ries at high k, so we treat kmax to be the energy scale at which we remain confidentin our theory. This is commonly chosen based on the Planck energy E∗≈ 1019GeV ,so kmax = E∗/h¯. This indicates thatρvac ≈ 1074 GeV 4 , h¯−3. (1.5)However, this value is approximately 120 orders of magnitude higher than the valueexpected from observation [8].For cosmologists of the late twentieth century, this represented a problem asboth theory and evidence from the cosmic microwave background indicated that thetotal energy density of the universe was close to critical. Thus, theorists concluded3that the missing energy density, about 2/3 of critical, must be some kind of smoothunknown energy called “dark energy” [12].One way to describe the amount energy density in the universe allocated to darkenergy is by a cosmological constant ΩΛ, which should be related to the matterdensity of the universe by ΩΛ +Ωm = 1, where Ωm is the mass density of theuniverse.One issue with thsi ΩΛ description is that it fails to account for explanations ofdark energy apart from a cosmological constant. Therefore, we can also describedark energy starting from energy conservation in an expanding universe,∂ρ∂ t+a˙a[3ρ+3P] = 0, (1.6)and defining the quantityw =Pρ. (1.7)The value of w allows us to describe a variety of universes, not just one dominatedby a cosmological constant. For w = −1, we have a cosmological constant. Forw = 0, we have a flat, matter-dominated universe and for w = 1/3, we have aradiation dominated universe [12]. We can then write an equation for the evolutionof dark energy [12]ρde ∝ exp(−3a∫ da′a′[1+w(a′)]). (1.8)1.1.2 Experimental Probes of Dark EnergyType Ia SupernovaIn the last years of the twentieth century, new observational evidence began a newera in the cosmological constant discussion. Observations of Type Ia Supernovaein the late 1990s and early 2000s indicated that the universe was expanding atan accelerating rate, providing evidence that the vacuum energy density ΩΛ wasindeed greater than zero.Type Ia Supernova are considered standard candles, because their luminosity4is well-known. In cosmology, standard candles are highly useful, as they allowscosmologists to calculated the distance to the standard candle object. This pro-cess requires a quantity known as the luminosity distance. An object with a givenluminosity L a distance d from the observer has a fluxF =Lfpid2. (1.9)In comoving coordinates, the flux isF =L(χ)4piχ2(a), (1.10)where L(χ) is the luminosity in a comoving spherical shell of radius χ(a) and a isthe scale factor [12]. If all emitted photons have the same energy, L(χ) is the timesthe number of photons passing through the comoving spherical shell in a unit oftime. Due to the expansion of the universe, the energy of the photons will be lesstoday than when they were emitted and the energy per unit time in our comovingshell will be a2 smaller than the source luminosity. This means we can writeF =La24piχ2(a), (1.11)which makes apparent a way to return to the simple form of equation 1.9: definedL =χa(1.12)the luminosity distance [12]. If we know a luminosity distance and an absolutemagnitude of a source, we can find its apparent magnitude using the expressionm−M = 5log(dL10pc)+K, (1.13)where K is is correction for wavelength shift due to expansion, m is apparent mag-nitude, and M is absolute magnitude [12].Type Ia supernovae have almost identical absolute magnitudes, so any set ofType Ia supernova magnitudes can be easily compared. Scott Dodelson’s Modern5Cosmology textbook offers a brief illustrative example. Given two supernovae, oneat z = 0.83 with m = 24.32 and one at z = 0.026 with m = 16.08 and the fact thatthe absolute magnitudes are the same (so the difference in apparent magnitude isdue only to luminosity distance), we can write24.32−16.08 = 5log(dL(z = 0.83))−5log(dL(z = 0.026)). (1.14)The smaller luminosity distance is sufficiently low redshift to be written as dL(z =0.026) = z/H0 = 0.026/H0, making the larger distance our only unknown. Ourobservation indicates that H0 dL(z = 0.83) = 1.16, but for a matter dominated uni-verse (Ωm = 1, ΩΛ= 0), H0 dL(z = 0.83) = 0.95. Even our toy problem indicatesthat we must add a cosmological constant of some sort [12].This same conclusion was reached, much more rigorously by several supernovasurveys, beginning with Riess et. al. and Perlmutter et. al. in 1998 and 1999 [26,29]. These surveys sampled small numbers of high redshift supernovae (around10 each) and fit for cosmological parameters ΩM and ΩΛ, probing whether theuniverse was matter or radiation dominated. Each group found that there was asignificantΩΛ component, although the Riess et. al. value was significantly higher.Since the late 1990s, many more supernova surveys have been conducted,which have corroborated these initial findings and added more supernovae to thetotal sample. Some notable results include those of the Supernova CosmologyProject, the Sloan Digital Sky Survey, and the Hubble Space Telescope ClusterSupernova Survey, and the Supernova Legacy Survey [13, 15, 34, 39].Relatively recent supernova surveys have used improved datasets to find im-proved values for w andΩΛ. In 2011, Conley et. al. found w=−0.91+.16−.20(stat)+.07−.14(sys)using only supernova data, assuming a flat universe with constant w, marginalizingover Ωm [11].Supernova surveys are a foundational method of measuring ΩΛ, but there issignificant evidence that they are limited by systematic uncertainties. These arisefrom a variety of quarters including dust extinction, supernova colours, and photo-metric calibration [37].6Cosmic Microwave BackgroundWhile not very useful as a direct probe of dark energy, the cosmic microwavebackground (CMB) is important in that it constrains other cosmological param-eters precisely. In particular, the CMB closely constrains the values of Ωmh2 andΩbh2, which are other major components of the energy content in the universe, par-ticularly if the universe is assumed to be flat (i.e. Ωk = 0). Additionally, the CMBcan be used to probe cosmic acceleration models more directly via the IntegratedSachs-Wolfe effect [37].The CMB’s use as a provider of careful measurements is also important in thecontext of the baryon acoustic oscillations, which are another experimental probediscussed below. CMB anisotropy data allows careful measurement of the physicalscale of oscillations deriving from baryon density, which allows the oscillations tobe used as a standard ruler. Without the careful constraints from the CMB suchanalysis might be impossible [31].Baryon Acoustic OscillationsImmediately after inflation, the baryons and photons in the universe were lockedtogether, and reactions between the photon pressure and matter density fluctuationsproduced sound waves. At recombination, the baryons and photons decoupled.However, the previously generated sound waves remained frozen in the structure ofthe baryonic matter. These fluctuations are known as baryon acoustic oscillationsor BAO. The size of the BAO was established by the size of the sound horizon atrecombination, the distance that sound could travel by the time of last scattering.This created a characteristic, fixed scale which can be used as a “standard ruler” onthe universe [3, 24, 37].Use of BAO as a standard ruler is significant to the study of dark energy, astracing baryon acoustic oscillations provides a definite size scale, allowing deter-mination of the angular diameter distance and the Hubble parameter at a variety ofredshifts [3]. The comoving distance to an object in the line-of-sight and transversedirections can be written asr‖ =c∆zH(z)(1.15)7andr⊥ = (1+ z) DA(z) ∆θ , (1.16)where ∆θ and ∆z are the observed dimensions of the object [31]. If we know r⊥and r‖ as is the case when measuring the BAO scale, we are able to determineDA(z) and H(z).This method may be preferable to other methods such as supernovae and clus-tering because it is minimally affected by non-linear gravitational clustering, galaxybiasing, and redshift distortions [31]. Therefore, it could be a vey clean probe ofdark energy.The baryon acoustic oscillation can be detected in multiple ways. All meth-ods depend on a clean determination of the BAO scale by CMB experiments suchas Planck, currently estimated to be 147.50± 0.24 [2]. To date, there have beenseveral surveys to detect baryon acoustic oscillations from galaxies, many with aneye towards dark energy constraints. These include WiggleZ, BOSS, SDSS maingalaxy survey, and the 6dF survey [4, 7, 17, 30].Weak LensingGravitational lensing is a phenomenon in which mass between an astronomical ob-ject and its observer distorts the image of the background source, deflecting it fromits proper position by some angle. This angle is generally too small to observedirectly, but weak lensing analysis can observe the gradient of the angle, whichmakes circular galaxies appear elliptical [3]. Weak lensing is a small distortion(about 1%) in the size and shape of images of distant galaxies, generated by dis-tortion from light bending past galaxies or clusters of galaxies in the foreground ofthe lensed galaxy, It can be used to measure either the galaxy’s shearing (distortionin shape) or magnification (distortion in size), though shearing measurements arefar more common [37].Individual galaxies are not circular, so we cannot determine the deformationand thus the lensing signal from any individual galaxy. Instead, we compile largesamples of galaxies and detect a lensing signal as a pattern of aligned shapes ina region of galaxies. This means that weak lensing measurements require a verylarge galaxy survey [3].8Weak lensing can be used as a probe of dark energy because the deflectionangle from the expected position of the lensed object depends on the mass of theforeground lensing object and the distances between the observer, the lens, and thelensed source. With careful analysis, then, weak lensing can be used to constrainboth the angular-diameter distance as a function of redshift and the growth rate ofstructure [3].However, weak lensing measurements are challenging. They require a largesample of lensed objects. They also have very complicated error analyses, requir-ing the consideration of statistical errors arising from cosmological model, con-cerns about observational bias, and systematic errors arising from astrophysicalprocesses in the sources and intervening space [37].Galaxy ClustersMeasuring galaxy clusters is one of the oldest techniques for investigating darkenergy, as measurements from galaxy clusters indicated that Ωm < 1 as early asthe 1980s. Today, galaxy clusters continue to be important ways of understandingdark energy. They are the largest gravitationally collapsed objects in the universe,marking locations with large density fluctuations in the early universe. We cananalytically predict their mass function, the number of galaxy clusters per unitcomoving volume per cluster mass. We can then measure their actual abundancein a region of the sky and compare the two [3].Cluster results scale sensitively with both the comoving volume and the mat-ter density of the universe, meaning that cluster results are sensitive to changes ineither the matter density or the size of the universe, both of which are of relevantobservables for dark energy [3]. In particular, the halo mass function which servesas a predictor for cluster abundances is sensitive to the combination of cosmologi-cal parameters σ8Ωm, and is therefore very sensitive to the initial value of w. Smallchanges in w can alter the evolution of the universe at late times, in turn alteringthe development of clusters [37].Galaxy clusters can and have been detected in a variety of ways, including x-rayemissions the Sunyaev-Zeldovich effect, and even gravitational lensing. However,the primary challenge in using galaxy clusters to study dark energy is not detect-9ing the clusters but understanding the relationship between the observables of thegalaxy cluster and a cosmological model. Galaxy cluster measurements are not di-rectly measuring the mass of the cluster, but rather proxies like x-ray flux or galaxycounts [3]. Calibrating the observable-mass relationship is the largest challengefacing cluster analysis. It can be accomplished by simulations of galaxy clusters,by extrapolating the few direct mass measurements to larger samples, or by relyingon statistical methods involving either additional observables or weak lensing [37].1.2 21 cm Intensity MappingOne of the most exciting new categories of cosmology experiments is 21 cm hydro-gen intensity mapping experiments. While challenging, 21 cm experiments optionsfor examining a variety of cosmological questions, including both dark energy viathe universe’s accelerating expansion and the epoch of reionization.1.2.1 Understanding 21 cm Neutral HydrogenThe 21 cm line of hydrogen arises from the hyperfine transition of hydrogen, duringwhich the electron spin flips. It was a triumph of theoretical astrophysics whendiscovered in the 1940s, one of very few spectral lines discovered following aprecise theoretical prediction. At redshift 0, the line is at ν = 1420.4057 MHz.It is potentially useful to cosmology for a few reasons. First, it is a spectralline, so by mapping it at different frequencies (different redshifts), it is possibleto trace its full three dimensional evolution. For example, at the era when darkenergy becomes dominant (around z= 1−3), the 21 cm line is found at ν ≈ 400−800 MHz and at the Epoch of Reionization, it is found around ν ≈ 30−200 MHz[14]. Second, at higher redshift, it probes the IGM, which is the dominant locationof baryonic matter [14]. At lower redshifts, it probes the clustering of collapsedhalos and therefore the underlying matter density distribution [28]. At either epoch,it serves as a tracer of baryonic matter in the universe.Mapping 21 cm hydrogen emission is a potentially useful cosmological probefor several epochs. It may provide a way to examine the dark ages, between re-combination and reionization, which are difficult to observe, as they precede theformation of astrophysical objects. It may also be a useful way of examining the10epoch of reionization, with the advantage that appropriate frequency coverage al-lows maps at a progression of redshifts. Finally, at more recent times, it can beused to detect the BAO and therefore constrain dark energy, but without requiringsurveys to resolve individual galaxies [14, 28].1.2.2 21 cm Intensity Mapping as a Probe of Dark EnergyOne method for constraining dark energy with BAO is by using 21 cm intensitymapping. This method removes one of the significant challenges in measuring theBAO at appropriate redshifts: it does not require detection of individual galaxies,just large scale variations in HI mass [9]. Therefore, it requires less resolution thangalaxy surveys with the same goal. The smallest spatial scale such an experimentwould need to consider is the third BAO peak, past which nonlinear evolutionattenuates BAO structure. Its wavelength is 35h−1Mpc, meaning that a Nyquistsampled map need only have 18h−1 MPc sized pixels. For relevant redshifts suchas z = 1.5, this means an angular wavelength of 20 arcminutes, corresponding to a200 wavelengths or 100 m, is necessary to resolve BAO structure [9].The mean brightness temperature of the 21 cm line at redshift z = 1−3 can beestimated asTb = 0.3(ΩHI10−3)(Ωm+a3ΩΛ0.29)−1/2(1+ z2.5)1/2mK, (1.17)whereΩm andΩΛ are cosmological parameters andΩHI≈ 1×10−3 at z= 1 [9, 28].Calculating the mean brightness temperature for 21 cm hydrogen at 18h−1 Mpcscales indicates that signal should be expected to be about 150µK, dim relativeto foregrounds. This means that detecting BAO will require careful removal offoregrounds and Fourier analysis of large sections of sky. This foreground problemis common to all 21 cm experiments and has been thought about carefully by manyin the field, e.g. [18].Bright foregrounds, nonlinearity, and limits on observable volume all constrainthe redshift space in which 21 cm intensity mapping to detect BAO is useful.Though finite, this area is large enough to be cosmologically interesting. Inten-sity mapping is a viable approach between redshifts of about z = 0.5−2.5 and for11scales on the order of k = 0.01− 0.1. At low redshift, the boundary is drawn bylimited observable volume, at high k the limit arises from nonlinearity obscuringBAO wiggles, and the lower k limit arises from bright foregrounds making fore-ground removal infeasible [9].Figure 1.1: Achievable parameter space for BAO detection with 21 cm inten-sity mapping. The left exclusion arises from limited observable volume,the top exclusion from nonlinearity obscuring the BAO wiggles, andthe bottom exclusion from bright foregrounds exceeding the removablethreshold. Image from [9].After detecting the BAO in neutral hydrogen, the analysis proceeds as in otherdark energy experiments using BAO analysis. Calculations from Chang et.al. fore-cast that BAO from intensity mapping combined with Planck could constrain keydark energy parameters to the same level as other “Stage 2” methods, as outlinedin the Dark Energy Task Force Report [9]. See Figure 1.2 for a graphical represen-tation of the confidence intervals for potential IM + Planck results.CHIME, the Canadian Hydrogen Intensity Mapping Experiment, is one exam-12Figure 1.2: Constraints on the dark energy equation of state and its redshiftdependence from DETF stage I + Planck (outermost line), DETF stageI+ III + Planck (intermediate dotted line), DETF stage I + III + IV +Planck (inner dotted line), intensity mapping + Planck (inner solid linefor best case, outer solid line for worst case, and all options combined(dashed lines for best and worst - almost indistinguishable). Image from[9].ple of an experiment using intensity mapping to detect the BAO and constrain darkenergy. It will be discussed further in Section 1.3.1.2.3 Designing 21 cm ExperimentsWhile 21 cm intensity mapping can be used for more than one cosmological pur-pose, depending on a given experiment’s frequency range, there are common prin-ciples present in many 21 cm experiments. All 21 cm experiments are mappingexperiments. All are advantaged by having significant frequency coverage, cor-13responding to significant redshift coverage. All are concerned about removal offoregrounds.In principle, 21 cm signal can be observed with a single dipole. Indeed, struc-ture in the 21 cm signal has been detected using only single dish telescopes suchas Parkes Observatory and Green Bank Telescope [10, 25]. However, such surveysare not ideal for creating multi-frequency full sky maps of the 21 cm emission.As the objective is ultimately a full sky map, steerability is unimportant, so transittelescopes, in which antenna are in fixed positions and the sky rotates past them,are common. Additionally, heavily redundant array configurations are preferred,generally with close-packed feeds. Most 21 cm experiments do incorporate reflec-tors, either cylinders or dishes, but in principle a sufficiently large block of antennacould be used as a 21 cm experiment. Many attempt to make use of modern compu-tation efficiency to either rapidly process cross-correlations or to limit the amountof correlations necessary to proceed, as discussed in [35, 36].Though 21 cm intensity mapping is being used for a variety of cosmologicalapplications, the similarity in experimental design and challenges such as fore-ground removal allow experiments even with different science goals to share bestpractices in design, calibration, and data analysis.1.2.4 A Brief Census of 21 cm ExperimentsWhile 21 cm cosmology is still a new field, a few experiments have already beenconducted. These were predominantly smaller, less ambitious experiments de-signed to inform larger, upcoming experiments, but they were still worth notingas introductions to the field.In the higher frequency, late-time expansion focused range, there have beenboth dedicated experiments and surveys involving conventional radio telescopes.The first of these was the Pittsburgh Cylinder Telescope, a very early cylindertelescope prototype, was composed of two 10 m by 25 m cylinders 25 m apart,with a fixed, transit telescope design and sixteen dipoles per cylinder [27]. Earlyefforts at intensity mapping in this band also included survey at the Green BankTelescope and the Parkes Telescope, with both surveys detecting the 21 cm signal[10, 25]. At present, the CHIME Pathfinder is an active telescope in this frequency14range [5].There have also been several first generation or pathfinder experiments in thelower frequency, higher redshift range. Perhaps the most prominent is the Preci-sion Array for Probing the Epoch of Reionization, located in South Africa. PAPERpioneered many practical elements of EoR 21 cm analysis and also created a com-petitive map at their frequency range [23]. Other firsts generation telescopes in thisfrequency range include MITEoR, 21 CentiMeter Array, LOw Frequency ARray,the Giant Metrewave Radio Telescope EoR Experiment and the Murchison Wide-field Array [20, 22, 40–42]Additionally, upcoming years will see more 21 cm experiments coming on-line, investigating both the higher redshift EoR era and the lower redshift late-timeexpansion era. At higher frequencies, CHIME in the northern hemisphere andHIRAX in the southern hemisphere will focus on detecting BAO with 21 cm mea-surements. At lower frequency, HERA and SKA Low will be able to learn moreabout the Epoch of Reionization.1.3 An Introduction to CHIMEThe Canadian Hydrogen Intensity Mapping Experiment is a 21 cm intensity map-ping experiment designed to detect baryon acoustic oscillations and thereby con-strain dark energy as discussed in previous sections. CHIME is a collaborationbetween University of British Columbia, University of Toronto, McGill Univer-sity, and the Dominion Radio Astrophysical Observatory (DRAO). It is located atDRAO in Kaleden, BC.CHIME is a transit telescope, composed of cylindrical reflectors with dual po-larization feeds located along a focal line suspended above the cylinder surface.The CHIME Pathfinder, pictured in Figure 1.3, consists of two 20 m by 36 mcylinders with 128 dual-polarization feeds. It has been operational since 2014, sodata from the Pathfinder can be used to test analysis and calibration methods. FullCHIME, pictured in Figure 1.4, is currently under construction but expected to be-gin taking data later in 2017. It consists of four 20 m by 100 m cylinders with1024 dual-polarization feeds. On both telescopes, these dual-polarization feeds arespaced about 30 cm apart on the focal line, in a manner that is intentionally highly15redundant.Figure 1.3: The CHIME Pathfinder at DRAO. It consists of two 20 m by 36m cylinders and 256 total inputs, and operates between 400-800 MHz.Figure 1.4: CHIME under construction at DRAO. It consists of four 20 mby 100 m cylinders and 2028 total inputs. It will operate at the samefrequency range as the CHIME Pathfinder.The Pathfinder and CHIME both observe at a frequency range from 400-800MHz. This range is covered in 1024 390 kHz channels distributed evenly acrossthe band. As observing neutral hydrogen at varied frequencies is equivalent to ob-serving it at different redshifts, CHIME observes from z= 0.8−2.5, a range whichis of significant importance in observing baryon acoustic oscillations to constraindark energy.16Both telescopes see a narrow “hotdog shaped” primary beam, approximately100 degrees north-to-south and approximately two degrees east-to-west. As CHIMEis a transit telescope, it does not point at specific objects. Instead, the CHIME beampasses over the entire northern sky in these two degree strips as the earth rotates.CHIME’s frequency dependent resolution is approximately 0.25-0.5 [21].Like other 21 cm experiments, foreground removal is a major concern forCHIME. To successfully measure BAO, CHIME must be not allow the system-atic errors from foreground filtering or calibration to dominate statistical errors.The foreground signals in particular are quite bright, up to 700 K and generallybetween 10-20 K. In contrast, 21 cm signal from BAO is expected to be about 100µK [21]. The stringent foreground filtering necessary to detect the BAO requiresexcellent calibration techniques.CHIME will require precise calibration in several different instrumental com-ponents. First, CHIME must know the beam response precisely. Frequency de-pendent structure in the antenna beam will transform angular structure in the fore-grounds into spectral structure to CHIME visibilities. Polarized foregrounds willalso undergo Faraday rotation and introduce spectral structure to the visibilities.Simulations indicate that CHIME must understand the beam width to approxi-mately 0.1% to avoid biasing the derived power spectrum by an amount greaterthan statistical uncertainty [21, 32].CHIME must also calibrate the relative complex gain as a function of time toachieve acceptable brightness accuracy. End-to-end simulations again shine lighton the maximum random variations allowed in the complex gains. These simu-lations indicate that random gain variations must be less than 1% on 60 secondtimescales [21, 32]. CHIME is, however, saved from conducting an absolute cali-bration as our BAO measurements do not require an absolute sky brightness value.Cross-talk, coupling between channels is expected to occur at various pointson the CHIME system, including between feeds, cables, and digitzer and correla-tor boards. Cross-talk in cross-correlation measurements will add signal and noisewith stable coefficients. This cross-talk will not appear as a constant additive offsetbut as a signal at random phases. CHIME will have to measure cross-talk coeffi-cients and then extract the effects from the data [21].Finally, CHIME will need to understand the instrument passband to a part17within 105. CHIME will accomplish this by assuming that bright foreground re-gions of our maps have smooth frequency spectrums, but passband effects are in-trinsically tied to beam calibration and therefore cannot be fully estimated withouta solid estimate of beam calibration [21].18Chapter 2Implementing RedundantBaseline CalibrationAs discussed in Chapter 1 as well as [21], [32], and [18], 21 cm intensity map-ping experiments such as CHIME have stringent calibration requirements, to allowthem to carefully remove foregrounds and ultimately extract cosmological infor-mation. There are a myriad of potential calibration methods, but they can roughlybe combined into three categories.The first is point source or other sky based calibrations. Point source cali-bration involves using the telescope to view a source which is very well-knownand comparing the data to the known solution to extrapolate instrumental fac-tors. Point source calibration, however, is constrained by external knowledge ofthe point source in question.The second method is noise source calibration. In this method, an externalsource of noise is broadcast into the interferometer and switched off and on at shortintervals. Data from the intervals where the noise source is off is then comparedto data where the noise source is on, eliminating sky data and leaving only in-strumental information. This can be the most analytically straightforward method(although is not always, depending on the noise source configuration) but requiresexternal hardware and can therefore be more complicated than it initially appears.The third category of calibration method is self-calibration, wherein the intrin-sic properties of the telescope and the data ordinarily collected are used to calibrate19the telescope. Methods in this category, including redundant baseline calibration,are desirable because they require neither detailed knowledge of the sky nor exter-nal noise sources, but instead use the properties of the instrument itself to calibrate.This means it can be effective even in the absence of sufficiently well-known pointsources or external noise sources. However, self-calibration methods are generallyrelative and unable to determine the absolute calibration of an instrument, so theyare often used in conjunction with other methods.Redundant baseline calibration, specifically, exploits the property that there aremany baselines in CHIME which generate redundant information. To understandwhat we mean by redundant baselines and why they are potentially useful for cal-ibration, we must also understand the structure of data in CHIME, discussed inSection 2.1. In Sections 2.2 and 2.3 we discuss the structure of the redundant base-line algorithm and in Section 2.4 we discuss an implementation of the amplitudecalibration on data from a simulation of the CHIME Pathfinder.2.1 Constructing VisibilitiesAt its most fundamental level, a transit telescope like CHIME can be viewed as acollection of feeds measuring electric field from the sky. To keep track of polariza-tion information, we would like to know what the contribution to the electric fieldfrom every direction is and therefore define ε , which is an electric field density ina frequency interval and solid angle. We can then write the electric field asdE = (µ0c)1/2 ε (nˆ,ν)d2nˆ′dν (2.1)and subsequently write the Poynting Vector asSp =1µ0cE×H =∫d2nˆ d2nˆ′ dν dν ′ nˆ 〈ε(nˆ) · ε(nˆ′)〉 (2.2).The astronomical radio signals we are interested in are generally incoherent,20enabling us to define ε in terms of the Stokes parameters〈εa (nˆ,ν)ε∗b(nˆ′,ν)〉= 2kBλ 2δ(nˆ− nˆ′)δ (ν−ν ′)× [PIabI(nˆ)+PabQQ(nˆ)+PabUU(nˆ)+PabVV (nˆ)],(2.3)where indices ab represent basis vectors transverse to the line of sight. The polar-ization tensors PXab are known variations on the Pauli matrices.Any feed i on the telescope is collecting a weighted combination of the electricfields it receives, designated Fi and given byFi(φ) =∫d2nˆAai (nˆ,φ)εa(nˆ)e2piinˆ·ui(φ), (2.4)where φ is a rotation angle, and ui is a physical location. This beam has a solidangle Ωi =∫d2nˆ|A(nˆ)|2Real interferometric data , however, is not presented as individual feed re-sponses, but as visibilities, cross correlations between multiple feeds: Vi j = 〈FiF∗j 〉.These visibilities can be written asVi j(φ) =∫∑BSi j(nˆ,φ)S(nˆ), (2.5)where S represents each Stokes Parameter (I, Q, U, V) and BSi j is a beam transferfunction. These beam transfer functions compile all the necessary instrumentalinformation into one quantity, which is given byBSi j(nˆ,φ) =2Ωi jAi(nˆ,φ)A∗j(nˆ,φ)PSabe2piinˆ·uij(φ) (2.6)where Ai is the antenna response, Ωi j =√ΩiΩ j, PSab is the previously men-tioned polarization tensor, and uij is a vector of the length of the baseline betweenfeeds i and j [32].While a number of factors influence the actual value of the visibility, relativelyfew change within a given telescope. We assume, quite reasonably, that the skyvaries little relative to the spatial scale of our telescope in a given time step, so fora given rotation angle φ , the visibilities should vary based on beam transfer matri-21ces rather than sky variation. Within the beam transfer function, if we assume thatall feeds have the same beam pattern A, we are left with a function that varies onlybased on baseline distance uij. Therefore, any two sets of feeds with the same base-line distance uij should produce the same visibility, i.e. we expect the informationfrom these baselines to be redundant. One consequence of redundancy is that anyinformation that is not identical amongst redundant visibilties must originate frominstrumental factors, most prominently the instrumental gains.The visibility calculated above is not precisely the visibility measured by thetelescope. The measured visibility combines the actual visibility calculated abovewith both instrumental gains and noise. In other words,V measi j = gig∗j Vi j +ni j, (2.7)where gi, and g j are the complex gains of feeds i and j, Vi j is the actual visibility,and ni j is a noise term. We assume this noise term to be uncorrelated, and on theorder of Tsys/√τ∆ν , where Tsys is the system temperature, τ is integration time,and ∆ν is bandwidth [19].We linearize this equation by taking the logarithm and separating the real andimaginary parts so that we have two equations for each measured baseline:ln |V measi j |= ln |gi|+ ln |g j|+ ln |Vi j|+Re(ni j) (2.8)arg(V measi j)= arg(gi)+ arg(g∗j)+ arg(Vi j)+ Im(ni j). (2.9)This enables us to separately solve for the real and imaginary parts of the gain andthe actual visibility [19].222.2 Amplitude CalibrationOnce the set of measured visibilities has been linearized as in Equation 2.8, we canwrite a matrix equation:ln |V meas01 |ln |V meas12 |ln |V meas02 |...=1 1 0 0 · · · 1 0 0 · · ·0 1 1 0 · · · 0 1 0 · · ·1 0 1 0 · · · 0 0 1 · · ·.........ln |g0|ln |g1|ln |g2|...ln |V0|ln |V1|+Re(η01)Re(η12)Re(η02)...(2.10)More succinctly,d = Mx+η , (2.11)where d is a vector containing the measured visibilities, M is a matrix containingthe various combinations of gain and true visibility, and x is a vector consisting ofthe instrumental gains for all feeds and the actual visibilities as defined in equation2.5We can solve this matrix equation using least squares methods, so thatx =(MT N−1M)−1 MT N−1d (2.12)where N = 〈ηηT 〉 is the noise covariance matrix.This noise covariance matrix N can be set in a number of ways. In the simplestapproximation, we use the identity matrix as the noise covariance matrix. We aretherefore assuming that noise is uncorrelated (by selecting a diagonal matrix asN). This can be a valid way to think about the noise covariance, but is also anover-simplification of the scenario and therefore can be a detriment.As noted above, this procedure yields a matrix, x composed of the gain ampli-tudes for each feed and the amplitude of the actual visibility for each feed. How-ever, the amplitude calibration is affected by at minimum one degeneracy. This isapparent from calculating the null space of the M matrix, which is non-zero andarises because redundant baseline calibration is in itself a relative not absolute cal-23ibration technique. We therefore apply a gauge fixing condition of some form. Thesimplest is to impose a value for one gain in the set and set all gains relative to thatvalue. The primary disadvantage to this method is simple. We do not necessarilyknow a priori the value of any single gain. The concern is less that we might choosea wrong numerical value for this gain (as we will only consider other gains rela-tive to it in our analysis) but that we may accidentally choose an ill-behaved gain,one with very unusual structure. In the case that the reference gain experiencesdramatic fluctuations, this will induce dramatic fluctuations in our other gains aswell.We attempt to mitigate this risk by setting the gain relative to an expectedaverage point for the gains, rather than an individual gain. Specifically, our usualcondition is∑iln |gi|= 0. (2.13)This condition states that the sum of the natural logarithms of the gains must beequal to zero. This condition then places the gain amplitude values around one,which is a reasonable choice for instrumental gains on the CHIME Pathfinder.2.2.1 Complications to the Amplitude CalibrationSeveral complications exist in the amplitude calibration, both from the perspec-tive of non-idealities in a real telescope like the CHIME Pathfinder and from thecalibration method itself.Probably the most significant is the addition of non-identical primary beams be-tween feeds. In summary, if the beam amplitudes Ai for each feed are not identical,one of underlying assumptions of redundancy collapses and the redundant base-line calibration method performs in unexpected, though sometimes useful, ways.This issue will be discussed in greater detail in Chapter 4, where we will present adetailed simulation of such a telescope.At the simplest level, we can improve our noise covariance matrix by using adiagonal matrix with more carefully considered An individually generated approx-imation of the noise covariance is generally more exact, if available. One simpleoption is to use the radiometer noise test to generate a value for σ2. Another optionis to derive the noise covariance matrix directly from the variance of the data itself.24One major assumption made in simple implementations of redundant baselinecalibration is that noise is uncorrelated. However, this is quite often not the case,meaning that our diagonal matrix estimate for N is inaccurate. A common reasonfor correlated noise in systems like CHIME is cross-talk. Cross-talk arises whentwo feeds interact with one another and therefore skew the results from their mu-tual baseline. This is primarily a concern when dealing with short, intra-cylinderbaselines. Therefore, the simplest way to mitigate cross-talk is to ignore either allintra-cylinder baselines or intra-cylinder baselines shorter than a pre-determinedminimum baseline distance.This decision is however not without consequences. Working solely acrosscylinders in particular introduces an additional degeneracy into the problem. Thisis evidenced by an additional dimension in the null space of the coefficient matrix,M, indicating that the problem is no longer uniquely solved. In the absence ofinformation within a cylinder, the two cylinder’s absolute levels can change inde-pendently. The initial degeneracy fixing condition present in the algorithm pertainsto the overall level of the system’s gain. Without knowledge within a single cylin-der, the algorithm is free to distribute gain between the two cylinders in any way itpleases, e.g. assigning very high gain to one and a very low gain to the other, solong as the overall level remains consistent with the degeneracy fixing condition.This can be simply and effectively remedied by splitting the degeneracy fixingcondition into two complementary conditions, one which sets a level for the firstcylinder and one that sets a level for the second cylinder.2.3 Phase CalibrationOur earlier linearization of the visibility equation split the amplitude and phase ofthe visibility, allowing them to be solved for separately. Though this thesis’ focus ison amplitude calibration using redundant baselines, redundant baseline calibrationcan also be used as a relative phase calibrator, via a method outlined in MichaelSitwell’s PhD thesis [33]. At first glance, it would appear that we could applythe same process to solve for phase, but degeneracies in the problem make thisstraightforward approach intractable.To implement redundant baseline phase calibration, we first construct a matrix25G = |g〉〈g|, using an initial estimate Gˆi j = V measi j /V truei j for each component. Thisestimate for V truei j can be arbitrary, so one simple option is to use the Vtruei j calculatedfrom the amplitude calibration. With an estimate for G, we attempt to solve for thegain vector that minimizes chi-squared, which we can write as [33]χ2 =∑i jkl(Gˆi j−Gi j)C−1i j,kl(Gˆkl−Gkl). (2.14)If the covariance matrix C−1i j,kl is uncorrelated between baselines and gives eachbaseline the same variance σ2, we can reduce Equation 2.14 toχ2 =1σ2∑i j|Gˆi j−Gi j|2. (2.15)As a consequence of our definition of G, |g〉 is an eigenvector of G with theeigenvalue 〈g|g〉. We can therefore use the eigenvalue decomposition to find valuesof the gains. As Gˆ is Hermitian and positive semi-definite, its eigenvalue decompo-sition is the same as its singular value decomposition. This is convenient becauseit allows us to find which eigenvector we should use as our gain estimate. Specifi-cally, this SVD equivalent means that reducing Gˆ to a rank-1 matrix with the SVDis the equivalent to minimizing χ2. We are able to conclude that the eigenvectorcorresponding to the largest eigenvalue is our desired gain vector [33].With an estimate of the gains, we can re-estimate the true visibility for a givenbaseline b = i− j, writingVb =∑i gig∗i+bVmeasi,i+b∑i |gi|2|gi+b|2. (2.16)We then iterate this process until our results converge [33].There are a few caveats to this method. First, the assumption that covariancematrix Ci j,kl is proportional to the identity used to reach Equation 2.15 requires usto assume autocorrelations should receive the same weight as all other correlations,which is distinctly inaccurate. However, if we only need to solve for gain phases,we can replace Gˆi j → Gˆi j/|Gˆi j| which makes Gˆ a matrix with complex elementswith unit norms and thus the diagonal elements are all scaled to one. Second,26there are a number of degeneracies in the phase solution. It is insensitive to botha rotation of the sky (or equivalently a tilting of the array) as well as the absolutephase level of the system. These degeneracies can be fixed either by calculating thedeviation from expected phase at a point source or by simply fixing the first twogains in the gain vector to known values.Though this method has been previously shown to be successful in small cases(see [33]), in the remainder of this thesis we will focus solely on amplitude cali-bration.2.4 Redundant Baseline Calibration on Simulated DataAlthough we have at our disposal actual CHIME pathfinder data, we choose tofirst present redundant baseline calibration results from our carefully constructedsimulations, as this allows us to make decisions about most effective way to im-plement redundant baseline calibration without having our results clouded by theirregularities and complexities of real experimental data.2.4.1 Simulating the CHIME PathfinderAs we are interested only in testing a calibration method and not in testing 21cm hydrogen intensity detection, we use the software package COsmology in theRAdio Band (CORA) to generate a foreground map including the galaxy and pointsources. Our map includes the actual values of point sources brighter than 4 Jy aswell as a synthetic population of dimmer point sources (brighter than 0.1 Jy at 151MHz) and a Gaussian realization representing even dimmer unresolved sources.The galactic component of the map is extrapolated from the Haslam map at 408MHz with additional random fluctuations [16]. The map, with the colour bar scaledto make the galaxy visible, is shown in Figure 2.1.After simulating a test sky, we generate test beam transfer matrices. Whilewe use alternative beam models later, for this analysis we confine ourselves to thefiducial quasi-Gaussian beam introduced in [32]. This beam model varies slightlyfor the two polarizations present in CHIME, here designated x polarization (thefeed dipole pointing east-west) and y polarization (the dipole pointing north-south).The fiducial model is not a direct attempt to solve for the beam pattern but rather27Sky Map: Galaxy0 100Figure 2.1: The simulated sky map used for simulations in Section 2.4. Itcontains both point sources and galaxy, but the point sources appear asisolated pixels and are therefore almost invisible at this size. The colourbar has been scaled to show the galaxy, as it is by default saturated bythe brightest point sources. The map is created using a combination ofthe Haslam 408 MHz map, known bright point sources in the CHIMEfrequency band, and Gaussian realizations of dimmer point sources.an approximation composed of the product of a function describing response in theEW direction and a function describing response in the N-S direction. We writethe beam amplitude for an unfocused dipole, our model for the feed, asAD (θ ;θW ) = exp(− ln22tanθ 2tanθW 2), (2.17)with θW the beam’s full width at half-power. Given the height of our feed relativeto its conducting ground plane (i.e. the cylinder surface), we can calculate θ as2pi/3 in the H-plane and 0.675θh in the E-plane. We treat our problem then as aFraunhofer diffraction problem with our given amplitude. Therefore, our cylinderhas the amplitudeAF (θ ,θW ,W )∝∫ W/2−W/2AD(2arctan2piW;θW)e−ikxsinθdx∝∫ 1−1e− ln2tanθW 2u21−u2 ipiWλ usinθdu(2.18)28incorporating the fact that for a cylinder with f-ratio 1/4, a ray hitting a distance of xaway from the cylinder’s centre reflects at an angle θ = 2arctan(2x/W )= 2arctanu[32].We can create our overall beam for each polarization using these two A func-tions and an addition function pa, which is a unit vector in the polarization directionfor a dipole in a given direction. Therefore, the x feed has a beam amplitudeAXa (nˆ) = AF(arcsin(nˆ · xˆ);θE ,W )×AD (arcsin(nˆ · yˆ);θH) pa(nˆ; xˆ) (2.19)and the y feed has a beam amplitudeAYa (nˆ) = AF(arcsin(nˆ · xˆ);θH ,W )×AD (arcsin(nˆ · yˆ);θE) pa(nˆ; yˆ), (2.20)with xˆ and yˆ unit vectors pointing in the East (transverse) and North (parallel)directions respectively [32].It is critical to note that the fiducial beam model is a significant simplificationof the CHIME beam. It does not account for cylinder surface considerations orfor any complicated beam structure or non-identicalities between individual feeds.This model is also not informed by knowledge of the instrument’s specific charac-teristics.The simulation pipeline creates a model telescope using the layout of the CHIMEpathfinder at a provided time and the fiducial beam described above. It generatesbeam-transfer matrices then applies the input sky map to these beam transfer ma-trices to create a pseudo realistic realization of what the telescope might see.The simulation pipeline then separately adds noise and complex gain to thesimulated data. First, we add a constant amount of noise, corresponding to thereceiver temperature, modeled as 50 K. Then, we generate complex gains foreach input of the the simulated CHIME pathfinder. These gains have an averagevalue of one, but include long-timescale random fluctuations. We do not expectCHIME’s complex gains to change particularly rapidly (e.g. on few minute to hourtimescales) but we do expect variations over several hours or a day. The gain fluc-tuations in our initial implementation are likely also larger amplitude than realisticgain fluctuations. Thus, we could describe this as a pessimistic gain scenario. The29pipeline also calculates the appropriate combinations of feeds and applies thesegains. Finally, we add a small amount of Gaussian random noise to the samples.At this stage, our simulated data should look and behave exactly like realCHIME Pathfinder data, assuming that our initial model of the CHIME Pathfinderis correct. (We must particularly look out for inconsistencies between our assumedquasi-Gaussian beam model and the real telescope’s more complicated beam model.)2.4.2 First implementationWe first and most simply implement redundant baseline calibration assuming anidentity noise covariance matrix. We do ensure that we remove feeds which aremasked out in the simulation, as when they are left in the data set, the algorithmstruggles to find a solution for those inputs which matches our imposed degeneracyfixing condition and therefore degrades the entire solution set. We also include onlyone polarization at a time and therefore run our redundant baseline analysis twice toextract all relevant underlying visibilities and gains. We do not attempt to recoverthe cross-polarization correlations, as they are not meaningful in this context.Figure 2.2 shows the complete set of gains derived from redundant baselinecalibration on our simulated CHIME pathfinder data over one six hour time streamfile. While the plot is in many ways an overwhelming amount of information, thereare a few results we can deduce from looking at this compilation.First, notice the set of lines at exactly 1.0. These lines are the result of forcingthe gain for masked out feeds and feeds from the second polarization to be 0.Redundant baseline calibration actually calculates the natural log of the gains, andwe therefore exponentiate the entire data set before presenting it here. Therefore,our zeroed out gains become exactly 1.0. We can thus ignore those lines entirely.Second, notice the improved precision at about 2000 seconds into the timestream, which is a simulated TauA transit. Redundant baseline calibration is inprinciple sky independent. However, in practice, bright sources dramatically im-prove the signal-to-noise ratio in CHIME Pathfinder data. When this happens, theredundant baseline algorithm is able to provide a noticeably more precise solution,as it is much less strongly influenced by sources of noise in the data. For example,when looking at an effectively blank sky, the 50 K receiver temperature is signif-300 5000 10000 15000 20000 25000time +1.44396e90.40.60.81.01.21.41.61.8|gain|Identity noise covarianceFigure 2.2: We see here the complete gain results for redundant baseline cal-ibration implemented on a simulated CHIME Pathfinder data set, as-suming identity noise covariance. Notice the set of lines at exactly 1.0,showing feeds which are masked out and therefore forcibly set to 0 (thenexponentiated to become 1). Notice also the improved precision of re-sults at around 2000 seconds after the beginning of the file, a result ofimproved signal-to-noise ratio during point source transit. The long-term wavy structure in the gains closely trace the input gain fluctuation.icant. However, it is intuitively obvious that it is not nearly as significant whenlooking at a bright point source with a brightness temperature much greater than50 K.This is an interesting result, as it indicates that the assumption that redundantbaseline is sky independent is true only in a limited sense. While redundant base-line calibration is not sensitive to the exact sky model, it is sensitive to signalto noise ratio variations. This is potentially valuable, as it may allow improvedalgorithms in the future which incorporate some sky info and therefore improveperformance.Third, notice the long-term wavy structure in the gains. Initially, this may seemconcerning, as it seems that the solutions are floating around. However, whencreating the simulation, we incorporated long-term gain fluctuations. Figure 2.2is not the ideal figure to demonstrate that these variations are as expected, but thestructure of the fluctuations in Figure 2.2 should be considered an encouragingsign. We will investigate deviation from the input fluctuations more exactly later31in our analysis.0 5000 10000 15000 20000 25000time +1.44396e90.00.10.20.30.40.50.60.70.8Recovered Calibrated VisibilityRedundant Baseline Recovered VisibilityFigure 2.3: We see here the underlying visibilities derived from the redun-dant baseline calibration algorithm. We notice that the algorithm hascorrectly located the point source transits and some underlying featuresof the galaxy. The point source transits are the visible parabolic struc-tures located at the beginning and end of the file, which corresponds toabout 6 hours of time series data. We notice very little structure at timeswhen the input simulated visibilities are predominantly noise, which isencouraging.We can also generate a plot of the redundant baseline calibrated underlyingvisibilities. These visibilities should be those generated by Equation 2.5 whereasthe visibilities input to the algorithm were of the form of Equation 2.7. Calibratedvisibilities are, of course, desirable as our motivation for knowing the complexgains is ultimately to recover calibrated visibilities of the form of Equation 2.5.We notice that our recovered visibilities correctly locate our expected point sourcetransits and a few underlying features of the galaxy. As those are the only twoinputs to our map (see Figure 2.1), we consider that success. (Or at least concludethat any failures in the redundant baseline calibration will be visible only in thegain solutions.)2.4.3 Determining Ideal Noise CovarianceWhile at a base level, we are pleased with the performance of the identity noisecovariance version of the algorithm, we suspect that the performance could be im-proved by specifying the noise covariance more carefully. Therefore, we also tested32variations of the noise covariance matrix. First, we calculate the noise covariancedirectly from the input data. Second, we use the radiometer noise test to determinethe noise covariance. Note that while both of these methods attempt to representthe noise more accurately than the identity, both are constructed into diagonal ma-trices (just like the identity), representing a continued assumption that there is nocorrelated noise. This should be an accurate portrayal of our simulated data, wherethere genuinely is no correlated noise, but the assumptions may break down forreal CHIME Pathfinder data. Summary results from these methods, equivalent tothose presented in Figure 2.2 are presented in Figures 2.4 and 2.5.0 5000 10000 15000 20000 25000time +1.44396e90.40.60.81.01.21.41.6|Gain|Redundant Baseline Gain Solution: Noise Covariance from DataFigure 2.4: We show here the redundant baseline-derived gains, using a noisecovariance calculated directly from the data. Compare to Figure 2.2 andFigure 2.5. We again see the congregation of turned off or opposite po-larization inputs at 1, improved results at higher signal to noise, and ageneral correspondence with the shape of the gain fluctuations. How-ever, we also see decreased noise in the solution relative to the identitynoise covariance results.All three variations generate reasonable results and thus we are not able to tellwhich is superior purely by looking at these summary plots. We therefore moveforward with specific comparison plots. We take advantage of the fact that we areusing simulated data to directly compare our redundant baseline outputs to our sim-ulated inputs. We compare both in absolute difference and in percent difference.We find that on average the identity noise covariance gain results are within 15% ofthe input gains, while the radiometer and data derived noise covariances are withinapproximately 5% of the input gains. Figure 2.6 shows a sampling of percent dif-330 5000 10000 15000 20000 25000time +1.44396e90.40.60.81.01.21.41.6|gain|Radiometer noise covarianceFigure 2.5: We show here the redundant baseline-derived gains, using a noisecovariance calculated directly from the radiometer equation. Compareto Figure 2.2 and Figure 2.4. We again see the congregation of turnedoff or opposite polarization inputs at 1, improved results at higher signalto noise, and a general correspondence with the shape of the gain fluc-tuations. However, we also see decreased noise in the solution relativeto the identity noise covariance results. This decreased noise is verysimilar to the covariance from data results in Figure 2.4.ference results. We also look at absolute deviations, particularly for the radiometerand data-derived variations, see Figure 2.7 for a sampling. Finally, we find theaverage values and standard deviation of the gain differences for radiometer anddata-derived results.One important thing we notice is that the differences are consistently negative.We calculate our deviations as input minus recovered, and therefore deduce that ourredundant baseline gain solution is consistently higher than our input gain set(forboth data-derived and radiometer noise covariances). We suggest this is likely aneffect of bias in redundant baseline calibration or an indication that our degeneracyfixing condition is failing, but also suggest that this deserves further examination inimplementing redundant baseline calibration. One way to correct for this might beto relax the assumption that the noise is independent and thus to modify our noisecovariance matrix to not be diagonal.340 5000 10000 15000 20000 25000+1.44396e90.800.850.900.951.001.051.101.151.201.25|gain|Feed 65identity noise covradiometer noise covdata noise covinput0 5000 10000 15000 20000 25000+1.44396e90.60.81.01.21.41.61.8Feed 73identity noise covradiometer noise covdata noise covinput0 5000 10000 15000 20000 25000+1.44396e90.70.80.91.01.11.21.3|gain|Feed 81identity noise covradiometer noise covdata noise covinput0 5000 10000 15000 20000 25000+1.44396e90.80.91.01.11.21.31.4Feed 89identity noise covradiometer noise covdata noise covinput0 5000 10000 15000 20000 25000+1.44396e90.60.70.80.91.01.11.21.31.4|gain|Feed 97identity noise covradiometer noise covdata noise covinput0 5000 10000 15000 20000 25000+1.44396e90.60.81.01.21.4Feed 106identity noise covradiometer noise covdata noise covinput0 5000 10000 15000 20000 25000Time (seconds) +1.44396e90.80.91.01.11.2|gain|Feed 113identity noise covradiometer noise covdata noise covinput0 5000 10000 15000 20000 25000Time (seconds) +1.44396e90.70.80.91.01.11.21.31.4Feed 121identity noise covradiometer noise covdata noise covinputFigure 2.6: In this figure, we compare the input gains to the results for re-dundant baseline amplitude calibration for the identity noise covariance(cyan), data-derived noise covariance (magenta), and radiometer noisecovariance (blue) for a sampling of inputs on the west cylinder. We seethat the results trace the general structure of the gain at all times, thatthey are generally improved at times corresponding to point source tran-sits, and that results for the data and radiometer noise covariances haveless noisy solutions. Similar figures for other portions of the simulatedarray are shown in the Appendix.350 5000 10000 15000 20000 25000+1.44396e9201510505101520|gain|: actual to RB % diffFeed 65identity noise covradiometer noise covdata noise cov0 5000 10000 15000 20000 25000+1.44396e9252015105051015Feed 73identity noise covradiometer noise covdata noise cov0 5000 10000 15000 20000 25000+1.44396e92015105051015|gain|: actual to RB % diffFeed 81identity noise covradiometer noise covdata noise cov0 5000 10000 15000 20000 25000+1.44396e92015105051015Feed 89identity noise covradiometer noise covdata noise cov0 5000 10000 15000 20000 25000+1.44396e9201510505101520|gain|: actual to RB % diffFeed 97identity noise covradiometer noise covdata noise cov0 5000 10000 15000 20000 25000+1.44396e9201510505101520Feed 105identity noise covradiometer noise covdata noise cov0 5000 10000 15000 20000 25000Time (seconds) +1.44396e9201510505101520|gain|: actual to RB % diffFeed 113identity noise covradiometer noise covdata noise cov0 5000 10000 15000 20000 25000Time (seconds) +1.44396e9201510505101520Feed 121identity noise covradiometer noise covdata noise covFigure 2.7: In this figure, we compare the percent deviation from input gainsfor the identity noise covariance (cyan), data-derived noise covariance(blue), and radiometer noise covariance(magenta) for a sampling of in-puts on the west cylinder. The improved precision at higher signal tonoise regions is less obvious here, but the smaller scatter in radiometerand data derived noise covariances is present clear.362.4.4 Conclusions from initial implementationWe find that our initial implementation is largely successful, as we are able torecreate to within 5% the input gain amplitudes. However, we recognize there aresome concerns, such as the apparent bias of our results. Our first implementa-tion shows some encouraging results, but also leaves room for improvement. Oneobvious spot for improvement is to incorporate a careful implementation of the re-dundant baseline phase calibration, which is currently only implemented for smalltest cases.While we are encouraged by our success in recovering a simulated model, wealso realize that this chapter’s results are limited to cases that truly qualify as re-dundant baselines, which may not be realistic cases. Looking forward, in Chapters3 and 4, we will apply redundant baseline calibration to a perturbed simulation andto real CHIME Pathfinder data.37Chapter 3Using Redundant BaselineCalibration in CHIMEPathfinder DataOur analysis in Chapter 2 suggests redundant baseline calibration is a model worthpursuing for a telescope such as CHIME. As our eventual goal is to have calibratedCHIME data for use in intensity mapping, it is important to demonstrate that cali-bration methods can be used with real data as well as with simulations. Therefore,we apply redundant baseline calibration to real CHIME pathfinder data.However, our redundant baseline analysis in Section 3.1 indicates significantdeviation from the expected gain results. This could in principle be due to unex-pected but genuine structure in the telescope’s complex gains, but other calibra-tion methods applied to the CHIME Pathfinder do not support such a conclusion.Therefore, we argue that the deviation from expected gain results is attributable todeviations from redundancy in the CHIME Pathfinder. We then examine the extentto which nominally redundant baselines are actually redundant in a point-sourcecalibrated CHIME Pathfinder data set, and find that deviation from redundancy issignificant.383.1 Redundant Baseline Calibration on CHIMEPathfinder Data3.1.1 Modifying redundant baseline calibration for use on real dataCross-TalkBefore beginning our analysis, we make a few modifications to our redundant base-line approach to account for known sources of error in the CHIME Pathfinder data.Most importantly, we know that the individual antenna in the CHIME Pathfindersuffer from cross-talk. This renders intra-cylinder baselines suspect up to someminimum separation. This effect likely on applies for very short intra-cylinderbaselines (e.g. shortest or second shortest baselines), but we do not have a defi-nite metric to establish what the minimum length necessary to avoid cross-talk isin the CHIME Pathfinder. Therefore, for this analysis we exclude intra-cylinderbaselines.This straightforward method for eliminating cross-talk, however, has implica-tions for the rest of the analysis. In particular, removing intra-cylinder informa-tion introduces an additional degeneracy into the solution. Our usual degeneracyfixing condition sets ∑ ln |gi| = 0, but in the absence of intra-cylinder baseline in-formation, this condition fails to ensure that the algorithm does not actually set∑ ln |gi,cylinder 1|=−1 and ∑ ln |gi,cylinder 2|= 1.Therefore, we simply turn our overall degeneracy fixing conditions into twoidentical conditions:∑i, cylinderln |gi, cylinder|= 0, (3.1)one for each cylinder.Using TransitsIn the simulated data, we calculated redundant baseline solutions over an entiresix hour period of data, including both point transits and quiet portions of the sky.However, when using real CHIME Pathfinder data, we choose to confine ourselvesto point source transits. We saw in Chapter 2 that at higher signal to noise peri-39ods in the simulation, our redundant baseline solution was significantly less noisyand clearer than during quiet sections of the sky. CHIME Pathfinder data will nec-essarily be more complicated than simulated data, with less well-known systemtemperature and more complicated noise structure, so we choose to focus only onthe clearest sections of the more complicated system.We are also more confident as to the expected structure of the complex gainsduring the short period of time encompassed by a point source transit than we areover a longer period of time. While we expect time dependent gain fluctuations inCHIME Pathfinder data, we expect fluctuations during the tens of minutes encom-passed in a point source transit to be relatively small.Removing “Dead” FeedsIn real data, we are also confronted with the possibility of input channels which areabsent or being used for non-CHIME antenna electronics. Including such anoma-lous channels will wreak havoc on the redundant baseline analysis, as all correla-tions including that non-CHIME Pathfinder channel will no longer be redundantwith other correlations of the same baseline distance. For example, the RFI an-tenna will clearly not correlate with CHIME antennas at all like a CHIME antennawould.Additionally, due to changes in experiment configuration or electronics mal-function, the CHIME Pathfinder experiences intermittent drop-outs: feeds whichin principle should be CHIME antennas but are in fact either turned off or malfunc-tioning. Correlations involving such channels are also non-redundant.If the number of inactive feeds is relatively small, the redundant baseline algo-rithm generally recognizes that these feeds are different and attempts to set theirgains to 0 - effectively removing them from the final set. However, for best accu-racy, we should remove these feeds ourselves before running the redundant base-line calibration algorithm.In a small set of feeds, this can be done by manually checking each feed’sautocorrelation and simply excluding them from the data set requested from thebroader CHIME Pathfinder data repository. For a larger set of feeds (e.g. theCHIME Pathfinder), this method becomes laborious to follow. Therefore, we must40develop a method to ensure we know which feeds should be excluded from ouranalysis and to exclude all correlations including such feeds.The first step is fairly straightforward - we establish by loading and plotting ahandful of working inputs’ autocorrelations a threshold value (in correlator units)for a valid feed’s autocorrelation then confirm that each autocorrelation in our dataset remains above that threshold. (While we exclude autocorrelations from ourredundant baseline analysis due to their noise properties, we can use them here aswe are simply ascertaining whether the feed is functioning properly as a CHIMEantenna.) We create a list of feeds which are not above this threshold and designatethem “dead feeds.” “Dead feeds” is a mild misnomer - they may be mechanicallyimpaired, absent, or simply never present (i.e. the channel is designated for analternative input such as a noise source, RFI antenna, or the John A. Galt 26 mtelescope).The second is slightly more complicated. At the most basic level, we can re-place the columns in our degeneracy fixing corresponding to these feeds with zeros(instead of ones). This excludes those feeds from the degeneracy fixing condition,but may not fully solve our problem as the “dead feeds” are still in the data set.Thus, we also create a slightly more complicated, but more accurate version. Wecreate a full coefficient matrix, with a degeneracy fixing condition that excludesdead feeds. We then create a vector of the length of the number of correlationswhich is one for all correlations we want to include and zero for all correlationsinvolving “dead feeds.” We multiply our coefficient matrix by this vector, zero-ing out all rows corresponding to correlations involving “dead feeds.” Thus, weforce the algorithm to exclude those correlations from its calculation and our finalsolution excludes “dead feeds.”With these caveats in mind, namely our exclusion of intra-cylinder baselineswith modified degeneracy fixing condition; our use of transits; and our exclusionof “dead feeds,” we are ready to apply redundant baseline calibration to CHIMEPathfinder data.413.1.2 ResultsBesides the modifications above, the redundant baseline algorithm remains un-changed. Below we see the results for a Cygnus A transit, first the recovered un-derlying visibilities and second the recovered redundant baseline gains.0 500 1000 1500 2000 2500 3000 3500 4000 4500Time (seconds) +1.444529e90.00.20.40.60.81.01.21.4|Vis|1e 8 Cygnus A transit: True visibilityFigure 3.1: Redundant baseline true visibilities for a June 2015 CygA transit.In this analysis, we use only inter-cylinder baselines and exclude “deadfeeds” using our more complete algorithm.The two polarizations aresolved for separately. The recovered true visibilities certainly recoverthe existence of a point source transit and a reasonable shape estimatefor it. We do observe some spikes through the solution - these maybe attributable to the varying level of redundancy for short vs. longbaselines. The units on the y-axis are correlator units, as redundantbaseline is a purely relative calibrator. We could normalize this plot toknow values for CygA should we prefer a plot in Jansky.The true visibility plot resembles our expected result closely enough that we arereassured that our algorithm is likely functioning. However, the extent of structurein the plot of redundant baseline gains is cause for concern. It indicates somethingunexpected is occurring in our solution, something that justifies further analysis.It is immediately apparent that the gains have significant slope. We attempt toquantify this amount of slope in Figure 3.3. We anticipate that during the relatively420 500 1000 1500 2000 2500 3000 3500 4000 4500Time (seconds) +1.444529e90.00.51.01.52.02.53.03.5|gain|Cygnus A transit: Gain AmplitudeFigure 3.2: Redundant baseline gains calculated for a June 2015 CygA tran-sit. In this analysis, we use only east-west baselines, and exclude “deadfeeds.” As described previously, we solve for East and South polariza-tions separately and recombine the results after our calculations. Noticethe definite slope in the gain values during the point source transit. Thisis contrary to our expectation that gains would not vary much on a short,point source transit time scale.brief timescale of the point source transit, the complex gains will change relativelylittle. Instead, we see dramatic changes, from e.g. 1.5 to 0.5. This could be doto purely mistakes in our algorithm, but this is unlikely because the true visibilitygraph was reasonable and the algorithm is almost totally identical to that used in thesuccessful simulation analysis. It seems most likely that this represents a deviationfrom our assumptions of redundancy. Recall from Chapter 2 which showed thatthe visibility was determined by the beam functions Ai and A j, the sky, and a phasefactor dependent on baseline distance. The redundant baseline algorithm assumesthis description is precisely correct and that Ai = A j. If this is untrue, then wewould see failures in the calibration.Independent analysis within the CHIME collaboration suggests that deviationsin the feed locations are relatively small. This diminishes the probability that the43baseline distance is causing deviations in our calibration. Additionally, we have notassumed a sky value but know that it would be consistent for all feeds and thereforediscount the sky as a possible source of deviation.This indicates the beam function is likely the source of the problem. If thetwo beam functions Ai and A j in any given visibility Vi j are not identical, then theredundant baseline algorithm would not operate as intended and could generateslopes in the gains such as those observed in Figure 3.2. This could also accountfor the spikes and deviations in Figure 3.1. If this is the case, we would expectrelatively small deviations from day to day for a given point source.While beam values might be very different at different points on the sky, theyshould not be significantly time dependent. Therefore, we plot a sampling of re-dundant baseline gain amplitudes over two consecutive CygA transits in Figure3.4. Notice that for a given feed, the two curves have similar structure, though fordifferent feeds, the curves show a variety of behaviours. This result is also con-sistent with our hypothesis regarding beams, which would not significantly varyfrom day to day. The absolute values vary, but this is not a cause for concern as theredundant baseline solution is only relative, so the absolute levels on two separatesets of data may vary freely.3.2 Examining Redundancy in the CHIME PathfinderEvidence from the section 3.1.2 suggests variation from redundancy in the CHIMEPathfinder, likely due to feed-to-feed beam variations. If that conclusion is ac-curate, deviations from redundancy should be visible in other analyses. Directlycomparing measured visibilities to check for redundancy is not a useful technique,as those measured visibilities contain complex gain information and are thereforeobviously non-redundant. We therefore require a set of calibrated data, with com-plex gains removed.Thanks to the efforts of the CHIME Pipeline Processing Team, such a set ofdata exists. Members of the CHIME collaboration created a set of fully processed,point source calibrated results for CHIME Pathfinder data from the fall of 2015 andthe spring of 2016 and selected consistently good quality frequencies to be used astest frequencies. The analysis that follows uses a subset of that data, specifically4460 70 80 90 100 110 120 130feed number0.030.020.010.000.010.02Approximate slope of gainWest Cylinder190 200 210 220 230 240 250 260feed number0.030.020.010.000.010.020.03Approximate slope of gainEast CylinderFigure 3.3: In this figure, we calculate a rough measure of the slope of eachgain, obtained by taking the rise over run for an individual feed’s redun-dant baseline gain results between samples 60 and 100. The 40 samplerange represents approximately one standard deviation around transit,which is the period of time in which we are most confident in our re-sults. This figure indicates that the slope effect observed by eye doesappear to be significant.pass 1p, which occurred from October 9-22, 2015. This data set was calibrated toCygA.We began our analysis with the simplest possible process: selecting pairs ofbaselines which should be redundant and plotting them during a point source tran-sit. Results from this process for four of the eight test frequencies and for allinstances of a short inter-cylinder baseline during a Cassiopeia A transit are shownin Figure 3.5. Each of these plots shows all instances of the same baseline, whichshould be redundant. Therefore, we would naively expect the graph to show onesingle trace, repeatedly overplotted. Perhaps accounting for small deviations orshifts in gain from CygA to CasA, there would be some small range around anaverage value. However, that is not what is recorded in Figure 3.5. We see instead450 10 20 30 40 50 60 70 80 900.40.60.81.01.21.4feed 65 gainday 0day 10 10 20 30 40 50 60 70 80 900.40.60.81.01.21.41.61.8feed 73 gainday 0day 10 10 20 30 40 50 60 70 80 900.60.81.01.21.41.61.8feed 80 gainday 0day 10 10 20 30 40 50 60 70 80 900.40.60.81.01.21.41.6feed 89 gainday 0day 10 10 20 30 40 50 60 70 80 900.40.60.81.01.21.41.61.8feed 97 gainday 0day 10 10 20 30 40 50 60 70 80 900.60.81.01.21.41.6feed 105 gainday 0day 10 10 20 30 40 50 60 70 80 90time(samples)0.81.01.21.41.61.82.02.2feed 113 gainday 0day 10 10 20 30 40 50 60 70 80 90time(samples)0.60.81.01.21.41.61.82.02.2feed 121 gainday 0day 1Figure 3.4: We compare redundant baseline gain results for selected feeds onthe west cylinder calculated over two consecutive CygA transits. Wenotice that there is very little deviation between the two days and re-gard this as evidence that the cause of the slope in the gain solution isnot strongly time dependent and is likely a property of the array. Anidentical figure showing the east cylinder is included in the Appendix.significant scatter in the values. Discounting very high or very low curves whichappear to be outliers, there seems to be about a factor of two variation betweeninstances of the same nominally redundant baselines.We continued our analysis by viewing a slice of the visibilities at the peak ofa CasA transit, as seen in Figure 3.6. This allows us to compare the relative am-plitude of the transits in each nominally redundant instance and quickly examinewhether there is obvious north-south structure in the value of the nominally redun-dant baselines. We suggest that there is not a significant pattern along the cylinder,46340 345 350 355 3600100020003000400050006000|Vis|CasA at 769 MHz340 345 350 355 36001000200030004000500060007000CasA at 643 MHz340 345 350 355 360Right Ascension0100020003000400050006000|Vis|CasA at 518 MHz340 345 350 355 360Right Ascension01000200030004000500060007000CasA at 443 MHzFigure 3.5: We compile redundancy comparisons for a short intercylinderbaseline during a CasA transit in pass 1p for four test frequencies. Nom-inally, these are instances of the same redundant baseline, and we wouldtherefore expect there to be little to no variation between curves in agiven frequency. Instead we observe significant deviation. We proposethat this deviation is largely derived from variations in the beam patternbetween feeds.though there are regions that are systematically higher or lower.We also examined the effect of changing to a different point source, to under-stand the declination dependence of the deviation from redundancy. By examininga Taurus A transit in Figure 3.7, identical to the transit plot from CasA in Figure3.5, we found that the variation from redundancy is not significantly declinationdependent. The exact ordering and amplitude of deviations from redundancy varysomewhat between CasA and TauA, leading us to believe there may be a slightdeclination dependence in deviation from redundancy. However, the existence ofthe phenomenon is consistent across different declinations, leading us to believe470 10 20 30 40 50 60 7005001000150020002500300035004000|Vis|CasA at 769 MHz0 10 20 30 40 50 60 7001000200030004000500060007000|Vis|CasA at 643 MHz0 10 20 30 40 50 60 700100020003000400050006000|Vis|CasA at 518 MHz0 10 20 30 40 50 60 70First Feed Number in Pair01000200030004000500060007000|Vis|CasA at 443 MHzFigure 3.6: We compile slices at CasA transit peak for each nominally re-dundant instance of a short inter-cylinder baseline at each of four testfrequencies. We notice that there is not a defined pattern in visibilityvalue based on feed location, which indicates that the effect causing thedeviation from redundancy does not vary in a systematic way along theCHIME Pathfnder cylinder..48that is not the primary factor.70 75 80 85 9005001000150020002500|Vis|TauA at 769 MHz70 75 80 85 9005001000150020002500TauA at 643 MHz70 75 80 85 90Right Ascension05001000150020002500|Vis|TauA at 518 MHz70 75 80 85 90Right Ascension05001000150020002500TauA at 443 MHzFigure 3.7: We compile redundancy comparisons for a short intercylinderbaseline during a TauA transit in pass 1p in the same manner as Figure3.5. We note that while the exact magnitude of deviations from redun-dancy may differ, the general structure of the deviation is similar to thatpresent in the CasA data, indicating the existence of such deviations isnot declination-dependent although the values may be.Until this point, the redundancy analysis has focused on sidereal stacks andtherefore taken into account each day of data in the stack. One potential cause forthe deviation from redundancy could be deviations across days. Perhaps one verydeviant day influenced the rest of the stack and created the appearance of deviationsfrom redundancy.Therefore, we plotted comparisons between two consecutive sidereal days,CHIME Sidereal Day (CSD) 693 and CSD 694. However, each individual dayshows a similar pattern to the overall sidereal stack. Comparing the differencebetween the two days, it is clear that the deviation between the two days is signif-49icantly smaller than the deviation within a given baseline on a given day. (It is infact about 10% of the spread in each individual day).340 345 350 355 360Right Ascension (degrees)0100020003000400050006000700080009000|Visibility|CasA Transit CSD 693 Short EW Baseline340 345 350 355 360Right Ascension0100020003000400050006000700080009000CasA Transit CSD 694 Short EW BaselineFigure 3.8: In this figure, we observe all instances of a nominally redundant,short inter-cylinder baseline at frequency 518 MHz, compared betweenCSD 693 and CSD694. Notice that both individual days are signifi-cantly non-redundant, lessening the likelihood that the deviations fromredundancy present in the sidereal stack for pass1 p are caused by a de-viant day included in the pass. It appears further that deviation fromredundancy is not strongly time-dependent.Based on several different pathways of analysis, we conclude that there are sig-nificant deviations from redundancy in point source calibrated CHIME Pathfindervisibilities. Based on the consistent existence of these deviations across differentdeclinations and times, we assert that these deviations are probably due to feed-to-feed beam variations. This conclusion is strengthened when combined withevidence from our redundant baseline test, which suggested similar non-time de-pendent deviations from redundancy.We hypothesize that this lack of redundancy is arising from feed-to-feed beamvariations, as that would explain both the presence of the deviations across timeand declination as well as the deviations in redundant baseline results. Armedwith that knowledge, we postulate that the redundant baseline algorithm may beable to probe these feed-to-feed beam variations for periods where the complexgain is relatively constant (e.g. a point source transit). We will investigate thesepossibilities further in Chapters 4 and 5.50340 345 350 355 360RA0246810|Vis| Ratio between DaysRatio between CSD 693 and CSD 694340 345 350 355 360RA3000200010000100020003000|Vis| Diff between DaysDifference between CSD 693 and CSD 694Figure 3.9: The right hand panel shows the ratio between the two panels ofFigure 3.9 and the left hand panel shows the difference between them.Each is a short inter-cylinder baseline at frequency 518 MHz on CSD693 and CSD694. The deviation between days is on the order of 100,while the spread within a day is on the order of 1000, meaning the de-viation between instances is much larger. Though the ratio is relativelylarge for areas outside of the central transit, at the transit peak, it isapproximately 1.51Chapter 4Redundant Baseline Calibrationwith Perturbed Beams4.1 Creating a Simulation with Beam PerturbationsAs we saw in Chapter 3, the actual CHIME Pathfinder instrument shows significantperturbations in beams, and it is therefore valuable to have a method of simulatingsuch beam perturbations to test analysis and calibration techniques in a controlled,realistic setting. We have therefore constructed an extension to the existing CHIMEsimulation pipeline which can flexibly incorporate beam perturbations. The designof the beam perturbation simulation is described in greater detail in the Appendix.4.1.1 Design of beam perturbationWe treat the beam perturbation or perturbations as additional parameters in theexpression for beam amplitude, A. We then Taylor expand to first order and moveforward with a first order representation of the beam.In other words, we transform the beam function from A(nˆ;φ) to A(nˆ;φ ;α),where α is some perturbation to a parameter of the beam, such as full width athalf maximum or pointing. We could also use this same structure to introduce anarbitrary number of perturbations, designating each by a Greek letter: α , β , δ , etc.52Our expression for the visibility is thenVi j =∫d2n Ai (nˆ;φ ;αi)A∗j (nˆ;φ ;α j)exp(2pinˆ ·uij)S, (4.1)where αi and α j are perturbations unique to each feed i and j, and S represents thesky.One way to generate such visibilities would be to uniquely determine beamamplitudes A(nˆ;φ ;α). However, directly incorporating unique beam functionswould be computationally expensive, as we would have to generate beam transfermatrices for each individual input (256 for a CHIME Pathfinder sized simulation)and then would have no redundancy present in our modified pipeline, requiringalmost an entirely separate pipeline.We do not require an exact representation of the individual beam functions,but only a reasonable approximation. Provided the beams are different from oneanother in a way we can recreate, we have achieved our goal. Therefore, we Taylorexpand the beam functions as a function of α and keep only terms to first order.Thus, we expand each A asAi (nˆ;φ ;α)≈ A(0) (nˆ;φ ;0)+A(1) (nˆ;φ ;0)α+O(α2)(4.2)and rewrite the visibility expression asVi j =∫d2n(Ai(0)(nˆ;φ)+A(1)i (nˆ;φ)αi)×(A∗(0)j (nˆ;φ)+A∗(1)j (nˆ;φ)α j)×exp(−2pi i′ nˆ ·uij) S.(4.3)Keeping only terms to first order, we haveVi j =∫d2n[A(0)i (nˆ;φ)A∗(0)j (nˆ;φ)+A(0)i (nˆ;φ)A∗(1)j (nˆ;φ)α j +A(1)i (nˆ;φ)A∗(0)j (nˆ;φ)αi]×exp(−2pi i nˆ ·uij) S.(4.4)We can think of this visibility then as being composed of three separate compo-53nents: an unperturbed visibility with beam A(0)i A∗(0)j and two combinations of anunperturbed and perturbed beam, i.e. with beams A(1)i A∗(0)j αi and A(0)i A∗(1)j α j.Should we desire more than one perturbation per polarization, we would simplyrepeat the Taylor expansion and recombination process for each additional param-eter β , γ , δ , etc.A way to avoid creating a full N set of beam transfer matrices then becomesapparent. Ordinarily, we would generate a set of beam transfer matrices for allA. Now, we generate a set for all A(0) and A(1) for each feed. We then use thesebeam transfer matrices to create a sidereal stream as in the usual simulation code.However, the usual code creates three times the correct number of products, as itcreates separate products for each combination of beam amplitudes and derivatives.Until this point, we have not had to alter the simulation pipeline code, merelythe input. Following this point, we want to combine perturbed and unperturbedcomponents, so we must alter the pipeline code to accommodate the new productstructure.4.2 Redundant Baseline Calibration ResultsWe previously supposed that a deviation expected results in redundant baseline am-plitude calibration could arise from feed-to-feed beam variations. In the redundantbaseline algorithm, one of our key suppositions is that the beam function for eachfeed is identical. When that assumption fails and there is feed-dependent beam in-formation in the input visibilities, the redundant baseline algorithm will attempt toincorporate that information into the only solely feed-dependent output it has: thegain values. Therefore, we expect that a perturbed beam’s beam structure will bevisible in redundant baseline gain amplitude results. (This conclusion is supportedby a comparison with holography data in Chapter 5.)It can be slightly complicated to establish a direct probe of the simulatedbeam’s structure as we do not have the ability to do e.g. simulated holographymeasurements. Plotting slices of the beam map is also complicated by the factthat the perturbation values α are added not in the initial telescope definition butin the “ExpandPerturbedProducts” step. For the moment, then, we use a slightlyindirect probe. We run the redundant baseline algorithm on both the final simulated54data set with gains, receiver temperature, and noise and an intermediate timestreamproduct, prior to the addition of gains. This intermediate product has no feed-to-feed variation from complex gains, so the gain amplitude values that the redundantbaseline algorithm generates are solely driven by feed-to-feed beam variations.Our first analysis step is the simplest: we simply plot the beam-only redundantbaseline results, the full redundant baseline results results, and the input gains.We notice that the full analysis mostly tracks the input gains, but has noticeabledeviation from them, particularly near point source transit. We hypothesize thatthe full result minus input gains quantity should be equivalent to the beam onlyresult (up to some constant offset). We plot these results and find this to be thecase.Correspondingly, we find that if we take the ratio of the redundant baseline so-lution with gains and beam variations to the redundant baseline solution with onlybeam variations, we recover the input gain variations as well as in the unperturbedcase. This is also highly encouraging, as it indicates that the addition of beamvariations does not irreparably corrupt the gain solutions, provided the informationfrom beam and gain variations can be separated.4.3 Solving for Beam Perturbation ValuesGiven our conclusion that the redundant baseline gain solutions in the presence ofbeam perturbations trace the per-feed beam perturbation structure, we would likeuse those solutions to solve for quantitative information about the beam pertur-bations from the redundant baseline gain information. Here, we attempt to find amethod to quantify the information redundant baseline calibration preserves and/orto develop a scheme to solve for beam perturbations (especially width perturba-tions) using the nominal redundancy of an instrument like the CHIME Pathfinder.This analysis assumes two very important things. First, we assume that thebeam pattern can be modeled as a fiducial beam and an arbitrary number of per-turbations to the fiducial model. Second, we assume that the telescope in questionhas an alternative calibration method for complex gains such as a noise source or acarefully constructed thermal model.550 50 100 150 200 250 300 3500.981.001.021.041.06|gain|Input vs. recovered: input 128gain + beambeam onlygain only0 50 100 150 200 250 300 3500.960.970.980.991.001.011.021.03Diff gain vs. beam: input 128gainbeam0 50 100 150 200 250 300 3500.981.001.021.041.061.081.101.12|gain|Input vs. recovered: input 136gain + beambeam onlygain only0 50 100 150 200 250 300 3500.960.970.980.991.001.011.021.03Diff gain vs. beam: input 136 gainbeam0 50 100 150 200 250 300 3500.951.001.051.10|gain|Input vs. recovered: input 144gain + beambeam onlygain only0 50 100 150 200 250 300 3500.970.980.991.001.011.021.031.04Diff gain vs. beam: input 144gainbeam0 50 100 150 200 250 300 3500.940.950.960.970.980.991.001.011.021.03|gain|Input vs. recovered: input 152gain + beambeam onlygain only0 50 100 150 200 250 300 3500.970.980.991.001.011.021.03Diff gain vs. beam: input 152gainbeamFigure 4.1: The left panel displays perturbed beam redundant baseline gainamplitude results, for selected feeds of a given polarization in a sim-ulated perturbed beam telescope, compared with the input gains and aredundant baseline analysis conducted on data without gains added (i.e.an analysis that detects only beam effects). The right panel comparesthe beam only analysis to the full analysis with redundant baseline gainssubtracted. This figures shows only a small sampling of inputs; more areshown in the Appendix. We notice that the full redundant baseline so-lutions deviate from the input gain solution near the peak of the beamonly solutions. We infer that this deviation is caused by the beam per-turbations, and the right hand panel confirms this.560 50 100 150 200 250 300 3500.981.001.021.041.06|gain|Input vs. recovered: input 128gain + beambeam onlygain only0 50 100 150 200 250 300 3500.981.001.021.041.06gain+beam/beam: input 128gain+beam/beamgain only0 50 100 150 200 250 300 3500.981.001.021.041.061.081.101.12|gain|Input vs. recovered: input 136gain + beambeam onlygain only0 50 100 150 200 250 300 3501.011.021.031.041.051.061.071.081.09gain+beam/beam: input 136gain+beam/beamgain only0 50 100 150 200 250 300 3500.951.001.051.10|gain|Input vs. recovered: input 144gain + beambeam onlygain only0 50 100 150 200 250 300 3500.900.951.001.05gain+beam/beam: input 144gain+beam/beamgain only0 50 100 150 200 250 300 350Time (samples during transit)0.940.950.960.970.980.991.001.011.021.03|gain|Input vs. recovered: input 152gain + beambeam onlygain only0 50 100 150 200 250 300 350Time (samples during transit)0.930.940.950.960.970.980.991.001.01gain+beam/beam: input 152gain+beam/beamgain onlyFigure 4.2: The left panel shows perturbed beam redundant baseline gain am-plitude results, for selected feeds of a given polarization in a simulatedperturbed beam telescope, compared with the input gains and a redun-dant baseline analysis conducted on data without gains added (i.e. ananalysis that detects only beam effects. The right panel compares theratio of the beam and gain to beam only analysis and the input gainvariations. the full analysis with redundant baseline gains subtracted,showing that the ratio recovers the correct input gain. As in Figure 4.1,results for more inputs are shown in the appendix.574.3.1 Solving with One Perturbation of Known StructureBefore we begin, we apply redundant baseline calibration to data simulated withthese beam structures and subtract the known gains from the redundant baseline re-sults. Alternatively (and in practice, preferably) we can just run redundant baselinecalibration on a simulation with no gains added. Thus all the gain deviation from 1must be due to beam effects. We now have recovered visibilities, which are createdby recombining the recovered redundant baseline gains and true visibilities. This,mathematically, isVrecij = exp(Mxˆ), (4.5)where M is the coefficient matrix in the redundant baseline problem as in Chapter2 and xˆ is the vector of recovered “gains” and true visibilities.We can expand Vijrec in equation 4.5 asVrecij = Vij0+∑(i j)αi∫A1i A∗0j e−(2piinˆ·uij)T (nˆ)d2nˆ+∑(i j)α j∫A∗1j A0i e(−2piinˆ·uij)T (nˆ)d2nˆ(4.6)We rewrite this into an entirely matrix equation setupVrecij = Vij0+Wijα. (4.7)Here, Vrecij is a vector of the recovered visibilities, Vij0 is a vector of the 0thorder (unperturbed) visibilities, α is a vector of beam perturbation values, and Wijis a matrix containing the appropriate V 1i j terms. The W matrices must be designedto encapsulate the structure of the beam perturbations and be able to be combinedwith the only desired perturbation value. While W is similar to M in that it gener-ates combinations of different feeds, it is different in that its entries are either 0 ora perturbation structure not 0 or 1. W is N(N+1)/2 rows by N columns, where Nis the number of feeds in a given telescope. Each row corresponds to a correlationij. In a given row Wi j, the ith element is of the form V 1i j,i =∫A1i A∗0j T e−2piinˆ·uij d2nˆand the jth element is of the form V 1i j, j =∫A0i A∗1j T e−2piinˆ·uij d2nˆ. Then, α is freeto be a vector of length N where each element is the perturbation corresponding to58a given feed. For example, in a four feed one cylinder telescope, W would beW =V 1i j,i V1i j, j 0 0V 1i j,i 0 V1i j, j 0V 1i j,i 0 0 V1i j, j0 V 1i j,i V1i j, j 00 V 1i j,i 0 V1i j, j0 0 V 1i j,i V1i j, j(4.8)Then, if we presume we understand the structure of the beam (but not the per-turbation value), we know V0 and W. We can then solve our matrix equation forαW+(Vrecij −V0ij)= α. (4.9)This method works well if we are able to provide full information (phase andamplitude) about the recovered visibilities. However, we would like to attempt thissolution with just information about the recovered redundant baseline amplitudes.4.3.2 Solving for Beam Perturbations with only AmplitudeInformationIn our current redundant baseline solution, we do not have full phase and ampli-tude information about recovered visibilities or “gains,” but only have amplitude.Therefore, instead of evaluating Equation 4.5, we need to evaluateln |Vijrec|= ln |V0ij+Wijα|. (4.10)We can evaluate the absolute value function by multiplying Vij0+Wijα by itscomplex conjugate, giving usln |Vijrec|= 12 ln((V0ij+Wijα) (V0ij+Wijα)∗)(4.11)We multiply out and Taylor expand to first order and are left withln |Vrecij |= ln |V0ij|+12α(V1∗ijV0∗ij+V1ijV0ij)(4.12)59We then redefine the non-zero elements of |Wi j| to be|Wij| ≡ 12(V1∗ijV0∗ij+V1ijV0ij)(4.13)and revert to the simpler expressionln |Vrecij |= ln |V0ij|+α|Wij|. (4.14)Finally, we solve for α:α = |Wij|+(ln |Vrecij |− ln |V0ij|). (4.15)If we assume that we know V 0 and V 1, this procedure produces a consistentsolution. See Figure 4.3 for an example with a small two cylinder, sixteen inputtelescope. We test this solution both using “input visibilities” - the actual perturbedvisibilities generated by the simulation - and the “recovered visibilities” - the vis-ibilities recovered using the redundant baseline solution. For the “input” case wefind near perfect agreement. The absolute deviations from the correct solution re-main approximately the same size for all perturbation values. Due to the rangeof actual values for α , from about 0.01 to about 0.001, percent differences corre-sponding to feeds with smaller perturbation values due have muh larger percentdifferences. For the “recovered” case, we find the agreement is relatively poor, butis correct to the first significant digit or better for all feeds. This is not surprising,as any variation in the redundant baseline solution propagates to the beam pertur-bation solution. The process of applying redundant baseline calibration does resultin a loss of information, which cannot be regained for the perturbation solution.This method solves for alpha uniquely at each time point. This means we canlearn additional information by solving for a time series of α values. Based onthe design of our simulation, α should be constant over all time, so we would beable to improve our estimate by considering possible variation in the recoveredα values. We notice that α solution are not actually constant, but they seem toimprove in constancy in regions of higher signal to noise. As in the individual timepoint solution, we find that the beam perturbation values from the “recovered” data600 2 4 6 8 10 12 14 16Feed number0.0200.0150.0100.0050.0000.0050.0100.015Perturbation valueBeam Perturbation ValuesActual valueInput known visRB recovered visFigure 4.3: This figure shows beam perturbation solution for a 16 feed totaltelescope with random perturbations α applied in the beam width of allfeeds has a few noticeable features. The actual input α values used tocreate the simulation are plotted in blue, but are almost exactly over-plotted by the green values. The green values represent the result whenthe output of the simulation is used as the “recovered visibility,” and wetherefore expect this close correspondence. The red curve represents theresults using the redundant baseline calibration results as the “recoveredvisibility” and is noticeably less accurate than the green.set deviate slightly from the exact input α values used in the simulation, but thisremains consistent with loss of information in the redundant baseline calibrationalgorithm. One apparently odd characteristic is that the ordering of α values fromlargest to smallest is not necessarily constant at all times. This arises from theindependent solutions at each time point. This allows imprecise solutions for αvalues which are close in magnitude to cross one another at different time points.4.3.3 Relaxing Assumptions about V 0i j and V1i jIn our analysis thus far, we have assumed that we know the unperturbed first orderperturbed components of the visibility, V 0i j and V1i j. However, this assumes that we610 2 4 6 8 10 12 14 16Feed number1.51.00.50.00.51.0Percent differenceInput Known Vis0 2 4 6 8 10 12 14 16Feed number302010010203040RB Recovered VisFigure 4.4: We see here the percent difference between the actual simulatedα beam perturbation values and the recovered α values for both the ac-tual input visibility and the recovered redundant baseline solution as in-put visibility. We observe that the input visibility has quite good agree-ment with the actual perturbation values, but the recovered redundantbaseline solution is useful only for order of magnitude approximations.know the convolution of the zeroth and first order components of the beam withthe sky at all times. This is necessarily not the case for real data, so we would likea method to replace these parameters.One way forward is to take advantage of the near-redundancy of the baselines.Each instance of a baseline is composed of a zeroth order part, which is redundantwith other baselines of the same length, and a first order part which is not. There-fore, presuming the perturbation values are scattered about zero, we can take anaverage of all instances of a given nominally redundant baseline and substitute thisaverage for V 0i j. In other words,Vi− j ≈V 0i− j =1m(i j)m∑k=(i j)0V meask , (4.16)where m is the number of instances of a given nominally redundant baseline and Iam using the convention that Vi j is an individual correlation between feeds i and jand Vi− j is the redundant correlation for all i’s and j’s with the baseline spacing i-j.620 50 100 150 200 250 300 350 4000.040.030.020.010.000.010.02Beam PerturbationSim as Input: Alpha Time Series0 50 100 150 200 250 300 350 4000.040.030.020.010.000.010.02Beam PerturbationRB as Input: Alpha Time Series0 50 100 150 200 250 300 350 400Right Ascension (Degrees)10203040506070|Vis|Comparison Time SeriesFigure 4.5: This figure examines the beam perturbation solution for a 16 feedtotal telescope with all feed perturbed for a sidereal day. Each timepoint is solved independently, but time dependent features are consis-tent with the redundant baseline solution more generally, e.g. that solu-tion improves with improved signal to noise ratio. The top panel showsthe result for the solution using the simulation output as the “recoveredvisibility,” the middle panel shows the result for a solution using theredundant baseline calibration results as the “recovered visibility,” andthe final panel shows a single visibility’s time series during this siderealday.63Then, we can subtract the average Vi− j from each actual measured Vi j andare left with an approximation of the perturbed components, which we will call∆Vi j, for each instance of the baseline.We know in principle that if Vi− j is a goodapproximation for V 0i j, ∆Vi j should be the perturbed portion of Vi j or rather. Insummary,∆Vi j ≡V measi j −Vi− j ≈ αiV 01i j +α jV 10i j . (4.17)However, we do not have enough information to separate ∆Vi j into i and j perturbedportions, so we consider the entire perturbed portion as being∆Vi j =Wi− jα, (4.18)where Wi− j is a matrix which in some way encodes the structure of the perturbedbeam and α is the perturbation values as before. It is important to note that thisWi− j is not identical to W referenced previously. That W has independent entriesfor each component of the perturbation structure, whereas this Wi− j is a singlevalue per unique baseline.At first glance, we cannot uniquely solve for all values of Wi− j and α , but ifwe can solve for each α value in terms of one of the α values. In this case, wesolve for (α j +α0) for each α j. We can also solve for a combination of (α j +α0)and (α j−α0), and use the additional information from the (α j−α0) solutions todetermine the sign of each α j value. However, we focus on here on the (α j +α0)solutions because instances of (α j−α0) do not exist for all α j.We begin by writing ∆V0i for all i. These equations take the form∆V01 =W0−1(α0+α1)∆V02 =W0−2(α0+α2)...∆V0n =W0−n(α0+αn).(4.19)We will then chain together instances of these known 0− i baselines to createa system of equations which can be linearized and solved in much the same way asthe redundant baseline problem. This chaining process is simplest for the shortest64baseline, 0−1. For example, for V12, we write∆V12−∆V01 =W0−1(α2+α1)−W0−1(α1+α0) =W0−1 (α2−α0) , (4.20)and for V13 we write∆V13−∆V12+∆V01 =W0−1(α3+α2)−W0−1(α2+α1)+W0−1(α1+α0)=W0−1 (α3+α0) .(4.21)In chaining instances of the shortest baseline, we create a combination of(α j +α0) terms and (α j−α0) terms. We solve for both terms simultaneously,but focus on the (α j +α0) terms.For this shortest north-south baseline, we create this type of combination foreach subsequent instance by using the formulaW0−1 (αm±α0) =m∑i=1∆Vi,i−1(−1)(i+m), (4.22)where m is the feed we are isolating. Equations with −1m = −1 are (α j−α0)equations, while equations with −1m = 1 are (α j +α0) equations.We can write a general relationship analogous to Equation 4.22, but we must beslightly more creative. For baselines longer than the shortest north-south baseline,we will not be able to chain every instance, but will only be able to use a limitedset. Our more general relationship isWδαm·δ =m∑i=0∆Viδ ,(i+1)δ (−1)i+m, (4.23)where δ is the difference in feed indices, m is the instance number, and ∆V is asbefore. As before, odd m values lead to (α j−α0) equations and even m valueslead to (α j +α0) equations.For each baseline we will have a different number of instances, varying in-versely with the length of the baseline. For example, in our 32 feed one cylindertelescope, we will have 31 (m = 0 to m = 31) instances of the shortest baseline,but only 15 instances (m = 0 to m = 15) of the second shortest baseline and 1065instances (m = 0 to m = 10) for the third shortest baseline. This does imposes arelatively large minimum size on the telescope. For a one cylinder telescope, werequire 32 inputs per polarization to solve for both (α j +α0) and (α j−α0) values.We have now developed a system of equations of the form ∆Vi j =Wi− jα j. Welinearize these equations in the same manner as in the redundant baseline problemoutlined in Chapter 2, by taking the logarithm. Then, our equations are of the formlog |Vi j|= log |Wi− j|+ log |α j|. (4.24)Continuing as in the standard redundant baseline problem, we construct a matrixMαW reminiscent of the M matrix of Chapter 2 and re-write our problem as avector equation∆V = MαW xˆ, (4.25)and solving for xˆ, a vector of α and Wi− j values.This solution is underdetermined for our small test problems, meaning that theresult represents a minimum norm solution and not a unique determination of thevalues for (α j +α0), (α j−α0), and Wi− j. Therefore, we do not necessarily expectto see perfect recovery of the actual values for (α j +α0), (α j−α0), and Wi− j. Byexamining the null space of the MαW , we can make more precise determinationsof which values may deviate from expected. In future analysis, applying additionalindependent constraints may enable us to determine values more accurately, butthis underdetermined solution offers a starting point. A larger telescope modelwould create an over-determined problem instead of an underdetermined one, butis left for future work.By determining the ratio of (α j +α0) to a particular (α j +α0) value (and thesame for (α j−α0)) for both the original simulated α values and the recovered(α j +α0) and (α j−α0) results, we can begin to assess the effectiveness of thisapproach. Figures 4.6 and 4.7 show a first result, which correlates strongly withthe known values determined from the actual simulations.660 5 10 15 20 25 30 35Feed number0.51.01.52.02.53.0Ratio relative to feed 2Comparing alpha ratio values: plus alpha0Recovered alphaActual alphaFigure 4.6: We examine the ratio (α j +α0)/(α2+α0) for both the recov-ered (α j +α0) values and the actual (α j +α0) values for a 32 feed onecylinder telescope. The solution values are close to the expected val-ues, except at noticeable outlier feed 16. The imperfect correspondenceto the correct answers, in spite of the absence of noise, is due to theunderdetermined nature of the problem.4.4 ConclusionsWe simulated a CHIME Pathfinder like telescope with a beam perturbation in fullwidth half maximum, then applied the redundant baseline calibration method to it.By applying the redundant baseline calibration algorithm as outlined in Chapter2, we have confirmed our hypothesis that in non-ideal cases, redundant baselinecalibration is sensitive to the effects of non-identical beams.We also examined possible methods for extracting information about beamperturbations from either gain-calibrated data or from gain-calibrated, redundantbaseline calibrated data. However, our approach is limited in one case by the as-sumption of a known beam and sky model at all times. In both cases, our approachis limited by the assumption of noiseless data and perfect calibration.However, we conclude that there is strong evidence for the utility of redundantbaseline calibration and related methods in characterizing feed-to-feed beam per-turbations. In Chapter 5, we will demonstrate the usefulness of redundant baseline670 5 10 15 20 25 30 35Feed number024681012Ratio relative to feed 2Comparing alpha ratio values: minus alpha0Recovered alphaActual alphaFigure 4.7: We examine the ratio (α j−α0)/(α2−α0) for both the recov-ered (α j−α0) values and the actual (α j−α0) values for a 32 feedone cylinder telescope. Unlike the (α j +α0)/(α2+α0) solution, the(α j−α0)/(α2−α0) solution exists only for approximately every otherinput. As these results derive from the same underdetermined problem,they too deviate from the expected values in ways that can be examinedmore carefully using the null space.calibration in predicting beam perturbations in actual data by comparing to pointsource holography, a well established beam mapping approach.68Chapter 5Holography and RedundantBaseline Calibration as BeamProbesThough results from simulations are powerful evidence for the efficacy of redun-dant baseline calibration in finding feed-to-feed beam variations in the CHIMEPathfinder, this result is essentially worthless if it cannot be applied to real CHIMEPathfinder data. However, conducting such an analysis on real CHIME Pathfinderdata is challenging as we do not know the exact beam pattern of the telescope orthe actual complex gain. We must find an additional test to check the validity of ourredundant baseline analysis. We choose to compare our redundant baseline anal-ysis to holographic beam measurements conducted using the CHIME Pathfinderand the John A. Galt 26 m telescope at DRAO (hereafter the 26 m telescope). Asboth methods should measure beam amplitudes, if our redundant baseline analysisis working as we believe it is, the two methods will agree.5.1 Holography: A method for probing CHIME beamsThe importance of precise beam measurements for an instrument like CHIME haslong been known, and therefore the CHIME collaboration has been engaged inmapping the full two-dimensional primary beam of each feed with point-source69holography during the operation of the CHIME Pathfinder [6].Holographic beam mapping is a recognized technique to make beam measure-ments in radio telescopes. The maps are made by tracking a bright point sourcewith one reference telescope while correlating that telescope with the CHIMEPathfinder. During the transit, we measure the source’s track through the stationaryCHIME Pathfinder beam and are thus measuring the east-west track of the CHIMEPathfinder beam. Specifically, we introduce the 26 m telescope as an additionalinput, correlating it as if it were a CHIME pathfinder channel. Thus, we create aset of visibilitiesV26 i ∝ A26 A∗i (nˆps;φ)T (nˆps)exp[−2piinˆps ·uij] (5.1)As the 26 m telescope can only point at one point source at a time, we requiremultiple measurements at different declinations to properly measure the north-south shape of the beam. Holography efforts for the CHIME Pathfinder to dateuse Cygnus A, Taurus A, Virgo A, Hercules A, Hydra A, Perseus B, and 3C 295[6].5.2 Comparing redundant baseline beam measurementswith holography dataTo evaluate our redundant baseline beam probe’s efficacy on real data, we used datafrom the July 2015 Cygnus A holography run combined with a redundant baselinesolution for the same transit, applying our best practices, inter-cylinder only re-dundant baseline algorithm (as described in Chapters 2 and 3) to the uncalibratedCHIME pathfinder data from the same transit.The holography analysis generates absolute measurements of the beam pattern,whereas our redundant baseline results are both only relative and only applicableto feed-to-feed variations (e.g. not a complete trace of the beam). Therefore, wemust be a bit creative in comparing the two data sets. Specifically, both shouldrecord relative, feed-to-feed variations. Therefore, we primarily analyze the data bylooking at ratios between each CHIME Pathfinder feed and an arbitrarily selectedreference CHIME feed. In order to orient ourselves in analyzing the holography70data, we do a brief comparison by eye.5.2.1 Comparing by eyeIn the simplest method for comparing redundant baseline transits with hologra-phy transits, we simply plot both the redundant baseline gains and the holographytransits on the same axes and looks for correlation between the slope of the gain(positive or negative) and the shift in the peak of the transit (right or left) from theexpected point. To do this comparison, we must scale up the holographic beammeasurements, as their value in correlator units is of the order of 10−9 while thegains are of the order 1 by design.Figures 5.1 and 5.2 show a summary of these plots, showing every eighth feedon the CHIME Pathfinder, with the goal of acquiring a sampling of the cylinder.There does appear to be a correlation between the slope of the gains and the shiftin the peak from the average peak sample, but it is not easily quantified in this “byeye” analysis.5.2.2 Ratio analysisWe anticipate that the redundant baseline gains include information both about thecomplex gain and beam of each receiver. We further approximate that during abright point source transit, each feed sees approximately the same very bright sky,dominated by a source at one declination. Therefore, we can roughly approximatethe visibility asVi j = gi g∗j Ai A∗j T, (5.2)where T is a constant sky value. We further approximate that CHIME Pathfindergains are constant on the timescale of a single point source transit. Thus, relativeshape variations between two redundant baseline gain solutions necessarily arisefrom beam variations (since the sky and the gain are taken to be constant).We can simply represent the variations between each feed’s beam pattern bytaking the ratio between each calculated redundant baseline gain and an arbitrarilychose reference gain (in this case, that of feed 65). In other words the ratio RRBwe710 20 40 60 80 100 120 140 160 1800123456Feed 650 20 40 60 80 100 120 140 160 18001234567Feed 730 20 40 60 80 100 120 140 160 1800123456Feed 810 20 40 60 80 100 120 140 160 1800123456Feed 890 20 40 60 80 100 120 140 160 18001234567Feed 970 20 40 60 80 100 120 140 160 1800123456Feed 1050 20 40 60 80 100 120 140 160 180Time (samples)0123456Feed 1130 20 40 60 80 100 120 140 160 180Time (samples)012345678Feed 121Holography Trace/Two Day RB ComparisonFigure 5.1: Comparison between holography transit and calculated redundantbaseline gain for selected feeds on the west cylinder, looking only atresults from the east west polarization. The two gain traces representsolutions for CygA transits on consecutive days. The average transitpeak time is marked by the blue vertical line, and the shift to before orafter the average peak time appears to correlate with the slope of theredundant baseline gain.720 20 40 60 80 100 120 140 160 18001234567Feed 1920 20 40 60 80 100 120 140 160 1800123456Feed 2000 20 40 60 80 100 120 140 160 1800123456Feed 2080 20 40 60 80 100 120 140 160 1800123456Feed 2160 20 40 60 80 100 120 140 160 1800123456Feed 2240 20 40 60 80 100 120 140 160 1800123456Feed 2320 20 40 60 80 100 120 140 160 180Time (samples)01234567Feed 2400 20 40 60 80 100 120 140 160 180Time (samples)0123456Feed 248Holography Trace/Two Day RB ComparisonFigure 5.2: Comparison between holography transit and calculated redundantbaseline gain for the selected feeds on the east cylinder, EW polariza-tion. The average transit peak time is marked by the blue vertical line,and the shift to before or after the average peak time appears to correlatewith the slope of the redundant baseline gain.73are interested in isRRB =gRB,igRB,65(5.3)for each feed i. The absolute level of each ratio is irrelevant (and approximatelyone), but the shape of the ratio traces the ratio of the feed’s beam pattern to thereference feed’s beam pattern. In other words, we examine bothWe also replicate this ratio using the holography data. We take the ratio be-tween each channel’s holography cross-correlation and a reference channel’s holog-raphy cross-correlation. In other words the ratio Rholo we are interested in isRholo =Vholo,iVholo, 65, (5.4)for each feed i. In this ratio, we cancel out the common portions of the beam patternand are also left with a ratio describing feed-to-feed beam variations. Therefore,we can now make quantitative, head-to-head comparison between gain ratios andholography ratios to see whether the two show consistent beam variations.If the two methods of analyzing the beams are consistent, we would expect thedifference between them to be 0 during transit. It is however important to note thatwe would not necessarily expect consistency between the two methods before orafter the point source transit. Our approximation in Equation 5.2 would no longerapply, and the comparison would no longer be meaningful.For context, we first plot the redundant baseline measure of feed-to-feed beamvariation independently of the holography data. This is calculated by dividing eachredundant baseline gain by the value of a reference gain, from feed 65, at each timepoint. If our reference feed has anomalous structure in its gain, this is a dangerousmethod as it will propagate that structure into each ratio. Therefore, before calcu-lating these ratios, we established that feed 65 had no significant anomalies relativeto other feeds. The only traces at or near zero are those of “dead feeds,” which areinput channels that are either not functioning or not connected to CHIME channels.We then compared the two ratios by taking the difference between them. Thisgives us a sense of the deviation between feed-to-feed beam variation found by740 10 20 30 40 50 60 70 80 90Time (samples)0.00.51.01.52.02.5RB Gain amplitude relative to feed 65Per feed Redundant Baseline Gain VariationFigure 5.3: Redundant baseline gain amplitude results relative to a referenceredundant baseline gain solution (from feed 65). The variations in thisplot are expected to trace feed-to-feed beam amplitude variations. Thesefeed-to-feed beam amplitude variations will then be compared to holog-raphy results to verify the correspondence.holograph and by the redundant baseline gains. In other words, we examine∆RB-holo =Vholo,iVholo,65− gRB,igRB,65. (5.5)We quantified these results by we taking averages of the differences, both foreach feed during transit and for each time over all feeds. Figure 5.4 shows theaverage difference between redundant baseline and holography beam estimates ateach time point. Recall that the transit peaks at approximately sample 80 and that itreaches half its maximum value approximately 20 samples before and after transit,so the prime transit region is samples 60-100, with the region worth consideringstretching between samples 40 and 120. Notice that, while areas outside of the75transit in Figure 5.4 are quite erratic, areas in the sample 40-120 range are quitestable and close to 0, as expected.We also found the average difference between ratios for each feed during thereasonable region (samples 40-120); these are plotted in Figure 5.5. Feeds towardthe centre of the cylinder do seem to have slightly lower deviation from 0, but thereis not a definitive pattern. Notice that during transit, all the feeds average differ-ences of less than 0.1, and most average differences less than ±0.05. We note thatthe average deviations in the difference between the ratios, is approximately 10%of the actual range of the ratios. In other words, the difference between redundantbaseline beam analysis and holography beam analysis is significantly smaller thanthe variation in either method. We consider this to be strong evidence that the twomethods are equivalent.5.2.3 ConclusionsBased on our comparisons, we find that redundant baseline gains and holographicbeam mapping report feed-to-feed beam variations during a point source transitwhich are in agreement. Though there are definite deviations between the tworesults, we see by comparing Figure 5.3 to Figure 5.4 that these deviations fromeach other are small relative to the size of the feed-to-feed beam variations.We regard this analysis as observational evidence of the effect we observed insimulations in Chapter 4. It does seem that deviation from redundancy in CHIMEPathfinder data is generated by beam variations and these beam variations can bedetermined with redundant baseline calibration.7640 50 60 70 80 90 100 110 120Time (samples)0.40.30.20.10.00.10.20.30.4Mean difference: Holography Ratio - RB RatioComparison to Holographic Beam VariationFigure 5.4: In this figure, the points represent the average ratio differenceacross feeds at a given time point, while the shaded band represents onestandard deviation in the ratio difference. Notice that the deviation val-ues (and especially the standard deviation) are small nearest the transit(which peaks at about sample 100) and larger near the edges. Addi-tionally, notice that the average deviation at points nearest the transit(the center of the plot) are significantly smaller than the range in valueswithin the set.7760 70 80 90 100 110 120 130Feed number0.100.050.000.050.10average values: holography ratio - gain ratioWest cylinder190 200 210 220 230 240 250 260Feed number0.100.050.000.050.10average values: holography ratio - gain ratioEast cylinderFigure 5.5: In this figure, we plot the average deviation between holographyand redundant baseline gains for each feed in our sample, averaged overthe 80 time samples closest to the transit peak (about 25 minutes ofdata). Notice that for all feeds, the deviation from zero is less than±0.1, much smaller than the values of the redundant baseline gain ratiosthemselves.78Chapter 6Conclusions and Further Work6.1 ConclusionsIn this thesis, we have examined the potential for redundant baseline calibrationin CHIME, based on both simulations of and data from the CHIME Pathfinder.We find that, for an ideal version of a CHIME-like telescope, redundant baselineamplitude calibration is very effective.However, we observe that data from the CHIME Pathfinder is not well cali-brated by redundant baseline amplitude calibration, in a manner that is consistentbetween days. This analysis also led us to discover significant non-redundancyin point source calibrated CHIME Pathfinder results. As this non-redundancy isalso stable over time, we conclude both problems likely arise from beam variationsbetween feeds.We therefore develop and implement a perturbed beam simulation, and showthat it displays similarly deviant redundant baseline results. However, we also showthat redundant baseline amplitude calibration on such a system recovers informa-tion about feed-to-feed beam variations, both as a general trend and quantitativelyfor limited, noise-less cases.Finally, we compare redundant baseline amplitude results from real CHIMEPathfinder data to holographic beam measurements, showing that both reproducethe same feed-to-feed beam variations. Given the importance of beam calibrationfor successful CHIME analysis, this is encouraging.796.2 Future WorkIn many ways, though, this work raises more questions than it answers. Of particu-lar concern is the significant deviation from redundancy in point source calibratedvisibilities. Full N2 data for the final CHIME instrument cannot be stored, mean-ing the data must be compressed. At first glance, the logical way to do this is bycollapsing data across redundant baselines. If these baselines are not actually re-dundant, this may lead to significant signal loss in compressed data and decreasedability to attain CHIME’s science goals.Besides new questions regarding redundancy, there is still work to be done inimplementing the redundant baseline algorithm, in particular implementing phasecalibration on a CHIME Pathfinder sized telescope. Several paths forward presentthemselves based on the work discussed here. These can be classed into three maincategories. The first is extensions of the existing redundant baseline algorithm, thesecond is improvements of redundant baseline calibration based on our findings,and the third is further examination of redundant baselines as a probe of beamstructure.The first category is the most obvious, although not necessarily the simplest inpractice. The current work does not include implementations of the phase compo-nent of the redundant baseline calibration algorithm in the CHIME Pathfinder. Thisis an important step if we would like to seriously use redundant baseline calibra-tion to calibrate CHIME Pathfinder data. Additionally, we have not implementedany component of redundant baseline calibration for a full CHIME scale telescope.Because CHIME is a significant increase in scale beyond the CHIME Pathfinder,we will have to make this transition mindfully to avoid devouring computationalresources. Finally, we have done very little examination of redundant baseline cal-ibration across multiple frequencies. Preliminary examinations have not shownanything surprising, but a careful analysis should be done for the sake of complete-ness.The second category, improvements to the redundant baseline calibration algo-rithm, is likely the richest and certainly the most open-ended. We have definitivelyshown that redundant baseline calibration is affected by feed-to-feed beam varia-tions and that such beam variations are present in the CHIME Pathfinder. Prelim-80inary examinations of CHIME feeds and structure indicate that they will remain afactor in CHIME. Therefore, if we would like to use redundant baseline calibra-tion as a CHIME calibration method, we must find ways to use the algorithm inthe absence of identical beams. While it is too early to assert a particular methodas the definite path forward, approaches might either incorporate information fromsky maps via iterative Gibbs sampling or might solve for additional parameters asbasis functions of the non-identical beams.The final category, further use of redundant baselines as a beam probe, divergesin two directions, from the work in solving for beam perturbations in simulationsand from the work comparing to holography. The beam perturbation solutionspresented here are somewhat limited in that they are done on simulated data withno gain error or noise incorporated. It is obviously important to ascertain whetherthe methods put forward here are possible in more realistic scenarios, by testingthem with simulations with gain errors and noise. If it seems that the methods inChapter 4 are feasible for realistic scenarios, we should also try to apply them toreal CHIME Pathfinder data. 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APJ, 758:L24, Oct. 2012.doi:10.1088/2041-8205/758/1/L24. → pages 1588AppendicesAppendix ACHIME simulation pipelineThe CHIME simulation pipeline is composed of several git repositories (cora, ca-put, draco, driftscan, and ch-util), many of which are publicly available on github.The progression through the pipeline outlined in Chapter 2 is demonstrated inFigure A.1.A.1 Modification to pipeline codeA.1.1 Use of CylinderPerturbed Telescope ObjectTo create a telescope with perturbed beams, we must modify our starting point.The standard CHIME Pathfinder model as used in the previous chapter assumesa quasi-Gaussian beam which is identical for all inputs. (In other words, it lacksthe additional term α used in the beam model presented in 4.1.1.) To incorporatebeam perturbations, we use and later modify an existing class, PerturbedCylinder,written by Richard Shaw for the simulations discussed in [32].The class begins with the same quasi-Gaussian beam used in the standardCHIME Pathfinder simulation, but adds the structure for a perturbation in the widthof the beam, using the structure outlined in 4.1.1. PerturbedCylinder uses a finitedifference method to generate the first derivatives necessary for the perturbed beam890 50 100 150 200 250 300 350 400Right Ascension020406080100120|Visiblity|Sidereal Stream0 50 100 150 200 250 300 350 400Right Ascension010203040506070|Visiblity|Expanded Sidereal Stream0 5000 10000 15000 20000 25000Time +1.44396e9010203040506070|Visiblity|Timestream0 5000 10000 15000 20000 25000Time +1.44396e9020406080100120|Visiblity|Timestream with Receiver Temp0 5000 10000 15000 20000 25000Time +1.44396e9020406080100120140160180|Visiblity|Timestream with Receiver Temp & Gains0 5000 10000 15000 20000 25000Time +1.44396e9020406080100120140160180|Visiblity|Timestream with Receiver Temp, Gains, NoiseFigure A.1: In this figure, we see the progression of simulated data throughthe simulation pipeline. First, a sidereal stream is generated for eachunique baseline. Then, the sidereal stream is expanded so that thereis a representation of each individual baseline. At this stage of thepipeline process, these products do not incorporate complex gains ornoise and are therefore perfectly redundant. Third, we expand fromsidereal streams to individual time streams, with “20 second” samples,mimicking the actual CHIME data. Fourth, we add a constant receivertemperature to our timestream. Fifth, we add complex gains to thetimestream, and finally ,we add sample noise.90structure. This class can also easily be modified to account for other perturbations.A.1.2 ExpandPerturbedProductsAfter creating a sidereal stream, the simulation pipeline generally executes the task”ExpandProducts,” which expands the sidereal stream from having one instanceof each unique product to having all redundant instances of that unique product.To incorporate our perturbed beam, we insert one entirely new task and also mod-ify this task to both create all instances of a unique baseline and to combine theunperturbed component of the visibility with the two perturbed components.We first generate the actual perturbation values αi, which we disregarded increating the initial, pre-expansion sidereal stream, using a new task GeneratePer-turbations. This task, accompanied by the new container BeamPerturbations, usesthe numpy standard normal random number generator to generate small, randommultipliers which are scaled down to ensure this is in fact a small perturbation. Thescaling factor can be input as a parameter when setting up the simulation or is setto a default value of 0.01. (This is a somewhat arbitrary, but seems to satisfy ourgoal of having a small perturbation.)After generating perturbations, we turn our attention to modifying“ExpandProducts”to accommodate our perturbed simulation. We accomplish this by modifying theexisting perturbation expansion code. The unperturbed ”ExpandProducts” matchesa product index pi with the individual pairs of inputs fi and f j, then loops over each[pi,( fi, f j)] set. This is equivalent to forming the A(0)i A∗(0)j term in our perturbedsimulation. For each product pi, we follow a similar process to create the A(0)i A(∗1)jand A(1)i A∗(0)j terms. At this time, we also re-incorporate the previously calculatedvalues, α , multiplying each product of beam functions by the appropriate α value.Finally, we add the three terms together (the zeroth order term as well as the firstorder terms in fi and f j.) Should we prefer to assume the perturbations are not verysmall, we can also add the double perturbed term here αiα j A(1)i A∗(1)j ; this term isnormally neglected but the information for it is all present at this point.The initial sidereal stream (pre-expansion) created with the perturbed beam ap-pears to have a different number of inputs than the telescope actually has, Ninput(1+Npert) rather Ninput. Therefore, in the expansion step, we must not only write out a91new set of visibilities but must also write out a modified input map, which reflectsthe physical number of inputs in the telescope. (Failure to do this will confoundlater steps in the pipeline, such as adding random gains.)Once we have output our new expanded stream, we are able to follow the sim-ulation pipeline as in the unperturbed case, creating time streams and adding areceiver temperature, complex gains, and noise.A.2 Example resultsResults from a small scale example, with 16 total inputs, are shown in the figuresbelow. This telescope size is of course less interesting for further applications ofsimulations with perturbed beam models, but it represents a good test set and iseasier to compare between perturbed and unperturbed results.0 50 100 150 200 250 300 350 400RA020406080100120|Visibility|Pre-Expansion Perturbed Sidereal Stream0 50 100 150 200 250 300 350 400RA05101520253035|Visibility|Pre-Expansion Unperturbed Sidereal StreamFigure A.2: Sample sidereal stream prior to expansion of redundant baselinesand re-combination of perturbed components.920 50 100 150 200 250 300 350 400RA01020304050607080|Visibility|Expanded Sidereal Stream with Perturbation0 50 100 150 200 250 300 350 400RA05101520253035|Visibility|Expanded Sidereal Stream UnperturbedFigure A.3: The left hand panel shows sample sidereal stream after to expan-sion of redundant baselines, for an identical, unperturbed simulation.The right hand panel shows a sample perturbed sidereal stream afterthe expansion of redundant baselines and re-combination of perturbedcomponents. There are now as many different baselines as real CHIMEPathfinder data and redundancy has been broken by the addition of per-feed beam perturbations. However, at this stage, the simulated datadoes not include instrumental gains or any noise estimate and is there-fore not generally suitable for analysis tasks. In this particular figure,the perturbation is turned up to approximately 0.1 (from approximately0.01) to make its existence more obvious. Without the perturbation, thetwo panels would be identical, as the underlying telescope configura-tion is the same as is the input sky map.930 5000 10000 15000 20000 25000+1.44396e901020304050607080|Vis|tstream0 5000 10000 15000 20000 25000+1.44396e9020406080100120140|Vis|tstream2 (Receiver Temp)0 5000 10000 15000 20000 25000+1.44396e9050100150200|Vis|tstream3 (Complex Gains)0 5000 10000 15000 20000 25000time +1.44396e9050100150200|Vis|tstream4 (Sample Noise)Figure A.4: After the product expansion stage, the simulation pipeline shouldproceed as in a standard unperturbed version. First, we add the receivertemperature, then time dependent complex gains, then we add samplenoise. After adding the complex gains and sample noise, would-beredundant baselines are no longer redundant in either the perturbed orunperturbed case, but do resemble raw CHIME Pathfinder data94Appendix BSupplemental Figures950 5000 10000 15000 20000 25000+1.44396e90.70.80.91.01.11.21.31.41.51.6|gain|Feed 192identity noise covradiometer noise covdata noise covinput0 5000 10000 15000 20000 25000+1.44396e90.70.80.91.01.11.21.31.4Feed 200identity noise covradiometer noise covdata noise covinput0 5000 10000 15000 20000 25000+1.44396e90.80.91.01.11.21.31.4|gain|Feed 208identity noise covradiometer noise covdata noise covinput0 5000 10000 15000 20000 25000+1.44396e90.60.70.80.91.01.11.21.3Feed 216identity noise covradiometer noise covdata noise covinput0 5000 10000 15000 20000 25000+1.44396e90.40.60.81.01.21.4|gain|Feed 224identity noise covradiometer noise covdata noise covinput0 5000 10000 15000 20000 25000+1.44396e90.50.60.70.80.91.01.11.2Feed 232identity noise covradiometer noise covdata noise covinput0 5000 10000 15000 20000 25000Time (seconds) +1.44396e90.60.70.80.91.01.11.2|gain|Feed 240identity noise covradiometer noise covdata noise covinput0 5000 10000 15000 20000 25000Time (seconds) +1.44396e90.60.81.01.21.4Feed 248identity noise covradiometer noise covdata noise covinputFigure B.1: From Chapter 2. In this figure, we compare the percent deviationfrom input gains for the identity noise covariance, data-derived noisecovariance, and radiometer noise covariance for a sampling of inputson the east cylinder.960 5000 10000 15000 20000 25000+1.44396e9201510505101520|gain|: actual to RB % diffFeed 192identity noise covradiometer noise covdata noise cov0 5000 10000 15000 20000 25000+1.44396e925201510505101520Feed 200identity noise covradiometer noise covdata noise cov0 5000 10000 15000 20000 25000+1.44396e9201510505101520|gain|: actual to RB % diffFeed 208identity noise covradiometer noise covdata noise cov0 5000 10000 15000 20000 25000+1.44396e9201510505101520Feed 216identity noise covradiometer noise covdata noise cov0 5000 10000 15000 20000 25000+1.44396e92015105051015|gain|: actual to RB % diffFeed 224identity noise covradiometer noise covdata noise cov0 5000 10000 15000 20000 25000+1.44396e92015105051015Feed 232identity noise covradiometer noise covdata noise cov0 5000 10000 15000 20000 25000Time (seconds) +1.44396e92015105051015|gain|: actual to RB % diffFeed 240identity noise covradiometer noise covdata noise cov0 5000 10000 15000 20000 25000Time (seconds) +1.44396e92015105051015Feed 248identity noise covradiometer noise covdata noise covFigure B.2: From Chapter 2. In this figure, we compare the absolute devi-ation from input gains for the identity noise covariance, data-derivednoise covariance, and radiometer noise covariance for a sampling ofinputs on the east cylidner.970 5000 10000 15000 20000 25000Time (Seconds) +1.44396e90.150.100.050.000.050.100.15Average recovered - input gain differenceRedundant Baseline (Noise Cov from Data) vs. InputFigure B.3: From Chapter 2. Here, we view the average deviation betweenthe input and recovered gain. The pink band represents one standarddeviation above and one standard deviation below the average.980 10 20 30 40 50 60 70 80 900.51.01.52.02.53.0feed 192 gainday 0day 10 10 20 30 40 50 60 70 80 900.51.01.52.02.53.0feed 200 gainday 0day 10 10 20 30 40 50 60 70 80 900.51.01.52.02.53.0feed 208 gainday 0day 10 10 20 30 40 50 60 70 80 900.40.60.81.01.21.41.61.8feed 216 gainday 0day 10 10 20 30 40 50 60 70 80 900.81.01.21.41.61.82.02.22.42.6feed 225 gainday 0day 10 10 20 30 40 50 60 70 80 900.40.60.81.01.21.41.61.82.0feed 232 gainday 0day 10 10 20 30 40 50 60 70 80 90time(samples)0.60.81.01.21.4feed 240 gainday 0day 10 10 20 30 40 50 60 70 80 90time(samples)0.40.60.81.01.21.4feed 248 gainday 0day 1Figure B.4: From Chapter 3. We compare redundant baseline gain resultsfor selected feeds on the east cylinder calculated over two consecutiveCygA transits. We notice that there is very little deviation between thetwo days and regard this as evidence that the cause of the slope in thegain solution is not strongly time dependent and is likely a property ofthe array.990 50 100 150 200 250 300 3500.850.900.951.001.051.101.15|gain|Input vs. recovered: input 160gain + beambeam onlygain only0 50 100 150 200 250 300 3500.970.980.991.001.011.021.03Diff gain vs. beam: input 160gainbeam0 50 100 150 200 250 300 3500.960.970.980.991.001.011.021.031.04|gain|Input vs. recovered: input 168gain + beambeam onlygain only0 50 100 150 200 250 300 3500.970.980.991.001.011.021.031.04Diff gain vs. beam: input 168gainbeam0 50 100 150 200 250 300 3501.001.051.101.15|gain|Input vs. recovered: input 176gain + beambeam onlygain only0 50 100 150 200 250 300 3500.980.991.001.011.021.031.04Diff gain vs. beam: input 176gainbeam0 50 100 150 200 250 300 350Time (samples during transit)0.960.981.001.021.041.061.081.10|gain|Input vs. recovered: input 184gain + beambeam onlygain only0 50 100 150 200 250 300 350Time (samples during transit)0.950.960.970.980.991.001.011.02Diff gain vs. beam: input 184gainbeamFigure B.5: From Chapter 4. A continuation of Figure 4.11000 50 100 150 200 250 300 3500.920.940.960.981.001.021.041.061.08|gain|Input vs. recovered: input 192gain + beambeam onlygain only0 50 100 150 200 250 300 3500.960.981.001.021.04Diff gain vs. beam: input 192gainbeam0 50 100 150 200 250 300 3500.960.970.980.991.001.011.021.03|gain|Input vs. recovered: input 200gain + beambeam onlygain only0 50 100 150 200 250 300 3500.960.970.980.991.001.011.02Diff gain vs. beam: input 200 gainbeam0 50 100 150 200 250 300 3500.920.930.940.950.960.970.980.991.001.01|gain|Input vs. recovered: input 208gain + beambeam onlygain only0 50 100 150 200 250 300 3500.980.991.001.011.02Diff gain vs. beam: input 208gainbeam0 50 100 150 200 250 300 350Time (samples during transit)0.980.991.001.011.021.031.04|gain|Input vs. recovered: input 216gain + beambeam onlygain only0 50 100 150 200 250 300 350Time (samples during transit)0.970.980.991.001.011.021.03Diff gain vs. beam: input 216gainbeamFigure B.6: From Chapter 4. A continuation of Figure 4.11010 50 100 150 200 250 300 3500.981.001.021.041.06|gain|Input vs. recovered: input 224gain + beambeam onlygain only0 50 100 150 200 250 300 3500.970.980.991.001.011.021.03Diff gain vs. beam: input 224gainbeam0 50 100 150 200 250 300 3500.900.920.940.960.981.001.021.04|gain|Input vs. recovered: input 232gain + beambeam onlygain only0 50 100 150 200 250 300 3500.900.920.940.960.981.001.02Diff gain vs. beam: input 232gainbeam0 50 100 150 200 250 300 3500.880.900.920.940.960.981.001.021.04|gain|Input vs. recovered: input 240gain + beambeam onlygain only0 50 100 150 200 250 300 3500.920.930.940.950.960.970.980.991.001.01Diff gain vs. beam: input 240gainbeam0 50 100 150 200 250 300 350Time (samples during transit)0.860.880.900.920.940.960.981.00|gain|Input vs. recovered: input 248gain + beambeam onlygain only0 50 100 150 200 250 300 350Time (samples during transit)0.960.970.980.991.001.011.02Diff gain vs. beam: input 248gainbeamFigure B.7: From Chapter 4. A continuation of Figure 4.11020 50 100 150 200 250 300 3500.850.900.951.001.051.101.15|gain|Input vs. recovered: input 160gain + beambeam onlygain only0 50 100 150 200 250 300 3500.850.900.951.001.051.101.15gain+beam/beam: input 160gain+beam/beamgain only0 50 100 150 200 250 300 3500.960.970.980.991.001.011.021.031.04|gain|Input vs. recovered: input 168gain + beambeam onlygain only0 50 100 150 200 250 300 3500.950.960.970.980.991.001.011.02gain+beam/beam: input 168gain+beam/beamgain only0 50 100 150 200 250 300 3501.001.051.101.15|gain|Input vs. recovered: input 176gain + beambeam onlygain only0 50 100 150 200 250 300 3501.021.041.061.081.101.121.141.16gain+beam/beam: input 176gain+beam/beamgain only0 50 100 150 200 250 300 350Time (samples during transit)0.960.981.001.021.041.061.081.10|gain|Input vs. recovered: input 184gain + beambeam onlygain only0 50 100 150 200 250 300 350Time (samples during transit)1.001.021.041.061.081.101.12gain+beam/beam: input 184gain+beam/beamgain onlyFigure B.8: From Chapter 4. A continuation of Figure 4.21030 50 100 150 200 250 300 3500.920.940.960.981.001.021.041.061.08|gain|Input vs. recovered: input 192gain + beambeam onlygain only0 50 100 150 200 250 300 3500.920.940.960.981.001.021.041.06gain+beam/beam: input 192gain+beam/beamgain only0 50 100 150 200 250 300 3500.960.970.980.991.001.011.021.03|gain|Input vs. recovered: input 200gain + beambeam onlygain only0 50 100 150 200 250 300 3500.970.980.991.001.011.021.031.04gain+beam/beam: input 200gain+beam/beamgain only0 50 100 150 200 250 300 3500.920.930.940.950.960.970.980.991.001.01|gain|Input vs. recovered: input 208gain + beambeam onlygain only0 50 100 150 200 250 300 3500.910.920.930.940.950.960.970.98gain+beam/beam: input 208gain+beam/beamgain only0 50 100 150 200 250 300 350Time (samples during transit)0.980.991.001.011.021.031.04|gain|Input vs. recovered: input 216gain + beambeam onlygain only0 50 100 150 200 250 300 350Time (samples during transit)0.980.991.001.011.021.03gain+beam/beam: input 216gain+beam/beamgain onlyFigure B.9: From Chapter 4. A continuation of Figure 4.21040 50 100 150 200 250 300 3500.981.001.021.041.06|gain|Input vs. recovered: input 224gain + beambeam onlygain only0 50 100 150 200 250 300 3500.991.001.011.021.031.041.051.061.07gain+beam/beam: input 224gain+beam/beamgain only0 50 100 150 200 250 300 3500.900.920.940.960.981.001.021.04|gain|Input vs. recovered: input 232gain + beambeam onlygain only0 50 100 150 200 250 300 3500.960.981.001.021.041.06gain+beam/beam: input 232gain+beam/beamgain only0 50 100 150 200 250 300 3500.880.900.920.940.960.981.001.021.04|gain|Input vs. recovered: input 240gain + beambeam onlygain only0 50 100 150 200 250 300 3500.920.940.960.981.001.021.041.06gain+beam/beam: input 240gain+beam/beamgain only0 50 100 150 200 250 300 350Time (samples during transit)0.860.880.900.920.940.960.981.00|gain|Input vs. recovered: input 248gain + beambeam onlygain only0 50 100 150 200 250 300 350Time (samples during transit)0.860.880.900.920.940.960.981.00gain+beam/beam: input 248gain+beam/beamgain onlyFigure B.10: From Chapter 4. A continuation of Figure 4.21050 20 40 60 80 100 120 140 160 1801012345Feed 660 20 40 60 80 100 120 140 160 1801012345Feed 740 20 40 60 80 100 120 140 160 1801012345Feed 820 20 40 60 80 100 120 140 160 1801012345Feed 900 20 40 60 80 100 120 140 160 1801012345Feed 980 20 40 60 80 100 120 140 160 1801012345Feed 1060 20 40 60 80 100 120 140 160 180Time (samples)1012345Feed 1140 20 40 60 80 100 120 140 160 180Time (samples)1012345Feed 122Relative Holography - Relative RB GainFigure B.11: From Chapter 5. Differences between redundant baseline gainratios and ChIME-26 m cross correlation ratios; indicative of differ-ence between beam estimates for a given feed on the west cylinder.1060 20 40 60 80 100 120 140 160 1801012345Feed 1920 20 40 60 80 100 120 140 160 1801012345Feed 2000 20 40 60 80 100 120 140 160 1801012345Feed 2080 20 40 60 80 100 120 140 160 1801012345Feed 2160 20 40 60 80 100 120 140 160 1801012345Feed 2240 20 40 60 80 100 120 140 160 1801012345Feed 2320 20 40 60 80 100 120 140 160 180Time (samples)1012345Feed 2400 20 40 60 80 100 120 140 160 180Time (samples)1012345Feed 248Relative Holography - Relative RB GainFigure B.12: From Chapter 5. Differences between redundant baseline gainratios and CHIME-26 m cross correlation ratios; indicative of differ-ence between beam estimates for a given feed on the east cylinder.107