J. Pety and N. Rodríguez-Fernández: Revisiting the theory of interferometric wide-field synthesis<strong>tel</strong>-<strong>00726959</strong>, version 1 - 31 Aug 2012θ fwhm /δα s ) for different uv distances (in units of d prim ). We seethat we derive a 1% accuracy at all u when we samp<strong>le</strong> the imageplane at a rate of 5 dumps per primary beam. However getting a0.1% accuracy needs quite high sampling rates (about 15). Thismust be compared with the accuracy of know<strong>le</strong>dge of B.We note that if a better accuracy is needed than the oneachievab<strong>le</strong> with the highest sampling rate, it is in theory possib<strong>le</strong>to replace in the correlator software the rectang<strong>le</strong> apodizingfunction by another function which falls more smoothly. Toavoid the loss of sensitivity inherent to the use of such an apodizingfunction (by throwing away data at the edges of the time intervalof integration), would require, for instance, to half-overlapthe integration intervals. This would imply more book-keepingin the correlator software and some noise correlation betweenthe measured visibilities.ReferencesBevington, P. R., & Robinson, D. K. 2003, Data Reduction and Error AnalysisFor the Physical Sciences, 3rd edn. (McGraw-Hill)Bhatnagar, S., & Cornwell, T. J. 2004, A&A, 426, 747Bhatnagar, S., Cornwell, T. J., Golap, K., & Uson, J. M. 2008, A&A, 487, 419Bracewell, R. N. 2000, The Fourier Transform and its Applications, 3rd edn.(McGraw-Hill)Clark, B. G. 1980, A&A, 89, 377Cornwell, T. J. 1988, A&A, 202, 316Cornwell, T. J., Holdaway, M. A., & Uson, J. M. 1993, A&A, 271, 697Cornwell, T. J., Golap, K., & Bhatnagar, S. 2008, IEEE Journal of Se<strong>le</strong>ctedTopics in Signal Processing, 2, 647Cotton, W. D., & Uson, J. M. 2008, A&A, 490, 455D’Addario, L. R., & Emerson, D. T. 2000, On-The-Fly Fringe Tracking, ALMAmemo, 331Ekers, R. D., & Rots, A. H. 1979, in Image Formation from Coherence Fucntionsin Astronomy, ed. C. van Schoonedveld (D. Reidel), IAU Proc., 49, 61Frater, R. H., & Docherty, I. S. 1980, A&A, 84, 75Goldsmith, P. F. 1998, Gaussian Beam, Quasioptical Propagation andApplications (IEEE Press)Gueth, F., Guilloteau, S., & Vial<strong>le</strong>fond, F. 1995, in The XXVIIth YoungEuropean Radio Astronomers Conference, ed. D. A. Green, & W. Steffen,8Hamaker, J. P., Bregman, J. D., & Sault, R. J. 1996, A&AS, 117, 137Högbom, J. A. 1974, A&AS, 15, 417Holdaway, M. A., & Foster, S. M. 1994, On-The-Fly Mosaicing, ALMA memo,122Mangum, J. G., Emerson, D. T., & Greisen, E. W. 2007, A&A, 474, 679Narayan, R., & Nityananda, R. 1986, ARA&A, 24, 127Pety, J., Gueth, F., & Guilloteau, S. 2001, Impact of ACA on the Wide-FieldImaging Capabilities of ALMA, ALMA memo, 398Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992,Numerical Recipes in C, 2nd edn. (Cambridge University Press)Rodríguez-Fernández, N. J., Pety, J., & Gueth, F. 2008, Sing<strong>le</strong>-dish observationand processing to produce the short-spacing information for a millimeterinterferometer, IRAM memo, 2008-2Rodríguez-Fernández, N. J., Pety, J., & Gueth, F. 2009, Imaging ofinterferometric On-The-Fly observations: Context and discussion of possib<strong>le</strong>methods, IRAM memo, 2009-2Sault, R. J., Hamaker, J. P., & Bregman, J. D. 1996a, A&AS, 117, 149Sault, R. J., Stave<strong>le</strong>y-Smith, L., & Brouw, W. N. 1996b, A&AS, 120, 375Sault, R. J., Bock, D. C.-J., & Duncan, A. R. 1999, A&AS, 139, 387Schwab, F. R. 1984, AJ, 89, 1076Sramek, R. A., & Schwab, F. R. 1989, Synthesis Imaging in Radio Astronomy,Conf. Ser. (ASP), 117Thompson, A. R., Moran, J. M., & Swenson, G. W. J. 1986, Interferometry andSynthesis in Radio Astronomy (John Wi<strong>le</strong>y & Sons)Page 21 of 21
IRAM Memo 2011-2WIFISYN:The GILDAS imp<strong>le</strong>mentation of anew wide-field synthesis algorithm ∗J. Pety 1,2 , N. Rodriguez-Fernandez 11. IRAM (Grenob<strong>le</strong>)2. Observatoire de ParisJan, 19th 2011Version 0.1AbstractThe usual way to image wide-field interferometric observations is known as mosaicking.Different variants of mosaicking exist (e.g. Cornwell, 1988; Sault et al., 1996), including aninteresting recent imp<strong>le</strong>mentation of mosaicking in the uv plane (golap et al., priv. comm.).Pety & Rodríguez-Fernández (2010) revisited the theory of wide-field imaging to propose adifferent algorithm to image interferometric wide-field observations, based on the well-knownEkers & Rots’ scheme. This algorithm is named wide-field synthesis because it explici<strong>tel</strong>ysynthesizes the wide-field spatial frequencies throughout the uv plane. This memo describesthe current state of the imp<strong>le</strong>mentation of this algorithm in a new package, named WIFISYN,of the GILDAS software suit.<strong>tel</strong>-<strong>00726959</strong>, version 1 - 31 Aug 2012WIFISYNContentsContents1 Introduction 32 Theory 32.1 Observation setup and measurement space . . . . . . . . . . . . . . . . . . . . . . . 32.2 Processing by explicit synthesis of the wide-field spatial frequencies . . . . . . . . . 53 Practice 53.1 Simulating a wide-field observation . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Fourier transform along αs and βs . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2.1 Gridding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2.2 Reordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.3 Wide-field synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Shifting and averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 Getting the dirty image through inverse Fourier transform . . . . . . . . . . . . . . 114 Conclusion 12A Imp<strong>le</strong>mentation plan 14B WIFISYN Language Internal Help 17B.1 Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17B.2 COMPLEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17B.3 FFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17B.4 LOAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18B.5 READ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18B.6 SETUP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18B.7 UVBEAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18B.8 UVGRID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18B.9 UVMAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18B.10 UVSYMMETRY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19B.11 VARIABLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19B.12 WIFI2VISI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19B.13 WRITE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192∗ This work was mainly funded by the European FP6 “ALMA enhancement” grant. It was also partially fundedby the grant ANR-09-BLAN-0231-01 from the French Agence Nationa<strong>le</strong> de la Recherche as part of the SCHISMproject.WIFISYN1 Introduction11. introductionThe usual way to image wide-field interferometric observations is known as mosaicking. Differentvariants of mosaicking exist (e.g. Cornwell, 1988; Sault et al., 1996), including an interestingrecent imp<strong>le</strong>mentation of mosaicking in the uv plane (golap et al., priv. comm.). Pety &Rodríguez-Fernández (2010) revisited the theory of wide-field imaging to propose a different algorithmto image interferometric wide-field observations, based on the well-known Ekers & Rots’scheme. This algorithm is named wide-field synthesis because it explici<strong>tel</strong>y synthesizes the widefieldspatial frequencies throughout the uv plane. This memo describes the current state of theimp<strong>le</strong>mentation of this algorithm in a new package, named WIFISYN, of the GILDAS software suit.The first section summarizes the ideas underlying the proposed algorithm. The second sectiondemonstrates the different steps taken to imp<strong>le</strong>ment the algorithm (i.e. simulation, gridding,Fourier transform along the scanned sky coordinates to synthesize the wide-field visibilities, applicationof a shift-and-average operator to obtain the wide-field uv plane and inverse Fouriertransform to yield the dirty image). We conclude on the additional needed steps to use thisnew algorithm on a daily basis. Appendix A includes the imp<strong>le</strong>mentation plan written beforecoding WIFISYN. Appendix B includes the document of the current user interface of the WIFISYNpackage.2 TheoryFigure 1 illustrates the princip<strong>le</strong>s underlying 1) the setup to get interferometric wide-field observationsand 2) our proposition to process them. For simplicity, we display the minimum possib<strong>le</strong>comp<strong>le</strong>xity without loss of generality. The top row displays the sky plane. The midd<strong>le</strong> rowrepresents the 4-dimensional measurement space at different stages of the processing.2.1 Observation setup and measurement spacePanel a) displays the sky region for which we aim for estimating the sky brigthness, I(α). Thefield of view of an interferometer observing in a given direction of the sky has a typical size set bythe primary beam shape. In our examp<strong>le</strong>, this is illustrated by any of the circ<strong>le</strong>s whose diameteris θprim. As we aim at observing a wider field of view, e.g. θfield, the interferometer needs to scanthe targeted sky field. We assume that we scan through stop-and-go mosaicking, ending up witha 7-field mosaic.After calibration, the output of the interferometer is a visibility function, V (up,αs), whoserelation to the sky brightness is given by the measurement equation∫V (up, αs) = B(αp − αs) I(αp) e −i2παpup dαp, (1)αpwhere V is the visibility function of 1) up (the spatial frequency with respect to the fixed phasecenter) and 2) αs (the scanned sky ang<strong>le</strong>), I is the sky brightness, and B the antenna powerpattern or primary beam of an antenna of the interferometer. Panel b.1) shows the measurementspace as a mosaic of sing<strong>le</strong>-field uv planes: The uv plane coverage of each sing<strong>le</strong>-field observationis displayed as a blue sub-panel at the sky position where it has been measured and which isfeatured by the red axes. We assume 1) that the interferometer has only 3 antennas and 2)that only a sing<strong>le</strong> integration is observed per sky position. This implies only 6 visibilities persing<strong>le</strong>-field uv plane.3
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