[tel-00726959, v1] Caractériser le milieu interstellaire ... - HAL - INRIA

J. Pety and N. Rodríguez-Fernández: Revisiting the theory of interferometric wide-field synthesisTable 1. Definition of the symbols used to expose the wide-field synthesisformalism.Symbol & DefinitionPlane(s) aα s Scanned angle skyu s Scanned spatial frequency uvα p Phased angle skyu p Phased spatial frequency uvI Sky brightness skyB Primary beam skyV Visibility function uv &skyS Sampling function uv &skyΔ Set of single-field dirty beams sky & skyD Set of wide-field dirty beams sky & skyΩ Sky-plane weighting function sky & skyW uv-plane weighting function (Ω ⊃ W) uv & uvG Gridding function (=gγ) uv &skyg uv-plane gridding function uvγ Sky-plane gridding function skyI dirty Wide-field dirty image skyNotes. (a) Planes of definition of the associated symbols.Fig. 1. Visualization of the different angular scales relevant to widefieldinterferometric imaging. The notion of anti-aliasing scale (θ alias )isintroduced and discussed in Sect. 4.2.tel-00726959, version 1 - 31 Aug 2012Table 2. Definition of the uv and sky scales relevant to wide-field interferometricimaging.Symbol[λ,rad] ad max ,θ synd prim ,θ primd alias ,θ aliasd field ,θ fieldd image ,θ imageDefinitionConjugate uv and angular scaleMaximum baseline length & Synthesized beamAntenna diameter & Primary beamwidthMinimum image size for tolerable aliasingTargeted field of viewFinal image sizeNotes. (a) The chosen units (radians for θ and wavelength for d) implythat the conjugate scales are linked through θ = 1/d, instead of the usualθ = λ/d.For reference, Table 1 summarizes the definitions of the symbolsused most throughout the paper. With the one-dimensional notationused throughout the paper, the number of planes quoteddirectly gives the number of associated dimensions of the symbols.Generalization to images would require a doubling of thenumber of planes/dimensions. Table 2 defines the uv and angularscales that are relevant to wide-field interferometric imaging,and Fig. 1 sketches the different angular scales. Each angularscale (θ) is related to a uv scale (d) through θ = 1/d, whereθand d are measured in radians and in units of λ (the wavelengthof the observation). In the rest of the paper, we explicitely distinguishbetween θ prim ≡ 1/d prim , the angular scale associated tothe diameter of the interferometer antennas, and θ fwhm ,thefullwidth at half maximum of the primary beam. The relation betweenθ prim and θ fwhm depends on the illumination of the receiverfeed by the antenna optics. In radio astronomy, we typically haveθ fwhm ∼ 1.2 θ prim (see e.g. Goldsmith 1998, Chap. 6). Finally, thenotion of anti-aliasing scale (θ alias ) is introduced and discussedin Sect. 4.2.complexity without loss of generality. The top row displays thesky plane. The middle row represents the 4-dimensional measurementspace at different stages of the processing. As it is difficultto display a 4-dimensional space on a sheet of paper, thebottom row shows 2-dimensional cuts of the measurement spaceat the same processing stages.2.2.1. Observation setup and measurement spacePanel a) displays the sky region for which we aim for estimatingthe sky brigthness, I(α). The field of view of an interferometerobserving in a given direction of the sky has a typical size set bythe primary beam shape. In our example, this is illustrated by anyof the circles whose diameter is θ prim . As we aim at observing awider field of view, e.g. θ field , the interferometer needs to scan thetargeted sky field. We assume that we scan through stop-and-gomosaicking, ending up with a 7-field mosaic.After calibration, the output of the interferometer is a visibilityfunction, V(u p , α s ), whose relation to the sky brightness isgiven by the measurement equation (Eq. (1)). Panel b.1) showsthe measurement space as a mosaic of single-field uv planes:the uv plane coverage of each single-field observation is displayedas a blue sub-panel at the sky position where it has beenmeasured and which is featured by the red axes. We assume1) that the interferometer has only 3 antennas and 2) that onlya single integration is observed per sky position. This impliesonly 6 visibilities per single-field uv plane. In panel b.2), the uvplanes at constant α s are displayed as the blue vertical lines. Themeasured spatial frequencies belong to the [−d max , −d min ]and[+d min , +d max ] ranges, where d min and d max are respectively theshortest and longest measured baseline length. d min is related tothe minimum tolerable distance between two antennas to avoidcollision. Here, we chose d min ∼ 1.5 d prim . The grey zone between−d min ,and+d min displays the missing short spacings.2.2. Basic conceptsFigure 2 illustrates the principles underlying 1) the setup to getinterferometric wide-field observations and 2) our proposition toprocess them. For simplicity, we display the minimum possible2.2.2. Processing by explicit synthesis of the wide-fieldspatial frequenciesAll the information about the sky brightness, I(α), is somehowcoded in the visibility function, V(u p , α s ). The high spatialPage 3 of 21