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

A&A 517, A12 (2010)tel-00726959, version 1 - 31 Aug 2012where W is a normalized weighting function. Using these tools,we demonstrated that:1. The dirty image (I G dirty) is a convolution of the sky brightnessdistribution (I) with a set of wide-field dirty beams (D G )varying with the sky coordinate α, i.e.,I G dirty (α) = ∫α ′ D G ( α − α ′ ,α ) I ( α ′) dα ′ . (99)Compared to single-field imaging, the dependency on theprimary beam is transferred from a product of the sky brightnessdistribution into the definition of the set of wide-fielddirty beams.2. The set of gridded dirty beams (D G ) can be computed fromthe ungridded sampling function (S ), the transfer function(B, the inverse Fourier transform of the primary beam),and the gridding convolution kernel (see Eqs. (42), (50)and (51)).3. The dependence of the wide-field dirty beams on the skyposition is slowly-varying, with their shape varying on anangular scale typically larger than or equal to the primarybeamwidth.Adaptations of the existing deconvolution algorithms should bestraightforward.A comparison with standard nonlinear mosaicking showsthat it is not mathematically equivalent to the wide-field synthesisproposed here, though both methods do recover the skybrightness. The main advantages of wide-field synthesis overstandard nonlinear mosaicking are1. Weighting is at the heart of the wide-field synthesis becauseit is an essential part of the shift-and-average operation.Indeed, not only can a multiplicative weight be attributedto each visibility sample before any processing, but the uvplaneweighting function (W, seeEq.(98)) is also a degreeof freedom, which should be set according to the conditionsof the observation and the imaging goals, e.g. highest signalto-noiseratio, highest resolution, or most uniform resolutionover the field of view. The W weighting function thus enablesus to modify the wide-field response of the instrument.On the other hand, mosaicking requires a precise weightingfunction in the image plane, which freezes the wide-field responseof the interferometer.2. Wide-field synthesis naturally processes the short spacingsfrom both single-dish antennas and heterogeneous arraysalong with the long spacings. Both of them can then bejointly deconvolved.3. The gridding of the sky plane dimension of the measuredvisibilities, required by the wide-field synthesis, may potentiallysave large amounts of hard-disk space and cpuprocessing power relative to mosaicking when handling datasets acquired with the on-the-fly observing mode. Wide-fieldsynthesis could thus be particularly well suited to process onthe-flyobservations.The wide-field synthesis algorithm is compatible with the uvwunfacetingtechnique devised by Saultetal.(1996a) to dealwith the celestial projection effect, known as non-coplanar baselines(see Appendix B). Finally, on-the-fly observations implyan elongation of the primary beam along the scanning direction.These effects can be decreased by an increase in the primarybeam sampling rate. However, it may limit the dynamic rangeof the image brightness if the primary beam sampling rate is toocoarse (see Appendix C).Page 16 of 21Acknowledgements. This work has mainly been funded by the European FP6“ALMA enhancement” grant. This work was also funded by grant ANR-09-BLAN-0231-01 from the French Agence Nationale de la Recherche as part ofthe SCHISM project. The authors thank F. Gueth for the management of the onthe-flyworking package of the “ALMA enhancement” project. They also thankS. Guilloteau, R. Lucas and J. Uson for useful comments at various stages ofthe manuscript and D. Downes for editing their English. They finally thank thereferee, B. Sault, for his insightful comments, which challenged us to try to writea better paper.Appendix A: DemonstrationsA.1. Ekers & Rots schemeFourier-transforming the visibility function along the α s dimensionat constant u p , we derive with simple replacements∫V up (u s ) = V up (α s ) e −i2πα su sdα s(A.1)α∫∫ s= B ( ) ( )α p − α s I αp e−i2π(α p u p +α s u s) dαs dα p . (A.2)α s α pWe then use the following change of variables β ≡ α p − α s anddβ = −dα s ,toget∫∫V up (u s ) = B (β) I ( [ ( ) ])−i2π αp up +uα p e s −βusdα p dβ (A.3)α p β[∫∫= B (β) e dβ] ⎡⎢⎣−i2πβ(−us) I ( )α p e−i2πα p(u p +u s) dαp⎤⎥⎦ (A.4)βα p= B (−u s ) I ( )u p + u s . (A.5)A.2. Incomplete samplingWe here demonstrate that Eqs. (23) and(34) are equivalent. Todo this, we take the direct Fourier transform of I dirty (α)∫∫I dirty (u) = D ( α − α ′ ,α ) I ( α ′) e −i2παu dαdα ′ , (A.6)αα ′and we replace I(α ′ ) by its formulation as a function of itsFourier transformI ( α ′) ∫= I ( u ′) e +i2πu′ α ′ du ′ .(A.7)u ′We thus deriveI dirty (u) =[∫∫D∫u ( α − α ′ ,α ) ]e −i2π(αu−α′ u ′) dαdα ′ I ( u ′) du ′ . (A.8)′ αα ′Using the following change of variables α ′′ ≡ α−α ′ , α ′ = α−α ′′and dα ′′ = −dα ′ , the innermost integral can be written as∫∫D ( α − α ′ ,α ) e −i2π(αu−α′ u ′) dαdα ′ =αα ′ ∫Dα[∫α ( α ′′ ,α ) ]e −i2πα′′ u ′ dα ′′ e −i2πα(u−u′) dα (A.9)∫′′= D ( u ′ ,α ) e −i2πα(u−u′) dα(A.10)α= D ( u ′ , u − u ′) . (A.11)In the last two steps, we have simply recognized two differentsteps of Fourier transforms of D. Finally,∫I dirty (u) = D ( u ′ , u − u ′) I ( u ′) du ′ .(A.12)u ′