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

A&A 517, A12 (2010)tel-00726959, version 1 - 31 Aug 2012withD hyb( u ′ , u ′′) ≡∫( ) (W hyb up , u ′ +u ′′ −u p S hyb up , u ′′) (B hyb up , u ′) du p . (77)u pFor a multiplying interferometer,∀ ∣ ∣ ∣up∣ ∣∣ < dprim , S ( u p ,α s)= 0 and S(up , u s)= 0. (78)This implies that D sdcontributes at u p = 0inthesumoveru p inEq. (77), D 7mcontributes for 9 m < u p< ∼ 40 m and D 12 m(u ′ ,u ′′ ) contributes for 15 m < u p < 150 m in the most compactconfiguration of ALMA.7. Comparison with standard nonlinear mosaicking7.1. Mosaicking in a nutshellSeveral excellent descriptions of the mosaicking imaging and deconvolutionalgorithms can be found (see e.g. Cornwell 1988;Cornwell et al. 1993; Sault et al. 1996b). Here, we summarizethe approach implemented in the gildas/mapping softwareused to image and deconvolve the data from the Plateaude Bure Interferometer. This approach is based on original ideasby F. Viallefond in the early 90s (Gueth et al. 1995).The basic ideas of nonlinear mosaicking are 1) imaging thedifferent fields of the mosaic independently; 2) linearly addingthe single-field dirty images into a dirty mosaic; and 3) jointlydeconvolving the dirty mosaic.7.1.1. Single-field imagingFor simplicity, we skip the gridding convolution in the followingequations because the gridding step does not change the natureof the equations. Imaging the fields individually means that wewill work at constant α s . We first define the single-field dirtyimage of the α s -field as( ) α p ( )I sfd αp ,α s ⊃ Iu sfd up ,α s , (79)pwhere the Fourier transform of the single-field dirty image isthe product of the sampling function S (u p ,α s ) and the visibilityfunction V(u p ,α s ):I sfd(up ,α s)≡ SV(up ,α s). (80)From the previous equations, it is easily demonstrated that( )∫I sfd αp ,α s = Δ ( )[ ( ) ( )]α p − α ′ p,α s B α′p − α s I α′p dα′p , (81)α ′ pwhere the single-field dirty beam is defined asΔ ( ) α pα p ,α s ⊃ S ( )uu p ,α s . (82)pWe can rewrite the previous equation asI α ssfd(αp)= {Δα s⋆ (B α sI)} ( α p), (83)meaning that the single-field dirty images can be written as alocal convolution of B α sI and Δ α s, the single-field dirty beamassociated to the currently imaged field.Page 14 of 217.1.2. Mosaicking the dirty imagesIn gildas/mapping 7 , the single-field dirty images are formedon the same grid (in particular the same pixel size and the sameimage size covering about twice the mosaic field of view). Thesesingle-field dirty images are then linearly averaged asI mos(αp)≡∫α sΩ mos(αp ,α s)Isfd(αp ,α s)dαs , (84)whereΩ mos(αp ,α s)≡w (α s ) B ( α p − α s)∫α sw (α s ) B 2 ( α p − α s)dαs(85)and w(α s ) is the sky plane weighting function, i.e.,∑w (α s ) = δ (α s − α i ) 1 · (86)σ 2 iiIn the previous equation, the α i are the positions of each skyplanemeasurement, and σ i is the rms noise associated with I α isfd .Cornwell et al. (1993) demonstrates that the noise in the mosaicimage naturally varies across the field as( ) 1N mos αp = √ ∫w (αα s ) B ( ) · (87)2 α p − α s dαssIn particular, it rises sharply at the edges of the mosaic.7.1.3. Joint deconvolutionStandard algorithms of single-field deconvolution must beadapted to the mosaicking case because both the dirty beamand the noise vary across the mosaic field of view. We describehere the adaptations made in gildas/mapping of the simplestCLEAN deconvolution method, described in Högbom (1974).Adaptations of more evolved CLEAN deconvolution methodsare also implemented following the same basic rules.1. We first initialize the residual and signal-to-noise maps fromthe dirty and noise mapsR 0(αp)= Imos(αp)(88)andSNR 0(αp)=I mos(αp)N mos(αp) · (89)2. The kth CLEAN component is sought on the SNR k−1 map insteadof the R k−1 map to ensure that noise peaks at the edgesof the mosaic are not confused with the true signal of thesame magnitude.3. Using that the kth CLEAN component is a point source ofintensity I k at position α k , the residual and signal-to-noisemaps are then upgraded as follows:R k(αp)= Rk−1(αp)−γI k∫α sΩ mos(αp ,α s)Δ(αp −α k ,α s)B (αk −α s ) dα s , (90)7 See http://www.iram.fr/IRAMFR/GILDAS for more informationabout the GILDAS software.