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

A&A 517, A12 (2010)tel-00726959, version 1 - 31 Aug 2012we easily deriveΣ ( ) ( )u p ,α s , u ′′(s = S up ,α s B u′′) s e−i2πu ′′s α s, (45)andΣ ( ) ( )u p , u s , u ′′s = S up , u s + u ′′ (s B u′′) s . (46)Using these notations, we have before gridding,SV ( u p ,α s)=∫α pΣ ( u p ,α s ,α p)I(αp)e−i2πα p u pdα p , (47)andD ( u ′ , u−u ′) ∫= W ( ) (u p , u−u p Σ up , u−u p , u p −u ′) du p . (48)u p4.1.2. Conservation of the wide-field measurement equationAppendix A.3 demonstrates that the wide-field dirty image ishere again the convolution of the sky brightness I by a widefielddirty beam D α or, in the Fourier plane,I G dirty〈SV (u) ≡ G〉 ∫(u) = D G ( u ′ , u − u ′) I ( u ′) du ′ (49)u ′withD G ( u ′ , u−u ′) ∫≡ W ( ) Gu p , u − u ( p Σ u p , u − u p , u ′) du p , (50)u pwhereΣ G ( u p ,α s , u ′) ≡∫∫g ( u p − u p) ′ ( (γ αs − α ′ s) Σ u′p ,α ′ s , u′ p − u′) du ′ p dα′ s . (51)u ′ pα ′ sWe thus have equations that resemble those containing the samplingfunction alone, except for 1) the replacement of the generalizedsampling function Σ by its gridded version Σ G and 2) theway the variables are linked together both in the gridding of Σ(i.e., Eq. (51)) and in the averaging of Σ G (i.e., Eq. (50)).4.2. Regular resamplingIt is well known that too low a resampling rate in one spaceimplies power aliasing in the conjugate space (see e.g. Bracewell2000; Press et al. 1992). Aliasing must be avoided as much aspossible because it folds power outside the imaged region backinto it. Table 3 defines the intervals of definition of the differentfunctions we are dealing with (i.e., visibilities, primary beam,dirty image, and dirty beam), as well as the associated samplingrates needed to enforce Nyquist sampling. The boundary valuesof the definition intervals (|u| max and |α| max ) are related to thesampling rates (∂α and ∂u, respectively) through|u| max · ∂α = |α| max · ∂u = 1 , (52)n sampwhere n samp is an integer characterizing the sampling. Nyquistsampling implies n samp = 2. However, slight oversampling (e.g.n samp = 3) is often recommended because the measures sufferfrom errors and the deconvolution is a nonlinear process. In thissection, we examine the properties of the different functions todefine their associated sampling rates.Page 8 of 21Table 3. Interval ranges of definition and associated sampling rates forthe used functions.Functions Intervals SamplingsVisibilities∣∣∣u ∣∣p ≤ dmax ∂u p = 2 d alias /n samp∣∣∣α ∣∣p ≤ θalias /2 ∂α p = θ syn /n samp|u s |≤d prim ∂u s = 2 d image /n samp|α s |≤θ image /2 ∂α s = θ prim /n sampPrimary beam ∣ ∣∣u′ ∣∣s ≤ dprim ∂u ′ s = 2 d alias/n samp∣∣α ′ s∣ ≤ θ alias /2 ∂α ′ s = θ prim/n samp∣∣u ′′s∣ ≤ d prim ∂u ′′s = 2 d alias /n samp∣∣α ′′s∣ ≤ θ alias /2 ∂α ′′s = θ prim /n sampDirty image |u| ≤d max ∂u = 2 d image /n samp|α| ≤θ image /2 ∂α = θ syn /n sampDirty beam |u ′ |≤d max ∂u ′ = 2 d image /n samp|α ′ |≤θ image /2 ∂α ′ = θ syn /n samp|u ′′ |≤d prim ∂u ′′ = 2 d image /n samp|α ′′ |≤θ image /2 ∂α ′′ = θ prim /n samp4.2.1. The α s sampling rate of the visibility functionWhen Fourier transforming the measurement Eq. (1) along theα s axis, we derive the Ekers & Rots Eq. (15). This equation impliesthat V(u p , u s ) is bounded inside the [−d prim , +d prim ]spatialfrequency interval along the u s axis. As a result, the visibilityfunction needs to be regularly resampled at a rate of only0.5/d prim to satisfy the Nyquist theorem. This was first pointedout by Cornwell (1988). This sampling rate is equal to θ prim /2or∼θ fwhm /2.4. The “usual, wrong” habit of sampling at θ fwhm /2isindeed undersampling with aliasing as a consequence. Mangumet al. (2007) discuss the consequences of undersampling indepthin the framework of single-dish imaging.4.2.2. The U p sampling rate of the visibility functionNow, the Fourier transform of the measurement Eq. (1) along theu p axis givesV ∼(αp ,α s)= B(αp − α s)I(αp), (53)where( ) α pV αp ∼,α s ⊃ V ( )uu p ,α s . (54)pWe use the tilde sign under V to denote the inverse Fourier transformof V along its first dimension. A well-known Fourier transformproperty implies that B has infinite support because B isbounded. The resampling rate along the u p axis therefore dependson the properties of the product of B(α p −α s ) times I(α p )asa function of α p . While no unique answer exists, three facts helpus to find the right sampling rate: 1) B falls off relatively quickly;2) the result depends on the spatial distribution of the sky brightnessand in particular on the dynamic range in brightness neededto accurately image it; 3) the measure of V(α p , α s ) has a limited∼accuracy owing to thermal noise, phase noise, and other possiblesystematics (e.g. pointing errors). For simplicity, we quantifythe measurement accuracy by a single number, namely themaximum instrumental fidelity measured in the image plane asdefined in Pety et al. (2001). There are two cases:1. the maximum instrumental fidelity limits the dynamic rangein brightness. For instance, Pety et al. (2001) showed that the