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

J. Pety and N. Rodríguez-Fernández: Revisiting the theory of interferometric wide-field synthesistel-00726959, version 1 - 31 Aug 2012As B is bounded inside the [−d prim , +d prim ]interval,SV up (u s )isa local average, weighted by S up (u s − u ′ s ) B(−u′ s ), of I(u p+u ′ s )around the u p spatial frequency.As expected, we recover Eq. (15) for the ideal case (i.e., infinite,continuous visibility function) because then S up (u s − u ′ s) =δ(u s −u ′ s). A more interesting case arises when the visibility functionis continuously sampled over a limited sky field of view, i.e.,∀u p , S ( u p ,α s)= 1 if |αs |≤θ field /2, (28)∀u p , S ( u p ,α s)= 0 if |αs | >θ field /2. (29)After Fourier transform this gives∀u p , S ( u p , u s)=1d fieldsinc(usd field)· (30)In this case, the local average of the sky brightness Fourier componentshappens on a typical uv scale equal to d field .However,the sinc function is known to decay only slowly. Some observingstrategy (e.g. quickly observing outside the edges of the targetedfield of view to provide a bandguard) could be consideredto apodize the sky-plane dependence of the sampling function,resulting in faster decaying S functions, hence in less mixing ofthe wide-field spatial frequencies.3.2.2. uv-plane wide-field measurement equationBecause we aim at estimating the Fourier component of I, weintroduce the following change of variables u ′ ≡ u p + u ′ s anddu ′ = du ′ s ,toderive∫(SV up (u s ) = S up up + u s − u ′) B ( u p − u ′) I ( u ′) du ′ . (31)u ′We then shift-and-average SV(u p , u s ) to build the Fourier transformof a wide-field dirty imageI dirty (u) ≡ 〈 SV 〉 (u) , with u = u p + u s . (32)Substituting the shift-and-average operator by its definition andusing Eq. (31) to replace SV up (u s ), we deriveI dirty (u) =∫∫W ( ) (u p , u−u p S up , u−u ′) B ( u p −u ′) I ( u ′) du p du ′ . (33)u p u ′This uv-plane wide-field measurement equation can be written as∫I dirty (u) = D ( u ′ , u − u ′) I ( u ′) du ′ , (34)u ′if we enforce the following equalityD ( u ′ , u−u ′) ∫≡ W ( ) (u p , u−u p S up , u−u ′) B ( u p −u ′) du p . (35)u pThis is one way to define D, which is convenient though unusual.It is implicit in this definition that we need to make a change ofvariable (u ′′ = u − u ′ )toderiveD ( u ′ , u ′′) ∫≡ W ( ) (u p , u ′ +u ′′ −u p S up , u ′′) B ( u p −u ′) du p . (36)u pIn the following, we use either one or the other definition of D,depending on convenience.3.2.3. InterpretationAppendix A.2 demonstrates that the image and uv-plane widefieldmeasurement equations (Eqs. (23) and(34)) are equivalentifD ( ) (α p,α s)α p ,α s⊃ D ( )u(u p ,u s) p , u s . (37)The image-plane wide-field measurement equation (Eq. (23))can be written asI dirty (α) = {D α ⋆ I}(α) . (38)Its interpretation is straightforward: the sky brightness distributionis convolved with a dirty beam, D(α ′ , α ′′ ), which varies withthe sky coordinate α ′′ . This raises the question of the rate ofchange of the dirty beam with the sky coordinate. This questionis addressed in Sects. 4.2 and 5.4. Gridding by convolution and regular resamplingWe want to Fourier transform the raw visibilities along the skydimension (α s ) at some constant value in the u p dimension. Theraw data, however, is sampled on an irregular grid in both the uvand sky planes. We need to grid the measured visibilities in boththe uv and the sky planes before Fourier transformation for differentreasons. First, the gridding in the uv plane will handle thevariation in the spatial frequency as the sky is scanned, i.e., thedifficulty and perhaps the impossibility of Fourier-transformingat a completely constant u p value. Second, the gridding alongthe sky dimension allows the use of Fast Fourier Transforms. Asusual, we grid through convolution and regular resampling.4.1. Convolution4.1.1. DefinitionsWe first define a gridding kernel that depends on both dimensions,G(u,α s ). This gridding kernel can be chosen as the productof two functions, simplifying the following demonstrations:G ( u p ,α s)≡ g(up)γ (αs ) . (39)We then define the sampled visibility function gridded in boththe uv and sky planes asSV ( ) ( )G u p ,α s ≡{G⋆ SV} up ,α s (40)∫∫= g ( u p − u p) ′ ( (γ αs − α ′ s) SV u′p ,α s) ′ du′p dα ′ s. (41)u ′ pα ′ sFinally, when assessing the impact of the gridding on the measurementEq. (34), a new function,Σ ( ) ( )u p ,α s ,α ′′(s ≡ S up ,α s B α′′ )s − α s , (42)and its Fourier transforms naturally appear in the equations.Defining the following Fourier transform relationshipsΣ ( )u p ,α s ,α ′′ α ss ⊃usΣ ( )u p , u s ,α ′′s , (43)andΣ ( u p , u s ,α ′′s) α ′′s⊃u ′′sΣ ( )u p , u s , u ′′s , (44)Page 7 of 21