[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 2012frequencies (from d min to d max ) are clearly coded along the u pdimension. The uncertainty relation between Fourier conjugatequantities also implies that the typical spatial frequencyresolution along the u p dimension is only d prim because the fieldof view of a single pointing has a typical size of θ prim .However,wide-field imaging implies measuring all the spatial frequencieswith a finer resolution, d field = 1/θ field . The missing informationmust then be hidden in the α s dimension.In Sect. 3, we show that Fourier transforming the measuredvisibilities along the α s dimension (i.e. at constant u p ) can synthesizethe missing spatial frequencies, because the α s dimensionis sampled from −θ field /2to+θ field /2, implying a typicalspatial-frequency resolution of the u s dimension equal tod field .Conversely,theα s dimension is probed by the primarybeams with a typical angular resolution of θ prim , implying thatthe u s spatial frequencies will only be synthesized inside the[−d prim , +d prim ] range. Panels c.1) and c.2) illustrate the effectsof the Fourier transform of V(u p , u s ) along the α s dimension, in4 and 2 dimensions, respectively. The red subpanels or verticallines display the u s spatial frequencies around each constant u pspatial frequency.In panels d.1) and d.2) (i.e. after the Fourier transform alongthe α s dimension), V(u p , u s ) contains all the measured informationabout the sky brightness in a spatial frequency space.However, the information is ordered in a strange and redundantway. Indeed, we show that V(u p , u s ) is linearly related toI(u p + u s ). To first order, the information about a given spatialfrequency u is stored in all the values of V(u p , u s )whichverifiesu = u p + u s (black lines on panel c.2).A shift operation will reorder the spatial scale informationand averaging will compress the redundancy (illustrated by thehalving of the number of the space dimensions). The use of ashift-and-average operator thus produces a final uv plane containingall the spatial scale information to image a wide fieldin an intuitive form. We thus call this space the wide-field uvplane. Panels d.1) and d.2) display this space, where the minimumrelevant spatial frequency is related to the total field ofview, while the maximum one is related to the interferometerresolution.Sections 3 and 4 show that applying the shift-and-averageoperator to V produces the Fourier transform of a dirty image,which is a local convolution of the sky brightness by a slowlyvarying dirty beam. As a result, inverse Fourier transform of 〈 V 〉and deconvolution methods will produce a wide-field distributionof sky brightness as shown in panel e) at the top right ofFig. 2.3. Beyond the Ekers & Rots schemeIn the real world, the visibility function is not only sampled, butthis sampling is incomplete for two main reasons. 1) The instrumenthas a finite spatial resolution, and the scanning of thesky is limited, implying that the sampling in both planes has afinite support. 2) The uv coverage and the sky-scanning coveragecan have holes caused either by intrinsic limitations (e.g.lack of short spacings or small number of baselines) or by acquisitionproblems (implying data flagging). The incomplete samplingmakes the mathematics on the general case complex. Wethus start with the ideal case where we assume that the visibilityfunction is continuously sampled along the u p and α s dimension.We then look at the general case.3.1. Ideal case: infinite, continuous samplingStarting from the measurement Eq. (1), Ekers & Rots (1979)firstdemonstrated (see Sect. A.1)that 2∀ ( u p , u s), Vup (u s ) = B (−u s ) I ( u p + u s). (15)For each constant u p spatial frequency, the Fourier transformthus synthesizes a function, V up (u s ), which is simply related toI(u p + u s ), the Fourier components of the sky brightness aroundu p . V(u p , u s ) is only defined in the [−d prim , +d prim ] interval alongthe u s dimension because B(−u s ) is itself only defined inside thisinterval, since B(−u s ) is the autocorrelation of the antenna illumination.We search to derive a single estimate of the Fourier componentsI(u) of the sky brightness. Equation (15) indicates that thefraction V(u p , u s )/B(−u s ) gives us an estimate of I(u) for eachcouple (u p , u s ) that satisfies u = u p + u s . However, the informationabout I is strangely ordered. There are two possible ways tolook at this ordering. 1) Starting from the measurement space,the Ekers & Rots scheme synthesizes frequencies around eachu p measure inside the interval [u p − d prim , u p + d prim ]atthed fieldspatial frequency resolution. 2) Starting from our goal, we wantto estimate I at a given spatial frequency u with a d field spatialfrequency resolution. We thus search for all the couples (u p , u s )satisfying u = u p + u s , which are displayed in panel c.2) of Fig. 2as the diagonal black lines. It immediately results that 1) thereare several estimates of I for each spatial frequency u and 2) thenumber of estimates varies with u. We can average them to get abetter estimate of I(u).This last viewpoint thus suggests averaging in the (u p , u s )space along linepaths defined by u = u p + u s . Such an operatorcan mathematically be defined as∫∫〈F〉(u) ≡ δ [ u − (u p + u s ) ] W ( ) ( )u p , u s F up , u s dup du s , (16)u p u swhere F is the function to be averaged and W is a normalizedweighting function. Using the properties of the Dirac function,we can reduce the double integral to∫〈F〉(u) = W ( ) ( )u p , u − u p F up , u − u p dup . (17)u pIn this equation, we easily recognize a shift-and-average operator.The normalized weighting function plays a critical role inthe following formalism, and we propose clever ways to defineW in Sect. 5. In the ideal case studied here, W can be defined asW(u p , u s ) ≡ 1/(2 √ 2 d prim )foru s in [−d prim , +d prim ],W(u p , u s ) ≡ 0 for other values of u s .In other words, we have just normalized the integral by the constantlength (2 √ 2 d prim ) of the averaging linepath.2 The convolution theorem, which states that the Fourier transform ofthe convolution of two functions is the product of the Fourier transformof both individual functions, is a special case for Eq. (15): it canbe recovered by setting u p = 0. Indeed, as already mentioned in theintroduction, the ideal measurement Eq. (1) can be interpreted as a convolutionwith an additional phase term. By Fourier transforming alongthe α s dimension, the convolution translates into a product of Fouriertransforms B and I, while the phase term translates into a shift of coordinates:u p + u s .Page 5 of 21