WIFISYN2. theoryWIFISYN3. practice<strong>tel</strong>-<strong>00726959</strong>, version 1 - 31 Aug 2012Figure 1: Illustration of the princip<strong>le</strong>s of wide-field synthesis, which enab<strong>le</strong>s us to image widefieldinterferometric observations. The top row displays the sky plane. The bottom row displaysthe 4-dimensional visibility space at different stages of the processing. In panels b) to d), thescanned dimensions (αs and us) are displayed in blue whi<strong>le</strong> the phased spatial sca<strong>le</strong> dimensions(up) are displayed in red and the spatial sca<strong>le</strong> dimensions (u) of the final wide-field uv plane aredisplayed in black. In detail, panel a) shows a possib<strong>le</strong> scanning strategy of the sky to measurethe unknown brightness distribution at high angular resolution: For simplicity it is here just a7-field mosaic. Panel b.1) and b.2) sketch the space of measured visibilities: The uv plane at eachof the 7 measured sky positions is displayed as a blue square box in panel b.1) and a blue verticalline in panel b.2). For simplicity, only 6 visibilities are plotted in panel b.1). Panels c.1) andc.2) sketch the space of synthesized visibilities after Fourier transform of the measured visibilitiesalong the scanned coordinate (αs): At each measured spatial frequency up (displayed on the blueaxes) is associated one space of synthesized wide-field spatial frequencies displayed as one of thered squares in panel c.1) and the red vertical lines in panel c.2). The wide-field spatial sca<strong>le</strong>sare synthesized 1) on a grid whose cell size is related to the total field of view of the observationand 2) only inside circ<strong>le</strong>s whose radius is the primary diameter of the interferometer antennas.Panels d.1) and d.2) display the final, wide-field uv plane. This plane is built by application of ashift-and-average operator. Inverse Fourier transform and deconvolution methods then producea wide-field distribution of sky brightnesses as shown in panel e).WIFISYN43. practiceFigure 2: Sky and uv coverage of a simulated interferometric on-the-fly observation with thePlateau de Bure interferometer. Left: Each cross display the average sky position attributed toa set of visibilities recorded during one integration time. The center of the coordinate system isthe phase center. Right: Each cross display the uv position of a measured visibilities whateverthe associated sky position. The black crosses are measured. The red ones are deduced from theHermitian symmetry of the visibilities.2.2 Processing by explicit synthesis of the wide-field spatial frequenciesAll the information about the sky brightness, I(α), is somehow coded in the visibility function,V (up,αs). The high spatial frequencies (from dmin to dmax) are c<strong>le</strong>arly coded along the up dimension.The uncertainty relation between Fourier conjugate quantities also implies that the typicalspatial frequency resolution along the up dimension is only dprim because the field of view of asing<strong>le</strong> pointing has a typical size of θprim. However, wide-field imaging implies measuring all thespatial frequencies with a finer resolution, dfield = 1/θfield. The missing information must then behidden in the αs dimension.Pety & Rodríguez-Fernández (2010) show that Fourier transforming the measured visibilitiesalong the αs dimension (i.e. at constant up) can synthesize the missing spatial frequencies,because the αs dimension is samp<strong>le</strong>d from −θfield/2 to +θfield/2, implying a typical spatialfrequencyresolution of the us dimension equal to dfield. Conversely, the αs dimension is probedby the primary beams with a typical angular resolution of θprim, implying that the us spatialfrequencies will only be synthesized inside the [−dprim, +dprim] range. Panel c) illustrates theeffects of the Fourier transform of V (up,us) along the αs dimension. The red subpanels displaythe us spatial frequencies around each constant up spatial frequency.In panel d) (i.e. after the Fourier transform along the αs dimension), V (up,us) contains allthe measured information about the sky brightness in a spatial frequency space. However, theinformation is ordered in a strange and redundant way. Indeed, we show that V (up,us) is linearlyrelated to I(up+us). To first order, the information about a given spatial frequency u is storedin all the values of V (up,us) which verifies u = up + us.A shift operation will reorder the spatial sca<strong>le</strong> information and averaging will compress theredundancy (illustrated by the halving of the number of the space dimensions). The use of a shiftand-averageoperator thus produces a final uv plane containing all the spatial sca<strong>le</strong> informationto image a wide field in an intuitive form. We thus call this space the wide-field uv plane. Paneld) displays this space, where the minimum re<strong>le</strong>vant spatial frequency is related to the total fieldof view, whi<strong>le</strong> the maximum one is related to the interferometer resolution.Applying the shift-and-average operator to V produces the Fourier transform of a dirty image,which is a local convolution of the sky brightness by a slowly varying 〈 〉 dirty beam (Pety &Rodríguez-Fernández, 2010). As a result, inverse Fourier transform of V and deconvolutionmethods will produce a wide-field distribution of sky brightness as shown in panel e) at the topright of Fig. 1.3 PracticeIn the real world, the visibility function is not only samp<strong>le</strong>d, but this sampling is incomp<strong>le</strong>te fortwo main reasons. 1) The instrument has a finite spatial resolution, and the scanning of the skyis limited, implying that the sampling in both planes has a finite support. 2) The uv coverageand the sky-scanning coverage can have ho<strong>le</strong>s caused either by intrinsic limitations (e.g. lack ofshort spacings or small number of baselines) or by acquisition prob<strong>le</strong>ms (implying data flagging).3.1 Simulating a wide-field observationIn order to test the imp<strong>le</strong>mentation, we need a control<strong>le</strong>d data set. We simulated the simp<strong>le</strong>stmeaningful case, i.e., a point source at the phase center (at the position of the famous HorseheadWIFISYN53. practiceduring 9 consecutive hours alterning beta and lambda coverage in a loop with 5 minutes calibrationmeasurements intersperse every 20 minutes. Each lambda or beta coverage lasts about 16minutes. There were thus 47 coverages observed in total. Taking into account various overheads(e.g., s<strong>le</strong>w time between rows) and shadowing (about 13% of the visibilities lost), we estimatethe observing efficiency to be about 60%, i.e. 60% of the time is spent on-source. We end upwith a data set of 148 147 visibilities. If the data was imaged through standard mosaicking, itwould imply between 500 and 10 000 actual fields depending on the acquisition system. Indeed,it is difficult to ensure that each scanning line always starts at exactly the same position on-skyfor each coverage as it is ideally assumed in this simulation.Fig. 2 separa<strong>tel</strong>y displays the sky and uv coverage, each cross representing one measure. Inreality, the uv coverage a priori depends on the position of the sky because of the Earth rotation.Fig. 3 thus display the uv coverage as a function of the position on the sky through a chessboarddisplay. Each case of the chessboard displays the all the uv measures which falls in a 20 ′′ -rangearound the marked sky coordinate. The uv coverage is qualitatively similar over the full fieldof view as a result of the observing strategy. The <strong>le</strong>ft panel of Fig. 3 is a zoom of one of theuv coverages shown on the chessboard. The uv coverage displayed on Fig. 2 is much denselypopulated than any of the individual uv coverages.Figure 3: Chessboard display of the uv planes. Left: Each panel display the uv coverage of allthe visibilities whose position on the sky falls in a 20 ′′ -range around the displayed sky coordinate.Right: Typical zoom of one of the uv coverages displayed <strong>le</strong>ft.nebula in the Orion mo<strong>le</strong>cular cloud). We choose to simulate observations with the Plateau deBure Interferometer because it allows us to c<strong>le</strong>arly test the effect of incomp<strong>le</strong>te instantaneous uvsampling due to the limited number of availab<strong>le</strong> antennas. The most compact configuration (Dconfiguration)was se<strong>le</strong>cted offering baseline <strong>le</strong>ngth up to about 120 m. We simulated observationsat the frequency of the 13 CO (J=1–0) line, i.e. 110.201 GHz or 2.72 mm. At this frequency, theprimary beamwidth of 15m-antenna is 44.9 ′′ .We simulated On-The-Fly observations covering a total field of view of 300 ′′ × 100 ′′ , whichconverts into the observation of about 17 independent fields. The field of view is rectangular toexplore the effect of non-square pixels in the uv planes. We scanned the source along two perpendiculardirection, named hereafter lambda and beta. The distance between two consecutiverows ensures Nyquist sampling, i.e., λ/D = 18.7 ′′ . We samp<strong>le</strong>d along the scanning directionat 5 points per primary beamwidth along the scanning direction to decrease the effect of beamelongation. We dumps at a rate of 0.5 Hz, i.e., one dump every two seconds. These two lastconsiderations implies a scanning velocity of 4.5 ′′ per second. We observed one full track, i.e.,3.2 Fourier transform along αs and βsWe want to Fourier transform the raw visibilities along the sky dimension (αs and βs) at someconstant value of up and vp. The raw data, however, is samp<strong>le</strong>d on an irregular grid in boththe uv and sky planes. We need to grid the measured visibilities in both the uv and the skyplanes before Fourier transformation for different reasons. First, the gridding in the uv plane willhand<strong>le</strong> the variation in the spatial frequency as the sky is scanned, i.e., the difficulty and perhapsthe impossibility of Fourier-transforming at a comp<strong>le</strong><strong>tel</strong>y constant (up,vp) values. Second, thegridding along the sky dimension enab<strong>le</strong>s the use of Fast Fourier Transforms.3.2.1 GriddingAs usual, we grid through convolution and regular resampling. The visibilities must be convolvedin 4 dimensions (αs, βs, up and vp). For simplicity sake’s, the convolution kernel is just theproduct of 4 one-dimensional functions along each convolution dimension. We used the standardspheroidal functions (cf. Fig. 4), whose Fourier transforms die off as quickly as themselves. Thisis a desirab<strong>le</strong> mathematical property in processing interferometry data because it limits aliasing.Fig. 5 displays the amplitude of the grided 4D visibility cube using a chessboard display, i.e.a set of uv images positionned at their right place in the (δRA,δDec) sky coordinate system.The first obvious visual impression when the color sca<strong>le</strong> is linear is that the visibility amplitudedecreases isotropically as the distance from the point source position (at the center of the coordinatesystem) increases. Indeed, the decrease in amplitude mimic the primary beam shape whichwas used to similate the visibilities, i.e. a Gaussian of 44.9 ′′ -beamwidth. The same chessboardis also displayed with a logarithmic color sca<strong>le</strong> to emphasize edge effects. The gridded visibilities“abruptly” go to zero following an almost rectangular pattern. This is just the consequence ofgridding along rectangular axes a limited field of view (the observed one) whi<strong>le</strong> the Gaussianprimary beam has an unlimited support.67
WIFISYN3. practiceWIFISYN3. practiceFigure 4: Convolution kernels used to grid the visibilities along the up (first column), vp (secondcolumn), αs (third column) and βs (fourth column) dimension. The top row displays the usedkernels whi<strong>le</strong> the bottom row displays their Fourier transform. The vertical red lines show thesize of the support over which the kernels are computed and used.The <strong>le</strong>ft panel is a zoom of the image displayed at the point source position on the chessboard.The visibility amplitude results here from the combination of the true visibility function(i.e., a Gaussian for a point source at the phase center) and of the density and quality of themeasurements at the considered position on the sky (because we grid the weighted visibilities,wV , and not just the visibilities themselves, V ). The images on the chessboard ressemb<strong>le</strong>s eachother to first order because the uv coverage is similar over the who<strong>le</strong> field of view.<strong>tel</strong>-<strong>00726959</strong>, version 1 - 31 Aug 20123.2.2 ReorderingThe chessboards in Fig. 5 display the up and vp visibility planes at constant values of αs and βs.However, we want to Fourier transform the grided visibilities along the sky dimensions (αs andβs) at constant value of up and vp. We thus reorder in Fig. 6 the chessboards so that they displaythe αs and βs visibility planes at constant values of up and vp. Nothing is actually done on thevisibility cube itself, it is just a change in the way it is displayed. We note that we now recognizeon the chessboard display the footprint of the uv plane sampling. Each image of the chessboardslooks like the two zooms, which c<strong>le</strong>arly show the Gaussian shape of the beam centered on thepoint source position. In particular, the images of the chessboards here are rectangular 1 (to keepthe ratio aspect of the αsβs planes).1 This is better seen on the chessboard with a logarithmic color sca<strong>le</strong>.WIFISYN83. practiceFigure 5: Chessboard display of the visibility amplitudes after the 4-dimensional gridding. Theamplitudes were arbitrarily normalized to get a maximum value of 1 because this allows thereader to easily quantify the dynamic at which potential artifacts arise. Left: Each panel displaythe image of the visibility amplitude at constant value of the sky coordinates (αs,βs) and asa function of the uv coordinates (up,vp). Right: Typical zoom of one of visibility amplitudeimages. Top: The color sca<strong>le</strong> is linear. Bottom: The color sca<strong>le</strong> is logarithmic.3.2.3 Wide-field synthesisThe chessboards in Fig. 9 display the visibility amplitude after Fourier transform along the (αs,βs)dimensions at constant (up,vp). We thus obtain a chessboards of (us,vs) planes at their positionsWIFISYN3.3 Shifting and averaging93. practiceFigure 6: Same as Fig. 5 after reordering of the axes, i.e. each panel display the image of thevisibility amplitude at constant value of the uv coordinates (up,vp) and as a function of the skycoordinates (αs,βs).Figure 7: Chessboard display of the amplitudes of the synthesized wide-field visibilities afterdirect Fourier transform along the αs and βs axes at constant (up,vp) values. The amplitudeswere arbitrarily normalized to get a maximum value of 1. Left: Each panel display the imageof the visibility amplitude at constant value of the wide sca<strong>le</strong> uv coordinates (us,vs) and as afunction of the narrow sca<strong>le</strong> uv coordinates (up,vp). Right: Typical zoom of one of visibilityamplitude images. Top: The color sca<strong>le</strong> is linear. Bottom: The color sca<strong>le</strong> is logarithmic.in the (up,vp) plane. The Fourier transform of a Gaussian is a Gaussian. The zoom panels thusdisplay to first order a Gaussian shape of full width at half maximum of about 15 m (i.e. thebure antenna diameter). The zoom panel using a logarithmic color sca<strong>le</strong> c<strong>le</strong>arly shows departurefrom the Gaussian shapes at the edges of the image because of the presence of “abrupt” edgesin Fig. 6. This is at this step that the wide-field visibility are synthesized, although they areordered in an unnatural way.Figure 8: Wide-field 2D uv plane after shifting-and-averaging the wide-field 4D uv plane displayedin Fig. 7. The amplitudes were arbitrarily normalized to get a maximum value of 1. Right: Thecolor sca<strong>le</strong> is linear. Left: The color sca<strong>le</strong> is logarithmic.After the wide-field synthesis step, we obtain a 4D wide-field visibility cube for which eachfinal uv spatial frequency (u,v) is measured for every (up,vp,us,vs) such that u = up + us andv = vp + vs. The use of a shift-and-average operator allows us to get an intuitive wide-fielduv plane because this first reorders the spatial frequencies at their right place and this thencompresses them.Fig. 8 display the wide-field uv plane, which results from this operation. It can be shown thatthe properties of the measurement equations imply that this wide-field uv plane must comply withthe Hermitian symmetry, i.e. V (u, v) = V ⋆ (−u, −v). This is equiva<strong>le</strong>nt to state that the dirtyimage must be real. However, this Hermitian symmetry is only approxima<strong>tel</strong>y enforced when theshift-and-average operator is used blindly over the 4D wide-field visibility cube, probably due torounding errors. We thus enforces the Hermitian symmetry through 1) computation of only halfof the wide-field uv plane (the one with negative values of v) and 2) deduction of the other halfusing the Hermitian symmetry.3.4 Getting the dirty image through inverse Fourier transformOnce the wide-field uv plane is availab<strong>le</strong>, the dirty image is obtained by taking the real part ofthe 2D Fourier transform of this plane. The top panel of Fig. 9 displays this dirty image, whichc<strong>le</strong>arly shows a point source at the phase center. As this image is the response of the wide-fieldsynthesized interferometer to a point source located at the phase center, it can be interpreted atthe dirty beam at the phase center. In this framework, the side-lobes peaks at about 10% andthere is a negative bowl surrounding it because of the missing zero-spacing. The bottom panelshows the absolute value of the dirty image with a logarithmic color sca<strong>le</strong> to display the <strong>le</strong>velsat which appear different kind of artefacts. Aliasing replications appear between 10 −2 and 10 −51011
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Table des matières1 Rapport de sou
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