Wide-Field Imaging: I

Transcription

1 Wide-Field Imaging: I S. Bhatnagar NRAO, Socorro Twelfth Synthesis Imaging Workshop 010 June 8-15

2 Wide-field imaging What do we mean by wide-field imaging W-Term: D Fourier transform approximation breaks down Full-beam imaging: Antenna Primary Beam (PB) effects cannot be ignored Mosaicking: Imaging fields with emission larger than the antenna Field-of-View (FoV)

3 The W-Term D approximation of the measurement equation (ME) breaks down ( The W-term problem ). Imaging dynamic range throughout the image is limited by deconvolution errors due to the sources away from the (phase) center.

4 Primary Beam Effects Antenna Primary Beam (PB) pattern cannot be approximated by unity ( Full Beam Imaging ). Imaging dynamic range throughout the image is limited by the deconvolution errors due to the sources in the half-power points and the side lobes.

5 Mosaicking (see later lectures) Imaging emission wider than the Field-of-View (FoV) ( Mosaicking ) Dominant sources of errors Antenna Pointing PB effects: rotation, multiple types of antenna in the array (ALMA) Deconvolution errors for extended emission Pointing centers (usually also the pointing phase center)

6 Theory re-cap: Measurement Eq. V Obs [ u ij l v ij m] u ij, v ij ; = I l, m, e dl dm van-cittert Zernike Theorem: Coherence function is a D Fourier Transform of the Sky Brightness distribution Full ME V Obs ij Data s.b ij S ij = J ij, t J s,, t I s, e Corruptions Sky s= l, m, n = l, m, 1 l m 1 : Directioncosins bij = ui u j, v i v j, wi w j = uij, v ij, wij :The Baseline vector ds Geometry

7 Theory re-cap: Measurement Eq. V Obs ij J ij = J i J j s.b ij S ij = J ij, t J s,, t I s, e ds Direction independent effects (e.g. Complex Gains a la SelfCal) Constant across the field of view J Sij = J Si J Sj Direction dependent effects (e.g. Antenna PB, ionosphere,...) Time variable direction dependent gains due to PB rotation for Az-El antennas Geometry: W-Term s.bij e [ u l v m w 1 l m 1 ] =e

8 The W-Term: Theory Ignore Jij (nominally calibrated data) and Jsij (ignore effects of PB) V = I s, [ e uij l v ij m w ij 1 l m 1 ] ds FoV is small: l m 1 Array is co-planar w Obs ij u max v max vcz: D Fourier transform works When FoV or wij is large, data and the image are not related by a simple D Fourier transform relationship.

9 The W-Term: Geometric interpretation l =sin u n=cos X w u X Phase of the visibilities for direction = u l For the interferometer in a plane: For the interferometer not in a plane: = u l w n 1 [ D approximation valid only when: (1) w is small compared to u, or () 0 ]

10 The W-Term: Optics interpretation Physically we measure... o ' V 1 = E1 u, v, w 0 E 0,0,0...and interpret as if it were V 1 = E1 u, v, w=0 E 0,0,0 E1 is equal to E'1 propagated using Fresnel diffraction theory V o1=v 1 u, v, w=0 G u, v, w, where G u, v, w =Fresnel P ropagater =FT [ e w 1 l m ] o 1 V = I l, m e [ u1 l v 1 m ] e w1 1 l m 1 dl dm A single interferometer is sensitive to multiple Fourier component Concept of redundant baselines is more restrictive than is usually thought!

11 Example: No W-Term Correction W-term is a phase error Sources move in the image in a systematic way Hermitian but a dispersive effect

12 Example: After W-Term Correction

13 Solutions: 3D Imaging Do a 3D inversion of the ME to make a 3D Image F l, m, n = V u, v, w e ] du dv dw Relate F(l,m,n) to the physical image as I l, m = [ u ij l v ij m w ij n F l, m,n 1 l m l m n 1 Interpretation Physical emission I(l,m) exists along the surface of a unit sphere inside the 3D-Image F(l,m,n) Resulting algorithm is not efficient Not used very often (read never used :-))

14 Solutions: Faceted Imaging Interpret I(l,m) as emission on the surface of Celestial Sphere of unit radius: l+m+n=1 Approximate the celestial sphere by a set of tangent planes a.k.a. facets Use D imaging on each facet Re-project the facet-images to a single D plane Number of facets required N Poly = f B max =B max D D Antenna diameter ; f =Full Antenna Beam

15 Solutions: Faceted Imaging Re-phase the entire data towards the direction of each facet center (tangent point) Since l+m is small (by construction), make a D image for each facet What happens when emission extends over multiple facets? Re-project the facet images to a single plane l R l' lp m m' mp Intuitively easy to understand Emission extending over multiple facets is an issue Multiple images Available in CASA and AIPS

16 Solutions: UV plane equivalent Since the facet-images are related to the single-plane image by a linear co-ordinate transformation, there must exist a equivalent operation in the visibility plane. I C l det C V C 1 1T u where C is the image domain co-ordinate transform l and u are the image and visibility plane co-ordinates 1 Projection error: =sin 1 1 cos 1 Error same as in image plane faceting! Produces a single image (no edge effects) Global deconvolution possible (extended emission) Use of advanced algorithms for extended emission possible Region definition as in the usual case Available in CASA and possibly in AIPS

17 Solutions: W-Projection Optics interpretation o 1 V = I l, m e [ u1 l v 1 m ] e w1 1 l m 1 dl dm V o u, v, w =V u, v, w=0 G u, v, w, where G u, v, w =FT [ e w 1 l m ] Size of the Fresnel Zone: rf w T If G G is unitary (or even approximately so) GT and G can be used in the forward and inverse transforms to produce distortion free images

18 The W-Projection Algorithm Optics interpretation V o u, v, w =V u, v, w=0 G u, v, w Algorithm Model prediction (major cycle) [Residual computation] Perform a D FFT of the model image (appropriately tapered) (this is V u, v, w=0 ) Evaluate the above convolution equation as part of Gridding to get V Compute the Dirty Image (minor cycle) [Deconvolution] Use G T u, v, w on each V o u, v, w during gridding to evaluate V u, v Perform a D FFT-1 of V u, v o u, v, w

19 W-Projection: Performance Scaling laws: Facet imaging: N Facets N GCF N vis W-Projection: N WPlanes N GCF N vis Ratio: N GCF for large number of facets/wplanes In practice WProjection algorithm is about 10x faster Size of G(u,v,w) increases with W

20 W-Projection: Performance WProjection about 10x faster than facet based imaging Algorithm complexity is significantly lower Users practically see no difference between wide field imaging and normal imaging Fits in the general mathematical framework of advanced imaging techniques This is more important than is realized! Works naturally with any minor cycle algorithm (Hogobm-, Clark-, CS-, MS-, AspClean, etc.) Naturally integrates with full-beam imaging/calibration Naturally includes multi-field imaging May not be useful for full-sky imaging telescopes with long baselines But can be combined with faceted imaging (implemented in CASA)

21 Examples: 74MHz, before correction Courtesy: K. Golap

22 Examples: WProjection imaging Courtesy: K. Golap Sub-image of an outlier field

23 A small digression Linear optimization view of deconvolution V o= A I o N M V =A I M N is Gaussian Random Variable in the data domain Is the optimal estimator. Deconvolution is then equivalent to o M minimize: = V AI M where I = k Pk ; Dirty Image Pixel Model P k is the Pixel Model M M I i =I i 1 Various algorithms differ in (1) parametrization of Pk, () types of constraints, and (3) how the constraints are applied

24 A small digression: ME entirely in the visibility domain: Projection methods V ij = Eij [ V Obs o ] where Eij represents a direction dependent (DD) effect Construct a Kij which models the desired DD effect If KTij Eij ~ 1 (Unitary Operator), compute update direction (Dirty Image) as FT [ K ij V ij ] I T Res o =I PSF Deconvolution: Minor Cycle Accurate residual computation (Chisq) as Res V ij =K ij FT Dirty 1 [ I M ] V Obs ij Deconvolution: Major Cycle Iterations will converge if the operator is approximately unitary

25 Image domain vs. data domain Projection methods utilize the available data optimally Non-linear operations (e.g. deconvolution) are done on images constructed using the entire data Errors are corrected using parametrized models Global corrections rather than local correction Since DD effects are typically also time varying, image domain based correction are non-optimal Necessarily require data partitioning Signal-to-noise available to solvers corresponds to a fraction of the total data Require many more DoF: At least one per direction of interest Local corrections ==> Could lead to Closure errors

26 Examples: D Imaging

27 Examples: Facet Imaging

28 Examples: WProjection Imaging

29 Full beam imaging Now ignore Jij (i.e., using calibrated data). Ignore also the w-term for the moment: Obs S u ij l v ij m V ij = J ij s,, t I s, e ds Some observations: J Sij is direction dependent: complex gain potentially different for different direction in the sky J Sij = J Si J Sj : This is true for most instrumental, atmospheric /ionospheric corruptions (all effects that obey closure relationship ) S S When J i =J j and stationary in time (e.g. PB of ideal, identical antennas), its effects can be corrected in the image domain I Obs =I s, J s s,

30 More observations... Jsi are in general complex (complex Primary Beams!) S S In real life, J i J j In real life, Jsi vary with time... Gain change at first side lobe due to rotation Source of time variability Cross hand power Pointing errors Geometric distortions pattern

31 Frequency dependence... Jsi vary with frequency... To the first order, PBs scale with frequency

32 Polarization dependence... Jsi vary with polarization Parallel Hand Pattern: PBRR [ R FT E E R ] Cross Hand Pattern: PBRL [ R FT E E L ]

33 Error propagation Azimuthal cuts at 50%, 10% and 1% of the Stokes-I error pattern AvgPB - PB(to) R o I = P S F [ P B I ]

34 Examples: Stokes-I and -V imaging W-Term errors! 3C147, EVLA, L-Band High DR 700,000:1 Errors due PB side-lobes?

35 Examples: Stokes-V Stokes-V = IRR-ILL Stokes-V Power pattern

36 Theory: Full-beam imaging Re-cap the simplified ME V Obs ij S ij = J s,, t I s, e Re-write it as o V Obs =E [ V ] =E E [ V ] ij ij i j uij l v ij m ds S S S J ij =J i J j E i : Antenna Aperture Illumination Pattern [ ] E i=ft J is Vobs is equal to true visibilities convolved with the auto-correlation of antenna Aperture Illumination pattern. If there exists a function Kij such that KTij * Eij ~ Delta Function T o o V Gij =K Tij V Obs = K E [ V ] [ V ]oij ij G ij ij T T o Vdij =K ij V ij =K ij E ij [ V ] [ V ] ij o FFT [ V ] I Gridding: Imaging: M M Prediction: V ij =K ij FFT [ I ]

37 Solution: A-Projection algorithm ℜ K ir : Antenna Aperture Illumination Pattern K ij phase K ij

38 Solution: A-Projection algorithm [ ] FT K ij K E in the image domain ij ij

39 Stokes-I: Before correction

40 Stokes-I: After correction

41 Stokes-V: Before correction

42 Stokes-V: After correction

43 Examples: EVLA Imaging

44 Examples: Time varying PBs Simulations for (Masaya Kuniyoshi (LWA/NRAO)) Model for EVLA PB at L-Band

45 Thesis Review Articles Papers Books References: Interferometry and Synthesis in Radio Astronomy, nd Ed.: Thompson, Moran and Swenson Synthesis Imaging in Radio Astronomy: II The White Book W-Projection: IEEE Journal of Selected Topics in Signal Processing, Vol., No. 5, 008 A-Projection: A&A, 487, 419, 008 (arxiv: ) Scale sensitive deconvolution of astronomical images: A&A, 46, 747, 004 (astro-ph/04075) MS-Clean: IEEE Journal of Selected Topics in Signal Processing, Vol., No.5,008 Advances in Calibration and Imaging in Radio Interferometry: Proc. IEEE, Vol. 97, No. 8, 008 Calibration and Imaging challenges at low frequencies: ASP Conf. Series, Vol. 407, 009 High Fidelity Imaging of Moderately Resolved Source; PhD Thesis, Briggs, NMT, 1995 Web Pages Parameterized Deconvolution for Wide-band Radio Synthesis Imaging; PhD Thesis, Rao Venkata; NMT, Home pages of SKA Calibration and Imaging Workshops (CALIM), 005, 006, 008, 009 Home Pages of: EVLA, ALMA, ATA, LOFAR, ASKAP, SKA, MeerKat