Challenges in advanced imaging with radio interferometric telescopes. S.Bhatnagar NRAO, Socorro April 08, 2005

Transcription

1 Challenges in advanced imaging with radi interfermetric telescpes S.Bhatnagar NRAO, Scrr April 08, 2005

2 Basic Interfermetry Terminlgy: Visibility: V=<EiE*j>, Ei is the electric field Baseline: Length f the prjected separatin between the antennas (B). Only relative separatin matters fr incherent emissin C rdinates: (u,v,w ) u=ui uj <EiE*j> An N element array instantaneusly measures N (N 1)/2 baselines (cmplex values)

3 Basic Interfermetry In terms f the sky brightness distributin (I(l,m)) V u, v, w = I l, m S u, v, w e dl dm N n 2 u l v m w n 1 S : uv sampling functin, (l,m): directin in the sky 2 2 n= 1 l m Fr l2+m2<<1 r small w, sky is the 2D Furier transfrm f the Visibility functin (van Cittert Zernike Therem) I =FT [V ] and V =S V I d =PSF I PSF =FT [ S ]: The Dirty Beam

4 Sampling in the Furier plane N element array instantaneusly measures O(N2) Furier cmpnents. As earth rtates the prjected baseline changes ==> each baseline measures a new Furier cmpnent V = S. V

5 Image plane PSF =FT [ S ] I =FT [V ] Sampling functin d I =FT [V ]=PSF I

6 Imaging The easurement Eq.: V = AI AN where A is the measurement matrix, V, I and N are the Visibility, Image and nise vectrs. A=FS where F is the Furier Transfrm peratr and S is a diagnal matrix f weights (w). Due t finite Furier plane sampling, A is singular and in general rectangular Image recnstructin: Slve fr I T T A V=BI BN where B= A A B: The Teplitz Beam atrix ATA=Id: The Dirty Image vectr

7 Data crruptin The full easurement Eq.: V, t =G, t [ X, t I l, m e Data Crruptins 2 lu mv dl dm] Sky G are the directin independent crruptins (e.g., multiplicative cmplex gains, etc.) X are the directin dependent crruptins (e.g., inspheric/atmspheric effects, Primary beam effects, etc.) Often G, X are separable int antenna based quantities as G=GiG*j V =G.[E V ] j where E =E i E and E i =FT [ X i ]

8 Data analysis peratins V =J AI Calibratin: Keeping I fixed, determine Ji's. Calibrate the data: V Crrected 1 =J V Fr directin independent crrectins: Incrprate directin dependent crrectins in imaging Imaging: Keeping J fixed Estimate I such that V Crrected AI is nise like

9 Calibratin V =J.V Fr full plarizatin treatment: V=[VRR,VRL,VLR,VLL] Image using Crrected V 1 g R i d d R i g =J V Fr single plarizatin: min:{ V L i L i V J.V is nise like Given I, slve fr Jis such that J i= J =J i J j V i j g g }

10 Imaging Use f FFT requires re sampling the Vis. (V(u,v) is nt measured n a regular grid) G crrected V = GCF V d G ake the Dirty Image as: I =FFT [V ] Cnvlutinal gridding: k u Decnvlve the PSF t estimate I General structure f decnvlutin algrithms: ajr cycle: Cmpute the mdel and the residual vis as AI and V AI inr cycle: Update the mdel image

11 Decnvlutin: 2D gemetry General frmulatin T 2 =[V AI ] W [V AI ] 2 I = k a k P p k R = I P The Clean algrithm: I = k a k x x k Image representatin in the pixel basis. 2 = I R Steepest descent inr cycle: Step size: s=max{ 2 } I i =I i 1 s ajr cycle: Update using FFT: V ir=v AI i

12 Example I AT(V AI) I V AI

13 Sme bservatins Expensive functin evaluatin is simplified: Cnvlutin becmes a shift and add peratin. Clean (and its variants) ignre inherent pixel level crrelatin in I (each pixel is a DOF) errrs are crrelated with extended emissin. Search space is assumed t be rthgnal. T many DOFs fr extended emissin (n. f pixels). Search space cnstrained by user defined bxes.

14 Image plane crrectins V u, v, w = X l, m, w I l, m e 2 lu mv dl dm V u, v, w =E u, v, w V u, v A is mre cmplicated (nt a FT peratr) General apprach: ajr cycle invlves: V AI and AT V AI Use E as the GCF t predict the mdel data (AI) Cmpute VR at high accuracy. Use an apprximatin fr AT: Use ET as the GCF

15 Primary Beam Effects E as functin f directin is measured a priri V =G.[E V ] where E l i,l j,u ; pi, p j Primary beam effects i E =E E i where E =FT [ easured PB i ] Plarized primary beam: Beam squint j Fr full beam plarimetry (EVLA) Pinting ffset calibratin Fr msaicing (EVLA, ALA) GCF different fr each visibility!

16 Example Residual image befre and after pinting crrectin. IV=PB(IRR ILL) Peak ~4% IV=PBRIRR PBLILL Peak ~0.2%

17 Scale sensitive decnvlutin Pixel t pixel nise is crrelated at the scale f the PSF supprt I d =BI BI N (Pixn methds nt applicable they are patented anyway!). The scale f the emissin fundamentally separates the signal (I) frm nise (IN). Use a pixel mdel with finite supprt Cnvlutin is n mre a shift n add peratin. Functin evaluatin becmes very expensive. V R=FT [I d PSF I ]

18 Scale sensitive decnvlutin Search space is nn rthgnal The ttal DOFs decrease, but cmputing cst is a strng functin f the number f parameters. Asp Clean (Adaptive Scale Pixel mdel) [A&A, 2004] Use I = k a k P p k Find the lcal scale by fitting e.g., where P p k =Gaussian Fr emissin cvering 100,000 pixels, required 1000 Pks Cmputing cst prhibitive fr mre ~100 active parameters. Sub spaces

19 Example

20 Challenges Fast and accurate frward and backward transfrm in the presence f image plane effects min:v E [ AI ] Slve fr directin dependent effects: Primary beam effects (pinting, squint, variable beam shape,...) [EVLA, ALA] Wide band imaging: E =E [EVLA] Need mre sphisticated parametrizatin f E [EVLA,ALA,LWA,SKA...] E =E,t Inspheric/atmspheric calibratin [LWA, SKA] Simple functinal representatin may be difficult

21 Challenges Cmpnent based image mdel Gridding errrs limit the highest dynamic range achievable. FFT based inversin f the E may nt be usable. Image sizes and visibility data sets will be large 100s f GB 1TB f data, 15K X 15K images transfrms between image and data Parallelizatin f cmputing as well as I/O

22 Challenges Scale sensitive decnvlutin Pixel mdel parameterized by the amplitude, (lcal) scale, psitin, frequency, and perhaps plarizatin. Ttal n. f parameters: ~10K Fast algrithm fr parameter estimatin Handle the cupling between pixels (nn rthgnal search space) Adaptively cntrl n the dimensinality f the search space.

23 References Interfermetry and Synthesis in Radi Astrnmy Thmpsn, ran and Swensn, 2001, Jhn Wiley&Sns Clean: Hgbm, J.A., 1974, A&AS,15,417 E: Crnwell,T.J, &Evans, K.J.,1985,A&A,143,77 Review f varius methds: Narayan, R. & Nityananda, R.,1986, ARA&A,24,127 ath. analysis f the Clean Alg.: Schwarz, U.J., 1978, A&A, 65,345 Asp Clean: Bhatnagar,S., & Crnwell, T.J., 2004,A&A,436,747 W prjectin: Crnwell,T.J., Glap,K., & Bhatnagar,S.,2005,A&A,in press Pinting cal: Bhatnagar,S., Crnwell,T.J.,&Glap, K., 2004, EVLA em #84 Ninth Synthesis Imaging Summer Schl lectures,