Interacademiaal Final Lecture 1.Mosaicing 2.The Measurement Equation

Interacademiaal Mosaicing

Interacademiaal Why mosaic???? Historically division between:  Single dish for imaging “large” sources or surveys. Scan sky to make image  Interferometers for imaging “small” sources. Single pointing making an image Optimum obviously is to combine single dish and interferometer (but it took some time to develop this)

Interacademiaal Example that is works VLA mosaic of W50 50 VLA D-array pointings (Dubner et al 1998) See entire structure of the remnant, not just the fine-scale features Primary beam

Interacademiaal LMC ATCA Stanimirovic et al. (1999).

Interacademiaal Large mosaic of Bootes region at 21 cm WSRT De Vries, Morganti, Rottgering et al. Just image a large region of the sky..... (WENSS, NVSS)

Interacademiaal Problems with large sources 1.Most of source missing 2.Deconvolution not good 3.Non-uniform noise, high noise at edges....

Interacademiaal What is a large source? “Large” can mean two things: 1.Large compared to shortest baselines Structures larger than /b min not detected by interferometer 2.Large compared to primary beam (field of view) Source is very extended:   /D. Mosaicing (i.e. combine several pointings) is solution to both problems (2 nd is obvious, will explain 1 st in a minute) Lack of short baselines often becomes a problem before source structure is larger than the primary beam: Mosaicing is almost always about Total Power!

Interacademiaal Interferometer does not measure large scales a lot of information missing measured uv points u Without single-dish data one would have to interpolate in uv plane

Interacademiaal Ekers & Rots Since dishes have finite size: “a uv point is not a point”

Interacademiaal Primary beam convolves in uv plane Information is not directly available because of convolution

Interacademiaal Mosaic unravels this convolution By doing mosaic, one obtains different combinations of the same information Works to some extent, in general not sufficient to recover total power

Interacademiaal Effective Fourier plane coverage for ALMA snapshot

Interacademiaal A simulation of VLA mosaicing Model brightness distribution Image smoothed with 6” Gaussian (VLA D config. resolution at 15 GHz)

Interacademiaal Primary beam application Primary beam used for simulations Model multiplied by primary beam & smoothed with 6” Gaussian. This is the best we can hope to reconstruct from a single pointing.

Interacademiaal Single field imaging Visibilities constructed with thermal Gaussian noise. Image Fourier transformed & deconvolved with MEM Primary beam-corrected image. Blanked for beam response < 10% peak. Need to Mosaic!

Interacademiaal Joint deconvolution of all 9 pointings Nine VLA pointings deconvolved via a non-linear mosaic algorithm (AIPS VTESS). No total power included. Same mosaic with total power added.

Interacademiaal Joint deconvolution of all 9 pointings Mosaic with total power added. Image smoothed with 6” Gaussian (VLA D config. resolution at 15 GHz)

Interacademiaal Linear mosaic of deconvolved images  Deconvolve pointings separately and then combine using the least squares estimate  Works ok if separate deconvolutions are accurate  As for point sources  Deconvolution is non-linear, so is better to deconvolve all data together  “Joint deconvolution” Weighted linear sum of deconvolved images Images have to be regridded to common grid

Interacademiaal Linear mosaic of dirty images Generalize by a least squares fit for a given position on the sky (optimum combination) Weighted linear sum of dirty images Images have to be made on common grid

Interacademiaal Primary beams and limited Fourier plane sampling Two problems: 1.Convolution and division do not commute  position dependent dirty beam 1.The primary beam goes to zero far from the pointing center

Interacademiaal Joint deconvolution via non-linear optimisation  Find a model image that fits all the observed data  e.g. using Maximum Entropy

Interacademiaal Joint deconvolution algorithms based on linear mosaics  Use any suitable deconvolution algorithm to solve the approximate convolution equation  Use alternating major/minor cycles  In minor cycle, solve approximate convolution equation to some level of accuracy  In major cycle, recalculate linear mosaic from residual images, subtract model in uv plane. This takes care of position-dependent dirty beam

Interacademiaal Sault & Brouw (Miriad) Dirty image is linear mosaic Dirty beam at position l o For each pointing: 1. Multiply model with primary beam A 2. Convolve with dirty beam (FFT) 3. Make linear mosaic of “dirty model” 4. Subtract from dirty image SDI clean more efficient

Interacademiaal Addition of total power  Conceptual + practical problem  Conceptual - what are we trying to do?  Practical - synthesis and single dish have been two separate worlds  Three approaches  Make single dish data look like synthesis data  Add synthesis and single dish images after deconvolution  Add synthesis and single dish data during deconvolution

Interacademiaal Combining in uv plane Kitt Peak 12m image convolved with BIMA primary beam, converted to uv data with sampling density similar to BIMA uv coverage, scaled & combined with BIMA data, inverted with a taper, joint deconvolution (MIRIAD). Kitt Peak 12mBIMA12m + BIMA Shepherd, Churchwell, & Wilner (1997)

Interacademiaal Merging of dirty images, joint deconvolution If there is significant overlap in uv- coverage: images can be “feathered” together in the Fourier plane. Overlap allows to determine relative scaling Merged data Parkes Single dish Interferometer ATCA mosaic

Interacademiaal ATCA merged data ATCA observations of HI in the SMC. Dirty mosaic, interferometer only. Deconvolved mosaic, interferometer only. Stanimirovic et al. (1999).

Interacademiaal ATCA merged data Total power image from Parkes.Interferometer plus single dish feathered together (immerge). Stanimirovic et al. (1999).

Interacademiaal Deconvolved mosaic, interferometer only. Stanimirovic et al. (1999). Interferometer plus single dish feathered together (immerge). Stanimirovic et al. (1999).

Interacademiaal VLA + GBT merged data GBT On-the-fly map of the large field, (AIPS++). 90” resolution. GBT+VLA mosaic using AIPS++ image.feather. Shepherd, Maddalena, McMullin, VLA mosaic of central region, 9 fields. Deconvolved with MEM in AIPS ” resolution.

Interacademiaal Mosaic observations Nyquist sample the sky: pointing separation Hexogonal grid vs for Cartesian grid More area for same number of pointings Uniform noise Observe extra pointings in a guard band around source. Get total power information. Have good uv overlap between single dish and interferometer (big single dish, good pointing/low sidelobes & short baselines). Observe short integrations of all pointing centres, repeat mosaic cycle to get good uv coverage and calibration until desired integration time is achieved.

Interacademiaal Causes of errors in mosaic images  Missing total power information  Calibration inconsistency between synthesis and total power observations  Lack of a guard band  Insufficient image or Fourier plane sampling  Errors in primary beam model  Pointing errors  Especially important at high frequencies

Interacademiaal The Measurement Equation

Interacademiaal Remember: The Stationary, Monochromatic Interferometer Assume small frequency width, and no motion. Consider radiation from a small solid angle d , from direction l. X ll Another antenna b multiply average An antenna But: s corrupted (by J)

Interacademiaal What signal is really collected?  The net signal delivered by antenna i, x i (t), is a combination of the desired signal, s i (t,l,m), corrupted by a factor J i (t,l,m) and integrated over the sky, and noise, n i (t):  J i (t,l,m) is the product of a host of effects which we must calibrate  J i (t,l,m) is a complex number  J i (t,l,m) is antenna-based

Interacademiaal Correlation of realistic signals  On the timescale of the averaging, the only meaningful average is of the squared signal itself (direction-dependent), which is just the image of the source  If J position independent, take outside integral  If all J=1, we of course recover the Fourier transform expression Calibration equation

Interacademiaal Full-Polarisation Formalism: Signal Domain  But radiation is vector, not scalar, so substitute: where the Jones matrix corrupts a signal as follows: Sofar we used:

Interacademiaal Full-Polarisation Formalism: Correlation  Four correlations are possible from two polarisations. The outer product (a ‘bookkeeping’ product) represents correlation in the matrix formalism:  A very useful property of outer products: outer product: AB : every element of A multiplied by B

Interacademiaal Full-Polarisation Formalism: Correlation (cont)  The outer product for Jones matrices:  J ij is a 4x4 Mueller matrix  Antenna and array design driven by minimising off-diagonal terms! outer product: AB : every element of A multiplied by B

Interacademiaal Signal Correlation and Matrices (cont)  And finally, for fun, the correlation of corrupted signals:  UGLY, but we rarely need to worry about the details at this level---just let this occur “inside” the matrix formalism, and work with the notation

Interacademiaal Correlation of realistic signals  On the timescale of the averaging, the only meaningful average is of the squared signal itself (direction-dependent), which is just the image of the source  If J position independent, take outside integral  If all J=1, we of course recover the Fourier transform expression

Interacademiaal Write down the calibration in matrix form becomes scalar:

Interacademiaal Convert from Stokes to instrument  Need dual-polarisation basis (p,q) to fully sample the incoming EM wave front, where p,q = R,L (circular basis) or p,q = X,Y (linear basis):  Devices can be built to sample these basis states in the signal domain (Stokes Vector is defined in “power” domain)  Some components of J i involve mixing of basis states, so dual- polarisation matrix description desirable or even required for proper calibration

Interacademiaal Really Finally: The Measurement Equation  We can now really write down the calibration in a general way:  …and consider how to solve it! The Measurement Equation

Interacademiaal The Measurement Equation - Simplified  First, isolate non-direction-dependent effects, and factor them from the integral:  Next, we recognize that it is sometimes possible to assume J sky =1, and we have a relationship between ideal and observed Visibilities: The old calibration equation. Recovered if all matrices are diagonal

Interacademiaal Old vs New Matrix Full polarisation Scalar New Old Other difference: Physical vs parametric

Interacademiaal Solving the Measurement Equation  The J terms can be factored into a series of components representing physical elements along the signal path:  Depending upon availability of estimates for various J terms, we can re-arrange the equation and solve for any single term, if we know V ideal :  After obtaining estimates for all relevant J, data can be corrected: Allows physical description of observing process!!

Interacademiaal A Dictionary of Calibration Components  J i contains many components:  F = ionospheric Faraday rotation  T = tropospheric effects  P = parallactic angle  E = antenna voltage pattern  D = polarisation leakage  G = electronic gain  B = bandpass response  K = geometric compensation  Order of terms follows signal path (right to left)  Each term has matrix form of J i with terms embodying its particular algebra (on- vs. off-diagonal terms, etc.)  Direction-dependent terms involve FT in solution  The full matrix equation (especially after correlation!) is daunting, but usually only need to consider the terms individually or in pairs, and rarely in open form (matrix formulation = shorthand)

Interacademiaal “Electronic” Gain, G  Catch-all for most amplitude and phase effects introduced by antenna electronics (amplifiers, mixers, quantizers, digitisers)  Most commonly treated calibration component  Dominates other effects for standard VLA observations  Includes scaling from engineering (correlation coefficient) to radio astronomy units (Jy), by scaling solution amplitudes according to observations of a flux density calibrator  Often also includes ionospheric and tropospheric effects which are typically difficult to separate unto themselves  Excludes frequency dependent effects (see B)

Interacademiaal Bandpass Response, B  G-like component describing frequency- dependence of antenna electronics, etc.  Filters used to select frequency passband not square  Optical and electronic reflections introduce ripples across band  Often assumed time-independent, but not necessarily so  Typically (but not necessarily) normalised

Interacademiaal Ionospheric Faraday Rotation, F  The ionosphere is birefringent; one hand of circular polarisation is delayed w.r.t. the other, introducing a phase shift:  Rotates the linear polarisation position angle  More important at longer wavelengths ( 2 )  More important at solar maximum and at sunrise/sunset, when ionosphere is most active and variable  Beware of direction-dependence within field-of-view

Interacademiaal Tropospheric Effects, T  The troposphere causes polarisation-independent amplitude and phase effects due to emission/opacity and refraction, respectively  Typically 2-3m excess path length at zenith compared to vacuum  Higher noise contribution, less transmission: Lower SNR  Most important at > 15 GHz where water vapour absorbs/emits  More important nearer horizon where tropospheric path length greater  Clouds, weather = variability in phase and opacity; may vary across array

Interacademiaal Parallactic Angle, P  Orientation of sky in telescope’s field of view  Constant for equatorial telescopes  Varies for alt-az-mounted telescopes:  Rotates the position angle of linearly polarised radiation (c.f. F)  Analytically known, and its variation provides leverage for determining polarisation-dependent effects

Interacademiaal Antenna Voltage Pattern, E  Antennas of all designs have direction-dependent gain  Important when region of interest on sky comparable to or larger than /D  Important at lower frequencies where radio source surface density is greater and wide-field imaging techniques required  Beam squint: E p and E q not parallel, yielding spurious polarisation  For convenience, direction dependence of polarisation leakage (D) may be included in E (off-diagonal terms then non-zero)

Interacademiaal Polarisation Leakage, D  Orthogonal polarisations not perfectly isolated  Well-designed feeds have d ~ a few percent or less  A geometric property of the feed design, so frequency dependent

Interacademiaal Geometric Compensation, K  Must get geometry right for Synthesis Fourier Transform relation to work in real time; residual errors here require “Fringe-fitting”  Antenna positions (geodesy)  Source directions (time-dependent in topocenter!) (astrometry)  Clocks  Electronic pathlengths  Importance scales with frequency and baseline length

Interacademiaal All there is to know about radio telescopes Full polarisation, physically motivated matrix description of entire observing process

Interacademiaal Acknowledgements For this lecture I made extensive use of material prepared by other people, in particular Tim Cornwell, George Moellenbrock and Crystal Brogan