Basics of Interferometry and Interferometric Calibration

1
Basics of Interferometry and Interferometric
Calibration

• Claire Chandler
• NRAO/Socorro
• (with thanks to Rick Perley and George
  Moellenbrock)

2
Overview

• Aperture synthesis
• A simple 2-element interferometer
• The visibility
• The interferometer in practice
• Calibration of interferometric data
• Baseline-based and telescope-based calibration
• Calibration in practice
• VLA example
• Non-closing errors
• Tropospheric phase fluctuations

3
Response of a parabolic antenna
Power response of a uniformly-illuminated,
circular parabolic antenna (D 25m, n 1GHz)
4
Origin of the beam pattern

• An antennas response is a result of coherent
  phase summation of the electric field at the
  focus
• First null will occur at the angle where one
  extra wavelength of path is added across the full
  width of the aperture q l/D

5
The beam pattern (cont.)

• A voltage V(q) is produced at the focus as a
  result of the electric field
• The voltage response pattern is the Fourier
  Transform of the aperture illumination for a
  uniform circle this is J1(x)/x
• The power response, P(q) µ V2(q)
• P(q) is the Fourier Transform of the
  autocorrelation function of the aperture, and for
  a uniformly-illuminated circle is the familiar
  Airy pattern, (J1(x)/x)2
• FWHP 1.02 l/D
• First null at 1.22 l/D

6
Aside Fourier Transforms

• A function or distribution may be described as
  the infinite sum of sines and cosines with
  different frequencies (Fourier components), and
  these two descriptions of the function are
  equivalent and related by
• Where x and s are conjugate variables, e.g.
• Time and frequency, t and 1/t n
• Distance and spatial frequency, l and 1/l u

7
Some useful FT theorems

• Addition
• Shift
• Convolution
• Scaling

8
Output for a filled aperture

• Signals at each point in the aperture are brought
  together in phase at the antenna output (the
  focus)
• Imagine the aperture to be subdivided into N
  smaller elementary areas the voltage, V(t), at
  the output is the sum of the contributions DVi(t)
  from the N individual aperture elements

9
Aperture synthesis basic concept

• The radio power measured by a receiver attached
  to the telescope is proportional to a running
  time average of the square of the output voltage
• Any measurement with the large filled-aperture
  telescope can be written as a sum, in which each
  term depends on contributions from only two of
  the N aperture elements
• Each term áDViDVkñ can be measured with two small
  antennas, if we place them at locations i and k
  and measure the average product of their output
  voltages with a correlation (multiplying) receiver

10
Aperture synthesis basic concept

• If the source emission is unchanging, there is no
  need to measure all the pairs at one time
• One could imagine sequentially combining pairs of
  signals. For N sub-apertures there will be
  N(N-1)/2 pairs to combine
• Adding together all the terms effectively
  synthesizes one measurement taken with a large
  filled-aperture telescope
• Can synthesize apertures much larger than can be
  constructed as a filled aperture, giving very
  good spatial resolution

11
A simple 2-element interferometer

• What is the response of the interferometer as a
  function of position on the sky, l sina ?

• In direction s0 (a 0) the wavefront arriving at
  telescope 1 has an extra path bs0 bsinq to
  travel compared with telescope 2
• The time taken to traverse this path is the
  geometric delay, tg bs0/c
• Compensate by inserting a delay in the signal
  path for telescope 2 equivalent to tg

12
Response of a 2-element interferometer

• At angle a relative to s0 a wavefront has extra
  path x usina ul to travel

• Expand to 2D by introducing b orthogonal to a, m
  sinb, and v orthogonal to u, so that in this
  direction the extra path y vm
• Write all distances in units of wavelength, x º
  x/l, u º u/l, etc. so that x and y are now
  numbers of cycles
• Extra path is now ul vm
• V2 V1 e-2pi(ulvm)

13
Correlator output

• The output from the correlator (the multiplying
  and time-averaging device) is
• For (l1¹l2, m1¹m2) the above average is zero
  (assuming mutual incoherence of the sky), so

14
The visibility

• Thus the interferometer measures the complex
  visibility, V, of a source, which is the FT of
  its intensity distribution on the sky
• u,v are spatial frequencies in the E-W and N-S
  directions, and are the projected baseline
  lengths measured in units of wavelength, B/l
• l,m are direction cosines relative to a reference
  position in the E-W and N-S directions
• (l0,m0) is known as the phase centre

15
Comments on the visibility

• This FT relationship is the van Cittert-Zernike
  theorem, upon which synthesis imaging is based
• It means there is an inverse FT relationship that
  enables us to recover I(l,m) from V (u,v)
• The visibility is complex because of the FT
• The correlator measures both real and imaginary
  parts of the visibility to give the amplitude and
  phase

16
More comments

• The visibility is a function of the source
  structure and the interferometer baseline
• The visibility is not a function of the absolute
  position of the telescopes (provided the emission
  is time-invariant, and is located in the far
  field)
• The visibility is Hermitian V (-u,-v) V
  (u,v). This is because the sky is real
• There is a unique relation between any source
  brightness distribution and the visibility
  function
• Each observation of the source with a given
  baseline length, (u,v), provides one measure of
  the visibility
• With many measurements of the visibility as a
  function of (u,v) we can obtain a reasonable
  estimate of I(l,m)

17
Some 2D FT pairs

• Image Visibility amp

18
Some 2D FT pairs

• Image Visibility amp

19
Some 2D FT pairs

• Image Visibility amp

20
(u,v) coverage

• A single baseline provides one measurement of V
  (u,v) per time-averaged integration
• Build up coverage in the uv-plane by
• having lots of telescopes
• moving telescopes around
• waiting for the Earth to rotate to provide
  changing projected baselines, u bcosq (note
  tg changes continuously too)
• some combination of the above
• Telescope locations
  (u,v) coverage
• 
  for 6 hour track

21
The measured visibility

• Resulting measured visibility
• (u,v) coverage V
  Vmeasured
• 
  
• Note because the visibility is Hermitian each
  measurement by the interferometer results in two
  points in the (u,v) plane, one for baseline 1-2,
  the other for baseline 2-1, etc. Earth rotation
  traces out two arcs per baseline
• Recovering I(l,m) from Vmeasured is the topic of
  the next lecture

22
Picturing the visibility fringes

• The FT of a single visibility measurement is a
  sinusoid with spacing l/B between successive
  peaks, or fringes
• Build up an image of the sky by summing many such
  sinusoids (addition theorem)
• Scaling theorem shows
• Short baselines have large
  fringe spacings
  and measure
  large-scale structure on the
  sky
• Long baselines have small
  fringe spacings
  and measure
  small-scale structure on the sky

23
The primary beam

• The elements of an interferometer have finite
  size, and so have their own response to the
  radiation from the sky
• This results in an additional factor, A(l,m), to
  be included in the expression for the correlator
  output, which is the primary beam or normalized
  reception pattern of the individual elements
• Interferometer actually measures the FT of the
  sky multiplied by the primary beam response
• Need to divide by A(l,m), to recover I(l,m)
• The last step in the production of the image

24
The delay beam

• A real interferometer must accept a range of
  frequencies (amongst other things, there is no
  power in an infinitesimal bandwidth)! Consider
  the response of our interferometer over frequency
  width Dn centred on n0
• At angle a away from the phase centre the excess
  path for the centre of the band in cycles is u0l
• At this point the excess path (in cycles) for the
  edges of the band, n n0 Dn/2, is ul
  u0l(n/n0)

25
The delay beam (cont.)

• Wavefronts are out of phase at the edges of the
  band compared with the centre where one extra
  wavelength of path is added across the entire
  band, or ul - u0l 0.5
• This occurs where
• Now l sin a a and fringe spacing qres
  1/u0
• So width of delay beam is
• ALMA example n 100 GHz, Dn 8 GHz, D 12m
  (qpb 53²) and qres 1², Þ qdb 12.5²
• ALMA will have to divide bandwidth into many
  channels to avoid loss of sensitivity due to
  delay beam!

26
The interferometer in practice the need for
calibration

• For the ideal interferometer the phase of a point
  source at the phase centre is zero, because we
  can correct for the known geometric delay
• However, there are other sources of path delay
  that introduce phase offsets that are
  telescope-dependent most importantly at
  submillimeter wavelengths these are water (vapour
  and liquid) in the troposphere, and electronics
• The raw amplitude of the visibility measured on a
  given baseline depends on the properties of the
  two telescopes (gains, pointing, etc.), and must
  be placed on a physical scale (Jy)
• There may be frequency-dependent amplitude and
  phase responses of the electronics
• There may also be baseline-based errors
  introduced due to averaging in time or frequency
• We must observe calibration sources to derive
  corrections for these effects, to be applied to
  our program sources

27
Correlation of realistic signals

• The signal delivered by telescope i to the
  correlator is the sum of the voltage due to the
  source corrupted by a telescope-based complex
  gain gi(t) ai(t)eifi(t), where ai(t) is a
  telescope-based amplitude correction and fi(t) is
  the telescope-based phase correction, and noise
  ni(t)
• The correlator output is then

28
realistic signals (cont.)

• The noise does not correlate, so the noise term
  áninjñ integrates down to zero
• Compare with a single dish, which measures the
  auto-correlation of the signal
• This is a total power measurement plus noise
• Desired signal is not isolated from noise
• Noise usually dominates
• Single dish calibration strategies dominated by
  switching schemes to isolate the desired signal

29
Telescope- and baseline-based errors

• Write the output from the correlator as
• where
• and we have introduced a factor gij(t) to
  take into account any residual baseline-based
  gain error
• If Gij(t) is slowly-varying this can be written
  as
• and we see that the correlator output Vijobs
  is the true visibility Vij modified by a complex
  gain factor Gij(t)

30
Contributions to Gij

• Gij comprises telescope-based components from
• Ionospheric Faraday rotation (important for cm,
  not for submm)
• Water in the troposphere (important for submm)
• Parallactic angle rotation
• Telescope voltage pattern response
• Polarization leakage
• Electronic gains
• Bandpass (frequency-dependent amplitudes and
  phases)
• Geometric (delay) compensation (important for
  VLBI)
• Baseline-based errors due to
• Correlated noise (e.g., RFI important for cm)
• Frequency averaging
• Time averaging (important for submm because of
  the troposphere)

31
Solving for Gij

• Can separate the various contributions to Gij to
  give
• Initially use estimates (if you have them), or
  assume Gijn 1, and solve for the dominant
  component of the error, assuming we know Vijtrue
  (note we observe calibration sources for which
  we do know Vijtrue)
• Iterate on the solutions for Gijn

32
Baseline-based calibration

• Since the interferometer makes baseline-based
  measurements, why not just use the observed
  (baseline-based) visibilities of calibrator
  sources to solve for Gij?
• We have to know that the calibrator is a point
  source (i.e., should have f 0 at the phase
  centre) or know its structure
• If we do not, using baseline-based calibration
  will absorb structure of the calibrator into the
  (erroneous) solution for Gij
• In general we have to be able to assume that
  baseline-based effects are astronomical in
  origin, i.e., tell us about the source visibility
  (there are some calibrations that are
  baseline-based more on these later)
• Most gain errors are telescope-based
• It is better to solve for N telescope-based
  quantities than N(N-1)/2 baseline-based
  quantities (fewer free parameters)

33
Telescope-based calibration closure relationships

• Closure relationships show that telescope-based
  gain errors do not irretrievably corrupt the
  information about the source (indeed, in the
  early days of radio interferometry, and today for
  optical interferometry, the systems were so
  phase-unstable that these closure quantities were
  all that could be measured)
• Closure phase (3 baselines) let fij fi-fj,
  etc.
• Closure amplitude (4 baselines)

fiDfi
fjDfj
fkDfk
34
Calibration in practice

• Observe nearby point sources against which
  calibration can be solved, and transfer solutions
  to target observations
• Choose appropriate calibrators usually strong
  point sources because we can predict their
  visibilities
• Choose appropriate timescales for calibration
• Typically need calibrators for
• Absolute flux density (constant radio source,
  planet)
• Nearby point source to track complex gain
• Nearby point source for pointing calibration
• Strong source for bandpass measurement
• Source with known polarization properties for
  instrumental polarization calibration

35
VLA example

• Science
• HI observations of the galaxy NGC2403
• Sources
• Target source NGC2403
• Near-target calibrator 0841708 (8 deg from
  target unknown flux density, assume 1 Jy
  initially)
• Flux density calibrators 3C48 (15.88 Jy), 3C147
  (21.95 Jy), 3C286 (14.73 Jy)
• Signals
• RR correlation, total intensity
• 1419.79 MHz (HI), one 3.125 MHz channel
• (continuum version of a spectral line observation)

36
Observing sequence
37
UV-coverages
38
Views of the raw data
39
Uncalibrated images
40
Gain errors are telescope-based
41
The telescope-based calibration solution - I

• Solve for telescope-based gain factors on 600s
  timescale (1 solution per scan on near-target
  calibrator, 2 solutions per scan on flux-density
  calibrators)
• Bootstrap flux density scale by scaling mean gain
  amplitudes of near-target (nt) calibrator
  (assumed 1Jy above) according to mean gain
  amplitudes of flux density (fd) calibrators

42
The telescope-based calibration solution - II
43
Calibrated calibrator phases - I
44
Calibrated calibrator phases - II
45
Calibrated visibilities
46
Calibrated images
47
Evaluating the calibration

• Are solutions continuous?
• Noise-like solutions are just thatnoise
• Discontinuities indicate instrumental glitches
• Any additional editing required?
• Are calibrator data fully described by
  antenna-based effects?
• Phase and amplitude closure errors are the
  baseline-based residuals
• Are calibrators sufficiently point-like? If not,
  self-calibrate
• Any evidence of unsampled variation? Is
  interpolation of solutions appropriate?
• Reduce calibration timescale, if SNR (and data)
  permits
• Evidence of gain errors in your final image?
• Phase errors give asymmetric features in the
  image
• Amplitude errors give symmetric features in the
  image

48
Typical calibration sequence, spectral line

• Preliminary solve for G on bandpass calibrator
• Solve for bandpass on bandpass calibrator
• Solve for G (using B) on gain calibrator
• Flux density scaling
• Correct the data
• Image!

49
Non-closing errors

• Baseline-based errors are non-closing errors
  (they do not obey the closure relationships)
• The three main sources of non-closing errors
  arise from
• Source structure!
• Frequency averaging
• Time averaging
• Frequency averaging
• Instrument has a telescope-based bandpass
  response as a function of frequency

• If all telescopes have the same response then
  averaging in frequency will preserve the closure
  relationships
• Different bandpass responses will introduce
  baseline-based gain errors (e.g., VLA-EVLA
  baselines)

50
Non-closing bandpass errors

• Solutions
• Design your instrument to match bandpasses as
  closely as possible
• Divide the bandpass into many channels, so that
  the closure relationships are obeyed on a
  per-channel basis
• Apply a baseline-based calibration
• BUT BEWARE OF FREQUENCY-DEPENDENT STRUCTURE IN
  THE CALIBRATION SOURCE

51
Time averaging tropospheric phase fluctuations

• Averaging over time-variable phase and amplitude
  fluctuations also introduces non-closing errors
• Particularly important in the submm, where
  variations in the water vapour content of the
  troposphere introduce phase fluctuations on very
  short timescales
• Precipitable water vapour, pwv (typically of
  order 1mm), results in excess electrical path
  
  and phase change
• Variations in the amount of pwv cause
• pointing offsets, both predictable and anomalous
• delay offsets
• phase fluctuations, worse at shorter wavelengths,
  resulting in low coherence and radio seeing,
  typically 1-3² at l 1 mm

52
Phase fluctuations loss of coherence

• Imag. thermal noise only
  Imag. phase noise thermal noise
• 
  Þ
  low vector average
• (high s/n)
  frms
• 
  Real
  Real
• Measured visibility V V0eif
• áVñ V0 áeifñ V0 e-f2rms/2 (assuming
  Gaussian phase fluctuations)
• If frms 1 radian,

53
Solutions to tropospheric phase fluctuations

• Use short integration times OK for bright
  sources that can be detected in a few seconds
• Fast switching calibrate in the normal way using
  a calibration cycle time, tcyc, short enough to
  reduce frms to an acceptable level effective for
  tcyc lt B/vwind
• Radiometry measure fluctuations in water vapour
  emission from the troposphere above each
  telescope with a radiometer, derive the
  fluctuations in pwv, and convert into a phase
  correction using
• Measure the amplitude decorrelation using a
  strong calibrator and very short integration
  times, assume áfñ0, and apply a baseline-based
  coherence correction to the amplitudes

54
Final remarks

• I have not covered polarimetry or many other
  subtleties for further reading see Synthesis
  Imaging in Radio Astronomy II, ASP Vol. 180
  (1998)
• Presentations from NRAO Synthesis Imaging Summer
  Schools, 2002, 2004, 2006
• http//www.nrao.edu/meetings/synthimwksp.shtml
• Interferometry and synthesis imaging requires you
  to think in FT space! Dont despair, this takes
  practice