Radio Astronomy 6

Transcription

Radio Astronomy 6

9/29/15 Interferometry
and
Synthesis Imaging
1 Arrays of Antennas
Arrays of Antennas
Arrays of Antennas
The Fringe Pattern
N-element Arrays
Phase Offsets
Phase Offsets
Coherent and
Incoherent Sums
Beam Steering
Alternative Usage of
Arrays
Cross Correlation
Interferometric Imaging
Complications in
Interferometry:
Summary
Image formation from
Interferometry: I
Interferometry: II
Interferometry: III
Synthesis Imaging
7 Oct 2009 – 2
2 1 9/29/15 3 4 2 9/29/15 Arrays
• 
• 
• 
• 
• 
Beam forming vs. interferometry.
Array factor contribution to the PSF.
Visibility functions.
Van Cittert-Zernike theorem.
Synthesis imaging and calibration.
5 Main Points for Beam Forming
!  The voltages (e.g. the baseband signals) from N
antennas can be summed to provide an effective area "
= N x (single-antenna area).
!  The PSF of the array can be shaped and directed
through appropriate complex weights for each signal "
in the sum.
!  With digital beam forming, multiple beams can be
formed (multiple source sampling).
!  All operations must occur at rates ~ 1 / bandwidth "
" enormous processing requirements.
6 3 9/29/15 Main Points for Interferometry
•  The cross correlation of voltages between antennas
(the visibility Γ(u,v) ) in an array is related to the mutual
coherence of the radiation field.
•  Cross correlations between polarizations and antennas
yield Stokes visibilities.
•  Local oscillator stability + ionosphere + atmosphere
limit the coherence time of the cross correlations.
7 Main Points for Interferometry
•  Advantages of aperture synthesis over single dish
observations:
•  Spatial filtering.
•  RFI rejection "
(RFI doesn’t correlate over long baselines).
•  Flexibility in image construction.
•  Disadvantages:
•  Limited collecting area (so far).
•  No ‘zero-spacing’ flux density.
•  Computation intensive "
(data sets ~ Terabytes with VLA).
8 4 9/29/15 Main Points for Interferometry
•  “Phase center”.
•  Effects due to finite bandwidths and integration times
(distortion of visibility function + reduction of field of view).
•  Calibration methods:
•  Amplitude, phase, and polarization calibration.
•  Image formation:
•  Van Cittert-Zernike theorem: "
Image = Fourier transform( visibility function ).
•  Sidelobes and dynamic range: "
CLEAN = deconvolution of PSF from raw image.
•  Self calibration (hybrid mapping).
9 Arrays of Antennas
Arrays of Antennas
Arrays of Antennas
The Fringe Pattern
Pair of antennas with incident plane wave:
Phase difference due to path length difference (PLD).
N-element Arrays
Phase Offsets
Phase Offsets
Coherent and
Incoherent Sums
∆φ = 2π
P LD
= k P LD = kb sin θ.
λ
Beam Steering
Arrays
Cross Correlation
Complications in
Interferometry:
Summary
Interferometry: I
Interferometry: II
Interferometry: III
Synthesis Imaging
7 Oct 2009 – 2
10 5 9/29/15 Arrays of Antennas
Arrays of Antennas
Arrays of Antennas
The Fringe Pattern
N-element Arrays
Phase Offsets
Phase Offsets
Coherent and
Incoherent Sums
Beam Steering
Arrays
E = E1 + E2
= E0 (eiφ1 + eiφ2 )
= E0 (eiφ1 + ei(φ1 +∆φ) )
= E0 eiφ1 (1 + ei∆φ ).
Cross Correlation
Complications in
Interferometry:
Summary
Interferometry: I
Interferometry: II
Interferometry: III
I = |E|2 = |E0 |2 2(1 + cos ∆φ) = 2I0 (1 + cos ∆φ)
⎧
⎪
⎨4I0 if ∆φ = 0,
⇒ I = 2I0 (1 + cos ∆φ) 2I0 if ∆φ = π/2,
⎪
⎩
0
if ∆φ = π .
Synthesis Imaging
7 Oct 2009 – 3
11 The Fringe Pattern
Arrays of Antennas
Arrays of Antennas
I(θ) = 2I0 (1 + cos(kb sin θ)).
The Fringe Pattern
N-element Arrays
Phase Offsets
Phase Offsets
Coherent and
Incoherent Sums
Beam Steering
Arrays
Cross Correlation
Complications in
Interferometry:
Summary
Fringe spacing ∆θ ≈
λ
b
for ∆θ ≪ 1.
Interferometry: I
Interferometry: II
Interferometry: III
Synthesis Imaging
12 7 Oct 2009 – 4
6 9/29/15 The Fringe Pattern
Arrays of Antennas
Arrays of Antennas
The Fringe Pattern
N-element Arrays
Phase Offsets
Phase Offsets
I(θ) = 2I0 (1 + cos(kb sin θ)).
Fringe spacing ∆θ ≈ λb for ∆θ ≪ 1.
Also need to include antenna power pattern Pn (θ):
Coherent and
Incoherent Sums
I(θ) → I(θ)Pn (θ).
Beam Steering
Arrays
Cross Correlation
Complications in
Interferometry:
Summary
Interferometry: I
Interferometry: II
Interferometry: III
Synthesis Imaging
Relative widths: λ/b vs. λ/D
• VLA: D = 25m; b ∼ 1–30 km.
• VLBA: D = 25m; b ∼ 6000 km.
7 Oct 2009 – 4
13 N-element Arrays
Arrays of Antennas
For an N-element array,
Arrays of Antennas
The Fringe Pattern
N-element Arrays
Phase Offsets
Phase Offsets
Coherent and
Incoherent Sums
E=
N
!
Ej eiφj =
j=1
Beam Steering
Arrays
Cross Correlation
Complications in
Interferometry:
Summary
Interferometry: I
Interferometry: II
Interferometry: III
Synthesis Imaging
I=|
N
!
Ej eik.Xj .
j=1
N
!
Ej eik.Xj |2
j=1
⇒ N “auto” terms, N(N-1) “cross” terms, N(N-1)/2 fringe patterns.
The array response to the sky brightness distribution is,
for a given pair of antennas,
the integral of the product of
the fringe pattern and the sky brightness.
This is ≈ the Fourier transform of the sky.
7 Oct 2009 – 5
14 7 9/29/15 Phase Offsets
Arrays of Antennas
Arrays of Antennas
Add a (controllable) phase difference at each antenna, φ̂j .
The Fringe Pattern
N-element Arrays
Phase Offsets
Phase Offsets
Coherent and
Incoherent Sums
Beam Steering
Arrays
Cross Correlation
Complications in
With a suitable φ̂j we can control the directionality of the array.
Interferometry:
Summary
Interferometry: I
Interferometry: II
Interferometry: III
E=
N
!
Ej ei(φj −φ̂j ) =
j=1
N
!
Ej ei(k.Xj −φ̂j ) .
j=1
Let Ej = E0 for all j , and ∆φj = (k.Xj − φ̂j ).
Synthesis Imaging
7 Oct 2009 – 6
15 Phase Offsets
Arrays of Antennas
Arrays of Antennas
The Fringe Pattern
N-element Arrays
Phase Offsets
Then, with E = E0
!N
j=1
ei∆φj ,
• Case 1: Perfect phase compensation, ∆φj = 0.
Then E = N E0 and I = N 2 I0 .
Phase Offsets
Coherent and
Incoherent Sums
Beam Steering
Arrays
Cross Correlation
• Case 2: Random ∆φj (why?).
"
I = I0 |
ei∆φj |2
j
Complications in
Interferometry:
Summary
Interferometry: I
Interferometry: II
= I0
""
j
ei(∆φj −∆φk ) |2
k
= N I0 + I0
""
j
k ̸= j)ei(∆φj −∆φk ) |2
(
For large N , final N (N − 1) terms sum ≈ 0; I = N I0 .
Interferometry: III
Synthesis Imaging
7 Oct 2009 – 7
16 8 9/29/15 Coherent and Incoherent Sums
Arrays of Antennas
With E = E0
Arrays of Antennas
The Fringe Pattern
Phase Offsets
j=1
ei∆φj ,
“Incoherent” case: I = N I0 .
“Coherent” case: I = N 2 I0 .
!
!
N-element Arrays
!N
Phase Offsets
If the phase offset at each antenna is noise-like, incoherent sum.
If the phase offset precisely compensates for a given direction, we
have a coherent sum in that direction: beam steering.
Coherent and
Incoherent Sums
Beam Steering
Arrays
Cross Correlation
Note that each antenna has a power pattern, and we end up with a
net power pattern for the array:
Complications in
(A)
Interferometry:
Summary
Pn
≡ the Fourier transform of the antenna locations.
Interferometry: I
Interferometry: II
Interferometry: III
Synthesis Imaging
7 Oct 2009 – 8
17 Beam Steering
Arrays of Antennas
Arrays of Antennas
The Fringe Pattern
N-element Arrays
Phase Offsets
Phase Offsets
Coherent and
Incoherent Sums
Beam Steering
Arrays
Cross Correlation
Control the direction of the main lobe of the array
by the control phases φˆj .
!
!
!
!
!
Complications in
Interferometry:
Summary
Interferometry: I
!
Reference direction: direction of main lobe
Array beam can be steered to any location that is within the
primary beam of each antenna.
How? Analog delay lines, or digital delays.
If ∃ multiple sets of control phases,
then multiple beams on the sky! Cost: Extra processing.
If each antenna has a wide FoV
(e.g., dipoles θF W HM ∼ 1 radian)
then much of the sky can be viewed simultaneously.
Steerable antennas: entire sky is accessible. (cf., SKA).
Interferometry: II
Interferometry: III
Synthesis Imaging
7 Oct 2009 – 9
18 9 9/29/15 Murchison Widefield Array
(MWA) – array of dipoles
that view half the sky.
19 Alternative Usage of Arrays
Arrays of Antennas
Arrays of Antennas
The Fringe Pattern
N-element Arrays
Rather than summing antenna outputs in real time,
process each pair of antennas separately,
and do all beam forming offline in software.
Phase Offsets
Phase Offsets
Coherent and
Incoherent Sums
Beam Steering
Arrays
Cross Correlation
⇒ Fourier transformation vs. correlation (convolution).
→ Cost: much, much more processing.
(Also, not usually real-time.)
Schematically:
Complications in
Interferometry:
Summary
Interferometry: I
Interferometry: II
Cross correlate pairs
Array(Xj ) −−−−−−−−−→ Γij
!
⏐
F vs. Xj ⏐
(A)
Pn (θ)
Convolve with sky
⏐
⏐
#F vs. b
−−−−−−−−→ I(θ)
Many directions
Interferometry: III
Synthesis Imaging
7 Oct 2009 – 10
20 10 9/29/15 Visibilities and Sky Images
F
F
-1
UV Coverage
Synthesized ("Dirty") Beam
v
on
d
ve
ol
C
Sky brightness distribution
Output dirty map
21 Cross Correlation
Arrays of Antennas
Arrays of Antennas
The Fringe Pattern
N-element Arrays
The “visibility” Γ is function of (bij ) = Xj − Xi .
Γij ≡ ⟨Ei Ej⋆ ⟩ = ⟨E(Xi )E ⋆ (Xj )⟩
Phase Offsets
Phase Offsets
Coherent and
Incoherent Sums
Beam Steering
Arrays
Cross Correlation
Complications in
Interferometry:
Summary
Interferometry: I
Simpler notation: Γ(b), where
⎛ ⎞
⎛ ⎞
bx
u
⎝
⎠
⎝
b = by = λ v ⎠
bz
w
For small fields of view, the “w” coordinate can be ignored
if the z -direction ≈ the reference direction.
(Assumption breaks down for very wide-field imaging.)
Interferometry: II
Interferometry: III
Synthesis Imaging
7 Oct 2009 – 11
22 11 9/29/15 Interferometric Imaging
Arrays of Antennas
Arrays of Antennas
The Fringe Pattern
The visibility function then depends on the “u” and “v ” coordinates:
Visibility function Γ(u, v).
N-element Arrays
Phase Offsets
I(θ) = I(θx , θy ) =
Phase Offsets
Coherent and
Incoherent Sums
!!
du dv e2πi(uθx +vθy ) Γ(u, v)
(The Van Cittert-Zernike theorem.)
Beam Steering
Arrays
(1)
Note: The primary antenna pattern Pn (θ) is implicit.
Cross Correlation
Interferometry:
Complications in
Interferometry:
Summary
!
!
!
Measure Fourier components of I(θ)
≡ Visibilities Γ(b, τ ).
ˆ .
Transform to get image estimate I(θ)
Interferometry: I
Interferometry: II
Interferometry: III
Synthesis Imaging
7 Oct 2009 – 12
23 Interferometric Imaging
Arrays of Antennas
Arrays of Antennas
The Fringe Pattern
N-element Arrays
Phase Offsets
Phase Offsets
Coherent and
Incoherent Sums
Beam Steering
Arrays
Cross Correlation
Complications in
Interferometry:
Summary
Interferometry: I
Interferometry: II
Interferometry: III
Synthesis Imaging
7 Oct 2009 – 13
24 12 9/29/15 Interferometric Imaging
Arrays of Antennas
Arrays of Antennas
!
I(θ) ≡ Kλ db eikb.θ Γ(b; τ )
!!
→
du dv e2πi(uθx +vθy ) Γ(u, v; τ )
The Fringe Pattern
N-element Arrays
Phase Offsets
Phase Offsets
Coherent and
Incoherent Sums
Beam Steering
Arrays
Cross Correlation
After sampling the Fourier components, transform to get:
Complications in
ˆ =
I(θ)
Interferometry:
Summary
"
e2πi(uθx +vθy ) Γ(u, v; τ )
u,v
Interferometry: I
Interferometry: II
where u, v, θ are discrete.
Interferometry: III
Synthesis Imaging
7 Oct 2009 – 14
25 Complications in Interferometric Imaging
Arrays of Antennas
Arrays of Antennas
The Fringe Pattern
N-element Arrays
Phase Offsets
Phase Offsets
Coherent and
Incoherent Sums
Beam Steering
Arrays
Cross Correlation
• Control phases are based on a reference direction
(the “phase center”), as in beam forming.
• Phases are corrupted by:
!
!
!
!
!
Intervening media (differential refraction, scattering, ...);
Imperfect knowledge of coordinate system (Earth rotation);
Imperfect knowledge of antenna locations (tectonics; geodesy);
Imperfect clocks and oscillators;
Variable phases in electronics (temperature).
Complications in
Interferometry:
Summary
Interferometry: I
Interferometry: II
Interferometry: III
Synthesis Imaging
7 Oct 2009 – 15
26 13 9/29/15 Interferometry: Summary
Arrays of Antennas
!
Arrays of Antennas
The Fringe Pattern
!
!
N-element Arrays
Phase Offsets
Phase Offsets
Coherent and
Incoherent Sums
!
Beam Steering
Arrays
Cross Correlation
!
!
Correlation function of voltages is related to mutual coherence
of the radiation field.
Cross-correlations → Stokes parameter vsibilities.
Coherece time is limited by LOs, atmosphere, ...
Compared to single dish observations, advantages:
Spatial filtering, RFI rejection, flexibility in image estimation.
But:
Limited collecting area, compuation intensive.
Larger scales are filtered out! Zero spacing flux density.
Complications in
!
Interferometry:
Summary
Interferometry: I
!
Smearing effects due to finite bandwidth and integration time
(i.e., averaging over time and frequency).
Consider: calibration, imaging, sky mapping schemes.
Interferometry: II
Interferometry: III
Synthesis Imaging
7 Oct 2009 – 16
27 Image formation from Interferometry: I
Arrays of Antennas
Arrays of Antennas
The Fringe Pattern
N-element Arrays
How do we measure Γ, the visibility function?
!
Phase Offsets
Phase Offsets
Coherent and
Incoherent Sums
!
Beam Steering
Arrays
Cross Correlation
!
Γ(u, v) relates to the cross-correlation function of voltages in
two antenna/receiver systems.
Calibration: antenna gains, phase errors.
Closure phase: φ1,2 + φ2,3 + φ3,1 = 0.
Closure amplitude: Γ(1, 2).Γ(3, 4) = Γ(1, 3).Γ(2, 4)
Self calibration and hybrid mapping.
Field of view issues: primary beam and delay beam.
Complications in
Interferometry:
Summary
Interferometry: I
Interferometry: II
Interferometry: III
Synthesis Imaging
28 7 Oct 2009 – 17
14 9/29/15 Image formation from Interferometry: II
Arrays of Antennas
Arrays of Antennas
The Fringe Pattern
N-element Arrays
Phase Offsets
Phase Offsets
Coherent and
Incoherent Sums
Beam Steering
Arrays
Cross Correlation
How do we go from Γ to the sky brightness distribution B ?
!
!
!
Fourier relationship between Γ ↔ B .
Relationship is approximate in practice.
In some cases, better to use alternatives to go from Γ → B
e.g., Maximum Entropy Method: Use an a priori model of the sky
and maximize a pre-defined statistic.
(But where do you get the model? A blank sky works.)
Complications in
Interferometry:
Summary
Interferometry: I
Interferometry: II
Interferometry: III
29 Synthesis Imaging
7 Oct 2009 – 18
Image formation from Interferometry: III
Arrays of Antennas
Arrays of Antennas
The Fringe Pattern
Constructing an image from a set of visibilities:
FFT + C LEAN, or MEM.
N-element Arrays
Phase Offsets
Phase Offsets
Coherent and
Incoherent Sums
!
!
Beam Steering
Arrays
Cross Correlation
!
!
Sampling in uv plane is incomplete.
Dealing with missing data: MEM, C LEAN take different
approaches (smooth extended structure vs. collection of point
sources).
Calibration errors lead to image degradation.
But we can use knowledge about the sky to improve calibration:
self-calibration and hybrid mapping.
Complications in
Interferometry:
Summary
Interferometry: I
Interferometry: II
Interferometry: III
Synthesis Imaging
Dynamic range: Dirty maps →
C LEAN
→
Self-cal
→
SKA goal →
10:1
100s:1
104 :1
106 – 107 .
7 Oct 2009 – 19
30 15 9/29/15 Practical Synthesis Imaging
−1.2 sin (t−π/3)
Any continuous function can be
built up as a
superposition of
sinusoids.
This could
represent, e.g., a sky brightness
distribution.
+2.1 sin (2t+π)
+0.9 sin (5t−π/4)
2 parameters: A sin(ωt+ø) = (A cos ø) sin ωt + (A sin ø) cos ωt
31 Practical Synthesis Imaging
Interferometer: Sample spatial harmonics.
X
X
X
32 16 9/29/15 Practical Synthesis Imaging
33 U-V Coverage
34 17 9/29/15 The “Dirty Beam”
35 Calibration
Amplitude vs UV dist for VLB 0600+45.UVDATA.1 Source:LGM06+45
Ants * - * Stokes RR IF# 1 Chan# 1 - 16
Amplitude vs UV dist for VLB 0600+45.UVDATA.1 Source:LGM06+45
Ants * - * Stokes LL IF# 1 Chan# 1 - 16
6.5
6.0
6.0
5.5
5.5
Janskys
Janskys
5.0
4.5
5.0
4.0
3.5
4.5
3.0
2.5
0
10
20
30
Mega Wavlngth
Freq = 1.6650 GHz, Bw = 8.000 MHz
40
0
10
20
30
40
Mega Wavlngth
Freq = 1.6650 GHz, Bw = 8.000 MHz
Visibility amplitudes vs baseline – uncalibrated and calibrated.
36 18 9/29/15 Visibilities and Sky Images
F
F
-1
UV Coverage
Synthesized ("Dirty") Beam
v
on
d
ve
ol
C
Sky brightness distribution
Output dirty map
37 Deconvolution with CLEAN
38 19 9/29/15 Deconvolution with CLEAN
39 Deconvolution with CLEAN
CONT: LGM06+45 LL 1668.750 MHZ VLB 0600+45.LCLN.1
45 00 00.06
00.04
DECLINATION (J2000)
00.02
00.00
44 59 59.98
59.96
59.94
06 00 00.008
00.006
00.004
00.002
00.00005 59 59.998
RIGHT ASCENSION (J2000)
Cont peak flux = 4.9889E+00 JY/BEAM
Levs = 5.000E-03 * (-2, -1, 1, 2, 4, 8, 16, 32,
64, 128, 256, 512)
59.996
40 20 9/29/15 Notes
41 42 21 9/29/15 43 44 22 9/29/15 45 46 23 9/29/15 47 48 24 9/29/15 Extra
49 50
25 9/29/15 51
26

Radio Astronomy 6

Transcription

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