Shear Measurement Review

1
Shear Measurement Review

• Bluffers guide
• Shear measurement pipeline
• Shear estimation methods
• KSB, Shapelets, Model fitting
• STEP, GREAT08
• PSF uncertainties
• Propagation of PSF errors
• PCA
• Cross-correlation
• Bluffers guide test

2
Shear Measurement Review

• Bluffers guide
• Shear measurement pipeline
• Shear estimation methods
• KSB, Shapelets, Model fitting
• STEP, GREAT08
• PSF uncertainties
• Propagation of PSF errors
• PCA
• Cross-correlation
• Bluffers guide test

3
Bluffers Guide to Shear MeasurementCan you say
these phrases with confidence?

• The shear responsivity depends on the galaxy
  ellipticity distribution
• P gamma has to be averaged over galaxies with
  similar properties
• If the wrong galaxy profile is assumed, the
  shear is biased
• You should definitely participate in GREAT08
• Shear systematics depend on the square of the
  PSF size
• PSF PCA technique allows a larger effective star
  density
• Cross correlation between images averages out
  random PSF shear biases

4
Shear Measurement Review

• Bluffers guide
• Shear measurement pipeline
• Shear estimation methods
• KSB, Shapelets, Model fitting
• STEP, GREAT08
• PSF uncertainties
• Propagation of PSF errors
• PCA
• Cross-correlation
• Bluffers guide test

5
Typical galaxy used for cosmic shear analysis
Typical star Used for finding Convolution kernel
6
Typical image
7
Overview
Example The DES pipeline
1. Measure PSF at star positions
DB
DB
2. Interpolate PSF
DB
DB
3. Measure shears
DB
DB
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1. Measure PSF at star positions
Example The DES pipeline
Object catalogue From coadds All S/Ngt5
Single Boolean per object
1a. Identify Stars
DB
DB
Star catalogue
PSF coefs (per star)
Images Noise images
1b. Get PSF coefficients
DB
DB
9
1. Measure PSF at star positions
Example The DES pipeline
Object catalogue From coadds All S/Ngt5
Single Boolean per object
1a. Identify Stars
DB
DB
Star catalogue
PSF coefs (per star)
Images Noise images
1b. Get PSF coefficients
DB
DB
Shapelets 30 floating point s 120 Bytes per
star
10
2. Interpolate PSF step (a)
Example The DES pipeline
PSF coefs (per star) (all exposures)
2a. PSF Interpolator Step 1
PSF(space,time)
DB
DB
11
2. Interpolate PSF step (a)
Example The DES pipeline
PSF coefs (per star) (all exposures)
2a. PSF Interpolator Step 1
PSF(space,time)
DB
DB
20 PCA components (3rd order polynomial per
62 chips) 80 kB per PCA compt
12
2. Interpolate PSF step (b)
Example The DES pipeline
PSF(space,time)
DB
Parameters of PSF Interpolation
2b. PSF Interpolator Step 2
DB
PSF coefs (per star)
DB
13
2. Interpolate PSF step (b)
Example The DES pipeline
PSF(space,time)
DB
Parameters of PSF Interpolation
2b. PSF Interpolator Step 2
DB
PSF coefs (per star)
DB
PCA
3rd order polynomial 62 x 1.2 kB
14
3. Shear measurement
Example The DES pipeline
Object catalogue All S/Ngt5
DB
Shears per object Best Per filter Per
exposure Errors on above Flags on the above
Images Noise images
DB
3. Shear measurement
DB
Parameters of PSF Interpolation
DB
15
3. Shear measurement
Example The DES pipeline
Object catalogue All S/Ngt5
DB
Shears per object Best Per filter Per
exposure Errors on above Flags on the above
Images Noise images
DB
3. Shear measurement
DB
Parameters of PSF Interpolation
DB
We may want e.g. flexions too
16
Overview
Example The DES pipeline
1. Measure PSF at star positions
DB
DB
2. Interpolate PSF
DB
DB
3. Measure shears
DB
DB
17
Shear Measurement Review

• Bluffers guide
• Shear measurement pipeline
• Shear estimation methods
• STEP, GREAT08
• KSB, Shapelets, Model fitting
• PSF uncertainties
• Propagation of PSF errors
• PCA
• Cross-correlation
• Bluffers guide test

18
Shear TEsting Programme (STEP)

• Started July 2004
• Is the shear estimation problem solved or not?
• Series of international blind competitions
• Start with simple simulated data (STEP1)
• Make simulations increasingly realistic
• Real data
• Current status
• STEP 1 simplistic galaxy shapes (Heymans et al
  2005)
• STEP 2 more realistic galaxies (Massey et al
  2006)
• STEP 3 difficult (space telescope) kernel (2007)
• STEP 4 back to basics

See Konrads Edinburgh DUEL talk
19
STEP1 Results
? Existing results are reliable
The future requires 0.0003
Heymans et al 2005
-0.2
0.2
20
STEP1 Results - Dirty laundry
Require 0.0003
0
Accuracy on g
Average -0.0010
-0.005
High noise
Low noise
noise level of image
21
PASCAL Challenge
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Gravitational Lensing
Galaxies seen through dark matter distribution
analogous to Streetlamps seen through your
bathroom window
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Cosmic Lensing
gi0.2
Real data gi0.03
25
Atmosphere and Telescope
Convolution with kernel
Real data Kernel size Galaxy size
26
Pixelisation
Sum light in each square
Real data Pixel size Kernel size /2
27
Noise
Mostly Poisson. Some Gaussian and bad
pixels. Uncertainty on total light 5 per cent
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GREAT08 Data

• 10 000 images
• divided into 10 sets

One galaxy per image Kernel is given One shear
per set Noise is Poisson
100 000 000 images Divided into 1000 sets
31
GREAT08 Results
You submit g1, g2 for each set of images
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GREAT08 Timeline

• 8 Feb 2008 GREAT08 Handbook public
• Feb 2008 Produce simulations
• Mar 2008 Internal analysis of simulations
• May 2008 Release simulations
• Leaderboard starts containing best internal
  results
• Nov 2008 Competition deadline
• Dec 2008 Workshop Release final report
• Input shears public

www.great08challenge.info
33
tbc
Going to astro-ph this week Please advertise to
your computer science and statistics colleagues
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GREAT08 Summary

• 100 million images
• 1 galaxy per image
• De-noise, de-convolve, average ? shear
• gi 0.03 to accuracy 0.0003 ? Q1000 ? Win!

35
Shear Measurement Review

• Bluffers guide
• Shear measurement pipeline
• Shear estimation methods
• STEP, GREAT08
• KSB, Shapelets, Model fitting
• PSF uncertainties
• Propagation of PSF errors
• PCA
• Cross-correlation
• Bluffers guide test

36
Quadrupole moments the simplest possible shear
measurement method
37
Effect of shear on quadrupole moments
38
Ellipticity ? (a-b)/(ab) from quadrupole
moments
Nasty noise properties
Please correct this using the paper
provided!! (Taylor expand for small g)
Update after talk better to start from reversed
version of the above eqn (swap eu with el and
change g to -g. Then get el eu g - (
(e_1u2-e_2u2) g_1 e_1u e_2u g_2 ) i (
(e_2u2-e_1u2) g_2 e_1u e_22 g_1)
O(g2) Use lte_1ugt lte_2ugt 0 so lteugt0 and
assume symmetry lte_1u2gtlte_2u2gt and lte_1u
e_2ugt 0
Where
So ?li can be treated as a noisy estimate of gi
e.g. see Bartelmann Schneider 1999 review p59,
Bonnet Mellier 1995, Seitz Schneider 1997
39
Ellipticity ?(a2-b2)/(a2b2)from quadrupole
moments
40
Ellipticity ?(a2-b2)/(a2b2)from quadrupole
moments
Taylor expand and use
Please check this Using the paper provided
Depends on properties of galaxies
Shear responsivity
e.g. see Bartelmann Schneider 1999 review p59,
Schneider Seitz 1995
41
Bartelmann Schneider 1999 p 61
42
KSBAs above, with weight function
Herein lies significant complications
The last two are the same in the case W(x,y)1
43
Why do weighted quadrupole moments?
44
The KSB shear responsivity
Kaiser, Squires Broadhurst 1995, Luppino
Kaiser 1997, Hoekstra et al. 1998
45
Improvements beyond KSB

• Bernstein Jarvis
• Mandelbaum et al.
• Kaiser 2000

46
Shapelets

• Laguerre polynomials
• Polynomial times Gaussian
• Nice QM formalism
• Lensing distortion has simple effect
• psf convolution can be removed by matrix
  multiplication

Massey Refregier 2004
47
Shapelets

• Three main approaches
• Refregier, Bacon, Massey
• Use shapelet model to make perfect image. Measure
  Qij from this
• Bernstein, Nakajima, Jarvis
• Shear circular shapelet model to match the
  image
• Kuijken
• Shear exactly circular shapelet model to match
  image

Massey Refregier 2004
48
Shapelets

• Three main approaches
• Refregier, Bacon, Massey
• Use shapelet model to make perfect image. Measure
  Qij from this
• Bernstein, Nakajima, Jarvis
• Shear circular shapelet model to match the
  image
• Kuijken
• Shear exactly circular shapelet model to match
  image

• Pros
• - Compact basis set
• - Shearing and convolution v fast
• Cons
• - QM formalism blindingly seductive
• Gaussian envelope not well matched to galaxies
  or psf
• .. see Sechlets poster..

49
Model fitting

• Sheared shapelets (Nakajima Bernstein Kuijken)
• Sums of Gaussians (Voigt Bridle in prep)
• de Vaucouleurs profiles (see Kitching talk)
• Arbitrary radial profile (Irwin Shmakova 2005)

50
Is there a bias on gwhen fit 1 elliptical
Gaussian?
51
Modelling the galaxy with a single Gaussian
Simulated galaxy
Model
Exponential e0.2
Gaussian

• PSF perfectly known
• Best-fit Gaussian to exponential found by
  minimising the ?2 between the images with
  respect to the 6 model parameters
  x0,y0,e,phi,a,A
• No noise added to data!

Voigt Bridle in prep
52
Modelling the galaxy with a single Gaussian No
PSF, small pixels
Simulated galaxy
Model
Exponential e0.2
Gaussian

• 15 pixels per FWHM along minor axis
• Best-fit Gaussian to exponential found by
  minimising the ?2 between the images with
  respect to the 6 model parameters x0,y0,e,phi,a,A

Question Is the measured galaxy ellipticity
biased? (Bias measured true ellipticity)
53
Modelling the galaxy with a single Gaussian No
PSF, small pixels
Simulated galaxy
Model
Exponential e0.2
Gaussian

• 15 pixels per FWHM along minor axis
• Best-fit Gaussian to exponential found by
  minimising the ?2 between the images with
  respect to the 6 model parameters x0,y0,e,phi,a,A

Answer No - bias on ellipticity measured is lt
0.1 (Bias measured true ellipticity)
54
Modelling the galaxy with a single Gaussian
Gaussian PSF, small pixels
Simulated galaxy
True convolved image
PSF Gaussian
Best-fit model
Model Gaussian
PSF true PSF
Question Is the measured galaxy ellipticity
biased?
55
Modelling the galaxy with a single Gaussian
Gaussian PSF, small pixels
Simulated galaxy
True convolved image
PSF Gaussian
Best-fit model
Model Gaussian
PSF true PSF
Answer Yes! - bias on galaxy ellipticity
measured gt 1
56
Modelling the galaxy with a single Gaussian
Gaussian PSF, small pixels
Simulated galaxy
True convolved image
PSF Gaussian
Best-fit model
Model Gaussian
PSF true PSF
Answer Yes! - bias on galaxy ellipticity
measured gt 1 Need to model the galaxy with more
than 1 Gaussian!
57
Modelling the galaxy with a single Gaussian
Including the effects of pixellisation
58
Is there a bias on gwhen fit 1 elliptical
Gaussian?
No
No
No
Yes
No
Yes
Voigt Bridle in prep
59
Is there a bias on gwhen fit 1 elliptical
Gaussian?
No
No
Wider implicationsMust model galaxy profile
accurately
No
Yes
No
Yes
Voigt Bridle in prep
60
Model galaxy as multiple co-elliptical Gaussians
Residuals
Model Gaussian
Simulated galaxy exponential
1 G
2 G

• Tied parameters x0,y0,e,phi
• Free parameters a,A

3 G
Voigt Bridle in prep
61
Fitting the galaxy with multiple
GaussiansIncluding the PSF
62
Is there a bias on gwhen fit multiple
co-elliptical Gaussians?
No
No
No
No
No
Yes
Voigt Bridle in prep
63
Fitting multiple Gaussians to galaxies with
non-elliptical isophotes

• Simulate galaxies with a sum of 2 Gaussians, each
  with the same flux, but different axis ratios.
• First Gaussian represents the bulge e fixed at
  0.
• Second Gaussian represents the disk e varied up
  to 0.4
• Perform ring test to obtain bias on shear when
  galaxy is modelled with a sum of Gaussians.
• Plot m and c as a function of disk ellipticity.

PSF
PSF convolved image
Simulated galaxy
64
Fitting multiple Gaussians to Galaxies with
non-elliptical isophotes
65
Fitting multiple Gaussians to Galaxies with
non-elliptical isophotes
66
Is there a bias on gwhen fit multiple
co-elliptical Gaussians?
No
No
No
No
Wider implicationsCannot model galaxies as
having elliptical isophotes
No
Yes
Voigt Bridle in prep
67
Shear Measurement Review

• Bluffers guide
• Shear measurement pipeline
• Shear estimation methods
• KSB, Shapelets, Model fitting
• STEP, GREAT08
• PSF uncertainties
• Propagation of PSF errors
• PCA
• Cross-correlation
• Bluffers guide test

68
PSF errors

• Finite number of photons from finite number of
  stars

Paulin-Henriksson et al 2008
69
PSF errors

• Finite number of photons from finite number of
  stars
• Interpolation between stars
• Stability of instrument and atmosphere links the
  above
• Color dependent point spread function
• Errors in star selection (binary stars, galaxies)
• Oversimplistic modelling of PSF shape
• Detector non-linearity

70
Propagation of PSF errors

• Simple case No noise, so use Qij
• R Qxx Qyy
• e (Qxx Qyy) / (QxxQyy)
• R, e is star Rg, eg is galaxy
• Qijconv Qij Qijg
• Problem eobs is used instead of etrue
• de eobs etrue
• Question how does egobs relate to egtrue?
• deg egobs egtrue
• Answer deg (R/Rg)2 de

e.g. Paulin-Henriksson et al 2008 eqn 13
71
Ground versus space?

• Ground PSF FWHM 0.7 arcsec
• Space PSF FWHM 0.1 arcsec
• deg (R/Rg)2 de
• How much better is space than ground?
• Assuming same galaxies are used
• Factor of 50 improvement in PSF error
  propagation!
• Other factors to consider, including stability of
  optics, atmosphere, cost etc

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Shear Measurement Review

• Bluffers guide
• Shear measurement pipeline
• Shear estimation methods
• KSB, Shapelets, Model fitting
• STEP, GREAT08
• PSF uncertainties
• Propagation of PSF errors
• PCA
• Cross-correlation
• Bluffers guide test

74
Anisotropic PSF
Focus too low
Focus (roughly) correct
Focus too high

• Whisker plots for three BTC camera exposures
  10 ellipticity
• Left and right are most extreme variations,
  middle is more typical.
• Is there a correlated variation in the different
  exposures? Yes!

Jarvis Jain 2004
75
PCA Fitting for PSF

• PSF patterns dominated by 1-parameterfocus.
  This is the first principal component.
• Using Principal Component Analysis, we can
  estimate a focus number for each exposure, fi.
• Then rather than fitting the star shapes in each
  exposure separately, we use all the stars in all
  the images, along with the focus values, fi.
• In general, we allow for several Principal
  Components, fi,k .
• Big computational fitting problem fit for 2000
  polynomial coefficients (30 principal components,
  10th order polynomials in x,y) using 100,000
  stars.
• These need not correspond to any physical
  variable. As long as theres a correlation
  between exposures, PCA is a nearly optimal way of
  using the information to correct for PSF
  anisotropy.

Jarvis Jain 2004
see also the approaches of Hoekstra04, van
Waerbeke, Hoekstra, Mellier04
76
After Processing
Focus (roughly) correct
Focus too high
Focus too low

• Remaining ellipticities are essentially
  uncorrelated.
• Measurement error is the cause of the residual
  shapes.
• 1st improvement higher order polynomial means
  PSF accurate to below 1 arcmin.
• 2nd Much lower correlated residuals on all
  scales!

Jarvis Jain 2004
77
Shear Measurement Review

• Bluffers guide
• Shear measurement pipeline
• Shear estimation methods
• KSB, Shapelets, Model fitting
• STEP, GREAT08
• PSF uncertainties
• Propagation of PSF errors
• PCA
• Cross-correlation
• Bluffers guide test

78
Shear Measurement Review

• Bluffers guide
• Shear measurement pipeline
• Shear estimation methods
• KSB, Shapelets, Model fitting
• STEP, GREAT08
• PSF uncertainties
• Propagation of PSF errors
• PCA
• Cross-correlation
• Bluffers guide test

79
Bluffers Guide to Shear MeasurementCan you say
these phrases with confidence?

• The shear responsivity depends on the galaxy
  ellipticity distribution
• P gamma has to be averaged over galaxies with
  similar properties
• If the wrong galaxy profile is assumed, the
  shear is biased
• You should definitely participate in GREAT08
• Shear systematics depend on the square of the
  PSF size
• PSF PCA technique allows a larger effective star
  density
• Cross correlation between images averages out
  random PSF shear biases