paulin-henriksson stéphane, voigt lisa, amara adam, bridle sarah, réfrégier alexandre

Stéphane Paulin-Henriksson / CEA-Paris --- STEP meeting 20/08/07 / JPL-LA

Paulin-Henriksson Stéphane, Voigt Lisa, Amara Adam, Bridle Sarah, Réfrégier Alexandre

General Features of Fitting Methods

Systematics in shape measurement:the PSF calibration


1. non gaussianity of shape estimators

2. finite accuracy of the PSF estimation

Systematics in shape measurement

Several sources of systematics



non gaussianity of shape estimatorsExample with the ellipticity of horizontal exponential galaxies:

ab

Set-up of this simulation:• a=2 pixels ; b=1 pixel==> =1/3, rg 2.4 pixels• signal-to-noise ratio of PSF-convolved galaxies constant 60 (gaussian background noise)

• centroid uniformly distributed inside the central pixel

• fit (2 minimisation) with the true model€

Φpixel = Φ x,y( ) dx dypixel

∫

€

ε =a−b

a+ b

€

Φ x,y( ) =Φ tot

2π a bexp −

x 2

a2+y 2

b2

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


Ntot 500(way too small for final

conclusions)


true value: =1/3<1>=0.3244

skewness, likelihood < 5%



non gaussianity of shape estimatorsTo be done:

• increase the statistics• go through the preliminary conclusion: non-gaussianity increases with the SNR

. what are the effects of pixelisation ?

--> Preliminary conclusions: pixelisation makes the deconvolution noisier (obvious) AND increases the non-gaussianity.

--> Will probably lead to a lower limit on the pixel size.


1. non gaussianity of shape estimators

2. finite accuracy of the PSF estimation


Several sources of systematics



accuracy of the PSF estimation

area on which the PSF is interpolated

galaxy to be deconvolved

stars

*

*

*

*

*

*

**

The PSF is estimated at the position of the galaxy with a limited accuracy

€

δεgal =rgPSF

rggal

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2

F δεPSF[ ] +G δrgPSF

[ ]( )

Dilution factor




The accuracy of the PSF estimation is limited by:

1. limited accuracy of the star shape measurements: each star is a noisy and pixelised realisation of the

PSF

2. PSF variations with the position: necessary to introduce an interpolation scheme




The accuracy of the PSF estimation is limited by:

1. limited accuracy of the star shape measurements: each star is a noisy and pixelised realisation of the

PSF

2. PSF variations with the position: necessary to introduce an interpolation scheme

In the following I am always in the ideal case where the PSF is perfectly stable. Then I study the accuracy of the PSF calibration according to: the SNR of stars, the PSF model and the number of stars. Then I address the issue of pixelisation effects


Accuracy of the PSF estimationfitting a gaussian with a gaussian

Very simple case with a gaussian PSF:

€

Φ x,y( ) =1

2π a bexp −

x 2

2a2+y 2

2b2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

Shape parameters are analytically predicted (in the case of infinitely small pixels) and verified by a simulation:

€

ε =a−b

a+ b= 0.05

σ [ε] =0.70

SNRb[ε] = 0

€

R2 = a2 + b2 =1.82 × a2

σ [R2] =2.58 × a2

SNR

b[R2] =R2

SNR2

a is variable and rule the size

ab


||<

est

. >

-

SNR10 20 50 100 150

(<est

. R

2>

- R

2)/

R2

10-3

10-4

10-2

10-1

10-3

10-5


SNR10 50 100 1000

[R

2]/

(2.5

8a

2)

[]

/0.7

0

500

10-1

10-2

10-2

10-1


Accuracy of the PSF estimation:the choice of the PSF model

Intuitively, we see that the PSF modeling is a compromise:

1. a too poor model is unrealistic. It is unable to mimic the real PSF ==> the estimated PSF shape is biased

2. a too rich model is noisy ==> the estimated PSF shape is noisy

==> necessary to use a model complex enough to mimic the PSF but as simple as possible

The optimal compromise depends on the data set. But we can look for arguments leading to this compromise.

The following is an answer to point 2: we study what are the errors on PSF shape parameters according to the PSF model


N

Tru

nca

tion o

f hig

hsp

ati

al fr

equenci

es

http://www.astro.caltech.edu/~rjm/shapelets

8

6

4

2

0

-2

-4

-6

-8

8

6

4

2

0

-2

-4

-6

-8

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8

M

To look through this, shapelets are

very convenient

M=0 functions rule the total

flux

M=+-2 functions rule the ellipticity

What are the errors on PSF

shape parameters according to

the PSF model ?


N

Tru

nca

tion o

f hig

hsp

ati

al fr

equenci

es

http://www.astro.caltech.edu/~rjm/shapelets

8

6

4

2

0

-2

-4

-6

-8

8

6

4

2

0

-2

-4

-6

-8

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8

Msimulation:• of stars from SpaceSTEP in various bases

• with various SNR (gaussian backg.)

• in each basis, stars are simulated and then fitted (2 minimisation)

A given basis has an

effective number of coefficients Np

for each shape parameter. The

question is in fact: How does a

shape parameter

measurement depend on Np ?


Np[] Np[R]

SNR = 50SNR = 200SNR = 800

[] [R]

€

σ[ε]∝ N p[ε]( )1.7

×1

SNR

€

σ[R2]∝ N p[R2]( )

2.9×

1

SNR



accuracy of the PSF fitting

€

σ[ε] ≈ 0.8 × N p[ε]( )1.7

×1

SNR

€

σ[R2] ≈ 8.0 × N p[R2]( )

2.9×

1

SNR

for<0.1 and

SNR>50




€

σ[εPSF ] ≈ 0.8 × (N p[ε*])1.7 ×1

SNR*

×1

N*

Accuracy of the PSF calibration according to the number of stars (used to compute the calibration)

€

σ[RPSF2 ] ≈ 8.0 × N p[R*

2]( )2.9

×1

SNR*

×1

N*




€

N* ~ 50 ×rgPSF

rggal

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

4

×N p

1

⎛

⎝ ⎜

⎞

⎠ ⎟

3−>6

×1000

SNR*

⎛

⎝ ⎜

⎞

⎠ ⎟

2

×10−8

σ PSFrequired2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

[Paulin-Henriksson et al., en préparation]

For bright stars (SNR typically > 50) and good conditions,it is possible to invert the previous equations and give the minimum number of stars required to achieve a given precision

dilution factor

SNR of * scientific requiremen

t

Nb of efficient coefficients in

the PSF model Power (of 3 to 6)

depending on the shape parameter

paulin-henriksson stéphane, voigt lisa, amara adam, bridle sarah, réfrégier alexandre

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