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
Stéphane Paulin-Henriksson / CEA-Paris --- STEP meeting 20/08/07 / JPL-LA
1. non gaussianity of shape estimators
2. finite accuracy of the PSF estimation
Systematics in shape measurement
Several sources of systematics
Stéphane Paulin-Henriksson / CEA-Paris --- STEP meeting 20/08/07 / JPL-LA
Systematics in shape measurement
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
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Stéphane Paulin-Henriksson / CEA-Paris --- STEP meeting 20/08/07 / JPL-LA
Ntot 500(way too small for final
conclusions)
Stéphane Paulin-Henriksson / CEA-Paris --- STEP meeting 20/08/07 / JPL-LA
true value: =1/3<1>=0.3244
skewness, likelihood < 5%
Stéphane Paulin-Henriksson / CEA-Paris --- STEP meeting 20/08/07 / JPL-LA
Systematics in shape measurement
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.
Stéphane Paulin-Henriksson / CEA-Paris --- STEP meeting 20/08/07 / JPL-LA
1. non gaussianity of shape estimators
2. finite accuracy of the PSF estimation
Systematics in shape measurement
Several sources of systematics
Stéphane Paulin-Henriksson / CEA-Paris --- STEP meeting 20/08/07 / JPL-LA
Systematics in shape measurement
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
Stéphane Paulin-Henriksson / CEA-Paris --- STEP meeting 20/08/07 / JPL-LA
Systematics in shape measurement
accuracy of the PSF estimation
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
Stéphane Paulin-Henriksson / CEA-Paris --- STEP meeting 20/08/07 / JPL-LA
Systematics in shape measurement
accuracy of the PSF estimation
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
Stéphane Paulin-Henriksson / CEA-Paris --- STEP meeting 20/08/07 / JPL-LA
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
Stéphane Paulin-Henriksson / CEA-Paris --- STEP meeting 20/08/07 / JPL-LA
||<
est
. >
-
SNR10 20 50 100 150
(<est
. R
2>
- R
2)/
R2
10-3
10-4
10-2
10-1
10-3
10-5
Stéphane Paulin-Henriksson / CEA-Paris --- STEP meeting 20/08/07 / JPL-LA
SNR10 50 100 1000
[R
2]/
(2.5
8a
2)
[]
/0.7
0
500
10-1
10-2
10-2
10-1
Stéphane Paulin-Henriksson / CEA-Paris --- STEP meeting 20/08/07 / JPL-LA
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
Stéphane Paulin-Henriksson / CEA-Paris --- STEP meeting 20/08/07 / JPL-LA
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 ?
Stéphane Paulin-Henriksson / CEA-Paris --- STEP meeting 20/08/07 / JPL-LA
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 ?
Stéphane Paulin-Henriksson / CEA-Paris --- STEP meeting 20/08/07 / JPL-LA
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
Stéphane Paulin-Henriksson / CEA-Paris --- STEP meeting 20/08/07 / JPL-LA
Systematics in shape measurement
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
Stéphane Paulin-Henriksson / CEA-Paris --- STEP meeting 20/08/07 / JPL-LA
Systematics in shape measurement
accuracy of the PSF fitting
€
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*
Stéphane Paulin-Henriksson / CEA-Paris --- STEP meeting 20/08/07 / JPL-LA
Systematics in shape measurement
accuracy of the PSF fitting
€
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