Determining the PSF over the Full FoV of LSST using Anisotropic - - PowerPoint PPT Presentation

Determining the PSF over the Full FoV of LSST using Anisotropic Gaussian Processes

Pierre-François Léget, Stanford University - KIPAC

• Gaussian processes are a non-parametric way to

interpolate data

• Interpolation is driven by the correlation function (aka

kernel) described by «hyperparameters»; for example,

• amplitude of the fluctuations
• correlation length
• …
• Basic steps in GP interpolation:
• Choose a correlation function (kernel)
• Fit hyperparameters
• Compute interpolated values
• Best Linear Unbiased Estimator.

➡ GP may be optimal interpolation method for atmospheric PSF parameters

Gaussian process (GP) interpolation:

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• Atmosphere introduces anisotropy

(preferential direction) in spatial variation

• f PSF’s parameters
• It means that the spacial variation in the

case of a anisotropic gaussian random field are characterized by a 2D 2-point correlation function

• For the anisotropic GP

, 4 hyperparameters:

• Amplitude of spatial fluctuations
• Correlation length
• « g1 »
• « g2 »
• Need to estimate those hyperparameters to

predict the PSF over the FoV using GP

Gaussian process (GP) interpolation:

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Same parametrization as in shape measurement

Davis et al. 2016

Anisotropic Gaussian random field Anisotropic Correlation function

• Generated realistic atmospheric PSFs across

FoV using GalSim.

• Phase screens generated with

Von Kármán power spectrum (Kolmogorov with finite

• uter scale).
• Each screen corresponds to different wind

speed & direction, & outer scale.

• 6 screens between 0.1 km & 15 km
• Wind speed increases in with altitude
• Dominant wind direction
• Assumes 30-sec exposures for an LSST
• like

aperture and obscuration (no optical PSF).

• 20 000 stars on LSST FoV:
• fit with elliptical Kolmogorov profile.
• Fit parameters: size, g1, g2.

Realistic simulation of atmospheric PSF

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• Two interpolations are made on simulation:
• GP interpolation with anisotropic

Von-Kármán kernel (over the full FoV)

• Second-order polynomial interpolation (for each CCD)
• 80% of stars are used for training (16 000).
• 20% of stars are kept for validation (4 000).

Interpolation of atmospheric PSF

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Results: Anisotropic 2-point correlation function fit on PSF parameters

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• Compute for each PSF’s parameter the

anisotropic 2-point correlation function (left plots)

• Estimate the covariance matrix with an

internal method (bootstrapping)

• Fit each measured anisotropic 2-point

correlation function with an anisotropic Von-Karman correlation function.

• For each interpolation, we compute “Rowe statistics”, which are ingredients

in the calculation of the systematic error in the shear correlation function due to errors in the PSF model.

• Rowe statistics computed for stars in validation sample (4 000 stars).
• Rowe statistics are lower for GP than for polynomial interpolation by a factor
• f ~3 for all 𝜍 statistics
• Next step: apply to real data (DES)

Results:

Gaussian processes with anisotropic Von-Kármán kernel (FoV) 2nd order polynomial interpolation (per CCD)

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