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*Page author and contact:* Rongpu Zhou

Starting with DR9, we use an extended PSF model to subtract the extended wings of bright stars from DECam images for DECaLS. The model is symmetric (which is a good approximation for DECam images), and is a linear combination of two components:

- A flexible inner PSF, which is a Moffat profile fit to the PSFEx image.
- A fixed outer PSF, which is either a power law (for \(g\hbox{-}\) and \(r\hbox{-}\mathrm{bands}\)) or a Moffat profile (for \(z\hbox{-}\mathrm{band}\)).

## The Inner PSF Profile

The inner PSF component is a Moffat profile:

- \(f_{\mathrm{Moffat}} = \frac{\beta-1}{\pi \alpha^2} \left[1 + \frac{r}{\alpha^2}\right]^{-\beta}\)

This formula gives a total flux of unity by definition. Its two free parameters, \(\alpha\) and \(\beta\), are obtained by fitting to the PSFEx image of each CCD. Only PSFEx pixels between 1.8 and 5 arcsec in radius are used in the fit.

The size of the DECam PSFEx image is only ~8.1 arcsec in radius, too small for implementing the extended PSF model. Nevertheless, we modify the the PSFEx image for the following reasons:

- It removes discontinuities when we subtract the outer wings of bright stars from the images
- The noisy parts of the PSFEx images are replaced by a noiseless model PSF. This is more of a cosmetic improvement and it should not
affect the
*Tractor*fitting of unsaturated stars or faint galaxies.

Here are illustrations of the weights for patching the PSFEx image and the weights of the stellar halo subtraction:

The (tentative) \(R\) values are \(R_1, R_2, R_3, R_4 = 5.0, 6.0, 7.0, 8.0\,(\mathrm{arcsec}\)). The transition radii \(R_1\) and \(R_2\) are chosen so
that beyond radius \(R_1\) there is little asymmetry in the PSFEx model (*i.e.* the symmetric Moffat profile is a good approximation).
For reference, the size of the PSFEx images is ~8.1 arcsec in radius (\(63 \times 63\) with pixel size \(0.262\,\mathrm{arcsec}\)) for DECam
and Mosaic, and ~6.8 arcsec (\(31 \times 31\) with pixel size \(0.454\,\mathrm{arcsec}\)) for 90Prime.

In the patched PSFEx image (the first eigen-image), the crossover between \(R_1\) and \(R_2\) marks the transition between the PSFEx
image and the model PSF. The flux gradually approaches zero between \(R_3\) and \(R_4\), and is strictly zero beyond \(R_4\). For the
other eigenimages, the values for each pixel gradually approach zero between \(R_1\) and \(R_2\) (*i.e.* the red curve in the above plot),
and they are strictly zero beyond \(R_2\). To preserve continuity, the outer PSF component is included for PSFEx patching, even though it's
not used in the fitting; it contributes a negligible fraction of the flux in the PSFEx image.

Here is an example of the Moffat profile fit to the PSFEx image, and the corresponding original and patched PSFEx image:

## The Outer PSF Profile

For \(g\hbox{-}\) and \(r\hbox{-}\mathrm{bands}\), the outer PSF profile is a fixed power law (normalized to a 22.5 magnitude star, in units of \(\mathrm{nanomaggies}/\mathrm{arcsec}^2\)):

- \(f_g = 0.00045 r^{-2}\)
- \(f_r = 0.00033 r^{-2}\)

For \(z\hbox{-}\mathrm{band}\), the outer PSF is a weighted Moffat profile. For most CCDs, the Moffat parameters (with radius in arcsec and surface brightness in \(\mathrm{nanomaggies}/\mathrm{arcsec}^2\)) and the weights are (for a 22.5 magnitude star):

- \([\alpha, \beta, weight] = [17.650, 1.7, 0.0145]\)

However, a small subset of DECam CCDs (N20, S8, S10, S18, S21 and S27) have a more compact outer PSF in the \(z\hbox{-}\mathrm{band}\), which is characterized using:

- \([\alpha, \beta, weight] = [16, 2.3, 0.0095]\)

## The Combined PSF model

Below is an example of the model PSF profile in \(g\hbox{-}\mathrm{band}\), with the pixel fluxes around a bright star overplotted. The inner PSF's are obtained by fitting the PSFEx image, and the outer PSF profile is fixed and normalized to a DECam magnitude predicted from Gaia magnitudes and colors. Also included are the corresponding original and halo-subtracted images.

## CCD-to-CCD variations in \(z\hbox{-}\mathrm{band}\)

A surprising finding from this work is that a small subset of DECam CCDs have different PSFs compared to the other CCDs in the \(z\hbox{-}\mathrm{band}\). These outlier CCDs
have consistently more compact PSFs at the \(r>\sim20\,\mathrm{arcsec}\) scale, and they can be approximated by a different Moffat profile. In DECaLS, this
difference only shows up in the \(z\hbox{-}\mathrm{band}\) images (although it's possible that such CCD-to-CCD variations might also appear in, *e.g.*,
\(i\hbox{-}\mathrm{band}\) or \(Y\hbox{-}\mathrm{band}\) images from DECam). The cause of these variations is unclear.

## File locations

The PSFEx files used in this analysis are available in the following directories at NERSC:

- Original PSF:
`/global/cfs/cdirs/cosmo/work/legacysurvey/dr9-archive/unpatched-psfex` - Patched PSF:
`/global/cfs/cdirs/cosmo/data/legacysurvey/dr9/calib/psfex`

## Code

The code used to analyze the DECam PSF for the Legacy Surveys is:

## Additional Information

Further extensive tests and plots characterizing the DECaLS DR9 PSF model is available for DESI collaborators on the wiki.