PSF photometry is one of the major ways to measure the flux of sources (detections) in PanSTARRS1 data. It is based on fitting a predefined analytical shape to the counts reported for each detection, and then applying a zero point conversion to the total object count rate thus obtained.

The current description is incomplete in some details, such as the degree of variation allowed in the PSF over a given warp, the exact definition of flux uncertainty, and the treatment of deviations ("residuals") from the chosen analytical formula.

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# PSF Photometry

Obtained from fitting a predefined PSF form to all detections. The quality of the fit can help determine whether a source is indeed a point source, it is extended, or it is spurious. PSF photometry is performed on warps by a module called PSFPHOT and on stacks by PSPHOTSTACK. The result of the fit is reported in the Detection table (or StackDetection table for stacks). Values reported include flux, uncertainty, position, elliptical size, and quality parameters.

## The PSF Model

The PSF model takes the form of an analytical function plus residuals. The fitted parameters and residuals vary with position, but are actually determined at 9 positions (in a regular 3x3 grid pattern) per skycell then interpolated to other positions.

Analytical functions tested include:

- GAUSS : exp (-z)
- PGAUSS : (1 + z + z
^{2}/2 + z^{3}/6)^{-1} - QGAUSS : (1 + kz + z
^{2.25})^{-1} - RGAUSS : (1 + z + z
^{k})^{-1} - PS1_V1 : (1 + kz + z
^{1.67})^{-1}

where z is the elliptical contour (akin to a radius squared):

- z = x
^{2}/(2σ^{2}_{xx}) + y^{2}/(2σ^{2}_{yy}) + σ_{xy}XY

The PS1_V1 model is the current default value for PS1 analysis.

Variability over the image is formally represented as:

- PSF = F[dx,dy;ai(x,y)] + R0[dx,dy] + x Rx[dx,dy] + y Ry[dx,dy]

Existing documentation states that a global linear fit is performed in which the fluxes of all objects is fitted for simultaneously with the following considerations:

- Simultaneous fit of fluxes for all objects in the image
- Chi-square fit:
- χ
^{2}= Σ(f_{i}- Σ (A_{j}F_{j)})^{2}W_{i}(i : pixels; j : objects) - W
_{i}– weighting function- now constant (from mid-2012), was inv variance
- using a constant weight removes a photometric bias found for faint sources

- χ
- minimization of A
_{j}requires inversion of large square matrix- N (number of objects) may be up to 100k
- but, highly diagonal, so inversion is actually fast

- ~ 1 second for 100k objects (unless they grow too large)

Unclear what constrains are placed on PSF parameters (other than flux) during the fitting.

How residuals are recorded and used to determine, e.g., aperture corrections is also unclear.

## Photometric and astrometric parameters from PSF fitting

The Detection table contains the following parameters related to PSF photometry:

xPos | raw pixels | REAL | 4 | -999 | PSF x center location. |

yPos | raw pixels | REAL | 4 | -999 | PSF y center location. |

xPosErr | raw pixels | REAL | 4 | -999 | Error in PSF x center location. |

yPosErr | raw pixels | REAL | 4 | -999 | Error in PSF y center location. |

psfFlux | Janskys | REAL | 4 | -999 | Flux from PSF fit. |

psfFluxErr | Janskys | REAL | 4 | -999 | Error on flux from PSF fit. |

psfMajorFWHM | arcsec | REAL | 4 | -999 | PSF major axis FWHM. |

psfMinorFWHM | arcsec | REAL | 4 | -999 | PSF minor axis FWHM. |

psfTheta | degrees | REAL | 4 | -999 | PSF major axis orientation. |

psfCore | dimensionless | REAL | 4 | -999 | PSF core parameter k, where F = F0 / (1 + k r^2 + r^3.33). |

psfQf | dimensionless | REAL | 4 | -999 | PSF coverage factor. |

psfQfPerfect | dimensionless | REAL | 4 | -999 | PSF weighted fraction of pixels totally unmasked. |

psfChiSq | dimensionless | REAL | 4 | -999 | Reduced chi squared value of the PSF model fit. |

psfLikelihood | dimensionless | REAL | 4 | -999 | Likelihood that this detection is best fit by a PSF. |

This information was obtained from the Schema Browser at http://web01.psps.ifa.hawaii.edu/PSI/schema_browser.php?describeTable=Detection (requires PSPS login) after selecting SAS42-PV3 in the PS1 Query Page. The STScI version is at http://plps3v.stsci.edu/PSI/schema_browser.php.

Note that psphot actually returns PSF_MAJOR and PSF_MINOR, whose relation to FWHM depends on the value of k for the PS1_V1 profile. For k=0, FWHM=PSF_MAJOR*2*sqrt(2)*pixel_size.

## Details needed and questions

- Is a multi-parameter fit used for each detection, or are the parameters (position, PSF shape) obtained from global fits (for PSF) and detection properties (position)? Above notes are not fully sufficient.
- How exactly is the flux error defined?
- How are the flags set?

## Source of information

Most of the information contained here is based on the PS1 Guide:

http://ps1sc.ifa.hawaii.edu/PS1wiki/index.php/PS1Guide#The_PSF_model

Other relevant papers include:

http://adsabs.harvard.edu/abs/2007ASPC..364..153M

http://adsabs.harvard.edu/abs/2012ApJ...756..158S

http://adsabs.harvard.edu/abs/2012ApJ...750...99T

These papers provide good information on the calibration process - correction for airmass, reduction to a standard magnitude system, other instrumental effects, etc - but none describe explicitly (or even implicitly) how the *flux* itself is determined. If you are aware of such a paper, please add a comment or email stefano@stsci.edu.