2002 January 28 AURA Software Workshop The MATPHOT Algorithm for Digital Point Spread Function CCD Stellar Photometry Kenneth J. Mighell National Optical Astronomy Observatory

2002 January 28 AURA Software Workshop This work is supported by a grant from the National Aeronautics and Space Administration (NASA), Order No. S F, which was awarded by the Long-Term Space Astrophysics (LTSA) program of NASA’s Office of Space Science.

2002 January 28AURA Software Workshop Analytical Point Spread Functions A simple model of a CCD observation of a single star on a non-flat background may be created with a tilted plane and some analytical function representing the sampled PSF (  ), where the sampled (pixelized ) PSF is defined as which is the volume integral over the i th pixel of the unsampled PSF (  ).

2002 January 28AURA Software Workshop Analytical Gaussian PSFs A Gaussian distribution is a good model for the PSF of a ground-based CCD observation since the central core of a ground-based stellar profile is approximately Gaussian (King 1971). The following is thus a good model of the unsampled PSF of a ground-based CCD observation of a single isolated star on a non-flat background:

2002 January 28AURA Software Workshop Levenberg-Marquardt Method Let us assume that we have a calibrated charge- coupled device (CCD) observation with N pixels and that z i is the intensity in electrons of the i th pixel at the location of (x i,y i ) with an error of  i. Let m(x,y;p 1,…,p M ) be a model of the intensity values that has two coordinates (x,y) and and M parameters. The measure of the goodness of fit between the data and the model, called chi-square, is defined as

2002 January 28AURA Software Workshop where  is a variable damping factor which is used to modify the principle diagonal of the Hessian matrix, The key idea of the Levenberg-Marquardt method of non- linear least-squares estimation consists of minimizing   (p) by (repeatedly) solving (and minimizing) the following equation for  the correction vector  and  ij is the Kronecker delta function (Levenberg 1944, practical algorithm for  Marquardt 1963).

2002 January 28AURA Software Workshop By defining  as a new matrix of order M x M,

2002 January 28AURA Software Workshop and  as a new vector of order M, one then has the computationally useful form of a set of M equations with M unknowns: The Levenberg-Marquardt method of non-linear least squares minimization performs an optimum interpolation between the Taylor series method and the gradient method (Marquardt 1963).

2002 January 28AURA Software Workshop The Levenberg-Marquardt method requires the computation of the partial derivatives of the observational model with respect to all free parameters. The partial derivatives of the simple Gaussian model are as follows:

2002 January 28AURA Software Workshop

2002 January 28AURA Software Workshop

2002 January 28AURA Software Workshop and finally,

2002 January 28AURA Software Workshop After the optimal parameter vector ( p 0 ) has been determined (with set to zero), the covariance matrix, may then be calculated. If the off-diagonal elements of the covariance matrix are negligibly small, then the standard (rms) errors can be estimated to be where  j is the standard error associated with the j th parameter.

2002 January 28AURA Software Workshop matphot1  =21.38

2002 January 28AURA Software Workshop matphot1  =21.38

2002 January 28AURA Software Workshop PSF-Fitting Performance Model where is “effective background area” defined as the reciprocal of the volume integral of the square of the PSF. By definition, the effective background area for a critically- sampled Gaussian PSF is  = 4  (1) 2 (  12.57) px 2. The WFPC2 sharpness parameter is the reciprocal of . Photometry: ^ important yet frequently ignored!

2002 January 28AURA Software Workshop King (1983) identifies  as the “equivalent-noise area” and notes that the numerical integration of a realistic ground- based stellar profile gives an equivalent area that is 2.45 times greater than that of a Gaussian stellar profile. Position Errors (Relative Astrometry): This astrometric performance model is more general than Irwin’s (1985) astrometric analysis of Gaussian PSFs.

2002 January 28AURA Software Workshop matphot1  =21.38

2002 January 28AURA Software Workshop matphot1  =21.38

2002 January 28AURA Software Workshop What do the partial derivatives look like? A constant of one … A tilted plane in x … A tilted plane in y … … the sampled PSF … difference of 2 Gaussians in x … difference of 2 Gaussians in y

2002 January 28AURA Software Workshop Position Partial Derivatives The mathematics of determining the position partial derivatives of the observational model with respect to the x and y direction vectors is exactly the same with analytical or digital PSFs. The implementation methodology, however, is significantly different. The position partial derivatives of digital PSFs can be determined using numerical differentiation techniques on the digital PSF. Numerical experiments have shown that the following five- point differentiation formula works well with digital PSFs: MATPHOT currently uses the above formula to compute position partial derivatives.

2002 January 28AURA Software Workshop matphot1  =21.38

2002 January 28AURA Software Workshop matphot1  =21.38

2002 January 28AURA Software Workshop matphot1  =21.38 The differentiation method error is < 1/16 th that due to photon noise!

2002 January 28AURA Software Workshop A Critique of Traditional PSF-Fitting… Most CCD stellar photometric reduction packages use analytical functions to represent the stellar PSF. These PSF-fitting programs generally compute all the major partial derivatives of the observational model by differentiating the volume integral of the analytical function representing the unsampled PSF over a CCD pixel. Deviations of the real-world PSF from the analytical PSF are typically stored in a residual matrix.

2002 January 28AURA Software Workshop Diffraction rings and spikes can provide a lot of valuable information about the position of a star, yet data about such common observational effects generally resides only in the residual matrix. The information in the residual matrix is typically ignored during in the PSF-fitting process except at the final step involving the determination of the chi-square goodness-of-fit between the data (the CCD observation) and the observational model (when the intensity–scaled residual matrix is added to the analytical PSF just before the goodness-of-fit is computed). What if we throw out the analytical PSF and only use the residual matrix to describe the Point Spread Function?

2002 January 28AURA Software Workshop Digital Point Spread Functions A digital Point Spread Function is a digital representation of the sampled PSF consisting of a numerical table (e.g., a matrix or a FITS image) instead of an analytical function. The MATPHOT algorithm for digital PSF-fitting CCD stellar photometry uses digital PSFs instead of analytical PSFs.

2002 January 28AURA Software Workshop How does one move a PSF? Analytical PSFs: Just compute the PSF at the desired location in the observational model. Digital PSFs: Take the reference digital PSF and shift it to the desired location using a perfect 2-d interpolation function. OK… but how is that done in practice? MATPHOT currently uses the following 21-pixel-wide damped sinc function to interpolate sampled PSFs: Note: The 2-d sinc function is separable in x and y.

2002 January 28AURA Software Workshop matphot2  =21.38

2002 January 28AURA Software Workshop Critically-sampled PSFs?

2002 January 28AURA Software Workshop matphot2  =13.58

2002 January 28AURA Software Workshop matphot2  =13.58 Yikes!

2002 January 28AURA Software Workshop TANSTAFL * * There Ain’t No Such Thing As (a) Free Lunch

2002 January 28AURA Software Workshop Over-sampled PSFs Accurate CCD stellar photometry may be obtained with under-sampled or critically-sampled data if over- sampled PSFs are used. For example, a 2x2 over-sampled PSF uses 4 subpixels to describe every true pixel of the CCD observation. Every subpixel of a 2x2 over-sampled PSF has twice the spatial resolution of the true pixels. MATPHOT uses over-sampled PSFs to do accurate CCD stellar photometry with under-sampled observations.

2002 January 28AURA Software Workshop matphot2  =13.58

2002 January 28AURA Software Workshop matphot2  =13.58 OK!

2002 January 28AURA Software Workshop Under-sampled PSFs?

2002 January 28AURA Software Workshop matphot2  =6.23 Yuk!

2002 January 28AURA Software Workshop matphot2  =6.23 OK!

2002 January 28AURA Software Workshop

2002 January 28AURA Software Workshop Ugly (real-world) PSFs?

2002 January 28AURA Software Workshop Next Generation Space Telescope 8-m TRW-concept arcsec/pixel  =31.05 PSF by John Krist

2002 January 28AURA Software Workshop matphot2  =31.05

2002 January 28AURA Software Workshop matphot2  =31.05

2002 January 28AURA Software Workshop How can I get MATPHOT? Get the 2001DEC27 release of my MXTOOLS package for IRAF at