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The MATPHOT Algorithm for Digital Point Spread Function CCD Stellar Photometry

The MATPHOT Algorithm for Digital Point Spread Function CCD Stellar Photometry. Kenneth J. Mighell National Optical Astronomy Observatory.

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The MATPHOT Algorithm for Digital Point Spread Function CCD Stellar Photometry

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

  2. This work is supported by a grant from the National Aeronautics and Space Administration (NASA), Order No. S-67046-F, which was awarded by the Long-Term Space Astrophysics (LTSA) program of NASA’s Office of Space Science. AURA Software Workshop 2002

  3. 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 sampledPSF (), where the sampled (pixelized ) PSF is defined as which is the volume integral over theith pixel of the unsampledPSF (). AURA Software Workshop 2002

  4. 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: AURA Software Workshop 2002

  5. Levenberg-Marquardt Method Let us assume that we have a calibrated charge-coupled device (CCD) observation with N pixels and thatziis the intensity in electrons of the ith pixel at the location of(xi,yi)with an error ofsi . Let m(x,y;p1,…,pM)be a model of the intensity values that has two coordinates (x,y) and and Mparameters. The measure of the goodness of fit between the data and the model, called chi-square, is defined as AURA Software Workshop 2002

  6. The key idea of the Levenberg-Marquardt method of non-linear least-squares estimation consists of minimizing c2(p) by (repeatedly) solving (and minimizing) the following equation for d (the correction vector): where l is a variable damping factor which is used to modify the principle diagonal of the Hessian matrix, and ij is the Kronecker delta function (Levenberg 1944, practical algorithm for l: Marquardt 1963). AURA Software Workshop 2002

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

  8. 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). AURA Software Workshop 2002

  9. 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: AURA Software Workshop 2002

  10. AURA Software Workshop 2002

  11. AURA Software Workshop 2002

  12. and finally, AURA Software Workshop 2002

  13. After the optimal parameter vector (p0) 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 jth parameter. AURA Software Workshop 2002

  14. matphot1 =21.38 AURA Software Workshop 2002

  15. matphot1 =21.38 AURA Software Workshop 2002

  16. PSF-Fitting Performance Model Photometry: ^ important yet frequently ignored! 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) px2 . The WFPC2 sharpness parameter is the reciprocal of . AURA Software Workshop 2002

  17. 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. AURA Software Workshop 2002

  18. matphot1 =21.38 AURA Software Workshop 2002

  19. matphot1 =21.38 AURA Software Workshop 2002

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

  21. 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. AURA Software Workshop 2002

  22. matphot1 =21.38 AURA Software Workshop 2002

  23. matphot1 =21.38 AURA Software Workshop 2002

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

  25. 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. AURA Software Workshop 2002

  26. 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? AURA Software Workshop 2002

  27. 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. AURA Software Workshop 2002

  28. 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 dampedsinc function to interpolate sampled PSFs: Note: The 2-d sinc function is separable in x and y. AURA Software Workshop 2002

  29. matphot2 =21.38 AURA Software Workshop 2002

  30. Critically-sampled PSFs? AURA Software Workshop 2002

  31. matphot2 =13.58 AURA Software Workshop 2002

  32. Yikes! matphot2 =13.58 AURA Software Workshop 2002

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

  34. 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. AURA Software Workshop 2002

  35. matphot2 =13.58 AURA Software Workshop 2002

  36. OK! matphot2 =13.58 AURA Software Workshop 2002

  37. Under-sampled PSFs? AURA Software Workshop 2002

  38. Yuk! matphot2 =6.23 AURA Software Workshop 2002

  39. OK! matphot2 =6.23 AURA Software Workshop 2002

  40. AURA Software Workshop 2002

  41. Ugly (real-world) PSFs? AURA Software Workshop 2002

  42. Next Generation Space Telescope 8-m TRW-concept 0.0128 arcsec/pixel=31.05PSF by John Krist AURA Software Workshop 2002

  43. matphot2 =31.05 AURA Software Workshop 2002

  44. matphot2 =31.05 AURA Software Workshop 2002

  45. How can I get MATPHOT? Get the 2001DEC27 release of my MXTOOLS package for IRAF at http://www.noao.edu/staff/mighell/mxtools AURA Software Workshop 2002

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