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The way in which the point-spread function is encoded is one of the principal details that distinguish the various existing photometric packages from one another. There are two general approaches which have been tried in the past:

(1) Some analytic function of suitable form is fitted to bright, isolated stars in the frame - much as we did in the second lecture. The assumed analytic form, together with the "best" values of the fitting parameters which result from these fits, are taken to define the PSF for the frame, which can then be evaluated at any distance (Deltax, Deltay) from the center of any particular star. Again, referring back to our fits in the second lecture, the hypothesis that the stellar profile is exactly described by a circular Moffat function, plus the derived values of the parameters Rhat and betahat, tell you everything you need to know about a frame's PSF. This PSF can then be fit to all the stars in the frame to determine their individual values of xhat0, yhat0, Chat, and Bhat. More complex analytic equations involving more shape parameters can easily be devised to model images that depart from simple circular symmetry. This is the approach adopted by ROMAFOT and STARMAN.

(2) The actual brightness values observed in pixels in and around the images of bright stars are assumed to define the frame's PSF empirically; subarrays containing bright stars are extracted from the main image, averaged together, and retained as look-up tables giving relative brightness as a function of (Deltax, Deltay) offset from a star's center. Since the positions of the stars within the coordinate grid of the original CCD frame will differ by different fractional-pixel amounts, if multiple stars are used to define the frame's PSF they must be shifted to a common centering within the coordinate grid by numerical interpolation techniques before they can be averaged together to form the frame's mean PSF - otherwise the model profile will be a bit broader than the true profile. When the resulting PSF is fitted to individual program stars, numerical interpolation techniques must again be used to shift that PSF to the actual fractional-pixel position of the program star. This is the approach used by RICHFLD and WOLF.

Each of these approaches has its advantages and disadvantages. The analytic PSF can be comparatively simple to program and use. However, it has the disadvantage that real stellar images are often quite far from circular - or even bilateral - symmetry due to optical aberrations or guiding errors, so it may become necessary to include a fairly large number of shape parameters in the model. ROMAFOT makes complex stellar profiles by adding up however many Moffat functions it needs; STARMAN employs the sum of a tilted Lorentz function and a circular Gaussian, involving some eight (by my count) shape parameters. When this many parameters are involved, they can become rather strongly intertangled: starting guesses for some of the more subtle parameters can be hard to come by, and some manual guidance or a fairly sophisticated set of clamps may be needed to help shepherd the shape determinations to final convergence. Even with all this sophistication, tractable analytic functions still may not provide enough flexibility to model pathological (but real) star images when the data are over-sampled. When a bright star has a full-width at half maximum of, say, 3 pixels, its profile is still often detectable out to a radius of 12 or more pixels: that means the star image may contain some 242 / 32= 64 potentially independent resolution elements. If the telescope tracking and aberrations are bad enough, it could require an analytic function with as many as 64 different shape parameters to fully describe such a profile.

The empirical point-spread function, on the other hand, has by definition essentially one free shape "parameter" for every pixel which is included in the look-up table. However bizarre the stellar profile, provided that it repeats from one star to the next, it will be accurately reproduced in the mean empirical PSF. With so many independent degrees of freedom, empirical PSFs tend to be noisy: if the PSF is derived from a single star, whatever noise existed in that star's own profile will be copied verbatim into the PSF. In contrast, With the analytic PSF the fact that a large number of pixels is used to define a small number of parameters tends to smooth out the noise; with the empirical PSF the noise can be suppressed only by averaging together many stars. A more serious problem for the empirical PSF is that it tends to fall apart when the data become critically sampled or undersampled. In the flanks of a star image in a critically sampled frame the relative intensity may change by nearly an order of magnitude from one pixel to the next. That being the case, it is extremely difficult for any cookbook interpolation scheme (RICHFLD uses splines, WOLF uses sinc interpolation) to achieve ~ 1% accuracy in predicting what the pixel values would have been had the star been centered differently. In contrast, knowledge of the analytic profile being continuous rather than discrete, an analytic profile can be evaluated for any pixel centering.

However, when an image is critically sampled or undersampled, it is no longer adequate to estimate the flux contained within a pixel by evaluating some analytic function at the center of that pixel: one must actually integrate the analytic function over the two-dimensional area of the pixel. In mathematical terms, the approximation

Equation 163

is no longer accurate at the ~ 1% level. Now for the empirical point-spread function, this is not an extra problem, because the observed brightness values that go into the look-up table are already de facto integrals over square subportions of the stellar profile. However, to fit an analytic PSF properly to such data, the analytic formula must be integrated over the area covered by each individual pixel before it can be properly compared to the brightness observed in that pixel. Since for complex formulae no simple analytic integral may exist, these integrations must in general be performed numerically. This can greatly increase the execution time.

In writing DAOPHOT, I decided to try to use that blend of analytic and empirical profiles which would best suit the kind of data that my colleagues and I were trying to reduce. Let me place things in context: some of our data were obtained with the Cerro Tololo and Kitt Peak telescopes. The CCDs we used there had scales of approx O".6 per pixel, and the seeing was seldom as good as 1"; 1".2 to 1".8 was more typical. Thus, these data ranged from critically sampled to slightly oversampled. The rest of our data were obtained at the cassegrain focus of the Canada-France-Hawaii telescope, with a scale of O".2 per pixel, and seeing typically O".6 - 1".2; these data were thus somewhat oversampled.

The obvious thing to do, then, was to use an empirical look-up table: this is the best way to map out any complexities in an oversampled profile that are produced by tracking errors and aberrations. However, on those occasions when our data did approach critical sampling, it was found that the inadequate interpolations added unacceptable noise to the fits. Therefore I adopted a hybrid point-spread function for DAOPHOT. First, the pixel-by-pixel integral of a simple analytic formula is fit by non-linear least squares to the data for the bright, isolated PSF stars. Then, all the stars' residuals from this fit are interpolated to a common grid, and averaged together into a look-up table. When this hybrid PSF is evaluated to fit some pixel in the image of a program star, first the best-fitting analytic formula is integrated over the area of that pixel, and then a correction from that analytic profile to the true model profile is obtained by interpolation in the look-up table of residuals.

What does this buy us? Well, as I said earlier, when the data are critically-to-over-sampled, it is difficult for a simple analytic formula to represent a real star image with an accuracy of order 1%. However, a well-chosen formula can represent a stellar profile with a pixel-to-pixel root-mean-square residual of order 5% (give or take, plus or minus, pro or con). By first fitting and subtracting this best analytic model, the values that I put into the look-up table are of order 20 times smaller than would have been the case had I put the raw observed intensities themselves into the table. As a result, the absolute errors caused by the inadequate interpolation are also of order 20 times smaller than they would have been.

There's another way of looking at this. When you use some numerical interpolation scheme, you are making implicit assumptions about the variations in the stellar profile between the sample points. If you use cubic-spline interpolation, you are saying that the fourth and all higher derivatives of the profile are identically zero. If you use sinc interpolation, you are saying that in the profile all spatial frequencies higher than the Nyquist frequency are identically zero. Neither of these assumptions is adequate for critically sampled star images. By combining an analytic first approximation with cubic-spline interpolation within a look-up table of corrections, I am saying that the first three derivatives of the stellar profile can be anything, but the fourth and all higher derivatives are precisely those of the analytic approximation. Had I used sinc interpolation within the table of corrections, I would have been saying that all spatial frequencies below the Nyquist frequency could be anything, but all frequencies above the Nyquist frequency would be those of the analytic approximation.

As I just implied, I adopted cubic-spline interpolation rather than sinc interpolation to evaluate the empirical correction from the analytic approximation to the true model profile. For the analytic approximation, I adopted the Gaussian function. A Gaussian function is not a particularly good match to a real stellar profile, and if you were to write a program to fit purely analytic model profiles to real star images, I would advise strongly against using Gaussians; a Moffat function, as used by ROMAFOT, or some superposition of Gaussian plus power law, as used by STARMAN, would be much better. However, as I will discuss in more detail tomorrow, the Gaussian function is believed to be a pretty good description of the central part of the stellar profile, where things are changing rapidly with position, while the non-Gaussian wings - which have small high-order derivatives (=> small amplitudes at high spatial frequencies) can be adequately mapped in the look-up table of corrections. Moreover, the Gaussian function does have one other - highly desirable - advantage: a two-dimensional Gaussian function can be separated into the product of two one-dimensional Gaussians:

Equation 164

Now, rather than having to compute a two-dimensional numerical integral over the area of each pixel, I can compute two one-dimensional integrals and multiply them together. This turns out to cut the reduction time in half. Maybe a mere factor of two doesn't seem like very much, but if you could get exactly the same answer with half the work, wouldn't you do it? And you do get effectively the same answer, as long as the data are at least critically sampled.

However, as the years have passed, we have started getting undersampled images. We've had some really good seeing at Cerro Tololo, and at the prime focus of the Canada-France-Hawaii telescope it's not rare to get 0".6 seeing, with 0".4 pixels. It turns out that when you are significantly undersampled (FWHM 1.5 pixels, say), the look-up table of corrections can no longer absorb all of the difference between a Gaussian function and a real stellar profile: the Gaussian is just too wrong. The easiest solution is simply to use an analytic function which looks more like a real star, so that the residuals the look-up table must handle are smaller. Accordingly, I am just starting work on a new generation of DAOPHOT which will offer the user a choice of analytic first approximations, including the old, standard Gaussian; a Moffat function, like ROMAFOT's; and a "Penny" function (a superposition of a Gaussian and a Lorentzian), like STARMAN's. The Gaussian can be used for critically sampled and oversampled data, where you get the factor of two increase in speed with no appreciable loss of precision. The Moffat function or the Penny function can be used for undersampled data; these reductions will take longer, but there should be minimal loss of precision due to inadequate interpolations.

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