Limits to the Precision of Joint Flux and Position Measurements on Array Data

Hans-Martin Adorf

Space Telescope--European Coordinating Facility, European Southern Observatory, D-85748 Garching b. München, F.R. Germany

Abstract:

1. Introduction

When preparing observing proposals the following questions frequently arise: ``Can I achieve such-and-such an RMS error with my planned observations?'' and ``What do I have to do to achieve a desired precision, simply expose longer, or use a different filter, or perhaps a different instrument configuration, or ...?'' It is well known how to carry out simple RMS error calculations for photometry and astrometry of isolated point sources (cf. King 1983). However, it is less obvious how to calculate errors for photometry and astrometry in crowded star fields, or for surface photometry and astrometry of extended sources. The problem becomes really intriguing when---transgressing imagery for a moment---the quantity of interest is a spectral line ratio, a temperature in an H2 region or even the Hubble constant. Furthermore, how can statistical errors be predicted when several physical quantities with interdependent uncertainties have to be estimated simultaneously?

2. The Cramér-Rao Minimum Variance Bound

Answers to the questions above can be derived using the Cramér-Rao minimum variance bound (MVB) theorem, a powerful statistical inequality noted already in the 1940's (Kendall & Stuart 1979; Adorf 1996b, and refs. therein). In case when a single quantity (e.g., a flux or a position) needs to be estimated, the Cramér-Rao theorem states that the variance of any unbiased estimator for is at least as large as the value given by the MVB.

The multi-parameter generalization of the one-parameter MVB-theorem addresses the problem of predicting the minimum statistical error in case where several quantities have to be estimated simultaneously. For instance in precision photometry, the object flux and position have to be estimated along with the background, i.e., four parameters in total. In this case the MVB becomes a matrix, and the MVB-theorem states that the difference between the variance-covariance matrix of an arbitrary unbiased estimator and the MVB-matrix is a positive semi-definite matrix.

The MVB is computable by inverting the so-called Fisher information , the statistical expectation of the square of the partial derivative of the log-likelihood function with respect to , the quantity of interest. Thus the MVB can, at least in principle, be computed, as soon as the likelihood function L is known as a function of the parameter(s) . In order to establish the likelihood function one needs a stochastic model of the observational process.

The deterministic part of that model describes the properties of the observables as if they were classical quantities measured with ideal ``noise-free'' equipment. When considering the limiting case of a linear detector, the expected intensity value at pixel i can be written as with , where the are the quantum efficiencies (``flat field''), is the effective point-spread function (PSF) incorporating the pixel response function (PRF), the denote some sky and/or detector background, and the are the unknown object intensities. The object positions are considered implicit parameters of the PSF.

Let us turn to the noise part of the stochastic model. When considering the important limiting case of a pure Poisson-noise process, the log-likelihood function, the Fisher-information and thus the MVB can be stated in closed form (Adorf 1996b). Given the intrinsically rather complex definition of the Fisher-information, the equations resulting from pure Poisson-noise assumptions are remarkably simple, as is evident from the fact that ten simple IDL statements (see Table 1) suffice to compute all the independent elements of the symmetric element Fisher information matrix for joint photometry and astrometry of an arbitrary source on top of a spatially stationary (non-variable) background.

Here the suffixes o, b, x, and y refer to the object flux, background flux, and object x- and y-coordinates, respectively.

The functions d_dx(phi) and d_dy(phi) represent spatial derivatives (``gradients'') of the expected image, which can be efficiently computed using high-fidelity image processing operators based on the discrete Fourier-transform (Adorf 1996a).

3. Application Areas

The applications of the Cramér-Rao MVB-theorem cover a wide range, from instrument design, to observational design, to data analysis. -- In instrument design the MVB-theorem can be used to predict how a particular design choice (e.g., the pixel size and implied sampling density) influences the photometric and astrometric performance of the planned instrument. -- In observational design the MVB permits predictions of e.g., (i) a lower bound to the photometric error for point sources and for surface photometry of extended sources (``relative photometry''), (ii) a lower bound to the precision with which the position of point sources can be measured, whether isolated or in a cluster (``relative astrometry''), or (iii) how different sub-pixel dither patterns affect the photometric and astrometric errors (Adorf 1995, 1996c). -- In data analysis, finally, the MVB permits checking the statistical efficiency of different data analysis algorithms: if one's favourite procedure cannot attain the MVB, it may not be optimum!

4. Example: Astrometry with WFPC2 On-board HST

The usefulness of the MVB can be demonstrated by addressing the question of how well one can carry out astrometry of point sources as a function of their brightness on a CCD-frame. Let us specifically consider the case where HST's WFPC2 camera observes an isolated A0 star for sec through the F555W filter. The WFPC2 exposure time calculator (Adorf et al. 1996) can be used to predict the object and background counts.

Assuming that the object image is centered on the pixel, neglecting flat-field variations, and using a TinyTim-generated optical PSF, minimum astrometric RMS errors can be predicted as shown in Figure 1.

Realistic calculations have to take into account the pixel response function (Adorf 1995), which describes the sensitivity variations of a CCD pixel across its area, and the important light scattering and electron diffusion into neighbouring pixels. In Figure 1 a model-PRF was used consisting of the convolution of an ideal PRF with a narrow Gaussian of mas. Note that for the brightest magnitudes the image will saturate and the predictions are too optimistic (i.e., the bound is too low), unless the exposure is split.

5. Conclusion

It has been shown how the Cramér-Rao minimum variance bound (MVB) theorem, which is based on a stochastic model of the observational process, can be turned into a useful predictive tool, which often obviates the need for Monte-Carlo simulations. The easy-to-use MVB code may specifically be used to predict minimum errors, e.g., for astrometric programmes using WFPC2 image data, and, more generally, to check the statistical efficiency of data analysis procedures. Most importantly perhaps, the MVB-theorem makes one think about the statistical efficiency of conventional data analysis methods (subtract background?, divide by flat-field??), and about statistically sound estimators for use in astronomical data analysis.

Acknowledgments:

I am grateful to my colleagues Bob Fosbury for persistent encouragement, and Jeremy Walsh for carefully reading the draft.

  
Figure 1: The minimum astrometric RMS error as predicted by the Cramér-Rao MVB theorem for an isolated A0 star in the brightness range of V=22--27 mag, observed with the WFC2 through the V(F555W) filter for sec. The computations for the upper curve (squares) include a realistic pixel response function (PRF), wheras the computations for the lower curve (triangles) do not. The difference in astrometric precision is due to the blurring and undersampling of the optical PSF by the WFC2 detector.
Figure 1: PS 11 Kb

References:

Adorf, H.-M. 1995, ``WFPC2 observations---when and how to dither?'' ST-ECF Newsl., 23, 19

Adorf, H.-M. 1996a, ``High-fidelity image processing operators'', in preparation

Adorf, H.-M. 1996b, ``Minimum errors in joint photometric and astrometric measurements'' or ``What the Cramér-Rao minimum variance bound can tell us'', in preparation

Adorf, H.-M. 1996c, ``Photometry and astrometry with WFPC2 and NICMOS data'', Proc. Conf. ``Science with the HST II'', 4.--8. Dec. 1995, Paris, in press

Adorf, H.-M., Biretta, J., & Suchkov, A. 1996, ``The WFPC2 exposure time calculator'', in preparation

Kendall, M., & Stuart, A. 1979, ``The Advanced Theory of Statistics'', Charles Griffin & Co. Ltd., London & High Wycombe, 748 pp.

King, I. R. 1983, PASP, 95, 163