Analysis of the Cramér-Rao Bound in the Joint Estimation of Astrometry and Photometry

Abstract

In this paper, we use the Cramér-Rao lower uncertainty bound to estimate the maximum precision that could be achieved on the joint simultaneous (or two-dimensional) estimation of photometry and astrometry of a point source measured by a linear CCD detector array. We develop exact expressions for the Fisher matrix elements required to compute the Cramér-Rao bound in the case of a source with a Gaussian light profile. From these expressions, we predict the behavior of the Cramér-Rao astrometric and photometric precision as a function of the signal and the noise of the observations, and compare them to actual observations—finding a good correspondence between them. From the Cramér-Rao bound, we obtain the well-known fact that the uncertainty in flux on a Poisson-driven detector, such as a CCD, goes approximately as the square root of the flux. However, more generally, higher-order correction factors that depend on the ratio B/F or F/B (where B is the background flux per pixel, and F is the total flux of the source), as well as on the properties of the detector (pixel size) and the source (width of the light profile), are required for a proper calculation of the minimum expected uncertainty bound in flux. Overall, the Cramér-Rao bound predicts that the uncertainty in magnitude goes as (S/N)-1 under a broad range of circumstances. As for the astrometry, we show that its Cramér-Rao bound also goes as (S/N)-1 but, additionally, we find that this bound is quite sensitive to the value of the background—suppressing the background can greatly enhance the astrometric accuracy. We present a systematic analysis of the elements of the Fisher matrix in the case when the detector adequately samples the source (oversampling regime), leading to closed-form analytical expressions for the Cramér-Rao bound. We show that, in this regime, the joint parametric determinations of photometry and astrometry for the source become decoupled from each other, and furthermore, it is possible to write down expressions (approximate to first order in the small quantities F/B or B/F) for the expected minimum uncertainty in flux and position. These expressions are shown to be quite resilient to the oversampling condition, and become thus very valuable benchmark tools to estimate the approximate behavior of the maximum photometric and astrometric precision attainable under prespecified observing conditions and detector properties.