Analysis of the Cramer-Rao lower uncertainty bound in the joint estimation of astrometry and photometry

TL;DR

This paper derives the Cramer-Rao lower bounds for joint photometry and astrometry estimation of point sources, providing analytical expressions and analyzing how observational parameters influence the achievable precision.

Contribution

It develops exact Fisher matrix expressions for Gaussian sources and analyzes the bounds in oversampling regimes, offering new analytical tools for maximum precision estimation.

Findings

Astrometric Cramer-Rao bound scales as (S/N)^{-1}

Background suppression significantly improves astrometric accuracy

Analytical expressions are robust across oversampling regimes

Abstract

In this paper we use the Cramer-Rao lower uncertainty bound to estimate the maximum precision that could be achieved on the joint simultaneous (or 2D) 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 Cramer-Rao bound in the case of a source with a Gaussian light profile. From these expressions we predict the behavior of the Cramer-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. We show that the astrometric Cramer-Rao bound goes as $(S/N)^{-1}$ (similar to the photometric bound) 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 Cramer-Rao bound. We show that, in this regime, the joint parametric determination 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 pre-specified observing conditions and detector properties.