Analysis and interpretation of the Cramer-Rao lower-bound in astrometry: One dimensional case

TL;DR

This paper derives the Cramer-Rao lower bound for the maximum achievable astrometric precision in 1D imaging systems, providing analytical formulas and insights for detector design and observational planning.

Contribution

It develops analytical expressions for the Cramer-Rao bound in 1D astrometry, linking precision to background, flux, and detector properties, and compares theoretical limits to real data.

Findings

Maximum precision scales as B/F^2 with background B and flux F.

When background is negligible, precision scales as F^{-1}.

Current techniques approach the theoretical Cramer-Rao limit closely.

Abstract

In this paper we explore the maximum precision attainable in the location of a point source imaged by a pixel array detector in the presence of a background, as a function of the detector properties. For this we use a well-known result from parametric estimation theory, the so-called Cramer-Rao lower bound. We develop the expressions in the 1-dimensional case of a linear array detector in which the only unknown parameter is the source position. If the object is oversampled by the detector, analytical expressions can be obtained for the Cramer-Rao limit that can be readily used to estimate the limiting precision of an imaging system, and which are very useful for experimental (detector) design, observational planning, or performance estimation of data analysis software: In particular, we demonstrate that for background-dominated sources, the maximum astrometric precision goes as $B/F^2$, where $B$ is the background in one pixel, and $F$ is the total flux of the source, while when the background is negligible, this precision goes as $F^{-1}$. We also explore the dependency of the astrometric precision on: (1) the size of the source (as imaged by the detector), (2) the pixel detector size, and (3) the effect of source de-centering. Putting these results into context, the theoretical Cramer-Rao lower bound is compared to both ground- as well as spaced-based astrometric results, indicating that current techniques approach this limit very closely. Our results indicate that we have found in the Cramer-Rao lower variance bound a very powerful astrometric "benchmark" estimator concerning the maximum expected positional precision for a point source, given a prescription for the source, the background, the detector characteristics, and the detection process.