Analysis and Interpretation of the Cramer-Rao Lower-Bound in Astrometry: One-Dimensional Case

作者：Mendez Rene A^*; Silva Jorge F; Lobos Rodrigo

来源：Publications of the Astronomical Society of the Pacific, 2013, 125(927): 580-594.

DOI：10.1086/671126

In this article 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 one-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 decentering. Putting these results into context, the theoretical Cramer-Rao lower bound is compared to both ground- as well as space-based astrometric results, indicating that current techniques approach this limit very closely. It is furthermore demonstrated that practical astrometric estimators like maximum likelihood or least-squares techniques cannot formally reach the Cramer-Rao bound, but that they approach this limit in the one-dimensional case very tightly, for a wide range of signal-to-noise ratio (S/N) of the source. Our results indicate that we have found in the Cramer-Rao lower variance bound a very powerful astrometric %26quot;benchmark%26quot; 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.

• 出版日期2013-5