天体测量与photometry联合估计的Cramer-Rao不确定度下界分析
Analysis of the Cramer-Rao lower uncertainty bound in the joint estimation of astrometry and photometry
摘要：
在本文中，我们使用的Cramer-Rao下限的不确定度估计的最大精度，可以达到联合同时（或二维）估计的测光和天体测量的点源测量的线阵CCD探测器阵列。我们开发的精确表达式的Fisher矩阵元素计算的Cramer-Rao界的情况下，一个源的高斯光剖面。从这些表达式中，我们预测的行为的Cramer-Rao天体测量和测光精度的信号和噪声的观测的函数，并将它们与实际观测-找到一个很好的对应关系。我们表明，天体测量的克拉美-拉奥界为$(S/N)^{-1}$（类似于光度界），但另外，我们发现，这个界是相当敏感的背景值-抑制背景可以大大提高天体测量的精度。我们提出了一个系统的分析的Fisher矩阵的元素的情况下，当检测器充分采样的源（过采样制度），导致封闭形式的Cramer-Rao界的解析表达式。我们表明，在这一制度中，联合参数测定的测光和天体测量的来源成为彼此解耦，而且，它是可能的写下来的表达式（近似一阶的小数量F/B或B/F）的预期最小的不确定性通量和位置。这些表达式被证明是相当有弹性的过采样条件下，因此成为非常有价值的基准工具，以估计在预先指定的观测条件和探测器属性下可达到的最大光度和天体测量精度的近似行为。
英文摘要：
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.
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