Minin Serge, Kamalabadi Farzad
Coordinated Science Laboratory, University of Illinois, 1308 West Main Street, Urbana, Illinois 61801, USA.
Appl Opt. 2009 Dec 20;48(36):6913-22. doi: 10.1364/AO.48.006913.
We derive analytical equations for uncertainties in parameters extracted by nonlinear least-squares fitting of a Gaussian emission function with an unknown continuum background component in the presence of additive white Gaussian noise. The derivation is based on the inversion of the full curvature matrix (equivalent to Fisher information matrix) of the least-squares error, chi(2), in a four-variable fitting parameter space. The derived uncertainty formulas (equivalent to Cramer-Rao error bounds) are found to be in good agreement with the numerically computed uncertainties from a large ensemble of simulated measurements. The derived formulas can be used for estimating minimum achievable errors for a given signal-to-noise ratio and for investigating some aspects of measurement setup trade-offs and optimization. While the intended application is Fabry-Perot spectroscopy for wind and temperature measurements in the upper atmosphere, the derivation is generic and applicable to other spectroscopy problems with a Gaussian line shape.
我们推导了在存在加性高斯白噪声的情况下，对具有未知连续背景分量的高斯发射函数进行非线性最小二乘拟合所提取参数的不确定性的解析方程。该推导基于最小二乘误差χ²在四变量拟合参数空间中的全曲率矩阵（等同于费舍尔信息矩阵）的求逆。所推导的不确定性公式（等同于克拉美 - 罗误差界）与大量模拟测量的数值计算不确定性高度吻合。所推导的公式可用于估计给定信噪比下的最小可实现误差，以及研究测量设置权衡和优化的某些方面。虽然预期应用是用于高层大气中风和温度测量的法布里 - 珀罗光谱学，但该推导是通用的，适用于具有高斯线形的其他光谱学问题。
