一种改进的超高能空气簇射尺寸不确定度评定方法
An Improved Method of Estimating the Uncertainty of Air-Shower Size at Ultra-High Energies
摘要：
要收集大量能量在$10^{17}$至$10^{18}$ eV以上的宇宙射线，需要在地面放置间距很大的粒子探测器，以探测大气中引发的大范围空气簇射。然后，通过将观测到的信号拟合到期望的函数形式来估计空气簇射的大小，初级能量的代理，以便在参考距离处内插信号。函数形式描述了预期信号随着与簇射核的距离的快速衰减，通常使用两个对数斜率来解释信号的短程和长程下降。与空气淋浴大小相关的不确定性是在最小值附近的拟合参数的对数似然二次依赖的假设下确定的，从而提供了有意义的方差-协方差矩阵。在本文中，我们表明，对于一个事件的拓扑结构，其中一个信号是远远大于其他的，最小值附近的拟合函数的二次依赖是一个很差的近似，导致不准确的估计的不确定性。为了恢复一个二次形状，我们建议使用极坐标周围的检测器记录最大的信号，投影到平面上的簇射阵面，定义的似然函数的对数极距，极角和对数簇射大小作为自由参数。我们表明，一个有意义的方差-协方差矩阵，然后恢复在新的坐标系，作为修改后的参数的拟合函数的依赖性是适当的近似由一个二次函数。使用的不确定性，在新的坐标系为随后的高层次的分析说明。
英文摘要：
The collection of a statistically significant number detected of cosmic rays with energy above $10^{17}$ to $10^{18}$ eV requires widely-spaced particle detectors at the ground level to detect the extensive air showers induced in the atmosphere. The air-shower sizes, proxies of the primary energies, are then estimated by fitting the observed signals to a functional form for expectations so as to interpolate the signal at a reference distance. The functional form describes the rapid falloff of the expected signal with the distance from the shower core, using typically two logarithmic slopes to account for the short-range and long-range decreases of signals. The uncertainties associated to the air-shower sizes are determined under the assumption of a quadratic dependence of the log-likelihood on the fitted parameters around the minimum, so that a meaningful variance-covariance matrix is provided. In this paper, we show that for an event topology where one signal is much larger than the others, the quadratic dependence of the fitted function around the minimum is a poor approximation that leads to an inaccurate estimate of the uncertainties. To restore a quadratic shape, we propose to use the polar coordinates around the detector recording the largest signal, projected onto the plane of the shower front, to define the likelihood function in terms of logarithmic polar distances, polar angles and logarithmic shower sizes as free parameters. We show that a meaningful variance-covariance matrix is then recovered in the new coordinate system, as the dependence of the fitted function on the modified parameters is properly approximated by a quadratic function. The use of the uncertainties in the new coordinate system for subsequent high-level analyses is illustrated.
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