Several interesting problems in remote sensing can be traced back to the question of the origin along the line of sight of the registered photons. In this paper we revive old concepts that directly follow from the equation of radiative transfer, namely the contribution and weighting functions. We give them, however, a new mathematical form by transforming them into a pair of probability density functions which have the advantage that they can be used in a more flexible manner. We derive these functions, demonstrate a simple relation between them and show how they can be used in principle. Then we proceed with simple applications to a case of upper-tropospheric humidity (UTH) retrieval. In particular, we show how the mean emission pressure level and mean emission temperature change with increasing UTH. We show that the mean emission pressure increases with increasing humidity and remains almost unchanged for UTH values greater than 50 %. The mean emission temperature is decreasing exponentially as UTH increases. The sensitivities of the mean emission pressure to various quantities, e.g. the temperature lapse rate, or retrieval situations, e.g. whether UTH or UTH with respect to ice is considered or which of two different versions of a receiver is used, is generally small compared to the 2σp$2\sigma_p$-width of the layer. The relation of the contribution and weighting functions to Jacobians is discussed as well. We note that the dependence of the mean emission pressure level and other statistical quantities can be formulated using the radiances or brightness temperatures directly. The new method thus offers additional possibilities for interpretation of data from passive remote sensing, and examples are given. In addition of deriving the desired product (for instance, UTH) one can derive and map the mean emission location, its width, and other physical properties like mean temperature of the emission layer. The necessary probability density functions are contained in the solution of the radiative transfer equation and can thus be obtained from runs of the corresponding models. We recommend that radiative transfer models be equipped with facilities to compute and output the contribution and weighting functions.

中文翻译：

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辐射传递的贡献和加权函数–在对流层上层湿度反演中的理论和应用

遥感中的几个有趣问题可以追溯到配准光子视线的起源问题。在本文中，我们重提了直接从辐射传输方程式遵循的旧概念，即贡献函数和加权函数。但是，通过将它们转换为一对概率密度函数，我们为它们提供了一种新的数学形式，其优点是可以更灵活地使用它们。我们导出这些功能，演示它们之间的简单关系，并说明如何在原理上使用它们。然后，我们对高对流层湿度（UTH）检索的情况进行简单的应用。特别是，我们显示了平均排放压力水平和平均排放温度如何随着UTH的增加而变化。我们表明，平均排放压力随着湿度的增加而增加，并且对于大于50％的UTH值几乎保持不变。随着UTH的增加，平均发射温度呈指数下降。平均排放压力对各种量（例如温度下降率）或取回情况（例如，是否考虑冰的UTH或UTH或使用两种不同版本的接收器）的敏感性通常小于层的2σp$ 2 \ sigma_p $-宽度。还讨论了贡献函数和加权函数与Jacobian的关系。我们注意到，平均发射压力水平与其他统计量之间的关系可以直接使用辐射度或亮度温度来表示。因此，新方法为解释来自被动遥感的数据提供了其他可能性，并给出了示例。除了导出所需的产品（例如UTH），还可以导出并绘制平均发射位置，其宽度以及其他物理属性（例如发射层的平均温度）。必要的概率密度函数包含在辐射传递方程的解中，因此可以从相应模型的运行中获得。我们建议辐射传递模型应配备计算和输出贡献和加权函数的工具。和其他物理特性，例如发射层的平均温度。必要的概率密度函数包含在辐射传递方程的解中，因此可以从相应模型的运行中获得。我们建议辐射传递模型应配备计算和输出贡献和加权函数的工具。和其他物理特性，例如发射层的平均温度。必要的概率密度函数包含在辐射传递方程的解中，因此可以从相应模型的运行中获得。我们建议辐射传递模型应配备计算和输出贡献和加权函数的工具。