Use of data assimilation to initialize hydrometeors plays a vital role in numerical weather prediction (NWP). To directly analyze hydrometeors in data assimilation systems from cloud-sensitive observations, hydrometeor control variables are necessary. Common data assimilation systems theoretically require that the probability density functions (PDFs) of analysis, background, and observation errors should satisfy the Gaussian unbiased assumptions. In this study, a Gaussian transform method is proposed to transform hydrometeors to more Gaussian variables, which is modified from the Softmax function and renamed as Quasi-Softmax transform. The Quasi-Softmax transform method then is compared to the original hydrometeor mixing ratios and their logarithmic transform and Softmax transform. The spatial distribution, the non-Gaussian nature of the background errors, and the characteristics of the background errors of hydrometeors in each method are studied. Compared to the logarithmic and Softmax transform, the Quasi-Softmax method keeps the vertical distribution of the original hydrometeor mixing ratios to the greatest extent. The results of the D’Agostino test show that the hydrometeors transformed by the Quasi-Softmax method are more Gaussian when compared to the other methods. The Gaussian transform has been added to the control variable transform to estimate the background error covariances. Results show that the characteristics of the hydrometeor background errors are reasonable for the Quasi-Softmax method. The transformed hydrometeors using the Quasi-Softmax transform meet the Gaussian unbiased assumptions of the data assimilation system, and are promising control variables for data assimilation systems.

使用数据同化来初始化水凝物在数值天气预报（NWP）中起着至关重要的作用。为了从对云敏感的观测中直接分析数据同化系统中的水凝物，水凝物控制变量是必要的。从理论上讲，通用数据同化系统要求分析，背景和观察误差的概率密度函数（PDF）应满足高斯无偏假设。在这项研究中，提出了一种高斯变换方法，可以将水汽流星转换为更多的高斯变量，该方法是从Softmax函数进行修改并重命名为Quasi-Softmax变换。然后将准Softmax变换方法与原始水凝物混合比及其对数变换和Softmax变换进行比较。空间分布 研究了背景误差的非高斯性质，以及每种方法中水汽凝结器背景误差的特征。与对数变换和Softmax变换相比，拟Softmax方法最大程度地保持了原始水凝物混合比的垂直分布。D'Agostino测试的结果表明，与其他方法相比，通过Quasi-Softmax方法转化的水凝物更具有高斯分布。高斯变换已添加到控制变量变换中，以估计背景误差协方差。结果表明，对于准Softmax方法，水凝流星背景误差的特征是合理的。使用Quasi-Softmax变换转换后的水凝物满足数据同化系统的高斯无偏假设，