Background Error Covariance Statistics of Hydrometeor Control Variables Based on Gaussian Transform

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.