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

Abstract

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.

摘 要

水凝物变量的初始化在数值模式预报中起到了非常重要的作用. 资料同化理论上要求分析场、背景场以及观测误差概率密度函数满足高斯无偏分布的假定. 针对水凝物变量的非高斯问题, 本研究提出了一个适用于水凝物变量的高斯转换方法, 并分别对不同转换方法获得的水凝物变量的空间分布特征、 误差的高斯性以及背景误差协方差的特征进行了分析. 高斯诊断分析表明, 与其它转换方法相比, 本方法转换后的水凝物变量比其它方法转换的变量更接近高斯分布, 同时该方法保持了水凝物变量本身的空间分布特征. 进一步地, 作者将该方法引入到控制变量转换中, 并统计分析了控制变量转换后的水凝物变量的背景误差协方差, 结果表明, 转换后的水凝物变量背景误差协方差的结构特征是合理的. 本研究表明, 经过本文方法高斯转换后的水凝物变量可以满足资料同化系统中背景误差高斯无偏分布的假定, 可以作为资料同化系统的控制变量.