Analyzing the Impacts of Non-Gaussian Errors in Gaussian Data Assimilation Systems

世界上大多数变分数据同化（DA）的操作和研究的基础是所有误差都是高斯分布的。对于天气尺度的天气系统，这个假设是一个很好的近似值，然而，即使在这些大尺度上，也有一些变量（例如，正定量，如相对湿度）不能用假设的高斯误差分布来适当地表征。对诸如湿度之类的正定变量施加负值的影响是，操作系统数据分析系统可能无法收敛或产生不稳定的数值预报或非物理模式状态。这种高斯假设也存在于基于最大似然估计（MLE）贝叶斯方法的检索系统中。在其他一些贝叶斯系统中，湿度变量被假定为对数正态分布，因此提取的变量是湿度变量的自然对数。这两种方法通过寻找不正确的统计量来描述随机变量的概率行为，从而在分析中引入偏差。在本项目中，将应用并评估非高斯变量的另一种方法，该方法将对数正态分布与高斯分布相结合，称为混合分布。这种混合分布允许同时检索和同化高斯和对数正态分布的变量。混合方法与其他两种方法的不同之处在于，它寻找随机变量之间正确协方差的分析模式，而不是最佳高斯近似的模式或对数正态分布的中位数。在这项工作的第一阶段，将开发确定在数据分析方案中可以和不可以对湿度施加高斯假设的方法，以及量化这种假设对检索量的影响。第二阶段将研究将高斯假设方法与混合分布方法中检索到的数据同化为更大的3D或4D-VAR（基于变分的）方法的影响，这适用于正在评估的特定模型系统[例如，NSF/ ncar支持的天气预报和研究（WRF） DA系统]。这项工作的智力价值将追溯到更好地观察和吸收湿度场的能力，并对其与其他模式预测场的相互作用有了更好的理解，以进行各种尺度的大气预测：天气、中尺度和云解析。这项工作预计将产生更广泛的影响，这将通过对大型数值天气预报系统中采用的数据分析方案进行谨慎的改变，从而有望促进对严重和/或极端天气事件的改进预测。通过对博士后研究助理的指导和早期职业发展，教育将产生更广泛的影响，他们将接受非高斯数据分析方法的培训，并获得检索和接近操作的大型数据分析系统的经验。

The basis for most of the world's operational and research variational data assimilation (DA) is that all errors are Gaussian distributed. For synoptic-scale weather systems this assumption is a good approximation, however even at these large scales there some variables (e.g., positive-definite quantities such as relative humidity) cannot be properly characterized by assumed Gaussian error distributions. The impact of an imposed negative value for a positive-definite variable such as moisture is that an operational system DA system could fail to converge or otherwise yield an unstable numerical forecast or an unphysical model state. This Gaussian assumption is also present in retrieval systems that are based upon a maximum likelihood estimation (MLE) Bayesian-type approach. In some other Bayesian systems the moisture variable is assumed to be lognormally distributed and so the retrieved variable is the natural logarithm of the moisture variable. Both approaches introduce a bias into the analysis by finding the incorrect statistic to describe the probabilistic behavior of the random variable. In this project an alternative approach for non-Gaussian variables that combines a lognormal distribution with a Gaussian distribution, referred to as a mixed distribution, will be applied and evaluated. This mixed distribution allows for the retrieval and assimilation of Gaussian- and lognormally-distributed variables simultaneously. The mixed approach is different to the other two approaches in that it is finding the analysis mode with the correct covariances between the random variables, and not the mode of the best Gaussian approximation or the median of the lognormal distribution.In the first stage of this effort, methods for determining where one can and cannot impose a Gaussian assumption for humidity within DA schemes, as well to quantify the impacts of such assumptions on retrieved quantities, will be developed. The second stage will investigate the impacts of assimilating retrieved data from the Gaussian assumption approach against the mixed distribution approach into a larger 3D- or 4D-VAR (variationally-based) approach as appropriate to the particular model system being evaluated [e.g., the NSF/NCAR-supported Weather Forecasting and Research (WRF) DA system]. The intellectual merit of this work will trace to the ability to better observe and assimilate moisture fields and develop an improved understanding of their interactions with other model-prognostic fields for a variety of dimensions of atmospheric prediction: synoptic, mesoscale and cloud resolving. Anticipated Broader Impacts of this effort will come through the ability to exert carefully motivated changes in DA schemes employed in larger numerical weather prediction systems, which would in-turn be expected to foster improved predictions of severe and/or extreme weather events. Broader impacts through education will occur through the mentorship and early-career development of a supported postdoctoral research associate, who will be trained in non-Gaussian DA methods as well as gain experience with retrievals and near-operational large DA systems.