Initial field optimization of Global Barotropic Model based on Analytical Four Dimensional Ensemble Variational
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摘要: 解析四维集合变分数据同化方法是一种无需伴随的、完全继承四维变分非线性处理能力的、“流依赖”的非顺序数据同化方法。本研究基于解析四维集合变分，开展全球正压谱模式初始场优化研究，验证解析四维集合变分的同化能力，并构建了高效的扰动集合成员生成方案，将解析四维集合变分降维到了样本空间，最后检验了解析四维集合变分对同化积分窗口长度和观测采样间隔的敏感性。试验结果表明，解析四维集合变分能优化全球正压谱模式的初始场，降维到样本空间后，只需要80个集合成员就可以取得很好的同化效果，在较长的同化积分窗口和观测采样间隔的条件下也可以达到理想的同化效果。Abstract: The Analytical Four Dimensional Ensemble Variational is a “flow-dependent” and non-sequential data assimilation method without adjoint models and completely inherits the nonlinear processing ability of Four Dimensional Variational. In this study, based on the Analytical Four Dimensional Ensemble Variational, the initial field optimization of the Global Barotropic Model was carried out, the assimilation ability of Analytical Four Dimensional Ensemble Variational in the medium complex model was verified, the efficient perturbation ensemble member generation scheme was explored, and Analytical Four Dimensional Ensemble Variational was reduced to the sample space, and finally the sensitivity to the assimilation time window length and the observation sampling interval was verified. The experiments results show that the Analytical Four Dimensional Ensemble Variational can optimize the initial field of the Global Barotropic Model, and after reducing the dimensionality to the sample space, only 80 ensemble members are required to achieve good assimilation effect, and ideal assimilation can also be achieved under the conditions of a long assimilation integration window and observation sampling interval.
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图 2 真实场（a）和背景场（b）的大气流函数空间分布
Fig. 2 Spatial contributions of the stream function of the true field (a) and the background field (b)
图 3 正态分布的随机扰动集合成员
本文构造了3 456个这样的集合成员，a，b仅是其中两个
Fig. 3 Ensemble members of normally distributed random perturbations
This paper constructs 3 456 such ensemble members, of which a, b are just two
图 5 A-4DEnVar目标函数（a）和初始场相对真实场的均方根误差（b）的迭代过程
Fig. 5 Iterative process of A-4DEnVar cost function (a) and RMSE of the initial field relative to the true field (b)
图 7 初始场相对真实场的大气流函数的误差分布
a. 背景场误差；b. 分析场误差
Fig. 7 Distribution of errors in atmospheric stream function relative to the true initial field
a. Background field error; b. analysis field error
图 8 改变同化积分窗口长度和观测采样间隔之后均方根误差变化
Fig. 8 RMSE changes after changing integral time window length and observation sampling interval
图 9 不同的同化积分窗口和观测采样间隔组合下收敛后的均方根误差
Fig. 9 Convergent RMSE under the combination of different integral time window length and observation sampling interval
表 1 经验正交函数分解的累计方差占比
Tab. 1 Cumulative percentage of variance for EOF
模态数量 累计方差占比/% 20 76.2 40 83.9 60 88.0 80 90.7 100 92.6 
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表 2 A-4EnVar全球正压谱模式初始场优化试验设计
Tab. 2 Experiments design of Global Barotropic Model initial field optimization based on A-4EnVar
试验 变量替换方式 同化积分窗口长度/h 观测采样间隔/h 集合成员数量 是否降维 A $ {{\boldsymbol{B}}_{00}} $变换、$ {{\tilde{\boldsymbol x}}_0} $变换 24 6 3 456 否 B $ {{\tilde{\boldsymbol x}}_0} $变换 24 6 20、40、60、80、100 是 C $ {{\tilde{\boldsymbol x}}_0} $变换 24、48、72、96 4、6、8、12 80 是
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表 3 收敛后A-4DEnVar目标函数和均方根误差减小的百分比（%）
Tab. 3 The reduction percentage of the cost function of A-4DEnVar and RMSE (%)
变量替换方式 目标函数 均方根误差 $ {{\boldsymbol{B}}_{00}} $变换 84.0 62.2 $ {{\tilde{\boldsymbol x}}_0} $变换 98.8 88.8
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表 4 优化后目标函数和均方根误差减小的百分比（%）
Tab. 4 The reduction percentage (%) of the cost function and RMSE after optimization
集合成员数量 是否降维 目标函数 均方根误差 3456 否 98.8 88.8 100 是 94.8 78.3 80 是 94.2 77.1 60 是 92.9 75.2 40 是 88.8 68.6 20 是 85.3 64.7
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