Effects of mis‐specified time‐correlated model error in the (ensemble) Kalman Smoother,Quarterly Journal of the Royal Meteorological Society

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Effects of mis‐specified time‐correlated model error in the (ensemble) Kalman Smoother
Quarterly Journal of the Royal Meteorological Society ( IF 3.0 ) Pub Date : 2020-10-11 , DOI: 10.1002/qj.3934
Haonan Ren ₁ , Javier Amezcua ₁ , Peter Jan van Leeuwen _{1,

2}

Affiliation

Data assimilation is often performed under the perfect model assumption. Although there is an increasing amount of research accounting for model errors in data assimilation, the impact of an incorrect specification of the model errors on the data assimilation results has not been thoroughly assessed. We investigate the effect that an inaccurate time correlation in the model error description can have on data assimilation results, deriving analytical results using a Kalman Smoother for a one‐dimensional system. The analytical results are evaluated numerically to generate useful illustrations. For a higher‐dimensional system, we use an ensemble Kalman Smoother. Strong dependence on observation density is found. For a single observation at the end of the window, the posterior variance is a concave function of the guessed decorrelation time‐scale used in the data assimilation process. This is due to an increasing prior variance with that time‐scale, combined with a decreasing tendency from larger observation influence. With an increasing number of observations, the posterior variance decreases with increasing guessed decorrelation time‐scale because the prior variance effect becomes less important. On the other hand, the posterior mean‐square error has a convex shape as a function of the guessed time‐scale with a minimum where the guessed time‐scale is equal to the real decorrelation time‐scale. With more observations, the impact of the difference between two decorrelation time‐scales on the posterior mean‐square error reduces. Furthermore, we show that the correct model error decorrelation time‐scale can be estimated over several time windows using state augmentation in the ensemble Kalman Smoother. Since model errors are significant and significantly time correlated in real geophysical systems such as the atmosphere, this contribution opens up a next step in improving prediction of these systems.

中文翻译：

（集成）卡尔曼平滑器中错误指定的时间相关模型误差的影响

资料同化通常是在完美模型假设下进行的。尽管越来越多的研究解释资料同化中的模型误差，但模型误差的错误指定对资料同化结果的影响尚未得到彻底评估。我们研究了模型误差描述中不准确的时间相关性对数据同化结果的影响，并使用一维系统的卡尔曼平滑器得出分析结果。对分析结果进行数值评估以生成有用的说明。对于高维系统，我们使用集成卡尔曼平滑器。发现对观察密度的强烈依赖性。对于窗口末尾的单个观察，后验方差是数据同化过程中使用的猜测去相关时间尺度的凹函数。这是由于先验方差随着时间尺度的增加而增加，加上较大的观测影响导致的下降趋势。随着观测数量的增加，后验方差随着猜测的去相关时间尺度的增加而减小，因为先验方差效应变得不那么重要。另一方面，后验均方误差具有作为猜测时间尺度的函数的凸形状，其中猜测时间尺度等于真实去相关时间尺度的最小值。随着观察的增多，两个去相关时间尺度之间的差异对后验均方误差的影响会减小。此外，我们表明，可以使用集成卡尔曼平滑器中的状态增强在多个时间窗口上估计正确的模型误差去相关时间尺度。由于模型误差在大气等真实地球物理系统中非常显着且与时间显着相关，因此这一贡献为改进这些系统的预测开辟了下一步。

更新日期：2020-10-11

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