What is the correct cost functional for variational data assimilation?
What is the correct cost functional for variational data assimilation?
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DOI：
10.1007/s00382-018-4146-y
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发表时间：
2018-04
期刊：
影响因子：
4.6
通讯作者：
J. Bröcker
中科院分区：
文献类型：
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作者：
J. Bröcker
数据同化的变分方法，特别是弱约束四维变分(WC-4DVar)，在地球科学中很重要，在其他社区(通常以不同的名称)也很重要。例如，通过将数据同化与最大后验概率(MAP)估计联系起来，成本函数和所产生的最优轨迹可能具有概率解释。这是可能的，特别是如果未知轨迹被建模为随机微分方程(SDE)的解，就像天气预报和气候建模中越来越多的情况一样。在这种情况下，通过最小化Onsager-Machlup泛函来获得MAP估计器(或SDE的“最可能路径”)。虽然这一事实是众所周知的，但文献中似乎有一些混淆，能量泛函有时被声称产生最可能的路径。本文的第一个目的是解决这一困惑，并证明能量泛函一般不提供最可能的路径。第二个目的是讨论在实践中的影响。虽然上述结果与连续时间的随机模型有关，但它们在实际中确实会产生影响，其中SDE是用离散时间格式来近似的。事实证明，使用SDE的近似值并计算其最可能路径并不一定会产生SDE本身最可能路径的良好近似值。这表明，即使在离散时间，也应该使用Onsager-Machlup泛函的一个版本，而不是能量泛函，至少如果要将解解释为MAP估计器的话。
Variational approaches to data assimilation, and weakly constrained four dimensional variation (WC-4DVar) in particular, are important in the geosciences but also in other communities (often under different names). The cost functions and the resulting optimal trajectories may have a probabilistic interpretation, for instance by linking data assimilation with maximum aposteriori (MAP) estimation. This is possible in particular if the unknown trajectory is modelled as the solution of a stochastic differential equation (SDE), as is increasingly the case in weather forecasting and climate modelling. In this situation, the MAP estimator (or “most probable path” of the SDE) is obtained by minimising the Onsager–Machlup functional. Although this fact is well known, there seems to be some confusion in the literature, with the energy (or “least squares”) functional sometimes been claimed to yield the most probable path. The first aim of this paper is to address this confusion and show that the energy functional does not, in general, provide the most probable path. The second aim is to discuss the implications in practice. Although the mentioned results pertain to stochastic models in continuous time, they do have consequences in practice where SDE’s are approximated by discrete time schemes. It turns out that using an approximation to the SDE and calculating its most probable path does not necessarily yield a good approximation to the most probable path of the SDE proper. This suggest that even in discrete time, a version of the Onsager–Machlup functional should be used, rather than the energy functional, at least if the solution is to be interpreted as a MAP estimator.