Ensemble Kalman methods are widely used for state estimation in the geophysical sciences. Their success stems from the fact that they take an underlying (possibly noisy) dynamical system as a black box to provide a systematic, derivative-free methodology for incorporating noisy, partial and possibly indirect observations to update estimates of the state; furthermore the ensemble approach allows for sensitivities and uncertainties to be calculated. The methodology was introduced in 1994 in the context of ocean state estimation. Soon thereafter it was adopted by the numerical weather prediction community and is now a key component of the best weather prediction systems worldwide. Furthermore the methodology is starting to be widely adopted for numerous problems in the geophysical sciences and is being developed as the basis for general purpose derivative-free inversion methods that show great promise. Despite this empirical success, analysis of the accuracy of ensemble Kalman methods, in terms of their capabilities as both state estimators and quantifiers of uncertainty, is lagging. The purpose of this paper is to provide a unifying mean field based framework for the derivation and analysis of ensemble Kalman methods. Both state estimation and parameter estimation problems (inverse problems) are considered, and formulations in both discrete and continuous time are employed. For state estimation problems, both the control and filtering approaches are considered; analogously for parameter estimation problems, the optimization and Bayesian perspectives are both studied. The mean field perspective provides an elegant framework, suitable for analysis; furthermore, a variety of methods used in practice can be derived from mean field systems by using interacting particle system approximations. The approach taken also unifies a wide-ranging literature in the field and suggests open problems.

翻译：集成卡尔曼方法在地球科学领域被广泛用于状态估计。其成功源于它们将底层（可能含噪声的）动力系统视为黑箱，从而提供了一种系统性的、无导数的方法论，用于融合含噪声的、局部的以及可能间接的观测数据来更新状态估计；此外，集成方法允许计算敏感性和不确定性。该方法于1994年在海洋状态估计的背景下被提出。此后不久，它被数值天气预报界采纳，现已成为全球最佳天气预报系统的关键组成部分。此外，该方法正开始被广泛用于地球科学中的众多问题，并正在被发展为通用无导数反演方法的基础，展现出巨大潜力。尽管取得了这些经验上的成功，对于集成卡尔曼方法在作为状态估计器和不确定性量化器方面的准确性分析仍然滞后。本文旨在为集成卡尔曼方法的推导和分析提供一个统一的、基于平均场的框架。文中同时考虑了状态估计和参数估计问题（反问题），并采用了离散时间和连续时间两种表述形式。对于状态估计问题，同时考虑了控制和滤波两种方法；类似地，对于参数估计问题，则研究了优化和贝叶斯两种视角。平均场视角提供了一个优雅的、适合分析的框架；此外，实践中使用的多种方法可以通过使用交互粒子系统近似从平均场系统中推导出来。所采用的方法也统一了该领域广泛的文献，并提出了有待解决的问题。