Analysis and design of covariance inflation methods using inflation functions. Part 1: Theoretical framework,Quarterly Journal of the Royal Meteorological Society

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Analysis and design of covariance inflation methods using inflation functions. Part 1: Theoretical framework
Quarterly Journal of the Royal Meteorological Society ( IF 3.0 ) Pub Date : 2020-07-03 , DOI: 10.1002/qj.3864
Le Duc _{1,

2} , Kazuo Saito _{1,

2,

3} , Daisuke Hotta ₂

Affiliation

We propose a unifying theory for covariance inflation (CI) in the Ensemble Kalman Filter (EnKF) that encompasses all existing CI methods and can explain many open problems in CI. Each CI method is identified with an inflation function that alters analysis perturbations through their singular values. Inflation functions are usually considered as functions of singular values of background or analysis perturbations. However, we have shown that it is more fruitful if inflation functions are viewed as functions of reduction factors of background singular values after assimilation. These factors indeed comprise the spectra of linear transformations between background and analysis perturbations. To be an inflation function, a function has to satisfy three conditions: (a) the functional condition: all reduction factors must increase, (b) the no‐observation condition: when no observations are assimilated, analysis perturbations are identical to background perturbations, and (c) the order‐preserving condition: inflated analysis singular values must have the same order as background singular values. If the upper‐bound condition, that is, inflated analysis error variances must be less than observation error variances, is imposed, the resulting inflation functions are shown to be equivalent to prior inflation functions which are functions of singular values of background perturbations. This condition is necessary if we want to inflate analysis increments in posterior CI. It turns out that the relaxation‐to‐prior‐spread method and the relaxation‐to‐prior‐perturbation method belong to the class of linear inflation functions. In this class, we also have constant inflation functions, multiplicative inflation functions and parameter‐varying linear inflation functions. More interesting, the Deterministic EnKF is found to belong to the class of quadratic inflation functions. This quadratic class introduces an elegant form for computing analysis perturbations through the Kalman gain. Higher‐order polynomial and non‐polynomial forms of inflation functions are less appealing in practice due to high computation cost and difficulty in determining free parameters.

中文翻译：

使用膨胀函数的协方差膨胀方法的分析和设计。第1部分：理论框架

我们在Ensemble Kalman滤波器（EnKF）中提出了协方差膨胀（CI）的统一理论，该理论涵盖了所有现有的CI方法，并且可以解释CI中的许多开放性问题。每种CI方法都带有一个膨胀函数，该函数通过其奇异值改变分析扰动。通货膨胀函数通常被视为背景或分析扰动的奇异值的函数。但是，我们已经表明，如果将通货膨胀函数视为同化后背景奇异值的折减因子的函数，则会更加富有成果。这些因素的确包含了背景和分析扰动之间的线性变换光谱。要成为通货膨胀函数，函数必须满足三个条件：（a）功能条件：所有折减系数必须增加，（b）无观测条件：当没有观测值同化时，分析扰动与背景扰动相同；并且（c）保持阶数的条件：膨胀的分析奇异值必须与背景奇异值具有相同的阶数。如果强加了上限条件，即膨胀的分析误差方差必须小于观察误差方差，那么所得的膨胀函数将显示为等于先前的膨胀函数，后者是背景摄动的奇异值的函数。如果我们想增加后置CI的分析增量，则此条件是必要的。事实证明，先扩张松弛法和先扰动松弛法属于线性膨胀函数的类别。在本课程中，我们还具有恒定的通货膨胀功能，乘性膨胀函数和参数可变线性膨胀函数。更有趣的是，确定性EnKF被发现属于二次膨胀函数。这个二次类介绍了一种优雅的形式，用于通过卡尔曼增益来计算分析扰动。由于高计算成本和确定自由参数的困难，通货膨胀函数的高阶多项式和非多项式形式在实践中吸引力较小。

更新日期：2020-07-03

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