基于无迹/群组变换变分滤波的最新进展

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本文是一篇发表在Elsevier期刊上的研究论文，名为“Unscented/ensemble transform-based variational filter”。该论文主要关注于一种基于 Unscented（不香料）滤波方法和Ensemble Transform技术的变分滤波算法。这两种方法在气象学、地球观测以及相关领域的数据处理和预测中具有重要意义，特别是在处理高维非线性系统中的不确定性估计和优化问题。 Unscented滤波是一种用于卡尔曼滤波的改进方法，它通过设计一组近似的粒子样本集合来近似高斯分布，从而减少滤波过程中对协方差矩阵精确性的依赖。这种方法在处理非线性和非高斯噪声时展现出优越性，因为它能够更准确地估计系统的状态并减小滤波过程中的误差。 Ensemble Transform方法则是一种将随机扰动从一个分布转换到另一个分布的技术，通常用于数据同化（如大气模型中的资料同化）。这种转换有助于保持样本之间的相互独立性，同时在保持统计特性的同时改善了数值稳定性。结合这两种方法，变分滤波器能够在处理大规模复杂系统时提供更为精确和稳定的解决方案。文章作者Ming Le和Christophe Baehre分别来自上海交通大学航天航空学院和法国国家气象研究中心，他们可能在这篇论文中探讨了如何将这些滤波技术应用于实际气象预测模型，例如气候模拟、风速预测或天气模式分析。他们的工作不仅有助于提升气象预报的精度，还可能为其他领域，如遥感和海洋观测，提供新的数据处理策略。值得注意的是，根据版权政策，此篇文章仅供作者内部非商业研究和教育用途，如教学和与同事分享，不可进行复制、分发、销售或发布到个人、机构或第三方网站。作者如果需要了解关于Elsevier存档和稿件政策的更多信息，可以访问http://www.elsevier.com/copyright。这篇论文提供了深入理解如何在实际应用中使用Unscented滤波和Ensemble Transform技术来优化变分滤波算法的重要见解，对于那些关注数据分析和模型预测的科研人员来说，具有很高的参考价值。

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2 M. Lei, C. Baehr / Physica D 246 (2013) 1–14

Unscented Kalman Filter (EnUKF). The EnUKF is then able to treat

some high dimensional systems with a variable sample size, which

depend on the possible rank reduction and a pre-computed sam-

ple covariance. Although there exists a debate on the combination

method [20,21], the numerical results are convincing.

Our work takes place at this sensitive point. For all the KF-like

filters the estimate is assumed to be a linear regression with the

observation (see Section 2.2.1 for details). This is the core of KF

but this is hard to accept for general situations especially for a

dynamics with strong nonlinearities and non-Gaussian perturba-

tions. In a certain sense it becomes a bottleneck for some complex

and large dimension applications. Therefore we develop an assimi-

lation method with an update step that is based on variational min-

imization instead of the Kalman regression. Naturally the linear

regression is replaced by quadratic terms of observation. First a UT

is applied for the sample generation and the statistical mean es-

timation. Then an Ensemble Transform (ET) is used to empirically

compute the error covariance. To deal with the high dimensional-

ity, we incorporate a modified rank reduction into the UT scheme

in order to maintain a size-reduced ensemble generation. There-

fore the new scheme is called the Unscented/Ensemble transform-

based Variational Filter (UEVF). This filter is able to deal with the

nonlinear and chaotic systems with high dimensionality as the nu-

merical experiments show in Section 5.

It is clear that the suggested method, outlined in Section 3, is at

the junction of different technologies: The UT approach in order to

give an evolution to the error covariance matrices, the ensemble

transform to implement the covariance update, the variational

minimization to get a state update for a nonlinear dynamics, the

rank reduction to achieve a significant dimension reduction with

an adaptive fashion.

This paper is divided into 6 sections. In Section 2, we introduce

the background methodologies, including three update schemes.

In Section 3, we derive the UEVF and introduce the variational

minimization update, and the sigma-points generation with a size-

reduced dynamics. In Section 4, three implementation issues about

the UEVF are discussed. First we present the rank-reduced co-

variance approximation, then the conjugate-gradient method to

minimize the cost-function, and finally the computation of the

background error matrix. The Section 5 concerns the numerical ex-

periments. There are two examples, the chaotic high dimensional

Lorenz-95 model and a simulation based on the 2D shallow water

equation. Finally the conclusion and a discussion are provided in

Section 6.

2. Background methods

We assume that we have an n-dimensional discrete dynamical

system

k+1

= M

k,k+1

) + u

(1)

and to observe the state x

we could (partly) obtain a measurement

sequence y

using the observer H

: R

× R

→ R

, i.e.,

= H

) + v

, k = 1, 2, . . . (2)

where k ∈ [0, T] is the discrete-time index and T the total time

step, x

∈ R

and y

∈ R

are the state and the noisy observa-

tion at the time k, M

k,k+1

is the nonlinear transition operator with

k,k+1

: R

× R

→ R

. u

∈ R

is the additive dynamical noise

with a zero mean and non null covariance Q

, which includes both

the ordinary process noise and mismatch errors of the mathemat-

ical model. We assume that u

is independent of the additive ob-

servation noise v

∈ R

with a zero mean and known covariance

. u

is also supposed to be independent of the initial state x

The problem of data assimilation is equivalent to estimating the

Probability Density Function (PDF) η

= pdf(x

1:k

). Two PDF’s

are used to describe it completely: the first is the predictor PDF

using the Markov transition given by the model, η

= pdf(x

k−1

the second is the update PDF η

= pdf(x

, y

1:k

). Finding these

two PDF’s solves the assimilation or the filtering problem.

Actually the update process does not have a unique representa-

tion as a Markov transition. The transformation η

may be repre-

sented by different transportation processes that are equivalent in

mean. This can be seen with the Feynman–Kac formulas [22].

2.1. Unscented transform

The scheme of unscented transform (UT) is designed to solve

the following estimation problem [19,23,24]: at time k − 1, we

have a Gaussian random variable x

k−1

∈ R

with mean x

k−1

and

covariance P

xx,k−1

, and Gaussian perturbations u

k−1

with zero-

mean and covariance Q

k−1

, which are assumed to be independent

of each other. Without loss of generality, we can apply a nonlin-

ear transition M on the x

k−1

to obtain a new random variable

= M(x

k−1

The interesting thing is to estimate the mean and

covariance of the transformed random variable x

The UT generates a set of 2n +1 state points {X

k−1,i

}

i=0

, which

are called sigma-points and defined by

k−1,i

}

i=0



k−1

, x

k−1





(n + λ)P

xx,k−1



i = 1, . . . , n



, (3)

where (



(n + λ)P

xx,k−1

)

denotes the i-th column of the square

root matrix



(n + λ)P

xx,k−1

. Incidentally in the implementation,

by using P

xx,k−1

= S

x,k−1

)

, a n ×n square root matrix S

x,k−1

can be obtained with a numerical decomposition algorithm. There-

fore (



(n + λ)P

xx,k−1

)

is rewritten as

√

n + λ



x,k−1



where λ

is a constant used for scaling adjusting and n is the dimension of

state x

Associated to the sigma-points, a set of weights {W

k−1,i

}

i=0

allocated by

k−1,0

n + λ

k−1,i

2(n + λ)

, i = 1, . . . , 2n.

(4)

It can be proved that the weighted sample mean

k−1

and

sample covariance

xx,k−1

of the finite set {X

}

i=0

perfectly match

the mean x

k−1

and the covariance P

xx,k−1

of x

k−1

respectively,

k−1



i=0

k−1,i

= x

k−1

, (5a)

xx,k−1



i=0

k−1,i

− x

k−1

)(X

k−1,i

− x

k−1

)

= P

xx,k−1

. (5b)

The above identities are independent of the choice of parameters

λ and n [23] and this feature will be employed to implement

a strategy of PDF re-approximation in construction of UEVF in

Section 3.

Being different from the dynamical equations (1)(2), we can assume that x

k−1

and u

k−1

are coupled together by x

= M(x

k−1

, u

k−1

). This expression can always

be converted into the simpler form x

= M(z

k−1

) introducing the augmented state

k−1

= [x

k−1

, u

k−1

]

, where T denotes the transpose operation. Consequently the

covariance of augmented state z

k−1

is denoted P

zz,k−1

= diag



xx,k−1

, Q

k−1



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