# Localization Analysis of Data Assimilation Methods Coupled with Fuzzy Control Algorithms

Chang Mingheng, Bai Yulong*, Ma Xiaoyan, Meng Ruoyu, Wang Lili

College of Physics and Electrical Engineering, Northwest Normal University, Lanzhou 730070, China

First author:Chang Mingheng(1992-), male ,Qin'an County, Gansu Province, Master student. Research areas include data assimilation observation error. E-mail: changmingheng@126.com

Abstract

Due to the fake correlation between distance-observations and assimilation-states during data assimilation, more attention has been paid to the localization method. Meanwhile, in the case of assimilation with a small number of sets, the observation data is difficult to be used effectively, which makes the assimilation effect not good enough. Therefore, a new fuzzy control was proposed to analyze the local method. The fuzzy control algorithm was used to judge the distance between the observation point and the status update point and to construct the fuzzy weight of the observation position. The study aimed to make use of the nonlinear Lorenz-96 model to compare the Fuzzy control combine Local Analysis algorithm (FLETKF) and Fuzzy control combine Ensemble Transform Kalman Filter method (FETKF), local Ensemble Transform Kalman Fliter (LETKF) and Ensemble Transform Kalman Filter algorithm (ETKF) when the nonlinear forced parameter changed. In addition, the strengths and weaknesses of four algorithms were discussed by different intensities. The results show that the new method can obtain more effective observation weights, avoiding the false correlation between long-distance observations and state variables, reducing the errors caused by the observation data which is difficult to be used effectively. Under different assimilation strength, FLETKF can maintain good robustness. However, in terms of assimilation time, the construction of the equivalent weight of the observation position requires additional time because the localization assimilation method of fuzzy control determines the distance between the observation point and the status update point. Parallel computing performance needs further study.

Keywords： FLETKF ; Fuzzy Control ; Localization methods ; ETKF

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Chang Mingheng, Bai Yulong, Ma Xiaoyan, Meng Ruoyu, Wang Lili. Localization Analysis of Data Assimilation Methods Coupled with Fuzzy Control Algorithms[J]. Advances in Earth Science, 2018, 33(8): 874-883 https://doi.org/10.11867/j.issn.1001-8166.2018.08.0874

## 2 理论与算法

### 2.1 ETKF

Bishop等[9]于2001年提出ETKF方法,是针对适应性观测问题提出的一种数据同化方法,其理论推导是基于Kalman滤波理论和集合变换思想得出的,是一种次优的数据同化方法,与EnKF方法[17]不同之处在于:ETKF方法利用集合变换和无量纲化的思想,求解与观测有关的预报误差协方差矩阵[19]。ETKF方法是通过将预报扰动乘以一个变换矩阵T来得到分析扰动。如下式:

Aa=TAf ,(1)

Pa=ZfTTf(Zf)T ,(2)

Zf=Af/$m-1$,(3)

T=C[Γ+I$]-12$,(4)

### 2.2 局地化方法——LETKF

LETKF通过裁剪相关系数矩阵[23]得到,其理论部分推导公式如下:

$xia$= $xif$+ $Ki,:i$( $di$- $Hixfi$) , (5)

$Si$$Ri-1/2HiAif$/$m-1$,(6)

$Kii,:$= $Ai,:fSiT$(I+ $SiSiT$)-1$Ri-1/2$/$m-1$,(7)

$Ti$=(I+ $SiTSi$)-1/2 , (8)

$Ai.:a$= $Ai.:fTi$,(9)

### 2.3 耦合模糊控制的局地化方法

2.3.1 模糊控制方法

Fig.1   Schematic diagram of assimilation system coupled with the fuzzy control

2.3.2 观测位置信息模糊控制方法

Fig.2   Diagram of assimilation system coupled with the fuzzy control between observation position and status[22]

2.3.3 构造模糊控制过程

(1)模糊机理

(2)数据库

(3)规则库

……

……

(4)模糊规则表的建立

Rk: if distk and Vk then coeffsk k=1,2,…,40,

$V=(I1×O1)+(I2×O2) +…+(I20×O20),(10)$

Table 1   The rule table of fuzzy control

disti
coeffsi
Vi
NBNMNSZOPSPMPB
NBPBPBPMPMPSPSZO
NMPBPMPMPSPSZONS
NSPMPMPSPSZONSNS
ZOPMPSPSZONSNSNM
PSPSPSZONSNSNMNM
PMPSZONSNSNMNMNB
PBZONSNSNMNMNBNB

(5)三角隶属度函数

$f(x)=0,xc(11)$

Fig.3   The graph of triangle membership function

(6)控制量的反模糊化

$coeffs=dist\circ V$。 (12)

Table 2   The fuzzy control response table of FLETKF

△dist0~2.30~3.21.1~1.41.5~5.21.6~5.615.25~20
△coeffs1~0.9750.975~0.950.95~0.9250.925~0.950.9~0.8750.025~0

### 2.4 耦合FETKF

2.4.1 背景知识

xa=xf+K(d-Hxf),(13)

Pa=(I-KH)Pf, (14)

K=PHT(HPHT+R)-1 ,(15)

x= $1m$E1,(16)

P= $1m-1$AAT,(17)

A=E$I-1m11T$,(18)

2.4.2 模糊控制与ETKF的结合

(1)模糊机理

$g=exp-12(disti/r)2,(19)$

(2)数据库

(3)规则库

Rk:if disti and Mi then gi, i=1,2,…,20。

$M=(i1×o1)+(i2×o2) +…+(i20×o20)。(20)$

(4)控制量的反模糊化

$g=dist\circ M$。 (21)

Table 3   The fuzzy control response table of FETKF

dist0~2.50~3.51.5~22.5~4.53~5.517.35~20
g1~0.950.925~0.90.9~0.8750.875~0.850.85~0.8250.025~0

### 2.5 FLETKF和FETKF的区别

FLETKF的模糊控制过程:FLETKF是加入局地化分析的方法后得到输入集dist,同样进行三角形隶属度函数离散化,得到的离散化后的值通过模糊规则输出等价权重;然后将离散后的dist和等价的权重进行笛卡尔乘积得到的值集合称为模糊关系V;最后将输入集dist和模糊关系V进行点积,采用最大隶属度函数原则去模糊化。

FETKF的模糊控制过程:FETKF是对观测系数不需要进行局地化,直接将输入集dist通过三角形隶属度函数离散化,通过模糊规则,得到的离散化后的值经过标准正态分布函数进行计算出相应的权重;然后将离散后的dist和等价的权重进行笛卡尔乘积得到的值集合称为模糊关系M;最后将输入集dist和模糊关系M进行点积,采用最大隶属度函数原则去模糊化。

## 3 非线性数值实验

### 3.1 概述

FLETKF和FETKF算法有效利用40维的非线性混沌系统Lorenz-96验证,由于经典四阶Runge-Kutta可以获得Lorenz-96的数值解。

$dXkdt=(Xk+1-Xk-2)Xk-1-Xk+F,(22)$

### 3.2 性能指标

$RMSEa=1n(∑t=1n(x-xt)2),(23)$

### 3.3 数值实验

Lorenz-96模型为强迫耗散模型,F为控制混沌系统的强迫参数,当F>4时,其解为混沌状态[30]。改变强迫参数F的值,改变模型的非线性程度,对比4种方法在不同强迫参数下的均方根误差。为了验证4种算法的性能,在实验中,选取强迫参数F为4~12,间隔为1。

Fig.4   The influence of forcing parameters F for assimilation results

Fig.5   The variable trend of RMSE with forcing parameters F

### 3.4 4种算法在不同强度下的性能比较

kσ=σfa ,(24)

σ= $tr(HPHTR-1)$,(25)

P= $1m-1$AAT,(26)

Fig.6   The four algorithms vary with the Ensemble number when assimilation intensity R is 0.001

R=1,也就是“中”同化条件下,图7中ETKF算法随着集合数的增加表现平稳的趋势,表现出ETKF对同化效果不敏感性,相对于R=0.001,4种算法的RMSE整体较小,说明了在不同强度下,对4种算法的影响比较明显。但相对而言,FLETKF的RMSE总能小于其他3种算法,表现出较好的收敛性。

Fig.7   The four algorithms vary with the Ensemble number when assimilation intensity R is 1

R=10,也就是“弱”同化条件下,图8中ETKF算法在N>20时,呈现滤波发散状态,但在整体效果上,相对于R=0.001,4种算法的RMSE较小,同时FLETKF呈收敛趋势,表现出良好的鲁棒性。

Fig.8   The four algorithms vary with the Ensemble number when assimilation intensity R is 10

Table 4   Comparison of performance of four algorithms at different assimilation strengths

R=0.001ETKF0.0123.735
LETKF0.2793.621
FETKF6.9133.527
FLETKF7.823.237
R=1ETKF0.0124.397
LETKF0.2754.03
FETKF7.0123.971
FLETKF132.983
R=10ETKF0.0124.616
LETKF0.2694.021
FETKF8.7233.985
FLETKF12.53.912

### 3.5 FETKF与FLETKF的RMSE值减小百分比性能比较

Table 5   The percentage reduction of RMSE of two algorithms

R=0.001R=1R=10
FLETKF3.2372.9833.912
FETKF3.5273.9713.985

## 4 结 论

The authors have declared that no competing interests exist.

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