﻿ 一种基于鲁棒集合滤波的资料同化方法
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 高原气象  2017, Vol. 36 Issue (4): 1052-1059  DOI: 10.7522/j.issn.1000-0534.2016.00072 0

### 引用本文 [复制中英文]

[复制中文]
Bai Yulong, Zhang Zhuanhua, You Yuanhong, et al. 2017. A New Data Assimilation Method Based on Robust Ensemble Filter[J]. Plateau Meteorology, 36(4): 1052-1059. DOI: 10.7522/j.issn.1000-0534.2016.00072.
[复制英文]

### 文章历史

1 引言

2 资料同化方法

 ${x_i} = {M_{i,i - 1}}\left( {{x_{i - 1}}} \right) + {u_i},$ (1)
 ${y_i} = {H_i}\left( {{x_i}} \right) + {v_i},$ (2)

2. 1 集合转换卡尔曼滤波(ETKF)

(1) 计算第i时刻的预报值及预报协方差矩阵

 $x_{i,j}^b = {M_{i,i - 1}}\left( {x_{i - 1,j}^a} \right),$ (3)
 $\bar x_i^b = \frac{1}{n}\sum\limits_{j = 1}^n {x_{i,j}^b} ,$ (4)
 $X_i^b = \frac{1}{{\sqrt {n - 1} }}\left[ {x_{i,1}^b - \bar x_i^b, \cdots ,x_{i,n}^b - \bar x_i^b} \right],$ (5)
 $P_i^b = X_i^b{\left( {X_i^b} \right)^T},$ (6)

(2) 计算第i时刻的分析值及分析协方差矩阵

 ${K_i} = P_i^bH_i^T{\left( {{H_i}P_i^bH_i^T + {R_i}} \right)^{ - 1}},$ (7)
 $\bar x_i^a = \bar x_i^b + {K_i}\left( {{y_i} - {H_i}\bar x_i^b} \right),$ (8)
 $X_i^a = X_i^b{T_i}{U_i},$ (9)
 $P_i^a = X_i^a{\left( {X_i^a} \right)^T},$ (10)

 $x_{i,j}^a = \bar x_i^a + \sqrt {n - 1} {\left( {X_i^a} \right)_j},$ (11)

2. 2 时间局地化的H滤波(TLHF)

HF(Simon, 2006)是鲁棒滤波的一种, 其解不需要对模型和观测误差作任何的假设。无论模型误差和观测误差的大小, 只要HF范数小于预先设定的正值$\frac{1}{\gamma }$, HF估计器的性能就能有所保证。利用最小最大准则计算在背景误差、模型误差和观测误差这些不确定条件下的估计值。

 $J_{x,i}^{HF} = \frac{{\left\| {{x_i} - x_i^a} \right\|_{{S_i}}^2}}{{\left\| {{x_i} - x_i^b} \right\|_{{{\left( {\Delta _i^b} \right)}^{ - 1}}}^2 + \left\| {{u_i}} \right\|_{Q_i^{ - 1}}^2 + \left\| {{v_i}} \right\|_{R_i^{ - 1}}^2}},$ (12)
 $\left\| {{x_i} - x_i^a} \right\|_{{S_i}}^2 \le \frac{1}{{{\gamma _i}}}\left( {\left\| {{x_i} - x_i^b} \right\|_{{{\left( {\Delta _i^b} \right)}^{ - 1}}}^2 + \left\| {{u_i}} \right\|_{Q_i^{ - 1}}^2 + \left\| {{v_i}} \right\|_{R_i^{ - 1}}^2} \right),$ (13)

 $\frac{1}{{{\gamma _i}}} > \frac{1}{{\gamma _i^ * }} \equiv \mathop {\inf }\limits_{x_i^a} \mathop {\sup }\limits_{{x_i},{u_i},{v_i}} J_{x,i}^{HF},$ (14)

TLHF的滤波过程如下:

 $x_i^b = {M_{i,i - 1}}x_{i - 1}^a,$ (15)
 $\Delta _i^b = {M_{i,i - 1}}\Delta _{i - 1}^aM_{i,i - 1}^T + {Q_i},$ (16)

 $x_i^a = x_i^b + {G_i}\left( {{y_i} - {H_i}x_i^b} \right),$ (17)
 ${\left( {\Delta _i^a} \right)^{ - 1}} = {\left( {\Delta _i^b} \right)^{ - 1}} + {\left( {{H_i}} \right)^T}{\left( {{R_i}} \right)^{ - 1}}{H_i} - {\gamma _i}{S_i},$ (18)
 ${G_i} = \Delta _i^a{\left( {{H_i}} \right)^T}{\left( {{R_i}} \right)^{ - 1}},$ (19)

 ${\left( {\Delta _i^a} \right)^{ - 1}} = {\left( {\Delta _i^b} \right)^{ - 1}} + {\left( {{H_i}} \right)^T}{\left( {{R_i}} \right)^{ - 1}}{H_i} - {\gamma _i}{S_i} \ge 0.$ (20)

2.3 集合时间局地化的H滤波(EnTLHF)

γiSi=α(Pib)-1时:

 ${\left( {\Delta _i^a} \right)^{ - 1}} = \left( {1 - \alpha } \right){\left( {P_i^b} \right)^{ - 1}} + {\left( {{H_i}} \right)^T}{\left( {{R_i}} \right)^{ - 1}}{H_i},$ (21)

γiSi=α[(Δib)-1+(Hi)T(Ri)-1Hi]=α(Pia)-1时:

 ${\left( {\Delta _i^a} \right)^{ - 1}} = \left( {1 - \alpha } \right){\left( {P_i^a} \right)^{ - 1}},$ (22)

Si=Im时:

 ${\left( {\Delta _i^a} \right)^{ - 1}} = {\left( {X_i^b{C_i}} \right)^{ - T}}\left( {{\mathit{\Gamma }_i} + I} \right){\left( {X_i^b{C_i}} \right)^{ - 1}} - {\gamma _i}{I_m}.$ (23)

 $\begin{array}{l} {\left( {\Delta _i^a} \right)^{ - 1}} = {\left( {X_i^b{C_i}} \right)^{ - T}}\left( {{\mathit{\Gamma }_i} - \alpha {\sigma _{i,n - 1}}{I_m}} \right){\left( {X_i^b{C_i}} \right)^{ - 1}}\\ \;\;\;\;\;\;\;\;\;\; = {\left( {X_i^b{C_i}} \right)^{ - T}}{\rm{diag}}\left( {{\sigma _{i,1}} - \alpha {\sigma _{i,n - 1}}, \cdots ,} \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\left( {1 - \alpha } \right){\sigma _{i,n - 1}}} \right){\left( {X_i^b{C_i}} \right)^{ - 1}}. \end{array}$ (24)

3 数值实验 3.1 Lorenz-96高维混沌系统

Lorenz-96系统(Lorenz and Emanuel, 1998)是一个由微分方程主导的二阶非线性动力系统, 即:

 $\frac{{{\rm{d}}{X_k}}}{{{\rm{d}}t}} = \left( {{X_{k + 1}} - {X_{k - 2}}} \right){X_{k - 1}} - {X_k} + F,$ (25)

3.2 观测系统

 ${y_i} = H\left( {{x_i}} \right) + {v_i},$ (26)

3.3 实验设计

 $RMSE = \frac{{\left\| {{X^t} - {{\bar X}^a}} \right\|}}{{\sqrt m }},$ (27)

4 数值实验结果及分析

4.1 背景协方差放大实验

α=0. 4, 强迫参数F分别为6, 8, 9时, 图 1给出了ETKF与EnTLHF的均方根误差(RMSE)均值(20次实验取平均)变化图。由图 1可知, ETKF方法中估计误差没有明显的减少。然而, 采用基于鲁棒集合滤波理论的EnTLHF资料同化方法后, 随着同化演进, 估计误差及误差波动明显小于ETKF方法, 说明采用EnTLHF方案得到的分析值精度高。与图 1b1c相比较, 图 1a中EnTLHF方法的估计误差均小于1;随着强迫参数F的增加, 即模拟增加模型误差, ETKF方法和EnTLHF方法的估计误差都增大, 但EnTLHF方案的性能较为优良, 验证了鲁棒集合资料同化方法的有效性, 同时说明EnTLHF方法对不同程度的模型误差具有较好的鲁棒稳健性。

 图 1 驱动参数F=6(a)、8(b)和9(c)时, ETKF与背景协方差放大条件下EnTLHF的RMSE均值 Figure 1 Time mean RMSE of ETKF and EnTLHF in the inflation condition of background covariance when the values of the forcing parameters of F are 6 (a), 8 (b) and 9 (c)

4.2 分析协方差放大实验

 图 2 驱动参数F=6(a)、8(b)和9(c)时, ETKF与分析协方差放大条件下EnTLHF的RMSE均值 Figure 2 Time mean RMSE of ETKF and EnTLHF in the inflation condition of analysis covariance when the values of the forcing parameter of F are 6 (a), 8 (b) and 9 (c)

4.3 转移矩阵放大实验

 图 3 在驱动参数F=6(a)、8(b)和9(c)及不同的α值下, 转移矩阵特征值放大条件下EnTLHF方法的RMSE时间均值 Figure 3 Time mean RMSE of EnTLHF in the inflation condition of transform matrix eigenvalues as PLC changes when the values of the forcing parameter of F are 6 (a), 8 (b) and 9 (c)
5 结论

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A New Data Assimilation Method Based on Robust Ensemble Filter
BAI Yulong , ZHANG Zhuanhua , YOU Yuanhong , LIU Yingjuan
College of physics and electronic engineering, Northwest Normal University, Lanzhou 730070, China
Abstract: The background error covariance matrix based on properties of the ensemble prediction statistics play an important role in the ensemble Kalman filter data assimilation. However, data assimilation divergence occurs from the inaccurate estimate of the covariance matrix and the limited ensembles. In this study, based on an ensemble time-local H-infinity filter which inflates the eigenvalues of the analysis error covariance matrix, a new data assimilation filter method is proposed, referred to as the inflation transform matrix eigenvalues algorithm, in order to improve properties of the estimation. The properties of data assimilation is improved in the framework of ensemble filters according to the min-max criterion of robust filtering theory. Using the nonlinear Lorenz-96 chaos system, we investigate how the ensemble time-local H-infinity filter methods impacts the robustness of the assimilation systems under the selected change conditions, such as initial background conditions, force parameters, and performance level coefficients. It is show that the ensemble time-local H filter has good robustness to the change of above parameters. Compared with traditional filter methods, robust filter methods can improve the assimilation effect.
Key Words: Ensemble transfer Kalman filter    Time-local H filter    Lorenz-96 chaos system    Robustness