1.安徽省气象信息中心 安徽省大气科学与卫星遥感重点实验室, 安徽 合肥 230031
2.中国气象局沈阳大气环境研究所, 辽宁 沈阳 110000
3.安徽省气候中心, 安徽 合肥 230031
4.国家气象中心, 北京 100081

Discontinuous Data 3D/4D Variation Fusion Based on the Constraint of L1 Norm Regularization Term
Wang Gen1,2, Sheng Shaoxue1, Liu Huilan1, Wu Rong3, Yang Yin4
1.Anhui Meteorological Information Centre Anhui Key Laboratory of Atmospheric Science and Satellite Remote Sensing, Hefei 230031, China
2.The Institute of Atmospheric Environment, China Meteorological Administration, Shenyang 110000, China
3.Anhui Climate Center, Hefei 230031, China
4.National Meteorological Center of China, Beijing 100081, China

First author:Wang Gen(1983-),male,Taizhou City, Jiangsu Province, Engineer. Research areas include satellite data assimilation, numerical simulation of GRAPES and multi-source data fusion.E-mail:203wanggen@163.com

Abstract

Classical 3D/4D variation fusion is based on the theory that error follows Gaussian distribution. When using minimization iteration, the gradient of objective function is involved, and the solution of which requires the continuity of data. This paper adopted the extended classical 3D/4D variation fusion method, and explicitly applied the prior knowledge, which was based on L1-norm, as regularization constraint to the classical variation fusion method. Original data was firstly projected into the wavelet domain during the implementation process, and new fusion model was adopted for data fusion in wavelet space, then inverse wavelet transform was used to project the result to the observation space. Ideal experiment was carried out by using linear advection-diffusion equation as four-dimensional prediction model, which made a hypothesis of the discontinuity with the data between background and observation, and that meant the derivatives between left and right were not equal on some points. The result of the experiment showed that the method adopted here was practicable. A further research was also done for multi-source precipitation fusion. Firstly, CMORPH inversion precipitation data were corrected through PDF (Probability Density Function, PDF) matching method based on GAMMA fitting function. Then corrected data was fused with the observation one. By comparison with the reference field, the result showed that this method can keep some outliers better, which might represent certain weather phenomenon. The L1-norm regularization variation fusion in this paper provided a possible way to deal with discrete data, especially for jump point.

Keyword: Discrete data; Variation fusion; L1-norm; Regularization term; Wavelet space.
1 引言

2 经典3DVar/4DVar变分融合理论分析

yi=H(xi)+vi (1)

3DVar/4DVar变分融合归结为在初始时刻极小化以下目标泛函[13, 22]:

J3D, 4D(x0, x1, …, xk)= $∑i=0k12‖yi-H(xi)‖Ri-12$+ $12$$x0b$-x0 $‖B-12$(2)

xi=M0, i(x0), i=0, …, k (3)

J4D(x0)= $∑i=0k12‖yi-HM0, ix0‖Ri-12$+ $12‖x0b-x0‖B-12$(4)

$H¯$= $HM0, 0T, …, (HM0, k)TT$;

$R¯$= $R00…00R1…⋮⋮⋮ 00…0Rk$

J4D(x0)

= $12$$y¯$- $H¯$x0 $‖R¯-12$+ $12$$x0b$-x0 $‖B-12$(5)

$x0a$=( $H¯TR¯-1H¯$+B-1)-1( $H¯TR¯-1y¯$+B-1 $x0b$) (6)

3 基于小波域的资料“ 不连续” 变分融合理论分析

$x0a$= $argminx0${JR4D(x0)+λ τ (x0)} (7)

xp=(∑ |xi|p)1/p(p> 0)。

$x0a=argminx0{JR4D(x0)}Φx0‖pp≤c8$

$x0a$= $argminx012‖y¯-H¯x0‖R¯-12$+ $12$$x0b$-x0 $‖B-12$Φ x0 $‖pp$(9)

λ 为拉格朗日乘子, 也称正则化参数, 公式(9)中要求λ 非负[27]。当λ =0时, 公式(9)变成了经典变分融合法。较小的λ 值能够减小先验知识对结果的影响; 较大的λ 值能够引入更多的先验知识。因此λ 起着重要的平衡作用, 能同时保持足够地接近观测和背景状态。通常λ 参数的确定是通过统计交叉验证获得[27]

$x0a$= $argminx012‖y¯-H¯x0‖R¯-12$+ $12$$x0b$-x0 $‖B-12$Φ x01 (10)

$x0a=argminx0{‖Φx0‖1}JR4D(x0)≤c11$

4 基于线性平流扩散方程的“ 不连续” 4DVar变分融合理想试验
4.1 线性平流扩散方程及变分融合观测算子介绍

4.1.1 4DVar预报模式— 线性平流扩散方程

$∂x(s, t)∂t+a(s, t)∇x(s, t)=θ∇2x(s, t)x(s, 0)=x0(s)$(12)

$D(s, t)=(4πθt)-1/2exp-|s|24θt$（13）

$A(s-at)=1, s=at0, otherwise$（14）

$xi=A0, iD0, ix015$

4.1.2 变分融合观测算子介绍

H= $181111111100000000…000000000000000011111111…00000000⋮⋮ ⋮0000000000000000…11111111$Rn× m(16)

4.2 变分融合误差协方差和观测值介绍

4.2.1 观测和背景误差协方差介绍

Cb= $ρ0ρ1…ρ(m)ρ1ρ0…⋮⋮… ρ1ρ(m)…ρ1ρ0$Rm× m

4.2.2 背景和观测值介绍

$x0b=x0t+∑iεiζi12Li$（17）

 Figure Option 图1 构建“ 真实状态” (a)分段函数(记为PF); (b)二次函数(记为QF); (c)正弦函数(记为SF); (d)平方指数函数(记为SEF)Fig.1 Building true state (a) Piecewise Function (PF); (b) Quadratic Function (QF); (c) Sine Function (SF); (d) Square Exponential Function (SEF)

y=H(M( $x0t$))oR1/2 (18)

4.3 “ 不连续” 变分融合理想试验

MSEr=$x0t$- $x0a$2/‖ $x0t$2

MAEr=$x0t$- $x0a$1/‖ $x0t$1

BIASr=$(x0t-x0a)¯$‖ /‖ $x¯0t$‖ (19)

 Figure Option 图2 经典变分融合和耦合L1范数正则化的融合效果对比分析Fig.2 Fusion effect analysis of classic variational fusion and coupling L1 norm regularization method

5 基于“ 不连续” 降水资料3DVar变分融合试验

5.1 CMORPH反演降水资料订正介绍

5.2 多源降水资料融合试验

SSIM(u, v)= $2u¯v¯+c1)(2(u-u̅)(v-v̅)¯+c2)(u¯2+v¯2+c1)((u-u̅)2¯+(v-v̅)2¯+c2)$(20)

 Figure Option 图3 代表站累积概率匹配图和安徽国家站偏差分布图(a)CMORPH和雨量计累积概率匹配; (b)CMORPH降水资料对应的国家站偏差值Fig.3 Cumulative probability matching of representative station and bias distribution of national station in Anhui Province (a) Cumulative probability matching of CMORPH and rain gauge; (b) Bias value of CMORPH of corresponding station

 Figure Option 图4 不同正则化参数的融合降水场与参考场的降水分布结构对比Fig.4 Comparison structure of precipitation fusion based on different regularization parameter and reference field

 Figure Option 图5 参数不同值融合场与参考场和CMORPH的结构相似性SSIM对比分析Fig.5 Comparative analysis of SSIM between fusion field based on different value of parameters and reference field and CMORPH

6 总结与展望

(1) 文中将背景资料特点(如不连续性)作为L1范数正则项耦合到3DVar/4DVar模型中。通过线性平流扩散方程作为预报模式, 平滑滤波和采样作为观测算子, 构建4个存在“ 拐点” 的试验, 得到文中使用的方法对这些点的融合具有较好的“ 保真性” 。经典4DVar在这些点处出现了“ Gibbs现象” 或者过度平滑了这些点的信息。由此试验得到此方法在融合一些存在天气现象的“ 离群点” 具有较好的效果。

(2) 在理论分析基础上开展了4组独立试验, 并采用相对均方误差、相对绝对误差均值和相对偏差进行度量, 得到文中使用的方法整体优于经典4DVar。

(3) 在多源降水资料融合时, 基于GAMMA拟合函数PDF法进行CMORPH反演降水资料订正。对比分析得到CMORPH反演降水资料整体比台站观测偏小。

(4) 将经典变分融合耦合L1范数正则项方法用于CMORPH反演降水和台站观测降水融合中。通过结构相似性SSIM度量不同正则化参数λ 值融合结果与参考场、背景场CMORPH的结构相似性, 得到该方法能较好地把背景场中的降水分布结构信息引入到模型, 并表现在融合场中。其融合结果与被融合资料数值量级相当, 而参考场量级偏小。

The authors have declared that no competing interests exist.