A Fuzzy-Logic-Based Covariance Localization Method in Data Assimilation
Abstract
:1. Introduction
2. Assimilation Background
2.1. Ensemble Kalman Filter
2.2. Traditional Covariance Localization (CL)
2.3. New Improved Covariance Localization Method: The CF Method
Algorithm 1: A cycle of the EnKF assimilation method with localization algorithm |
Require: is the assimilation step; initial background field matrix: , , , A, R; allocate space for observation data: y, HE, pos. m is the ensemble size. n is the state dimension. A is the ensemble anomalies. s is the scaled innovation vector. S is the scaled ensemble observation anomalies. dx is the ensemble mean correction. 1: for step = 1,… do 2: E = rk4step (@L40, dt, x, F) 3: 4: 5: 6: 7: 8: Localization: Algorithm 2 9: 10: 11: , and , 12: 13: 14: 15: 16: 17: 18: end for |
Algorithm 2: Get localization coefficients coupled with fuzzy logic control algorithm |
Require:d is the Euclidean distance between the observation point and the state update point. is the weight coefficient. and represent the center and width of the Gaussian membership function, respectively. is the position of the observation points. is the position of the state update points. nx is the model dimension. is the localization radius. is the scale range, . L is the fuzzy correlation coefficient matrix. is the so-called localization correlation matrix. denotes the elementwise product of matrices. /* abs is the absolute value function; hypot is the square root of sum of squares (hypotenuse). Gfuzzy refers to the local functions with fuzzy logic control; Gauss refers to the traditional Gaussian localization function; Gaspari_Cohn refers to the traditional Gaspari–Cohn (GC) localization function.*/ 1: if then 2: Lorenz 96: 3: else 4: QG model: 5: end if 6: switch tag 7: Case ‘Gfuzzy’ 8: 9: 10: 11: Case ‘Gauss’ 12: 13: Case ‘Gaspari_Cohn’ 14: if then 15: 16: else & 17: 18: … 19: end switch 20: 21: 22: 23: |
- (1)
- Create a fuzzy logic database
- (2)
- Fuzzy quantification
- (3)
- Define the membership function
- (4)
- Establish fuzzy control rules
- (5)
- Fuzzy inference and defuzzification
- (6)
- Model state updating
3. Configuration of Numerical Experiments
3.1. Experimental Model
- The Lorenz-96 model
- b
- The quasi-geostrophic (QG) model
3.2. Performance Index
- (1)
- The state variables and are generally obtained from the numerical solutions of differential equations. The RMSE of the analysis of the ensemble mean, at the specific time k, is defined as follows:
- (2)
- The power spectral density (PSD) is a probabilistic statistical value, the physical significance of which lies in the measure of the mean square value of random variables. Specifically, the area under the relation curve of the PSD–frequency value represents the variance, whereby a smaller area denotes better performance of the assimilation algorithm.
- (i)
- Given the forecast ensemble , the ensemble mean and the forecast ensemble anomalies are calculated; then, the forecast covariance matrix is calculated using Equation (6).
- (ii)
- The analysis ensemble is calculated using the Kalman analysis (see Equation (3)). Then, the analysis covariance matrix is calculated using Equation (5).
- (iii)
- The anomalies in the analysis ensemble and the Kalman gain are calculated.
- (iv)
- The eigenvalue of the PSD is calculated using the fast Fourier transform of the anomalies matrix for all ensembles , the analyzed anomalies , and the forecast anomalies .
3.3. Experimental Design and Preliminary Results
3.3.1. Influences of CL and CF on the Kalman Gain Matrix
3.3.2. Comparison of the Influence on the Two Localization Methods with Different Model Errors
4. Localization Behavior Using the Lorenz-96 Model
4.1. Change in Ensemble Numbers
4.2. Change in Covariance Inflation Factors
4.3. Change in Localization Radius
4.4. Performance Index PSD
5. Localization Behavior Using the Quasi-Geostrophic (QG) Model
5.1. Comparison of Assimilation Performance of Two Localization Methods
5.2. Comparison of Spatial Distribution Characteristics between Two Localization Methods
6. Conclusions
- Effectiveness of the new algorithm
- b.
- Sensitivity and robustness of the new algorithm in experimental settings
- c.
- Applicability in real weather models
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Bai, Y.; Ma, X.; Ding, L. A Fuzzy-Logic-Based Covariance Localization Method in Data Assimilation. Atmosphere 2020, 11, 1055. https://doi.org/10.3390/atmos11101055
Bai Y, Ma X, Ding L. A Fuzzy-Logic-Based Covariance Localization Method in Data Assimilation. Atmosphere. 2020; 11(10):1055. https://doi.org/10.3390/atmos11101055
Chicago/Turabian StyleBai, Yulong, Xiaoyan Ma, and Lin Ding. 2020. "A Fuzzy-Logic-Based Covariance Localization Method in Data Assimilation" Atmosphere 11, no. 10: 1055. https://doi.org/10.3390/atmos11101055