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The eddy viscosity is an important parameter in the atmospheric boundary layer meteorology, and we usually cannot determine their exact values by direct measurements, but we can only obtain an approximate range by indirect approximate method. In this paper, the eddy viscosity in the stochastic general Ekman momentum approximation model is used for the retrieval research and uncertainty analysis. The main purpose of retrieval is to reduce the uncertainty and narrow the approximate range of eddy viscosity. First, the polynomial chaos-ensemble Kalman filter and the wind observations are used for eddy viscosity retrieval and uncertainty reduction. The main idea of this method is to replace the Monte-Carlo method with polynomial chaos in the uncertainty quantification of ensemble Kalman filter, and thusavoiding the consumption of computing resources brought by massive samples. The goal of uncertainty quantification is to investigate the effect of uncertainty in the eddy viscosity on the model and to subsequently provide a reliable distribution of simulation results. Then two numerical experiments are implemented, i.e. experiment I in which the eddy viscosity is assumed to be constant, and experiment Ⅱ in which the eddy viscosity is assumed to be a vertically varying random parameter. The uncertainty of eddy viscosity in experiment I is reduced quickly, at the same time the mean of eddy viscosity can converge to a reference value. The effect in experiment Ⅱ is also remarkable after 16 data assimilation steps. These results show that the polynomial chaos-ensemble Kalman filter is an effective and fast method of solving the posterior distribution of eddy viscosity and reducing the uncertainty of eddy viscosity. Finally, we calculate the prior distribution of wind speed according to the prior distribution of eddy viscosity and identify the heavy uncertainty area in wind speed. The results indicate that the posterior distribution of eddy viscosity solved with wind observations in the big uncertainty area is more accurate, which provides an important guidance for selecting the location of observation points.
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Keywords:
- polynomial chaos expansion /
- ensemble Kalman filter /
- stochastic general Ekman momentum approximation model /
- uncertainty
[1] Baklanov A, Grisogono B, Bornstein R, Mahrt L, Zilitinkevich S, Taylor P, Larsen S, Rotach M, Fernando H 2011 Bull. Am. Meteorol. Soc. 92 123
[2] Tan Z M, Wang Y 2002 Adv. Atmos. Sci. 19 266
[3] Han Y Q, Zhong Z, Wang Y F, Du H D 2013 Acta Phys. Sin. 62 049201 (in Chinese) [韩月琪, 钟中, 王云峰, 杜华栋 2013 物理学报 62 049201]
[4] Xiu D 2010 Numerical Methods for Stochastic Computations:A Spectral Method Approach (Princeton:Princeton University Press) p126
[5] Li W X, Lin G, Zhang D X 2014 J. Comput. Phys. 258 752
[6] Yan B, Huang S X 2014 Chin. Phys. B 23 109402
[7] Metropolis N, Rosenbluth A W, Rosenbluth M N, Teller A H, Teller E 1953 J. Chem. Phys. 21
[8] Geman S, Geman D 1984 IEEE Trans. Pattern Anal. Mach. Intellig. 20 721
[9] Leng H Z, Song J Q, Cao X Q, Yang J H 2012 Acta Phys. Sin. 61 070501 (in Chinese) [冷洪泽, 宋君强, 曹小群, 杨锦辉 2012 物理学报 61 070501]
[10] Kalman R E 1960 J. Basic Engin. Trans. 82 35
[11] Gelb A 1974 Applied Optimal Estimation (Cambridge:MIT Press)
[12] Evensen G 2006 Data Assimilation:The Ensemble Kalman Filter (New York:Springer) p38
[13] Evensen G 1994 J. Geophys. Res.:Oceans 99 10143
[14] Ghanem R G, Spanos P D 1992 Stochastic Finite Element:A Spectral Spproach (New York:Springer) p214
[15] Xiu D, Karniadakis G S 2003 J. Comput. Phys. 187 137
[16] Schoutens W 2000 The Askey Scheme of Orthogonal Polynomials. In:Stochastic Processes and Orthogonal Polynomials (New York:Springer) pp1-13
[17] Sun N Z, Sun A 2015 Model Uncertainty Quantification. In:Model Calibration and Parameter Estimation:ForEnvironmental and Water Resource Systems (New York:Springer) pp407-458
[18] Isukapalli S S, Roy A, Georgopoulos P G 1998 Risk Anal. 18 351
[19] Tatang M A, Pan W, Prinn R G, McRae G J 1997 J. Geophys. Res. -Atmos. 102 21925 doi:101029/97jd01654
[20] Wang Y P, Cheng Y, Zhang Z Y, Lin G 2018 Math. Model. Nat. Phenom. 13 doi:101051/mmnp/2018023
[21] Whitaker J S, Hamill M 2003 Mon. Weather. Rev. 130 1913
[22] Matre O P L, Knio O M 2010 Spectral Methods for Uncertainty Quantification (Netherlands:Springer) p536
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[1] Baklanov A, Grisogono B, Bornstein R, Mahrt L, Zilitinkevich S, Taylor P, Larsen S, Rotach M, Fernando H 2011 Bull. Am. Meteorol. Soc. 92 123
[2] Tan Z M, Wang Y 2002 Adv. Atmos. Sci. 19 266
[3] Han Y Q, Zhong Z, Wang Y F, Du H D 2013 Acta Phys. Sin. 62 049201 (in Chinese) [韩月琪, 钟中, 王云峰, 杜华栋 2013 物理学报 62 049201]
[4] Xiu D 2010 Numerical Methods for Stochastic Computations:A Spectral Method Approach (Princeton:Princeton University Press) p126
[5] Li W X, Lin G, Zhang D X 2014 J. Comput. Phys. 258 752
[6] Yan B, Huang S X 2014 Chin. Phys. B 23 109402
[7] Metropolis N, Rosenbluth A W, Rosenbluth M N, Teller A H, Teller E 1953 J. Chem. Phys. 21
[8] Geman S, Geman D 1984 IEEE Trans. Pattern Anal. Mach. Intellig. 20 721
[9] Leng H Z, Song J Q, Cao X Q, Yang J H 2012 Acta Phys. Sin. 61 070501 (in Chinese) [冷洪泽, 宋君强, 曹小群, 杨锦辉 2012 物理学报 61 070501]
[10] Kalman R E 1960 J. Basic Engin. Trans. 82 35
[11] Gelb A 1974 Applied Optimal Estimation (Cambridge:MIT Press)
[12] Evensen G 2006 Data Assimilation:The Ensemble Kalman Filter (New York:Springer) p38
[13] Evensen G 1994 J. Geophys. Res.:Oceans 99 10143
[14] Ghanem R G, Spanos P D 1992 Stochastic Finite Element:A Spectral Spproach (New York:Springer) p214
[15] Xiu D, Karniadakis G S 2003 J. Comput. Phys. 187 137
[16] Schoutens W 2000 The Askey Scheme of Orthogonal Polynomials. In:Stochastic Processes and Orthogonal Polynomials (New York:Springer) pp1-13
[17] Sun N Z, Sun A 2015 Model Uncertainty Quantification. In:Model Calibration and Parameter Estimation:ForEnvironmental and Water Resource Systems (New York:Springer) pp407-458
[18] Isukapalli S S, Roy A, Georgopoulos P G 1998 Risk Anal. 18 351
[19] Tatang M A, Pan W, Prinn R G, McRae G J 1997 J. Geophys. Res. -Atmos. 102 21925 doi:101029/97jd01654
[20] Wang Y P, Cheng Y, Zhang Z Y, Lin G 2018 Math. Model. Nat. Phenom. 13 doi:101051/mmnp/2018023
[21] Whitaker J S, Hamill M 2003 Mon. Weather. Rev. 130 1913
[22] Matre O P L, Knio O M 2010 Spectral Methods for Uncertainty Quantification (Netherlands:Springer) p536
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