Data assimilation with Lorenz 3-variable model. Prepared by Shu-Chih Yang Modified by Juan Ruiz.
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1 Data assimilation with Lorenz 3-variable model Prepared by Shu-Chih Yang Modified by Juan Ruiz.
2 Governing equations dx dt dy dt dz dt = σ(y x) = rx y xz = xy bz Lorenz, E. N, 1963: Deterministic nonperiodic flow. J. Atmos Sci. 20,
3 Optimal interpolation scheme OI estimates the analysis state with the available observations and shortrange forecast (background state), given the time-independent error covariance. Analysis equation x a = x b + W(y o H (x b )) W = BH T (R+HBH T ) 1 B = (x b x t )(x b x t ) T ; R = R 0 I x b :background state x a : analysis state y o : observations H: observation operator B: background error covariance (averaged from the true background error collected for a very long period. R: observation error covariance
4 Package /home/$user/lmda LMOI ERRORSTAT LMETKF./src : matlab codes for model, data assimilation and plotting../input Background error covariance: B*_bst*.mat Observations: obs1_3dvar.mat./output Output states (truth, analysis, observations and background), plots
5 Matlab codes (I) main_drive.m: main driver Parameters control model and data assimilation: 1. bst: Observation/analysis interval (default=8) 2. R 0 : Observation error variance 3. iobs: observation locations (default [1;2;3], observing all variables) 4. (1) and (3) determine which file for background error covariance to load. (ex: With 8 time-step analysis interval and observing x and y, the corresponding file is Bxy_bst8.mat) To run the code in type main_drive in Matlab under./src
6 DA cycle First gues or background (8 time step forecast) 8 time steps truth Analysis Observations (truth + random error )
7 DA cycle Observation error Analysis error Background error Error growth during the forecast due to inestabilities and model error.
8 Matlab codes (II) Model L63eqs.m: governing equations stepit.m: forward integration by Runge-Kutta method DA procedure DA_init.m: initialize matrix operators (B, R, and H) for OI RUN_OI.m: compute OI analysis Plotting tools Daerrplt.m: plot the RMS analysis error stateplt.m: plot the analysis against truth by variable
9 Example of OI analysis error (observing x,y,z every 8 time-step)
10 Example: analysis vs. truth for x variable (observing x,y,z every 8 time-step)
11 Background error covariance estimation The NMC method (Parish and Deber, 1992) In the case of the 3 variable model, we need to estimate the error standard deviation for each variable and the error covariance between variables.
12 Package /home/$user/lmda LMOI ERRORSTAT LMETKF./src : matlab codes for model, correlation computation and plotting../input : A set of analysis to start the model integrations./output : Plots showing the components of the covariance matrix as a function of the position in the Lorenz atractor
13 Matlab codes (III) Error covariance estimation main_drive.m: Integrates the model for 8 and 16 time steps starting from the analysis available in the input folder. Estimates the background error as the difference between the 16 and 8 time steps forecasts. correlacion.m: Computes the correlation between two variables as a function of the position on the Lorenz atractor. promedio.m : Computes the mean of a variable as a function of the position on the Lorenz atractor. To run the code in type main_drive in Matlab under./src
14 Example of X error variance as a function of X and Y
15 Example of XZ error covariance as a function of X and Y
16 ETKF scheme Liu (2007)
17 ETKF equations: Hunt et. al The analysis is a weighed average of the background ensemble members. k is the number of ensemble members. X a, X b and W a are n*k matrices where n is the number of model variables.
18 Package /home/$user/lmda LMOI ERRORSTAT LMETKF./src : matlab codes for model, data assimilation and plotting../output Output states (truth, analysis, observations and background), plots
19 Matlab codes (IV) main_drive.m: main driver Parameters control model and data assimilation: 1. bst: Observation/analysis interval (default=8) 2. R 0 : Observation error variance 3. iobs: observation locations (default [1;2;3], observing all variables) 4. K: ensemble size 5. e(1) and e(2) parameters for multiplicative and additive covariance inflation. To run the code in type main_drive in Matlab under./src
20 Matlab codes (V) Model L63eqs.m: governing equations stepit.m: forward integration by Runge-Kutta method DA procedure DA_init.m: initialize matrix operators (B, R, and H) for OI RUN_ETKF.m: compute ETKF analysis Plotting tools Daerrplt.m: plot the RMS analysis error stateplt.m: plot the analysis against truth by variable
21 Example of ETKF analysis error (observing x,y,z every 8 time-step)
22 Example: analysis vs. truth for x variable (observing x,y,z every 8 time-step)
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