DART Tutorial Sec'on 9: More on Dealing with Error: Infla'on
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1 DART Tutorial Sec'on 9: More on Dealing with Error: Infla'on UCAR The Na'onal Center for Atmospheric Research is sponsored by the Na'onal Science Founda'on. Any opinions, findings and conclusions or recommenda'ons expressed in this publica'on are those of the author(s) and do not necessarily reflect the views of the Na'onal Science Founda'on.
2 Some Error Sources in Ensemble Filters 2. h errors; Representa'veness 3. Gross Obs. Error 4. Sampling Error; Gaussian Assump'on h h h t k+2 t k ** * 1. Model Error t k+1 5. Sampling Error; Assuming Linear Sta's'cal Rela'on DART Tutorial Sec'on 9: Slide 2
3 Dealing with Ensemble Filter Errors 2. h errors; Representativeness 3. Gross Obs. Error 4. Sampling Error; Gaussian Assumption Fix 1, 2, 3 independently, HARD but ongoing. O\en, ensemble filters... t k ** * h h h t k+2 1-4: Variance infla'on, Increase prior uncertainty to give obs more impact. 1. Model Error t k+1 5. Sampling Error; Assuming Linear Statistical Relation 5. Localiza'on : only let obs. impact a set of nearby state variables. O\en smoothly decrease impact to as func'on of distance. DART Tutorial Sec'on 9: Slide 3
4 Model/Filter Error: Filter Divergence and Variance Infla'on 1. History of observa'ons and physical system => true distribu'on. Probability 1.5 "TRUE" Prior PDF DART Tutorial Sec'on 9: Slide 4
5 Model/Filter Error: Filter Divergence and Variance Infla'on 1. History of observa'ons and physical system => true distribu'on. 2. Sampling error, some model errors lead to insufficient prior variance. 3. Can lead to filter divergence : prior is too confident, obs. Ignored. Probability 1.5 Variance Deficient PDF "TRUE" Prior PDF DART Tutorial Sec'on 9: Slide 5
6 Model/Filter Error: Filter Divergence and Variance Infla'on 1. History of observa'ons and physical system => true distribu'on. 2. Sampling error, some model errors lead to insufficient prior variance. 3. Can lead to filter divergence : prior is too confident, obs. Ignored. Probability 1.5 Variance Deficient PDF "TRUE" Prior PDF Naïve solu'on is variance infla'on: just increase spread of prior. For ensemble member i, inflate x i ( ) = λ ( x i x) + x DART Tutorial Sec'on 9: Slide 6
7 Model/Filter Error: Filter Divergence and Variance Infla'on 1. History of observa'ons and physical system => true distribu'on..8 "TRUE" Prior PDF Probability DART Tutorial Sec'on 9: Slide 7
8 Model/Filter Error: Filter Divergence and Variance Infla'on 1. History of observa'ons and physical system => true distribu'on. 2. Most model errors also lead to erroneous shi\ in en're distribu'on. 3. Again, prior can be viewed as being TOO CERTAIN. Probability "TRUE" Prior PDF Error in Mean (from model) DART Tutorial Sec'on 9: Slide 8
9 Model/Filter Error: Filter Divergence and Variance Infla'on 1. History of observa'ons and physical system => true distribu'on. 2. Most model errors also lead to erroneous shi\ in en're distribu'on. 3. Again, prior can be viewed as being TOO CERTAIN. Probability "TRUE" Prior PDF Error in Mean (from model) Variance Inflated Infla'ng can ameliorate this. Obviously, if we knew E(error), we d correct for it directly. DART Tutorial Sec'on 9: Slide 9
10 Physical Space Variance Infla'on Inflate all state variables by same amount before assimila'on. Capabili'es: 1. Can be effec've for a variety of models. 2. Can maintain linear balances. 3. Stays on local flat manifolds. 4. Simple and cheap. Liabili'es: 1. State variables not constrained by observa'ons can blow up. For instance unobserved regions near the top of AGCMs. 2. Magnitude of λ normally selected by trial and error. DART Tutorial Sec'on 9: Slide 1
11 Physical Space Variance Infla'on in Lorenz 63 Observa'on outside prior: danger of filter divergence Observa'on in red. Prior ensemble in green. 2 2 DART Tutorial Sec'on 9: Slide 11
12 Physical Space Variance Infla'on in Lorenz 63 A\er infla'ng, observa'on is in prior cloud: filter divergence avoided Observa'on in red. Prior ensemble in green. 2 2 Inflated ensemble in magenta. DART Tutorial Sec'on 9: Slide 12
13 Physical Space Variance Infla'on in Lorenz 63 Prior distribu'on is significantly curved Observa'on in red. Prior ensemble in green. 2 2 DART Tutorial Sec'on 9: Slide 13
14 Physical Space Variance Infla'on in Lorenz 63 Inflated prior outside alractor. Posterior will also be off alractor. Can lead to transient off-alractor behavior or 4 3 Model blow-up Observa'on in red. Prior ensemble in green. 2 2 Inflated ensemble in magenta. DART Tutorial Sec'on 9: Slide 14
15 Basic control of infla'on in DART is in &filter_nml Before Assimila'on A\er Assimila'on inf_flavor =,, inf_ini'al_from_restart =.false.,.false., inf_sd_ini'al_from_restart =.false.,.false., inf_determinis'c =.true.,.true., inf_ini'al = 1., 1., inf_sd_ini'al =.,., inf_damping = 1., 1., inf_lower_bound = 1., 1., inf_upper_bound = 1., 1., inf_sd_lower_bound =.,., Flavor: => NONE 1 => deprecated 2,3 => physical space Infla'on Value prior infla'on column posterior infla'on column Ini'ally, we ll change inf_flavor and inf_ini*al in first column. DART Tutorial Sec'on 9: Slide 15
16 Physical space variance infla'on in Lorenz 96 models/lorenz_96/work/ Try some values and see what happens to assimila'ons with Lorenz 96. Set inf_flavor to 3 to use state space infla'on. In the first column, set inf_ini*al to values like 1.5, 1.8, 1.1 &assim_tools_nml filter_kind = 1 cutoff = 1. spread_restoration =.false. sampling_error_correction =.false. &filter_nml ens_size = 2 perturb_from_single_instance =.false. inf_flavor = 3, inf_initial_from_restart =.false.,.false. inf_sd_initial_from_restart =.false.,.false. inf_initial = 1., 1. inf_sd_initial =.,. inf_damping = 1., 1. inf_lower_bound = 1., 1. inf_upper_bound = 1., 1. inf_sd_lower_bound =.,. DART Tutorial Sec'on 9: Slide 16
17 Variance infla'on in observa'on space Probability Not currently supported in DART Manhalan. Prior PDF Obs. Likelihood S.D. S.D For observed variable, have es'mate of prior-observed inconsistency. DART Tutorial Sec'on 9: Slide 17
18 Variance infla'on in observa'on space Probability Prior PDF Obs. Likelihood Actual SDs Expected Separation S.D. S.D For observed variable, have es'mate of prior-observed inconsistency Expected (prior_mean observa'on) = σ prior + σ obs Assumes that prior and observa'on are supposed to be unbiased. Is it model error or random chance? DART Tutorial Sec'on 9: Slide 18
19 Variance infla'on in observa'on space Probability Prior PDF Obs. Likelihood Actual SDs Expected Separation Inflated S.D. S.D For observed variable, have es'mate of prior-observed inconsistency Expected (prior_mean observa'on) = σ prior + σ obs 3. Infla'ng increases expected separa'on. Increases apparent consistency between prior and observa'on. DART Tutorial Sec'on 9: Slide 19
20 Variance infla'on in observa'on space: Lorenz 96 Variance infla'on in observa'on space not currently supported. Try some values and see what happens to Lorenz 96 assimila'on. Set inf_flavor=1, observa'on space infla'on in first column. Try some values and see what happens to Lorenz 96 assimila'on. Set inf_ini*al to values like 1.5, 1.8, 1.1 in first column. Make sure that cutoff=1 and ens_size=2. (These were serngs that diverged without infla'on) DART Tutorial Sec'on 9: Slide 2
21 DART Tutorial Index to Sec'ons 1. Filtering For a One Variable System 2. The DART Directory Tree 3. DART Run>me Control and Documenta>on 4. How should observa>ons of a state variable impact an unobserved state variable? Mul>variate assimila>on. 5. Comprehensive Filtering Theory: Non-Iden>ty Observa>ons and the Joint Phase Space 6. Other Updates for An Observed Variable 7. Some Addi>onal Low-Order Models 8. Dealing with Sampling Error 9. More on Dealing with Error; Infla>on 1. Regression and Nonlinear Effects 11. Crea>ng DART Executables 12. Adap>ve Infla>on 13. Hierarchical Group Filters and Localiza>on 14. Quality Control 15. DART Experiments: Control and Design 16. Diagnos>c Output 17. Crea>ng Observa>on Sequences 18. Lost in Phase Space: The Challenge of Not Knowing the Truth 19. DART-Compliant Models and Making Models Compliant 2. Model Parameter Es>ma>on 21. Observa>on Types and Observing System Design 22. Parallel Algorithm Implementa>on 23. Loca'on module design (not available) 24. Fixed lag smoother (not available) 25. A simple 1D advec>on model: Tracer Data Assimila>on DART Tutorial Sec'on 9: Slide 21
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