DART Tutorial Sec'on 1: Filtering For a One Variable System
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1 DART Tutorial Sec'on 1: Filtering For a One Variable System 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 Introduc'on This series of tutorial presenta'ons is designed to introduce both basic Ensemble Kalman filter theory and the Data Assimila'on Research Testbed Community Facility for Ensemble Data Assimila'on. There is significant overlap with the DART_LAB tutorial that is also part of the DART subversion checkout. If you have already studied DART_LAB, feel free to skip through the redundant theory slides. However, doing the exercises in all sec'ons of this tutorial is recommended in order to learn the best ways to use the DART system. DART Tutorial Sec'on 1: Slide 2
3 p( A BC) = Bayes Rule p(b C) = p(b x)p(x C)dx.2 Prior PDF A : Prior Es'mate based on all previous informa'on, C. B : An addi'onal observa'on. p(a BC) : Posterior (updated es'mate) based on C and B. DART Tutorial Sec'on 1: Slide 3
4 p( A BC) = Bayes Rule p(b C) = p(b x)p(x C)dx.2 Prior PDF Obs. Likelihood A : Prior Es'mate based on all previous informa'on, C. B : An addi'onal observa'on. p(a BC) : Posterior (updated es'mate) based on C and B. DART Tutorial Sec'on 1: Slide 4
5 p( A BC) = Bayes Rule p(b C) = p(b x)p(x C)dx.2 Prior PDF Product (Numerator) Obs. Likelihood A : Prior Es'mate based on all previous informa'on, C. B : An addi'onal observa'on. p(a BC) : Posterior (updated es'mate) based on C and B. DART Tutorial Sec'on 1: Slide 5
6 p( A BC) = Bayes Rule p(b C) = p(b x)p(x C)dx.2 Prior PDF Normalization (Denom.) Obs. Likelihood A : Prior Es'mate based on all previous informa'on, C. B : An addi'onal observa'on. p(a BC) : Posterior (updated es'mate) based on C and B. DART Tutorial Sec'on 1: Slide 6
7 p( A BC) = Bayes Rule p(b C) = p(b x)p(x C)dx.2 Prior PDF Posterior Normalization (Denom.) Obs. Likelihood A : Prior Es'mate based on all previous informa'on, C. B : An addi'onal observa'on. p(a BC) : Posterior (updated es'mate) based on C and B. DART Tutorial Sec'on 1: Slide 7
8 Color Scheme Green == Prior Red == Observa'on Blue == Posterior The same color scheme is used throughout ALL Tutorial materials. DART Tutorial Sec'on 1: Slide 8
9 Product of Two Gaussians p( A BC) = p(b C) = p(b x)p(x C)dx.6 This product is closed for Gaussian distribu'ons..4.2 Prior PDF Obs. Likelihood DART Tutorial Sec'on 1: Slide 9
10 Product of Two Gaussians p( A BC) = p(b C) = p(b x)p(x C)dx.6 This product is closed for Gaussian distribu'ons. Posterior PDF.4.2 Prior PDF Obs. Likelihood DART Tutorial Sec'on 1: Slide 1
11 Product of Two Gaussians Product of d-dimensional normals with means µ 1 and µ 2 and covariance matrices 1 and 2 is normal. N(µ 1, 1 )N(µ 2, 2 ) = cn(µ, ) DART Tutorial Sec'on 1: Slide 11
12 Product of Two Gaussians Product of d-dimensional normals with means µ 1 and µ 2 and covariance matrices 1 and 2 is normal. N(µ 1, 1 )N(µ 2, 2 ) = cn(µ, ) Covariance: Mean: = ( ) 1 µ = ( ) 1 ( 1 1 µ µ 2 ) DART Tutorial Sec'on 1: Slide 12
13 Product of Two Gaussians Product of d-dimensional normals with means µ 1 and µ 2 and covariance matrices 1 and 2 is normal. N(µ 1, 1 )N(µ 2, 2 ) = cn(µ, ) Covariance: Mean: = ( ) 1 µ = ( ) 1 ( 1 1 µ µ 2 ) Weight: c = 1 (2 ) d/2 1 + exp 1 1/2 2 2 ( µ 2 µ 1 ) T ( ) 1 µ 2 µ 1 ( ) We ll ignore the weight unless noted since we immediately normalize products to be PDFs. DART Tutorial Sec'on 1: Slide 13
14 Product of Two Gaussians Product of d-dimensional normals with means µ 1 and µ 2 and covariance matrices 1 and 2 is normal. N(µ 1, 1 )N(µ 2, 2 ) = cn(µ, ) Covariance: Mean: = ( ) 1 µ = ( ) 1 ( 1 1 µ µ 2 ) Weight: c = 1 (2 ) d/2 1 + exp 1 1/2 2 2 ( µ 2 µ 1 ) T ( ) 1 µ 2 µ 1 ( ) Easy to derive for 1-D Gaussians; just do products of exponen'als. DART Tutorial Sec'on 1: Slide 14
15 Product of Two Gaussians p( A BC) = p(b C) = p(b x)p(x C)dx.3 This product is closed for Gaussian distribu'ons..2.1 Prior PDF There are other families of func'ons for which it is closed But, for general distribu'ons, there s no analy'cal product. DART Tutorial Sec'on 1: Slide 15
16 Product of Two Gaussians p( A BC) = p(b C) = p(b x)p(x C)dx.3 This product is closed for Gaussian distribu'ons..2.1 Prior PDF Obs. Likelihood There are other families of func'ons for which it is closed But, for general distribu'ons, there s no analy'cal product. DART Tutorial Sec'on 1: Slide 16
17 Product of Two Gaussians p( A BC) = p(b C) = p(b x)p(x C)dx.3 This product is closed for Gaussian distribu'ons. Posterior PDF.2.1 Prior PDF Obs. Likelihood There are other families of func'ons for which it is closed But, for general distribu'ons, there s no analy'cal product. DART Tutorial Sec'on 1: Slide 17
18 Product of Two Gaussians p( A BC) = p(b C) = p(b x)p(x C)dx.6 Ensemble filters: Prior is available as finite sample..4.2 Prior Ensemble Don t know much about proper'es of this sample. May naively assume it is random draw from truth. DART Tutorial Sec'on 1: Slide 18
19 Product of Two Gaussians p( A BC) = p(b C) = p(b x)p(x C)dx How can we take product of sample with con'nuous likelihood? Prior PDF Prior Ensemble Fit a con'nuous (Gaussian for now) distribu'on to sample. DART Tutorial Sec'on 1: Slide 19
20 Product of Two Gaussians p( A BC) = p(b C) = p(b x)p(x C)dx Observa'on likelihood usually con'nuous (nearly always Gaussian) Prior PDF Obs. Likelihood Prior Ensemble If Obs. Likelihood isn t Gaussian, can generalize methods below. For instance, can fit set of Gaussian kernels to obs. likelihood. DART Tutorial Sec'on 1: Slide 2
21 Product of Two Gaussians p( A BC) = p(b C) = p(b x)p(x C)dx Product of prior Gaussian fit and Obs. likelihood is Gaussian..6 Posterior PDF.4.2 Prior PDF Prior Ensemble Obs. Likelihood Compu'ng con'nuous posterior is simple. BUT, need to have a SAMPLE of this PDF. DART Tutorial Sec'on 1: Slide 21
22 Sampling Posterior PDF.6 There are many ways to do this. Posterior PDF Exact proper'es of different methods may be unclear. Trial and error s'll best way to see how they perform. Will interact with proper'es of predic'on models, etc. DART Tutorial Sec'on 1: Slide 22
23 Sampling Posterior PDF.6 Ensemble Adjustment (Kalman) Filter.4.2 Prior Ensemble DART Tutorial Sec'on 1: Slide 23
24 Sampling Posterior PDF.6 Ensemble Adjustment (Kalman) Filter.4.2 Prior PDF Prior Ensemble Again, fit a Gaussian to sample. DART Tutorial Sec'on 1: Slide 24
25 Sampling Posterior PDF.6 Ensemble Adjustment (Kalman) Filter Posterior PDF.4.2 Prior PDF Prior Ensemble Obs. Likelihood Compute posterior PDF (same as previous algorithms). DART Tutorial Sec'on 1: Slide 25
26 Sampling Posterior PDF.6 Ensemble Adjustment (Kalman) Filter Posterior PDF Use determinis'c algorithm to adjust ensemble. 1. Shil ensemble to have exact mean of posterior. 2. Use linear contrac'on to have exact variance of posterior. DART Tutorial Sec'on 1: Slide 26
27 Sampling Posterior PDF Ensemble Adjustment (Kalman) Filter Posterior PDF Mean Shifted Use determinis'c algorithm to adjust ensemble. 1. Shil ensemble to have exact mean of posterior. 2. Use linear contrac'on to have exact variance of posterior. DART Tutorial Sec'on 1: Slide 27
28 Sampling Posterior PDF Ensemble Adjustment (Kalman) Filter Posterior PDF Variance Adjusted Mean Shifted Use determinis'c algorithm to adjust ensemble. 1. Shil ensemble to have exact mean of posterior. 2. Use linear contrac'on to have exact variance of posterior. DART Tutorial Sec'on 1: Slide 28
29 Sampling Posterior PDF Ensemble Adjustment (Kalman) Filter Posterior PDF Variance Adjusted Mean Shifted x u i = ( x p i x p )i σ u /σ p i = 1,..., ensemble size. ( ) + x u σ p is prior, u is update (posterior), is standard devia'on, overbar is ensemble mean. DART Tutorial Sec'on 1: Slide 29
30 Sampling Posterior PDF Ensemble Adjustment (Kalman) Filter Posterior PDF Variance Adjusted Mean Shifted Bimodality maintained, but not appropriately posi'oned or weighted. No problem with random outliers. DART Tutorial Sec'on 1: Slide 3
31 Sampling Posterior PDF Ensemble Adjustment (Kalman) Filter Posterior PDF Variance Adjusted Mean Shifted There are a variety of other ways to determinis'cally adjust ensemble. Class of algorithms some'mes called determinis'c square root filters. DART Tutorial Sec'on 1: Slide 31
32 1 st look at DART Diagnos'cs cd models/lorenz_63/work in your DART sandbox. csh workshop_setup.csh Does stuff you ll learn to do later. matlab -nodesktop Output from a DART assimila'on in 3-variable model. 2 member ensemble. Observa'ons of each variable once every 6 hours ; error variance 8. Observa'on ONLY impacts its own state variable. For assimila'on, looks like 3 independent single variable problems. Model advance between assimila'ons isn t independent. Ini'al ensemble members are random selec'on from long model run. Ini'al error should be an upper bound (random guess). DART Tutorial Sec'on 1: Slide 32
33 1 st look at DART Diagnos'cs Try the following Matlab commands. Each will ask you to: Input name of ensemble trajectory file: <cr> for preassim.nc Just select the default file by hipng carriage return for all Matlab exercises for now. plot_total_err plot_ens_mean_-me_series plot_ens_-me_series 'me series of distance between prior ensemble mean and truth in blue; spread, average prior distance between ensemble members and mean, in red. (Record total values of total error and spread for later.) 'me series of truth in blue; ensemble mean prior in red. Figure 1 is separate panel for each state variable. Figure 2 is three-dimensional plot. also includes prior ensemble members in green for Figure 1. DART Tutorial Sec'on 1: Slide 33
34 Simple Example: Lorenz 63 3-variable chao'c model Observa'on in red. Prior ensemble in green. Observing all three state variables. Obs. Error variance = 4.. Four 2-member ensembles. DART Tutorial Sec'on 1: Slide 34
35 Simple Example: Lorenz 63 3-variable chao'c model Observa'on in red. Prior ensemble in green. DART Tutorial Sec'on 1: Slide 35
36 Simple Example: Lorenz 63 3-variable chao'c model Observa'on in red. Prior ensemble in green. DART Tutorial Sec'on 1: Slide 36
37 Simple Example: Lorenz 63 3-variable chao'c model Observa'on in red. Prior ensemble in green. DART Tutorial Sec'on 1: Slide 37
38 Simple Example: Lorenz 63 3-variable chao'c model Observa'on in red. Prior ensemble in green. Ensemble is passing through an unpredictable region. DART Tutorial Sec'on 1: Slide 38
39 Simple Example: Lorenz 63 3-variable chao'c model Observa'on in red. Prior ensemble in green. Part of the ensemble heads for one lobe, the rest for the other.. DART Tutorial Sec'on 1: Slide 39
40 Simple Example: Lorenz 63 3-variable model Observa'on in red. Prior ensemble in green. DART Tutorial Sec'on 1: Slide 4
41 Using DART Diagnos'cs Using DART diagnos'cs from the simple Lorenz 63 assimila'on: Can you see evidence of enhanced uncertainty? Where does this occur? Does the ensemble appear to be consistent with the truth? (Is the truth normally inside the ensemble range?) DART Tutorial Sec'on 1: Slide 41
42 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 1: Slide 42
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