Aspects of the practical application of ensemble-based Kalman filters
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1 Aspects of the practical application of ensemble-based Kalman filters Lars Nerger Alfred Wegener Institute for Polar and Marine Research Bremerhaven, Germany and Bremen Supercomputing Competence Center BremHLR
2 Overview Ensemble generation Localization Covariance inflation Observations and their errors Model errors Bias correction Validation data Lars Nerger Application of Ensemble KFs
3 Data Assimilation - in short
4 System Information: Chlorophyll in the ocean mg/m 3 mg/m 3 Information: Model Generally correct, but has errors all fields, fluxes, Information: Observation Generally correct, but has errors sparse information (only surface, data gaps, one field) Combine both sources of information by data assimilation
5 Data Assimilation Optimal estimation of system state: initial conditions (for weather/ocean forecasts, ) trajectory (temperature, concentrations, ) parameters (growth of phytoplankton, ) fluxes (heat, primary production, ) boundary conditions and forcing (wind stress, ) Characteristics of system: high-dimensional numerical model - O(10 7 ) sparse observations non-linear Lars Nerger Application of Ensemble KFs
6 Data Assimilation Consider some physical system (ocean, atmosphere, ) time state Two main approaches: Variational assimilation model Sequential assimilation observation truth Optimal estimate basically by least-squares fitting Lars Nerger Application of Ensemble KFs
7 Zoo of ensemble-based/error-subspace Kalman filters A little zoo (not complete): EnKF(94/98) SEIK EAKF ETKF EnSQRTKF EnKF(2003) EnKF(2004) SEEK RRSQRT ROEK MLEF SPKF ESSE (Properties and differences are hardly understood) Lars Nerger Application of Ensemble KFs
8 Issues of the practical application No filter works without tuning Covariance inflation (forgetting factor) Localization Other issues Optimal initialization unknown (is it important?) Ensemble integration still costly Simulating model error Bias (model and observations) Observation errors are often unknown Nonlinearity Non-Gaussian fields or observations Lars Nerger Application of Ensemble KFs
9 Ensemble generation
10 What is the right ensemble? Ensemble represents state estimate and error covariance matrix uncertainty of (initial) state estimate correlations between observed and unobserved variables Methods (just a selection) Deviations between model and observations (not all variables/locations observed) Variability from long model integration (self-consistent; correct timing required; related to eigenvalues) random drawing vs. SVD-based selection Set of short-term model integrations Breeding Lars Nerger Application of Ensemble KFs
11 Sampling Example " % $ ' P t = $ ' ; x " t = 0.0 % $ ' # 0.0& # & Lars Nerger Application of Ensemble KFs
12 3D Box - interchanged intializations Ensemble size=10 Covariance matrix P from long model simulation MC: random sampling of P 2 nd : sample low-rank approximation of P Lars Nerger Application of Ensemble KFs
13 Localization
14 Domain localization - Local SEIK filter Analysis: Update small regions (e.g. single vertical columns) Consider only observations within cut-off distance neglects long-range correlations Re-Initialization: Transform local ensemble Use same transformation matrix in each local domain Nerger, L., S. Danilov, W. Hiller, and J. Schröter. Ocean Dynamics 56 (2006) 634
15 Local SEIK filter II Observation localization Localizing weight reduce weight for remote observations by increasing variance estimates use e.g. exponential decrease or polynomial representing correlation function of compact support similar, sometimes equivalent, to covariance localization used in other ensemble-based KFs Lars Nerger Application of Ensemble KFs
16 Example: Assimilation of pseudo sea surface height observations in the North Atlantic (twin experiment) Lars Nerger Application of Ensemble KFs
17 FEOM Mesh for North Atlantic finite-element discretization surface nodes: D nodes: z-levels: 23 eddy-permitting
18 Configuration of twin experiments Generate true state trajectory for 12/1992-3/1993 Assimilate synthetic observations of sea surface height (generated by adding uncorrelated Gaussian noise with std. deviation 5cm to true state) Covariance matrix estimated from variability of 9-year model trajectory ( ) initialized from climatology Initial state estimate from perpetual 1990 model spin-up Monthly analysis updates (at initial time and after each month of model integration) No model error; forgetting factor 0.8 for both filters Lars Nerger Application of Ensemble KFs
19 Modeled Sea Surface Height (Dec. 1992) Not aimed at oceanographic relevance! - large-scale deviations of small amplitude - small-scale deviations up to 40 cm
20 Improvement of Sea Surface Height (Dec. 1992) N=8 N=32 Improvement: red - deterioration: blue For N=8 rather coarse-scale corrections Increased ensemble size adds finer scales (systematically) Lars Nerger Application of Ensemble KFs
21 True and estimated errors (Dec. 1992) Correction only possible, if state error present!
22 Global vs. Local SEIK, N=32 (Mar. 1993) rrms = 83.6% rrms = 31.7% - Improvement regions of global SEIK also improved by local SEIK - localization provides improvements in regions not improved by global SEIK - regions with error increase diminished for local SEIK Lars Nerger Application of Ensemble KFs
23 Relative rms errors for SSH - global filter: significant improvement for larger ensemble - global filter with N=100: relative rms error localization strongly improves estimate - larger error-reduction at each analysis update - but: stronger error increase during forecast - very small radius results in over-fitting to noise
24 Covariance inflation
25 Covariance inflation True variance is always underestimated finite ensemble size sampling errors (unknown structure of P) model errors can lead to filter divergence Simple remedy Increase error estimate before analysis Possibilities Multiply covariance matrix by a factor (inflation factor, 1/forgetting factor) Additive error (e.g. on diagonal) Lars Nerger Application of Ensemble KFs
26 Impact of inflation on stability & performance Experiments with Lorenz96 model Global filter Localized, ensemble size 10 forgetting factor SEIK orig, random ensemble size forgetting factor LSEIK fix, obs. error= support radius Increased stability with stronger inflation (smaller forgetting factor) Optimal choice for inflation factor Lars Nerger Application of Ensemble KFs
27 Observations and their errors
28 Real observations They are not ideal Incomplete (space, time) Errors only estimated Errors can be correlated Can be biased Usual way of handling: pragmatism Lars Nerger Application of Ensemble KFs
29 Observation availability Surface temperature :00±6h :00±6h Strongly irregular data availability Frequent data gaps Assume constant error and homogeneous spatial influence S. Losa, Project DeMarine Environment
30 Satellite Ocean Color (Chlorophyll) Observations Natural Color 3/16/2004 Chlorophyll Concentrations Source: NASA Visible Earth, Image courtesy the SeaWiFS Project, NASA/GSFC, and Orbimage Lars Nerger Application of Ensemble KFs
31 Assimilated Observations mg/m 3 Daily gridded SeaWiFS chlorophyll data gaps: satellite track, clouds, polar nights ~13,000-18,000 data points daily (of 41,000 wet grid points) irregular data availability Nerger, L., and W.W. Gregg. J. Marine Systems 68 (2007) 237
32 Error Estimates Regional data errors from comparison with 2186 collocation points of in situ data Lars Nerger Application of Ensemble KFs
33 Observation errors II Account regionally for larger errors caused by aerosols (North Indian Ocean, tropical Atlantic) CDOM (Congo and Amazon) Error estimates adjusted for filter performance and stability
34 Model Errors
35 Model errors Representation of reality is not exact Incomplete equations (e.g. missing processes) Inexact forcing (e.g. wind stress on ocean surface) Accounting for model error Inflation (partly) Simulate stochastic part Bias estimation Lars Nerger Application of Ensemble KFs
36 Bias correction
37 Assimilation with global SEIK filter mg/m 3 mg/m 3 mg/m 3 some improvements of estimated total Chlorophyll Increased estimation errors in region with polar night SEIK assimilation crashes (earlier for larger ensemble sizes)
38 Bias Estimation un-biased system: fluctuation around true state biased system: systematic over- and underestimation (common situation with real data) 2-stage bias online bias correction 1. Estimate bias (using fraction of covariance matrix used in 2.) 2. Estimate de-biased state Forecast 1. forecast ensemble of biased states 2. no propagation of bias vector Nerger, L., and W.W. Gregg. J. Marine Systems, 73 (2008)
39 Estimated Chlorophyll - April 15, 2004 mg/m 3 mg/m 3 mg/m 3 strongly improved surface Chlorophyll estimate intended deviations (Arabian Sea, Congo, Amazon) other deviations in high- Chlorophyll regions
40 Comparison with independent data RMS log error In situ data from SeaBASS/NODC over (shown basins include about 87% of data) Independent from SeaWiFS data (only used for verification of algorithms) Compare daily co-located data points Assimilation in most regions below SeaWiFS error Bias correction improves almost all basins
41 Validation data
42 Validating a data assimilation system Need independent data for validation Necessary, but not sufficient: Reduction of deviation from assimilated data Required: - Reduction of deviation from independent data - Reduction of errors for unobserved variables Want to assimilate all available data Data-withholding experiments Twin experiments Validate with data of small influence Lars Nerger Application of Ensemble KFs
43 In-Situ chlorophyll data In situ data from SeaBASS/NODC over 1/1998-2/2004 Independent from SeaWiFS data (only used for verification of algorithms) North Central Pacific dominated by CalCOFI data North Central Atlantic dominated by BATS data
44 Summary Practical assimilation with ensemble-based Kalman filters Care and pragmatism required pure filter works suboptimal or not at all Theoretical foundation is incomplete Advancements in between Lars Nerger Application of Ensemble KFs
45 Thank you! Acknowledgements: W. Hiller, J. Schröter, T. Janjic, S. Losa (AWI) Watson Gregg, Nancy Casey (NASA/GSFC) Lars Nerger Application of Ensemble KFs
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