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Budgets and Bias in Data Assimilation Keith Haines, ESSC&DARC, Reading

Budgets and Bias in Data Assimilation Keith Haines, ESSC&DARC, Reading. Background: Marine Informatics Assimilation algorithms in Ocean circulation models Satellite and In Situ data sets Physically based covariances + simple errors in big and Biased models

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Budgets and Bias in Data Assimilation Keith Haines, ESSC&DARC, Reading

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  1. Budgets and Bias in Data Assimilation Keith Haines, ESSC&DARC, Reading • Background: Marine Informatics • Assimilation algorithms in Ocean circulation models • Satellite and In Situ data sets • Physically based covariances + simple errors in big and Biased models • Budget diagnostics based on assimilation • Met Office FOAM, ECMWF Seasonal Forecasting collaborations • DARC-NCOF Fellow Dan Lea based in NCOF group at Met Office • New project (Marine Quest) will look at assimilation constraints on Carbon within a coupled physics-biochemistry ocean model • e-Science/Grid: Model and Satellite data viewed in Google Maps/Earth • http://lovejoy.nerc-essc.ac.uk:8080/Godiva2 Carbon Fusion 9-11th May 2006

  2. Budgets and Ocean Thermohaline Circulation Ocean Box-Inverse solution Ganachaud and Wunsch (2000) After Broeker • Closed Budgets of .. • Heat, Salt, Mass/Volume, Tracers.. • Processes: Advection, Surface fluxes, Mixing, Data Assimilation Transport in Sverdrups 1Sv = 106 m3 s-1 Carbon Fusion 9-11th May 2006

  3. Ocean Box-Inverse Assimilation • Key assumption is for Steady State system • Therefore can use asynoptic data (different ocean sections observed at completely different times) • Try to correct for known variability eg. Seasonal cycle (surface properties and wind induced transports) • Deduce unknown box-exchanges (circulation and mixing rates) for closed system • Often problem underconstrained => use some Occams razor or conditioning assumption (smallest consistent flows/mixing rates) Carbon Fusion 9-11th May 2006

  4. Transport in Sverdrups 1Sv = 106 m3 s-1 Carbon Fusion 9-11th May 2006

  5. N. Atlantic Water Budgetby density class (11S-80N) 27.72 28.11 COADS surface fluxes CTD section at 11S Steady State (cf. Ocean Inverse) => Mixing Transformation Flux (Sv) Speer (1997) Carbon Fusion 9-11th May 2006

  6. Walin Budget diagnostics for HadCM3 climate model (100yr average) Transformation Flux (Sv) 27.72 28.11 Carbon Fusion 9-11th May 2006 Old and Haines 2006

  7. Data Assimilation in a time-evolving model? • Steady state box-inverse models estimate process rates or parametrisations like mixing from a 3D Variational problem • Similar “Parameter Estimation” while matching time–evolving data often uses 4DVar Assimilation • 4DVar very expensive computationally • The “budget within a box” concept is subsumed into seeking a solution to the temporal model equations • Parameter tuning assumes process representations are ‘structurally’ correct • Different approach: Assimilation corrects for model bias so evaluate assimilation as another process within Box Budgets • A posteriori “Process Estimation” Carbon Fusion 9-11th May 2006

  8. Process Estimation v. Parameter Estimation Parameter estimation 4DVar. Cost function containing fit to observations, a-priori info. Tune: initial state, sources/sinks, model parameters (diffusion)….. Carbon Fusion 9-11th May 2006

  9. Data Assimilation in a time-evolving model? • Steady state box-inverse models estimate process rates or parametrisations like mixing from a 3D Variational problem • Similar “Parameter Estimation” while matching time–evolvingdata often uses 4DVar Assimilation • 4DVar very expensive computationally • The “budget within a box” concept is subsumed into seeking a solution to the temporal model equations • Parameter tuning assumes process representations are ‘structurally’ correct • Different approach: Assimilation corrects for model bias so evaluate assimilation as another process within Box Budgets • A posteriori “Process Estimation” Carbon Fusion 9-11th May 2006

  10. OCCAM Assimilation Experiment Sea Level analysis 28th March 1996 RUN1 • 1993-96 • ECMWF 6hr winds • Monthly XBT assim. • 10-day-ly Altimeter assim. • SST weakly relaxed to Reynolds • SSS weakly relaxed to Levitus 1/4° x 36 levels Global Ocean Model Carbon Fusion 9-11th May 2006

  11. Process Estimation: Local Heat Budget Wm-2 Assimilation Advection • Bias • Patterns • Amplitudes • Space scales • Transients Trend 1993-96 Surface Flux Mixing Local Trend = Convergence + Assimilation + Surface Flux (+ Mixing) Carbon Fusion 9-11th May 2006 (Haines; 2003)

  12. Process Estimation: N Atlantic Box Budgets G = Volume Transformation Rate (Sv) (after Walin 1982) Thermodynamically Irreversible Processes -G/  = dV/dt -  G = (1) Surface Forcing, (2) Mixing, (3) Data Assimilation Fox and Haines (2003) JPO 16Sv Carbon Fusion 9-11th May 2006 Run1

  13. Process Estimation in the Ocean • Locally assimilation corrects for wrong Advection: eg. Gulf stream overshoots, Eastern Pacific thermocline • Basin average sense assimilation corrects for wrong forcing i.e. surface heat flux • Characteristic of certain processes can help to attribute assimilation contributions to box-budgets, eg. • Advection is conservative between regions (no sources or sinks) • Mixing also conservative AND always downgradient Carbon Fusion 9-11th May 2006

  14. Relevance to Carbon Budget Modelling and Assimilation? • Budget-box representation of terrestrial ecosystem • Conserved quantities: Carbon, Nitrogen/Nitrates?...... • Understand cycling rates in model control (seasonal etc.. dependencies) • Assimilation will try to constrain Amounts of conserved properties in each box. Unlikely to observe Transformation process rates? • Success of assimilation may depend on; • Frequency of assimilation • Rate at which model transformation processes act • Any feedback between Amounts of property and transformation rates • Generation of unwanted transient processes as model adjusts to new data Carbon Fusion 9-11th May 2006

  15. ERSEM - key features Carbon based process model Functional group approach Resolves microbial loop and POM/DOM dynamics Complex suite of nutrients Includes benthic system Explicit decoupled cycling of C, N, P, Si and Chl. Adaptable: DMS, CO2/pH, phytobenthos, HABs. Ecosystem Forcing Atmosphere Cloud Cover O2 DMS CO2 Wind Stress Si Irradiation Phytoplankton N u t r i e n t s Dino-f Pico-f Diatoms NO3 Flagell -ates Heat Flux Particulates CO2 NH4 Physics Bacteria PO4 Dissolved 0D Rivers and boundaries Hetero- trophs Micro- Meso- 1D Consumers Suspension Feeders D e t r i t u s Oxygenated Layer N u t r I e n t s Aerobic Bacteria Meio- benthos Deposit Feeders Redox Discontinuity Layer 3D Anaerobic Bacteria Reduced Layer UK MO GOTM POLCOMS Shelf Seas: Carbon+Biochemistry Modelling Carbon Fusion 9-11th May 2006

  16. Bias and Data Assimilation • Assimilation often correcting for Process Biases • In OCCAM model: • Locally assimilation corrects for wrong Advection: eg. mesoscale eddies in the wrong location or biased advection eg. Gulf stream overshoots • Basin average sense assimilation corrects for wrong forcing i.e. surface heat flux • Characteristics of certain processes can help to attribute assimilation contributions to box-budgets, eg. • Advection is conservative between regions (no sources or sinks) • Mixing also conservative AND always downgradient • May try to Account for bias when assimilating data as it should alter the error weighting between model and observations Carbon Fusion 9-11th May 2006

  17. Accounting for Bias in Data Assimilation • Dee (2006) Review in QJRMS • Variational formulation easiest to understand (derivable from Bayesian analysis; Drecourt et al; 2006) 2J(x,b,c) = (y-b-x)TR-1(y-b-x) + (x-xf+c)TB-1(x-xf+c) + (b-bf)TO-1(b-bf) + (c-cf)TP-1(c-cf) y =observation R =observation error covariance x =model state B =model background error covariance b =observation bias O =observation bias error covariance c =model forecast bias P =model forecast bias error covariance Superscript f are forecast values Observation operators have been omitted Carbon Fusion 9-11th May 2006

  18. Accounting for Bias in Data Assimilation • Solution (Analysed variables a) xa = (xf-cf) + K {(y-bf) – (xf-cf)} K = (B+P) [B+P+O+R]-1 ba = bf + F {(y-bf) – (xf-cf)} F = O [B+P+O+R]-1 ca = cf + G {(y-bf) – (xf-cf)} G = P [B+P+O+R]-1 or xa = (xf-ca) + K1{(y-ba) – (xf-ca)} K1 = B [B+R]-1 y =observation R =observation error covariance x =model state B =model background error covariance b =observation bias O =observation bias error covariance c =model forecast bias P =model forecast bias error covariance Usual problems are: (i) Knowing the Covariance errors (ii) Sequential 3DVar requires bias models for bf(t+1)= Mb[ba(t)]; cf(t+1)= Mc[ca(t)]; Carbon Fusion 9-11th May 2006

  19. Comments on Bias Modelling • Known Biases {bf (t); cf(t) known a priori eg. previous runs} • xa = (xf-cf) + K {(y-bf) – (xf-cf)} K = (B+P)[B+P+O+R]-1 • bf (t) = 0; cf(t) = 0 is particular case • (B+P) total model err cov.; (O+R) total obs. err. • Persistent Biases {bf(t+1)= ba(t); cf(t+1)= ca(t) } • xa = (xf-cf) + K {(y-bf) – (xf-cf)} K = (B+P)[B+P+O+R]-1 • ba = bf + F {(y-bf) – (xf-cf)} F = O[B+P+O+R]-1 • ca = cf + G {(y-bf) – (xf-cf)} G = P[B+P+O+R]-1 • If O,P i.e. F,G are small => may hope to converge to ~ constant b,c • Simplifications also arise if P=αB; O=βR => all Innovations proportional • Attribution of Bias: When are O,P sufficiently different to allow identification of misfits {(y-bf) – (xf-cf)} ? • Should always check misfits are consistent with B+P+O+R Carbon Fusion 9-11th May 2006

  20. Example: Bias Modelling applied toAltimeter Data Assimilation Mean Sea Level Bias Error Covariance O on Mean Sea Level Carbon Fusion 9-11th May 2006

  21. Example: Bias Modelling applied toAltimeter Data Assimilation Mean Sea Level Bias ba Corrected Mean Sea Level Carbon Fusion 9-11th May 2006

  22. CONCLUSIONS • Biased model parameterisations can be tuned through 4DVar but only as far as structural errors and computational resources allow • Alternatively build assimilation increments into box-budgets and seek to understand bias as process. Retains physically intuitive interpretation of Bias and Assimilation increments • Having identified bias it should be accounted for during assimilation as it impacts on error weighting of model and data. Will need a bias model eg. understand its persistence, spatial structure, diurnal/seasonal cycling. Carbon Fusion 9-11th May 2006

  23. Conservation properties of assimilation • Altimeter Assimilation • Displacement h => Gross Isopycnal geometry • + Currents (geostrophy) • Volume and T/S properties preserved on isopycnals • Adiabatic (Thermodynamically Reversible) • T Profile Assimilation • T(z) => Isothermal Water Volumes • T/S properties preserved (since salinity is not observed) • Volumes and T/S preserved below deepest observation • S(T) Assimilation • S(T) => Isopycnal Water Properties • Large scale, slow variations associated with ventilation and climatic change Carbon Fusion 9-11th May 2006

  24. Transformation (slow) Nutrient recycling fast Nutrient recycling fast Box Budgets and Assimilation Carbon Fusion 9-11th May 2006

  25. Example: Bias Modelling applied toAltimeter Data Assimilation Carbon Fusion 9-11th May 2006

  26. Schmitz (1996) Thermohaline Schematic Broeker Carbon Fusion 9-11th May 2006

  27. WOCE Atlantic Section A16 S N Note: Water mass origins AIW, NADW, ABW Currents, Circulation rates and Mixing rates not determined from Core method Long-lived Lagrangian properties of water used to trace spreading pathways. “Core method” Wust (1935) Carbon Fusion 9-11th May 2006

  28. Dissolved Inorganic Carbon Carbon Fusion 9-11th May 2006

  29. WOCE ComparisonN-S Pacific Temperature section P14TP+ERS1 data 1993 Simulation XBT Assimilation XBT and Altimeter Run available on Live Access Server www.nerc-essc.ac.uk/godiva WOCE Cruise How to quantify the role of assimilation in maintaining thermocline? Carbon Fusion 9-11th May 2006

  30. Relevant Ideas • Can we use assimilation methods to perform budgets? • Focus on conservative properties of system (total carbon?) and processes converting between reservoirs • Tune assimilation impact on processes rather than on model parameters Carbon Fusion 9-11th May 2006

  31. HadOCC Based on Web Services Carbon Fusion 9-11th May 2006

  32. MARQuest proposal • Assimilation of physical ocean data (temperature profiles, satellite data..) => constrain surface temperature and mixed layer depth to observations • Study different ecosystem models embedded into physical model with data assimilation. Compare carbon cycling processes! • Must develop treatment for ecosystem variables for when physical ocean data are assimilated. Careful attention to ecosystem and carbon budgets. • Work with Hadley centre/Met Office FOAM assimilation system. Carbon Fusion 9-11th May 2006

  33. Marine Assimilation in Global Ocean Models • Extensive experience developing • new assimilation algorithms eg. • most recently for ARGO data • Assimilation of hydrography • => vertical T gradients • Assimilation of altimetry • => horizontal T gradients • and currents • Algorithms used operationally at • Met Office, ECMWF, France,US • Assimilation control of surface T • and mixed layer depth will also • constrain Ecosystems 0m Assimilation results 500m Ship Validation WOCE Cruise Carbon Fusion 9-11th May 2006 15 S 55 N

  34. High resolution FOAM MarQuest: Assimilation impact on Ecosystems • Assimilation controls and corrects seasonal thermocline T and MLD • Biological production will be strongly influenced by assimilation HadOCC thermocline and chlorophyll conc. No Data Assimilation FOAM thermocline With Data Assimilation Carbon Fusion 9-11th May 2006 All data from www.nerc-essc.ac.uk/godiva

  35. Ideas • Get Icarus ERSEM pictures of carbon cycle • Get Oschlies results figures • More reference figure on inverse modelling • Contact new MIT woman about land surface assim Carbon Fusion 9-11th May 2006

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