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/Godiva 2 Carbon Fusion 9 -11 th May 2006
Budgets and Ocean Thermohaline Circulation After Broeker Ocean Box-Inverse solution Ganachaud and Wunsch (2000) • Closed Budgets of. . Transport in Sverdrups 1 Sv = 106 m 3 s-1 Heat, Salt, Mass/Volume, Tracers. . • Processes: Advection, Surface fluxes, Mixing, Data Assimilation Carbon Fusion 9 -11 th May 2006
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 -11 th May 2006
Transport in Sverdrups 1 Sv = 106 m 3 s-1 Carbon Fusion 9 -11 th May 2006
N. Atlantic Water Budget by density class (11 S-80 N) 27. 72 COADS surface fluxes CTD section at 11 S Steady State (cf. Ocean Inverse) => Mixing 28. 11 Transformation Flux (Sv) Speer (1997) Carbon Fusion 9 -11 th May 2006
Walin Budget diagnostics for Had. CM 3 climate model (100 yr average) Transformation Flux (Sv) Old and Haines 2006 Carbon Fusion 9 -11 th May 2006 27. 72 28. 11
Data Assimilation in a time-evolving model? • Steady state box-inverse models estimate process rates or parametrisations like mixing from a 3 D Variational problem • Similar “Parameter Estimation” while matching time–evolving data often uses 4 DVar Assimilation • 4 DVar 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 -11 th May 2006
Process Estimation v. Parameter Estimation Parameter estimation 4 DVar. Cost function containing fit to observations, a-priori info. Tune: initial state, sources/sinks, model parameters (diffusion)…. . Carbon Fusion 9 -11 th May 2006
Data Assimilation in a time-evolving model? • Steady state box-inverse models estimate process rates or parametrisations like mixing from a 3 D Variational problem • Similar “Parameter Estimation” while matching time–evolving data often uses 4 DVar Assimilation • 4 DVar 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 -11 th May 2006
OCCAM Assimilation Experiment RUN 1 • • • 1993 -96 ECMWF 6 hr winds Monthly XBT assim. 10 -day-ly Altimeter assim. SST weakly relaxed to Reynolds SSS weakly relaxed to Levitus Sea Level analysis 28 th March 1996 1/4° x 36 levels Global Ocean Model Carbon Fusion 9 -11 th May 2006
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 -11 th May 2006 (Haines; 2003)
Process Estimation: N Atlantic Box Budgets G = Volume Transformation Rate (Sv) (after Walin 1982) Thermodynamically Irreversible Processes - G/ = d. V/dt - G = (1) Surface Forcing, (2) Mixing, (3) Data Assimilation Fox and Haines (2003) JPO 16 Sv Run 1 Carbon Fusion 9 -11 th May 2006
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 -11 th May 2006
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 -11 th May 2006
Shelf Seas: Carbon+Biochemistry Modelling ERSEM - key features Forcing Ecosystem Carbon based process Cloud Cover model Wind Stress Atmosphere O 2 Irradiation Rivers and boundaries Pico-f MO POLCOMS NO 3 Diatoms CO 2 NH 4 Bacteria PO 4 Dissolved Heterotrophs Includes benthic system Adaptable: DMS, CO 2/p. H, phytobenthos, UK GOTM HABs. Flagell -ates Dino-f Particulates Complex suite of nutrients 1 D Explicit decoupled cycling of C, N, P, Si and Chl. 3 D Si Phytoplankton Heat Functional group Flux approach Physics Resolves microbial loop and POM/DOM dynamics 0 D DMS CO 2 Micro- Meso- Consumers Suspension Feeders D e t r i t u s Oxygenated Layer Aerobic Bacteria Meiobenthos Anaerobic Bacteria Carbon Fusion 9 -11 th May 2006 Deposit Feeders Redox Discontinuity Layer Reduced Layer N u t r I e n t s N u t r i e n t s
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 -11 th May 2006
Accounting for Bias in Data Assimilation • Dee (2006) Review in QJRMS • Variational formulation easiest to understand (derivable from Bayesian analysis; Drecourt et al; 2006) 2 J(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 -11 th May 2006
Accounting for Bias in Data Assimilation • Solution (Analysed variables a) xa = (xf-cf) + K {(y-bf) – (xf-cf)} ba = bf + F {(y-bf) – (xf-cf)} ca = cf + G {(y-bf) – (xf-cf)} K = (B+P) [B+P+O+R]-1 F = O [B+P+O+R]-1 G = P [B+P+O+R]-1 or K 1 = B [B+R]-1 xa = (xf-ca) + K 1{(y-ba) – (xf-ca)} y =observation x =model state b =observation bias c =model forecast bias R =observation error covariance B =model background error covariance O =observation bias error covariance P =model forecast bias error covariance Usual problems are: (i) Knowing the Covariance errors (ii) Sequential 3 DVar requires bias models for bf(t+1)= Mb[ba(t)]; cf(t+1)= Mc[ca(t)]; Carbon Fusion 9 -11 th May 2006
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 = b f + 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 -11 th May 2006
Example: Bias Modelling applied to Altimeter Data Assimilation Mean Sea Level Bias Error Covariance O on Mean Sea Level Carbon Fusion 9 -11 th May 2006
Example: Bias Modelling applied to Altimeter Data Assimilation Mean Sea Level Bias ba Corrected Mean Sea Level Carbon Fusion 9 -11 th May 2006
CONCLUSIONS • Biased model parameterisations can be tuned through 4 DVar but only as far as structural errors and computational resources allow • Alternatively build assimilation increments into boxbudgets 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 -11 th May 2006
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 -11 th May 2006
Box Budgets and Assimilation Transformation (slow) Nutrient recycling fast Carbon Fusion 9 -11 th May 2006
Example: Bias Modelling applied to Altimeter Data Assimilation Carbon Fusion 9 -11 th May 2006
Thermohaline Schematic Broeker Carbon Fusion 9 -11 th May 2006 Schmitz (1996)
WOCE Atlantic Section A 16 Note: Water mass origins AIW, NADW, ABW S N 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 -11 th May 2006
Dissolved Inorganic Carbon Fusion 9 -11 th May 2006
WOCE Comparison N-S Pacific Temperature section P 14 TP+ERS 1 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 -11 th May 2006
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 -11 th May 2006
Had. OCC Based on Web Services Carbon Fusion 9 -11 th May 2006
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 -11 th May 2006
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 0 m Assimilation results 500 m Ship Validation WOCE Cruise 15 S Carbon Fusion 9 -11 th May 2006 55 N
Mar. Quest: Assimilation impact on Ecosystems • Assimilation controls and corrects seasonal thermocline T and MLD • Biological production will be strongly influenced by assimilation Had. OCC thermocline and chlorophyll conc. No Data Assimilation FOAM thermocline With Data Assimilation High resolution FOAM th Carbon All data from www. nerc-essc. ac. uk/godiva Fusion 9 -11 May 2006
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 -11 th May 2006
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