Carbon Fusion 9-11th May 2006
Budgets and Bias in Data AssimilationKeith Haines, ESSC&DARC, Reading
Background: Marine Informatics• Assimilation algorithms in Ocean circulation models
Satellite and In Situ data setsPhysically based covariances + simple errors in big and Biased modelsBudget diagnostics based on assimilation
• Met Office FOAM, ECMWF Seasonal Forecasting collaborationsDARC-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/Earthhttp://lovejoy.nerc-essc.ac.uk:8080/Godiva2
Carbon Fusion 9-11th May 2006
Ocean Box-Inverse solution Ganachaud and Wunsch (2000)
Transport in Sverdrups 1Sv = 106 m3 s-1
Budgets and OceanThermohaline Circulation
After Broeker
• Closed Budgets of .. Heat, Salt, Mass/Volume, Tracers..• Processes: Advection, Surface fluxes, Mixing, Data Assimilation
Carbon Fusion 9-11th 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 9-11th May 2006
Transport in Sverdrups 1Sv = 106 m3 s-1
Carbon Fusion 9-11th May 2006
N. Atlantic Water Budgetby density class (11S-80N)
COADS surface fluxesCTD section at 11SSteady State (cf. Ocean Inverse) => Mixing
Transformation Flux (Sv)
Speer (1997)
27.72 28.11
Carbon Fusion 9-11th May 2006
Walin Budget diagnostics for HadCM3 climate model (100yr
average)
Transformation Flux (Sv)
Old and Haines 2006
27.72 28.11
Carbon Fusion 9-11th May 2006
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
Process Estimation v. Parameter Estimation
Parameter estimation4DVar. Cost function containing fit to observations, a-priori info.Tune: initial state, sources/sinks, model parameters (diffusion)…..
Carbon Fusion 9-11th May 2006
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
OCCAM Assimilation Experiment
• 1993-96• ECMWF
6hr winds• Monthly
XBT assim.
• 10-day-ly Altimeter assim.
• SST weakly relaxed to Reynolds
• SSS weakly relaxed to Levitus
Sea Level analysis 28th March 1996
1/4° x 36 levels Global Ocean Model
RUN1
Carbon Fusion 9-11th May 2006
Process Estimation: Local Heat Budget Wm-2
Local Trend = Convergence + Assimilation + Surface Flux (+ Mixing)
Assimilation Advection
Trend 1993-96 Surface Flux Mixing
•Bias•Patterns•Amplitudes•Space scales•Transients
(Haines; 2003)
Carbon Fusion 9-11th May 2006
Process Estimation: N Atlantic Box Budgets
-G/ = dV/dt -
G = (1) Surface Forcing, (2) Mixing, (3) Data Assimilation
G = Volume Transformation Rate (Sv) G = Volume Transformation Rate (Sv) (after Walin 1982)(after Walin 1982)
Thermodynamically Irreversible ProcessesThermodynamically Irreversible Processes
Fox and Haines (2003) JPO
16Sv
Run1
Carbon Fusion 9-11th 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 9-11th 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 9-11th May 2006
Shelf Seas: Carbon+Biochemistry Modelling
Hetero-trophs
Bacteria
Meso-Micro-
Particulates
Dissolved
Phytoplankton
Consumers
Pico-fDiatomsFlagell
-atesNO3
PO4
NH4
Si
CO2
Nutrients
Dino-f
Meio-benthos
AnaerobicBacteria
AerobicBacteria
DepositFeeders
SuspensionFeeders
Detritus
NutrIents
OxygenatedLayer
Reduced Layer
RedoxDiscontinuity
Layer
AtmosphereO2 CO2 DMS
3D
IrradiationWind Stress
Heat Flux
0D
Cloud Cover
Riv
ers
and
boundari
es
1D
Forcing Ecosystem
Physics
GOTMPOLCOMS
UKMO
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.
Carbon Fusion 9-11th May 2006
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
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 covarianceb =observation bias O =observation bias error covariancec =model forecast bias P =model forecast bias error covarianceSuperscript f are forecast valuesObservation operators have been omitted
Carbon Fusion 9-11th May 2006
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 covarianceb =observation bias O =observation bias error covariancec =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
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
Example: Bias Modelling applied toAltimeter Data Assimilation
Bias Error Covariance O on Mean Sea Level
Mean Sea Level
Carbon Fusion 9-11th May 2006
Example: Bias Modelling applied toAltimeter Data Assimilation
Mean Sea Level Bias ba Corrected Mean Sea Level
Carbon Fusion 9-11th May 2006
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
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
Conservation properties of assimilation
Carbon Fusion 9-11th May 2006
Box Budgets and Assimilation
Nutrient recyclingfast
Nutrient recyclingfast
Transformation(slow)
Carbon Fusion 9-11th May 2006
Example: Bias Modelling applied toAltimeter Data Assimilation
Carbon Fusion 9-11th May 2006
Thermohaline Schematic
BroekerBroeker
Schmitz (1996)
Carbon Fusion 9-11th May 2006
WOCE Atlantic Section A16
SS NNNote: Water mass origins AIW, NADW, ABW
Long-lived Lagrangian properties of water used to trace spreading pathways. “Core method” Wust (1935)
Currents,Circulationrates andMixing ratesnotdetermined from Coremethod
Carbon Fusion 9-11th May 2006
Dissolved Inorganic Carbon
Carbon Fusion 9-11th May 2006
WOCE ComparisonN-S Pacific Temperature section P14
TP+ERS1 data 1993
Simulation
XBT Assimilation
XBT and Altimeter
Run available onLive Access Serverwww.nerc-essc.ac.uk/
godiva
WOCE Cruise
How to quantify the role of assimilation in maintaining thermocline?
Carbon Fusion 9-11th 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 9-11th May 2006
Based on Web Services
HadOCC
Carbon Fusion 9-11th 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 9-11th May 2006
Assimilationresults
Ship ValidationWOCE Cruise
Marine Assimilation in Global Ocean Models
500m
0m
55 N15 S
•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
Carbon Fusion 9-11th May 2006
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 thermoclineWith Data Assimilation
High resolution FOAM
All data from www.nerc-essc.ac.uk/godiva
Carbon Fusion 9-11th 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